Modern Control Systems_P6
The response desired is deadbeat, so we use a third-order transfer function as
and the settling time (with a \(2\%\) criterion) is \(T_{s} = 4/\omega_{n}\). For a settling time of \(T = 0.5\text{ }s\), we use \(\omega_{n} = 8\).
The closed-loop transfer function of the system of Figure 12.20 with the appropriate \(G_{p}(s)\) is
We let \(K_{a} = 10,K_{b} = 2,K_{P} = 127.6,K_{I} = 527.5\), and \(K_{D} = 10.35\). Note that \(T(s)\) could be achieved with other gain combinations.
The step response of this system has a deadbeat response with a percent overshoot of P.O. \(= 1.65\%\) and a settling time of \(T_{s} = 0.5\text{ }s\). When the poles of \(G(s)\) change by \(\pm 50\%\), the percent overshoot changes to \(P.O. = 1.86\%\), and the settling time is \(T_{s} = 0.95\text{ }s\). This is an outstanding design of a robust deadbeat response system.
417.1. DESIGN EXAMPLES
In this section we present two illustrative examples. The first example illustrates the design of two degree-of-freedom controllers (that is, two separate controllers) for an ultra-precision diamond turning machine. In the second design example, we consider the practical problem of designing a controller in the presence of an uncertain time delay. The specific problem under investigation is a PID controller for a digital audio tape drive. The design process is highlighted with an emphasis on robustness.
418. EXAMPLE 12.9 Ultra-precision diamond turning machine
The design of an ultra-precision diamond turning machine has been studied at Lawrence Livermore National Laboratory. This machine shapes optical devices such as mirrors with ultra-high precision using a diamond tool as the cutting device. In this discussion, we will consider only the \(z\)-axis control. Using frequency response identification with sinusoidal input to the actuator we determined that
The system can accommodate high gains, since the input command is a series of step commands of very small magnitude (a fraction of a micron). The system has an outer loop for position feedback using a laser interferometer with an accuracy of 0.1 micron \(\left( 10^{- 7}\text{ }m \right)\). An inner feedback loop is also used for velocity feedback, as shown in Figure 12.21. FIGURE 12.21
Turning machine control system.
We want to select the controllers, \(G_{1}(s)\) and \(G_{2}(s)\), to obtain an overdamped, highly robust, high-bandwidth system. The robust system must accommodate changes in \(G(s)\) due to varying loads, materials, and cutting requirements. Thus, we seek a large phase margin and gain margin for the inner and outer loops, and low root sensitivity. The specifications are summarized in Table 12.4.
Since we want zero steady-state error for the velocity loop, we propose a velocity loop controller \(G_{2}(s) = G_{3}(s)G_{4}(s)\), where \(G_{3}(s)\) is a PI controller and \(G_{4}(s)\) is a phase-lead compensator. Thus, we have
and choose \(K_{P}/K_{I} = 0.00532,K_{4} = 0.00272\), and \(\alpha = 2.95\). We now have
The root locus for \(G_{2}(s)G(s)\) is shown in Figure 12.22. When \(K_{P} = 2\), we have the velocity closed-loop transfer function given by
| Specification | Transfer Function | |
| Velocity, \(V(s)/U(s)\) | Position \(Y(s)/R(s)\) | |
| Minimum bandwidth | $$950rad/s$$ | $$95rad/s$$ |
| Steady-state error to a step | 0 | 0 |
| Minimum damping ratio \(\zeta\) | 0.8 | 0.9 |
| Maximum root sensitivity \(\mid S_{K}^{r}\) | 1.0 | 1.5 |
| Minimum phase margin | $$90^{\circ}$$ | $$75^{\circ}$$ |
| Minimum gain margin | $$40\text{ }dB$$ | $$60\text{ }dB$$ |
419. Table 12.4 Specifications for Turning Machine Control System
FIGURE 12.22
Root locus for velocity loop as \(K_{p}\) varies.
Table 12.5 Design Results for Turning Machine Control System
Achieved Result
Velocity Position
Transfer
Closed-loop bandwidth
Steady-state error
Damping ratio, \(\zeta\)
Root sensitivity, \(\left| S_{K}^{r} \right|\)
Phase margin
Gain margin
0
1.0
0.92
Infinite
Position Transfer Function
0
1.0
1.2
\(76\text{ }dB\) which is a large-bandwidth system. The actual bandwidth and root sensitivity are summarized in Table 12.5. Note that we have exceeded the specifications for the velocity transfer function.
We propose a phase-lead compensator for the position loop of the form
and we choose \(\alpha = 2.0\) and \(K_{5} = 0.0185\) so that
We then plot the root locus for the loop transfer function
If we use the approximate \(T_{2}(s)\) of Equation (12.46), we have the root locus of Figure 12.23(a). Using the actual \(T_{2}(s)\), we get the close-up of the root locus shown in Figure 12.23(b). We select \(K_{P} = 1000\) and achieve the actual results for the total system transfer function as recorded in Table 12.5. The total system has a high phase margin, has a low sensitivity, and is overdamped with a large bandwidth. This system is very robust.
(a)
FIGURE 12.23
The root locus for \(K_{1} > 0\) for (a) overview and (b) close-up near origin of the s-plane.
(b)
420. EXAMPLE 12.10 Digital audio tape controller
Consider the feedback control system shown in Figure 12.24, where
The exact value of the time delay is uncertain, but is known to lie in the interval \(T_{1} \leq T \leq T_{2}\). Define
Then
FIGURE 12.24
A feedback system with a time delay in the loop.
FIGURE 12.25
Multiplicative uncertainty representation.
FIGURE 12.26
Equivalent block diagram depiction of the multiplicative uncertainty.
or
If we define
then we have
In the development of a robust stability controller, we would like to represent the time-delay uncertainty in the form shown in Figure 12.25 where we need to determine a function \(M(s)\) that approximately models the time delay. This will lead to the establishment of a straightforward method of testing the system for stability robustness in the presence of the uncertain time-delay. The uncertainty model is known as a multiplicative uncertainty representation.
Since we are concerned with stability, we can consider \(R(s) = 0\). Then we can manipulate the block diagram in Figure 12.25 to obtain the form shown in Figure 12.26. Using the small gain theorem, we have the condition that the closed-loop system is stable if
The challenge is that the time delay \(T\) is not known exactly. One approach to solving the problem is to find a weighting function, denoted by \(W(s)\), such that
If \(W(s)\) satisfies the inequality in Equation (12.48), it follows that
Therefore, the robust stability condition can be satisfied by
This is a conservative bound. If the condition in Equation (12.49) is satisfied, then stability is guaranteed in the presence of any time delay in the range \(T_{1} \leq T \leq T_{2}\) [5,32]. If the condition is not satisfied, the system may or may not be stable.
Suppose we have an uncertain time delay that is known to lie in the range \(0.1 \leq T \leq 1\). We can determine a suitable weighting function \(W(s)\) by plotting the magnitude of \(e^{- j\omega T} - 1\), as shown in Figure 12.27 for various values of \(T\) in the range \(T_{1} \leq T \leq T_{2}\). A reasonable weighting function obtained by trial and error is
FIGURE 12.27 Magnitude plot of \(\left| e^{- j\omega T} - 1 \right|\) for \(T = 0.1,0.5\), and 1.
FIGURE 12.28 Digital audio tape driver mechanism.
This function satisfies the condition
Keep in mind that the selection of the weighting function is not unique.
A digital audio tape (DAT) stores 1.3 gigabytes of data in a package the size of a credit card-roughly nine times more than a half-inch-wide reel-to-reel tape or quarter-inch-wide cartridge tape. A DAT sells for about the same amount as a floppy disk, even though it can store 1000 times more data. A DAT can record for two hours (longer than either reel-to-reel or cartridge tape), which means that it can run longer unattended and requires fewer changes and hence fewer interruptions of data transfer. DAT gives access to a given data file within 20 seconds, on the average, compared with several minutes for either cartridge or reel-to-reel tape [2].
The tape drive electronically controls the relative speeds of the drum and tape so that the heads follow the tracks on the tape, as shown in Figure 12.28. The control system is complex because motors have to be accurately controlled: capstan, take-up and supply reels, drum, and tension control. The elements of the design process emphasized in this example are highlighted in Figure 12.29.
Consider the speed control system shown in Figure 12.30. The motor and load transfer function varies because the tape moves from one reel to the other. The transfer function is
where nominal values are \(K_{m} = 4,p_{1} = 1\), and \(p_{2} = 4\). However, the range of variation is \(3 \leq K_{m} \leq 5,0.5 \leq p_{1} \leq 1.5\), and \(3.5 \leq p_{2} \leq 4.5\). Thus, the process belongs to a family of processes, where each member corresponds to different values of \(K_{m},p_{1}\), and \(p_{2}\). The design goal is
421. Design Goal
Control the DAT speed to the desired value in the presence of significant process uncertainties. FIGURE 12.30 Block diagram of the digital audio tape speed control system.
FIGURE 12.29 Elements of the control system design process emphasized in this digital audio tape speed control design.
If the performance does not meet the specifications, then iterate the configuration.
If the performance meets the specifications, then finalize the design.
Control the DAT speed e presence of significant plant uncertainties.
Ppecilcations: \(T_{s} < 2s\) DS2: Robust stability
See Figures 12.28 and 12.30
Associated with the design goal we have the variable to be controlled defined as
Variable to Be Controlled
DAT speed \(Y(s)\).
The design specifications are
422. Design Specifications
DS1 Percent overshoot of P.O. \(\leq 13\%\) and settling time of \(T_{s} \leq 2\text{ }s\) for a unit step input.
DS2 Robust stability in the presence of a time delay at the plant input. The time delay value is uncertain but known to be in the range \(0 \leq T \leq 0.1\). Design specification DS1 must be satisfied for the entire family of plants. Design specification DS2 must be satisfied by the nominal process \(\left( K_{m} = 4,p_{1} = 1,p_{2} = 4 \right)\).
The following constraints on the design are given:
$\square\ $ Fast peak time requires that an overdamped condition is not acceptable.
$\square\ $ Use a PID controller:
\(\square\ K_{m}K_{D} \leq 20\) when \(K_{m} = 4\).
The key tuning parameters are the PID gains:
423. Select Key Tuning Parameters
Since we are constrained to have \(K_{m}K_{D} \leq 20\) when \(K_{m} = 4\), we must select \(K_{D} \leq 5\). We will design the PID controller using nominal values for \(K_{m},p_{1}\), and \(p_{2}\). We then analyze the performance of the controlled system for the various values of the process parameters, using a simulation to check that DS1 is satisfied. The nominal process is given by
The closed-loop transfer function is
If we choose \(K_{D} = 5\), then we write the characteristic equation as
or
Per specifications, we try to place the dominant poles in the region defined by \(\zeta\omega_{n} > 2\) and \(\zeta > 0.55\). We need to select a value of \(\tau = K_{I}/K_{P}\), and then we can plot the root locus with the gain \(4K_{P}\) as the varying parameter. After several iterations, we choose a reasonable value of \(\tau = 3\). The root locus is shown in Figure 12.31. We determine that \(4K_{P} = 120\) represents a valid selection since the roots lie inside the desired performance region. We obtain \(K_{P} = 30\), and \(K_{I} = \tau K_{P} = 90\). The PID controller is then given by
The step response (for the process with nominal parameter values) is shown in Figure 12.32. A family of responses is shown in Figure 12.33 for various values of Robust Control Systems
FIGURE 12.31
Root locus for the DAT system with \(K_{D} = 5\) and \(\tau = K_{l}/K_{P} = 3\).
FIGURE 12.32
Unit step response for the DAT system with \(K_{P} = 30,K_{D} = 5\), and \(K_{l} = 90\).
FIGURE 12.33
A family of step responses for the DAT system for various values of the process parameters \(K_{m},p_{1}\), and \(p_{2}\).
\(K_{m},p_{1}\), and \(p_{2}\). None of the responses suggests a percent overshoot over the specified value of \(P.O. = 13\%\), and the settling times are all under the \(T_{s} \leq 2\text{ }sspec -\) ification as well. As we can see in Figure 12.33, all of the tested processes in the family are adequately controlled by the single PID controller in Equation (12.52). Therefore DS1 is satisfied for all processes in the family.
Suppose the system has a time delay at the input to the process. The actual time delay is uncertain but known to be in the range \(0 \leq T \leq 0.1\text{ }s\). Following the method discussed previously, we determine that a reasonable function \(W(s)\) which bounds the plots of \(\left| e^{- j\omega T} - 1 \right|\) for various values of \(T\) is
To check the stability robustness property, we need to verify that
The plot of both \(|W(j\omega)|\) and \(\left| 1 + \frac{1}{G_{c}(j\omega)G(j\omega)} \right|\) is shown in Figure 12.34. It can be seen that the condition in Equation (12.53) is indeed satisfied. Therefore, we expect that the nominal system will remain stable in the presence of time-delays up to 0.1 seconds. FIGURE 12.34
Stability robustness to a time delay of uncertain magnitude.
423.1. THE PSEUDO-QUANTITATIVE FEEDBACK SYSTEM
Quantitative feedback theory (QFT) uses a controller, as shown in Figure 12.35, to achieve robust performance. The goal is to achieve a wide bandwidth for the closedloop transfer function with a high loop gain \(K\). Typical QFT design methods use graphical and numerical methods in conjunction with the Nichols chart. Generally, QFT design seeks a high loop gain and large phase margin so that robust performance is achieved [24-26, 28].
In this section, we pursue a simple method of achieving the goals of QFT with an \(s\)-plane, root locus approach to the selection of the gain \(K\) and the compensator \(G_{c}(s)\). This approach, dubbed pseudo-QFT, follows these steps:
-
Place the \(n\) poles and \(m\) zeros of \(G(s)\) on the \(s\)-plane for the \(n\)th order \(G(s)\). Also, add any poles of \(G_{c}(s)\).
-
Starting near the origin, place the zeros of \(G_{c}(s)\) immediately to the left of each of the \((n - 1)\) poles on the left-hand \(s\)-plane. This leaves one pole far to the left of the lefthand side of the \(s\)-plane.
-
Increase the gain \(K\) so that the roots of the characteristic equation (poles of the closedloop transfer function) are close to the zeros of \(G_{c}(s)G(s)\).
This method introduces zeros so that all but one of the root loci end on finite zeros. If the gain \(K\) is sufficiently large, then the poles of \(T(s)\) are almost equal to the zeros of \(G_{c}(s)G(s)\). This leaves one pole of \(T(s)\) with a significant partial fraction residue and the system with a phase margin of approximately \(90^{\circ}\) (actually about \(85^{\circ}\) ). FIGURE 12.35
Feedback system.
EXAMPLE 12.11 Design using the pseudo-QFT method
Consider the system of Figure 12.35 with
where the nominal case is \(p_{1} = 1\) and \(p_{2} = 2\), with \(\pm 50\%\) variation. The worst case is with \(p_{1} = 0.5\) and \(p_{2} = 1\). We wish to design the system for zero steady-state error for a step input, so we use the PID controller
We then invoke the internal model principle, with \(R(s) = 1/s\) incorporated within \(G_{c}(s)G(s)\). Using Step 1, we place the poles of on the \(s\)-plane, as shown in Figure 12.36. There are three poles (at \(s = 0, - 1\), and -2 ), as shown. Step 2 calls for placing a zero to the left of the pole at the origin and at the pole at \(s = - 1\), as shown in Figure 12.36.
The compensator is thus
We select \(K = 100\), so that the roots of the characteristic equation are close to the zeros. The closed-loop transfer function is
This closed-loop system provides a fast response and possesses a phase margin of P.M. \(= 85^{\circ}\). When the worst-case conditions are realized $\left( p_{1} = 0.5 \right.\ $ and \(\left. \ p_{2} = 1 \right)\), the performance remains essentially unchanged. Pseudo-QFT design results in very robust systems.
423.2. ROBUST CONTROL SYSTEMS USING CONTROL DESIGN SOFTWARE
In this section, we investigate robust control systems using control design software. In particular, we will consider the commonly used PID controller in the feedback control system shown in Figure 12.16. Note that the system has a prefilter \(G_{p}(s)\).
The objective is to choose the PID parameters \(K_{P},K_{I}\), and \(K_{D}\) to meet the performance specifications and have desirable robustness properties. Unfortunately, it is not immediately clear how to choose the parameters in the PID controller to obtain certain robustness characteristics. An illustrative example will show that it is possible to choose the parameters iteratively and verify the robustness by simulation. Using the computer helps in this process, because the entire design and simulation can be automated using scripts and can easily be executed repeatedly.
424. EXAMPLE 12.12 Robust control of temperature
Consider the feedback control system in Figure 12.16, where
and the nominal value is \(c_{0} = 1\), and \(G_{p}(s) = 1\). We can design a compensator based on \(c_{0} = 1\) and check robustness by simulation. Our design specifications are
-
A settling time (with a \(2\%\) criterion) \(T_{S} \leq 0.5\text{ }s\), and
-
An optimum ITAE performance for a step input.
For this design, we will not use a prefilter to meet specification (2), but will instead show that acceptable performance (i.e., low percent overshoot) can be obtained by increasing a cascade gain.
The closed-loop transfer function is
FIGURE 12.37
Root locus for the PID-compensated temperature controller as \(\widehat{K}\) varies.
\(> > a = 16;b = 70;\) num=[1 a b]; den=[1 100 0 \(\rbrack\); sys=tf(num,den); \(>\) rlocus(sys)
\(>\) rlocfind(sys)
The associated root locus equation is
where
The settling time requirement \(T_{s} < 0.5\text{ }s\) leads us to choose the roots of \(s^{2} + as + b\) to the left of the \(s = - \zeta\omega_{n} = - 8\) line in the \(s\)-plane, as shown in Figure 12.37, to ensure that the locus travels into the required \(s\)-plane region. We have chosen \(a = 16\) and \(b = 70\) to ensure the locus travels past the \(s = - 8\) line. We select a point on the root locus in the performance region, and using the rlocfind function, we find the associated gain \(\widehat{K}\) and the associated value of \(\omega_{n}\). For our chosen point, we find that
Then, with \(\widehat{K},a\), and \(b\), we can solve for the PID coefficients as follows:
To meet the overshoot performance requirements for a step input, we will use a cascade gain \(K\) that will be chosen by iterative methods using the step function, as illustrated in Figure 12.38. The step response corresponding to \(K = 5\) has an FIGURE 12.38
Step response of the PID temperature controller.
acceptable percent overshoot of \(P.O. = 2\%\). With the addition of the gain \(K = 5\), the final PID controller is
We do not use the prefilter. Instead, we increase the cascade gain \(K\) to obtain satisfactory transient response. Now we can consider the question of robustness to changes in the plant parameter \(c_{0}\).
The investigation into the robustness of the design consists of a step response analysis using the PID controller given in Equation (12.57) for a range of plant parameter variations of \(0.1 \leq c_{0} \leq 10\). The results are displayed in Figure 12.39. The script is written to compute the step response for a given \(c_{0}\). It can be convenient to place the input of \(c_{0}\) at the command prompt level to make the script more interactive.
The simulation results indicate that the PID design is robust with respect to changes in \(c_{0}\). The differences in the step responses for \(0.1 \leq c_{0} \leq 10\) are barely discernible on the plot. If the results showed otherwise, it would be possible to iterate on the design until an acceptable performance was achieved. The interactive capability of the m-file allows us to check the robustness by simulation. FIGURE 12.39
Robust PID
controller analysis
with variations in \(c_{0}\).
\(c0 = 10\) Specify process parameter.
numg=[1]; deng=[1 \(\left. \ 2^{*}CO{cO}^{\land}2 \right\rbrack\);
numgc \(= 5^{\star}\lbrack 1161887\) 8260]; dengc \(= \lbrack 10\rbrack\);
sysg=tf(numg,deng);
sysgc=tf(numgc, dengc);
syso=series(sysgc,sysg);
sys=feedback(syso,[1]);
step(sys)
424.1. SEQUENTIAL DESIGN EXAMPLE: DISK DRIVE READ SYSTEM
In this section, we design a PID controller to achieve the desired system response. Many disk drive head control systems use a PID controller and use a command signal \(r(t)\) that utilizes an ideal velocity profile at the maximum allowable velocity until the head arrives near the desired track, when \(r(t)\) is switched to a step-type input. Thus, we want zero steady-state error for a ramp (velocity) signal and a step signal. Examining the system shown in Figure 12.40, we note that the forward path possesses two pure integrations, and we expect zero steady-state error for a velocity input \(r(t) = At,t > 0\).
The PID controller is
The motor field transfer function is
FIGURE 12.41
A sketch of a root locus at \(K_{D}\) increases for estimated root locations with a desirable system response.
FIGURE 12.40 Disk drive feedback system with a PID controller.
The second-order model uses \(G_{1}(s) = 5\), and the design is determined for this model.
We use the second-order model and the PID controller for the \(s\)-plane design technique illustrated in Section 12.6. The poles and zeros of the system are shown in the \(s\)-plane in Figure 12.41 for the second-order model and \(G_{1}(s) = 5\). Then we have the loop transfer function
We select \(- z_{1} = - 120 + j40\) and determine \(5K_{D}\) so that the roots are to the left of the line \(s = - 100\). If we achieve that requirement, then
and the percent overshoot to a step input is (ideally) P.O. \(\leq 2\%\) since \(\zeta\) of the complex roots is approximately 0.8 . Of course, this sketch is only a first step. As a FIGURE 12.42
Actual root locus for the secondorder model.
Table 12.6 Disk Drive Control System Specifications and Actual Performance
| $$\begin | ||
| \text{}\text{Performance}\text{} \ | ||
| \text{}\text{Measure}\text{} | ||
| \end{matrix}$$ | $$\begin |
\text{~}\text{Desired}\text{~} \\
\text{~}\text{Value}\text{~}
\end{matrix}$$ | $$\begin{matrix}
\text{~}\text{Response for}\text{~} \\
\text{~}\text{Second-Order}\text{~} \\
\text{~}\text{Model}\text{~}
\end{matrix}$$ |
| $$\begin{matrix}
\text{~}\text{Percent overshoot}\text{~} \
\begin{matrix}
\text{~}\text{Settling time for}\text{~} \
\text{~}\text{step input}\text{~}
\end{matrix}
\end{matrix}$$ | $$< 5%$$ | $$4.5%$$ |
| $$\begin{matrix}
\text{~}\text{Maximum response for}\text{~} \
\text{~}\text{a unit step disturbance}\text{~}
\end{matrix}$$ | $$< 50\text{ }ms$$ | $$6\text{ }ms$$ |
second step, we determine \(K_{D}\). We then obtain the actual root locus as shown in Figure 12.42 with \(K_{D} = 800\). The system response is recorded in Table 12.6. The system meets all the specifications.
424.2. SUMMARY
The design of highly accurate control systems in the presence of significant plant uncertainty requires the designer to seek a robust control system. A robust control system exhibits low sensitivities to parameter change and is stable over a wide range of parameter variations.
The PID controller was considered as a compensator to aid in the design of robust control systems. The design issue for a PID controller is the selection of the gain and two zeros of the controller transfer function. We used three design methods for the selection of the controller: the root locus method, the frequency response method, and the ITAE performance index method. An operational amplifier circuit used for a PID controller is shown in Figure 12.43. In general, the use of a PID controller will enable the designer to attain a robust control system. FIGURE 12.43
Operational amplifier circuit used for PID controller.
The internal model control system with state variable feedback and a controller \(G_{c}(s)\) was used to obtain a robust control system. Finally, the robust nature of a pseudo-QFT control system was demonstrated.
A robust control system provides stable, consistent performance as specified by the designer in spite of the wide variation of plant parameters and disturbances. It also provides a highly robust response to command inputs and a steady-state tracking error equal to zero.
For systems with uncertain parameters, the need for robust systems will require the incorporation of advanced machine intelligence, as shown in Figure 12.44.
FIGURE 12.44 Intelligence required versus uncertainty for modern control systems.
425. SKILLS CHECK
In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 12.45 as specified in the various problem statements.
FIGURE 12.45 Block diagram for the Skills Check.
In the following True or False and Multiple Choice problems, circle the correct answer.
- A robust control system exhibits the desired performance in the presence of significant plant uncertainty.
True or False
- For physically realizable systems, the loop gain \(L(s) = G_{c}(s)G(s)\)
must be large for high frequencies.
True or False
- The PID controller consists of three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term.
True or False
- A plant model will always be an inaccurate representation of the actual physical system.
True or False
- Control system designers seek small loop gain \(L(s)\) in order to minimize the sensitivity \(S(s)\).
True or False
- A closed-loop feedback system has the third-order characteristic equation
where the nominal values of the coefficients are \(a_{2} = 3,a_{1} = 6\), and \(a_{0} = 11\). The uncertainty in the coefficients is such that the actual values of the coefficients can lie in the intervals
Considering all possible combinations of coefficients in the given intervals, the system is:
a. Stable for all combinations of coefficients.
b. Unstable for some combinations of coefficients.
c. Marginally stable for all combinations of coefficients.
d. Unstable for all combinations of coefficients.
In Problems 7 and 8, consider the unity feedback system in Figure 12.45, where
-
Assume that the prefilter is \(G_{p}(s) = 1\). The proportional-plus-integral (PI) controller, \(G_{c}(s)\), that provides optimum coefficients of the characteristic equation for ITAE (assuming \(\omega_{n} = 12\) and a step input) is which of the following:
a. \(G_{c}(s) = 72 + \frac{6.9}{s}\)
b. \(G_{c}(s) = 6.9 + \frac{72}{s}\)
c. \(G_{c}(s) = 1 + \frac{1}{s}\)
d. \(G_{c}(s) = 14 + 10s\) -
Considering the same PI controller as in Problem 7, a suitable prefilter, \(G_{p}(s)\), which provides optimum ITAE response to a step input is:
a. \(G_{p}(s) = \frac{10.43}{s + 12.5}\)
b. \(G_{p}(s) = \frac{12.5}{s + 12.5}\)
c. \(G_{p}(s) = \frac{10.43}{s + 10.43}\)
d. \(G_{p}(s) = \frac{143}{s + 143}\) -
Consider the closed-loop system block-diagram in Figure 12.45, where
Determine which of the following PID controllers results in a closed-loop system possessing two pairs of equal roots.
a. \(G_{c}(s) = \frac{22.5(s + 1.11)^{2}}{s}\)
b. \(G_{c}(s) = \frac{10.5(s + 1.11)^{2}}{s}\)
c. \(G_{c}(s) = \frac{2.5(s + 2.3)^{2}}{s}\)
d. None of the above
- Consider the system in Figure 12.45 with \(G_{p}(s) = 1\),
and \(1 \leq a \leq 3\) and \(7 \leq b \leq 11\). Which of the following PID controllers yields a robustly stable system?
a. \(G_{c}(s) = \frac{13.5(s + 1.2)^{2}}{s}\)
b. \(G_{c}(s) = \frac{2(s + 40)^{2}}{s}\) c. \(G_{c}(s) = \frac{0.1(s + 10)^{2}}{s}\)
d. None of the above
- Consider the system in Figure 12.45 with \(G_{p}(s) = 1\) and loop transfer function
The sensitivity of the closed-loop system with respect to variations in the parameter \(K\) is
a. \(S_{K}^{T} = \frac{s(s + 3)}{s^{2} + 3s + K}\)
b. \(S_{K}^{T} = \frac{s + 5}{s^{2} + 5s + K}\)
c. \(S_{K}^{T} = \frac{s}{s^{2} + 5s + K}\)
d. \(S_{K}^{T} = \frac{s(s + 5)}{s^{2} + 5s + K}\)
- Consider the feedback control system in Figure 12.45 with plant
A proportional-plus-integral (PI) controller and prefilter pair that results in a settling time \(T_{s} < 1.8\text{ }s\) and an optimum ITAE step response are which of the following:
a. \(G_{c}(s) = 3.2 + \frac{13.8}{s}\) and \(G_{p}(s) = \frac{13.8}{3.2s + 13.8}\)
b. \(G_{c}(s) = 10 + \frac{10}{s}\) and \(G_{p}(s) = \frac{1}{s + 1}\)
c. \(G_{c}(s) = 1 + \frac{5}{s}\) and \(G_{p}(s) = \frac{5}{s + 5}\)
d. \(G_{c}(s) = 12.5 + \frac{500}{s}\) and \(G_{p}(s) = \frac{500}{12.5s + 500}\)
- Consider a unity negative feedback system with a loop transfer function (with nominal values)
Using the Routh-Hurwitz stability analysis, it can be shown that the closed-loop system is nominally stable. However, if the system has uncertain coefficients such that
the closed-loop system may exhibit instability. Which of the following situations is true:
a. Unstable for \(a = 1,b = 2\), and \(K = 4\).
b. Unstable for \(a = 2,b = 4\), and \(K = 4.5\).
c. Unstable for \(a = 0.25,b = 3\), and \(K = 5\).
d. Stable for all \(a,b\), and \(K\) in the given intervals. 14. Consider the feedback control system in Figure 12.45 with \(G_{p}(s) = 1\) and \(G(s) = \frac{1}{Js^{2}}\).
The nominal value of \(J = 5\), but it is known to change with time. It is thus necessary to design controller with sufficient phase margin to retain stability as \(J\) changes. A suitable PID controller such that the phase margin is greater than P.M. \(> 40^{\circ}\) and bandwidth \(\omega_{b} < 20rad/s\) is which of the following:
a. \(G_{c}(s) = \frac{50\left( s^{2} + 10s + 26 \right)}{s}\)
b. \(G_{c}(s) = \frac{5\left( s^{2} + 2s + 2 \right)}{s}\)
c. \(G_{c}(s) = \frac{60\left( s^{2} + 20s + 200 \right)}{s}\)
d. None of the above
- A feedback control system has the nominal characteristic equation
The process varies such that
Considering all possible combinations of coefficients \(a_{2},a_{1}\), and \(a_{0}\) in the given intervals, the system is:
a. Stable for all combinations of coefficients.
b. Unstable for some combinations of coefficients.
c. Marginally stable for all combinations of coefficients.
d. Unstable for all combinations of coefficients.
In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided.
a. Root sensitivity
b. Additive perturbation
c. Complementary sensitivity function
d. Robust control system
e. System sensitivity
f. Multiplicative perturbation
A system that exhibits the desired performance in the presence of significant plant uncertainty.
A controller with three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term.
A transfer function that filters the input signal prior to the calculation of the error signal.
A system perturbation model expressed in the additive form \(G_{C}(s) = G(s) + A(s)\) where \(G(s)\) is the nominal plant, \(A(s)\) is the perturbation that is bounded in magnitude, and \(G_{c}(s)\) is the family of perturbed plants.
The function \(G(s) = G_{c}(s)G(s)\left\lbrack 1 + G_{c}(s)G_{c}(s) \right\rbrack^{- 1}\) that satisfies the relationship \(C(s) + S(s) = 1\), where \(S(s)\) is the sensitivity function.
The principle that states that if \(G_{c}(s)G(s)\) contains the input \(R(s)\), then the output \(y(t)\) will track the input asymptotically (in the steady state) and the tracking is robust. g. PID controller
h. Robust stability criterion
i. Prefilter
j. Sensitivity function
k. Internal model principle
A system perturbation model expressed in the multiplicative form \(G_{m}(s) = G(s)\lbrack 1 + M(s)\rbrack\) where \(G(s)\) is the nominal plant, \(M(s)\) is the perturbation that is bounded in magnitude, and \(G_{m}(s)\) is the family of perturbed plants.
A test for robustness with respect to multiplicative perturbations.
A measure of the sensitivity of the roots (that is, the poles and zeros) of the system to changes in a parameter.
The function that \(S(s) = \left\lbrack 1 + G_{c}(s)G(s) \right\rbrack^{- 1}\) that satisfies the relationship \(C(s) + S(s) = 1\), where \(C(s)\) is the complementary sensitivity function.
A measure of the system sensitivity to changes in a parameter.
426. EXERCISES
E12.1 Consider a system of the form shown in Figure E12.1, where
Using the ITAE performance method for a step input, determine the required \(G_{c}(s)\). Assume \(\omega_{n} = 25\) for
Table 5.6. Determine the step response with and without a prefilter \(G_{p}(s)\).
E12.2 For the ITAE design obtained in Exercise E12.1, determine the response due to a disturbance \(T_{d}(s) = 0.5/s\).
(a)
FIGURE E12.1
Closed-loop control system. (a) Signal flow graph.
(b) block diagram.
(b) E12.3 A closed-loop unity feedback system has the loop transfer function
where \(b\) is normally equal to 4 . Determine \(S_{b}^{T}\), and plot \(20log10|T(j\omega)|\) and \(20log10|S(j\omega)|\) on a Bode plot.
Answer: \(S_{b}^{T} = \frac{- bs}{s^{2} + bs + 22}\)
E12.4 A PID controller is used in a unity feedback system where
The gain \(K_{D}\) of the controller
is limited to 500. Select a set of compensator zeros so that the pair of closed-loop roots is approximately equal to the zeros. Find the step response for the approximation
and the actual response, and compare them.
E12.5 A system has a process function
with \(K = 50\) and unity feedback with a PD compensator
The objective is to design \(G_{c}(s)\) so that the percent overshoot to a step is P.O. \(\leq 10\%\), and the settling time (with a \(2\%\) criterion) is \(T_{s} \leq 3\text{ }s\). Find a suitable \(G_{c}(s)\). What is the effect of decreasing process gain from \(K = 50\) to \(K = 25\) on the percent overshoot and settling time?
E12.6 Consider the control system shown in Figure E12.6 when \(G(s) = 2/(s + 3)^{2}\), and select a PID controller so that the settling time (with a \(2\%\) criterion) is less than 1.5 second for an ITAE step response. Plot \(y(t)\) for a step input \(r(t)\) with and without a prefilter. Determine and plot \(y(t)\) for a step disturbance. Discuss the effectiveness of the system.
Answer: One possible controller is
E12.7 For the control system of Figure E12.6 with \(G(s) = 1/(s + 6)^{2}\), select a PID controller to achieve a settling time (with a \(2\%\) criterion) of less than 1.0 second for an ITAE step response. Plot \(y(t)\) for a step input \(r(t)\) with and without a prefilter. Determine and plot \(y(t)\) for a step disturbance. Discuss the effectiveness of the system.
E12.8 Repeat Exercise 12.6, striving to achieve a minimum settling time while adding the constraint that \(|u(t)| < 20\) for \(t > 0\) for a unit step input, \(r(t) = 1,t \geq 1\).
Answer: \(G_{c}(s) = \frac{20s + 16}{s}\)
E12.9 A system has the form shown in Figure E12.6 with
where \(K = 1\). Design a PD controller such that the dominant closed-loop poles possess a damping ratio of \(\zeta = 0.6\). Determine the step response of the system. Predict the effect of a change in \(K\) of \(\pm 50\%\), on the percent overshoot. Estimate the step response of the worst-case system.
E12.10 A system has the form shown in Figure E12.6 with
where \(K = 1\). Design a PI controller so that the dominant roots have a damping ratio \(\zeta = 0.65\). Determine the step response of the system. Predict the effect of a change in \(K\) of \(\pm 50\%\) on the percent overshoot. Estimate the step response of the worstcase system.
FIGURE E12.6
System with controller.
E12.11 Consider a second-order system with the following state space representation
where $\mathbf{A} = \begin{bmatrix}
0 & 1 \
- p & - k
\end{bmatrix},p > 0,k > 0,\mathbf{B} = \begin{bmatrix}
0 \
1
\end{bmatrix}$, and \(\mathbf{C} = \begin{bmatrix} 1 & 0 \end{bmatrix}\).
a. What are the system's natural frequency \(\omega_{n}\) and damping ratio \(\zeta\) as functions of system parameters \(p\) and \(k\) ?
b. Given the parameter values of \(p\) and \(k\) vary in intervals of \(5 \leq p \leq 50\) and \(1 \leq k \leq 10\), what will be the ranges of variation of \(\omega_{n}\) and \(\zeta\) ?
For a nominal value of \(p = 20\), what is the range of the system damping ratio? Plot the root locus with variation of \(k\). If the system is required to have a percent overshoot of less than \(10\%\), a controller is added to improve damping capability by increasing parameter \(k\). What is the minimum value of \(k\) to maintain the required percent overshoot?
E12.12 Consider the second-order system
The parameters \(a,b,c_{1}\), and \(c_{2}\) are unknown \(a\) priori. Under what conditions is the system completely controllable? Select valid values of \(a,b,c_{1}\), and \(c_{2}\) to ensure controllability and plot the step response.
427. PROBLEMS
P12.1 Consider the uncrewed underwater vehicle (UUV) problem. The control system is shown in Figure P12.1, where \(R(s) = 0\), the desired roll angle, and \(T_{d}(s) = 1/s\). (a) Plot \(20log|T(j\omega)|\) and \(20log\left| S_{K}^{T}(j\omega) \right|\). (b) Evaluate \(\left| S_{K}^{T}(j\omega) \right|\) at \(\omega_{B},\omega_{B/2}\), and \(\omega_{B/4}\).
P12.2 Consider the control system is shown in Figure P12.2, where \(\tau_{1} = 10\text{ }ms\) and \(\tau_{2} = 1\text{ }ms\).
(a) Select \(K\) so that \(M_{p\omega} = 1.39\). (b) Plot \(20log|T(j\omega)|\) and \(20log\left| S_{K}^{T}(j\omega) \right|\) on one Bode plot.
(c) Evaluate \(\left| S_{K}^{T}(j\omega) \right|\) at \(\omega_{B},\omega_{B/2}\), and \(\omega_{B/4}\). (d) Let \(R(s) = 0\), and determine the effect of \(T_{d}(s) = 1/s\) for the gain \(K\) of part (a) by plotting \(y(t)\).
P12.3 Magnetic levitation (maglev) trains may replace airplanes on routes shorter than 200 miles. The maglev train developed by a German firm uses electromagnetic attraction to propel and levitate heavy vehicles, carrying up to 400 passengers at \(300 - mph\) speeds. But the \(\frac{1}{4}\)-inch gap between car and track is difficult to maintain \(\lbrack 7,12,17\rbrack\).
The block diagram of the air-gap control system is shown in Figure P12.3. The compensator is
(a) Find the range of \(K\) for a stable system. (b) Select a gain so that the steady-state error of the system is less than 0.2 for a step input command. (c) Find \(y(t)\) for the gain of part (b). (d) Find \(y(t)\) when \(K\) varies \(\pm 15\%\) from the gain of part (b).
FIGURE P12.1
Control of an underwater vehicle [13].
FIGURE P12.2
Remote-controlled TV camera. FIGURE P12.3
Maglev train
control.
P12.4 An automatically guided vehicle is shown in Figure P12.4(a) and its control system is shown in Figure P12.4(b). The goal is to track the guide wire accurately, to be insensitive to changes in the gain \(K_{1}\), and to reduce the effect of the disturbance \(\lbrack 15,22\rbrack\). The gain \(K_{1}\) is normally equal to 1 and \(\tau = 1/10\).
a. Select a compensator \(G_{c}(s)\) so that the percent overshoot to a step input is \(P.O\). \(\leq 15\%\), and the settling time (with a \(2\%\) criterion) is \(T_{S} \leq 0.5\text{ }s\).
b. For the compensator selected in part (a), determine the sensitivity of the system to small changes in \(K_{1}\) by determining \(S_{K_{1}}^{T}\).
c. If \(K_{1}\) changes to 2 while \(G_{c}(s)\) of part (a) remains unchanged, find the step response of the system and compare selected performance figures with those obtained in part (a).
p. Determine the effect of \(T_{d}(s) = 1/s\) by plotting \(y(t)\) when \(R(s) = 0\).
P12.5 A roll-wrapping machine (RWM) receives, wraps, and labels large paper rolls produced in a paper mill \(\lbrack 9,16\rbrack\). The RWM consists of several major stations: positioning station, waiting station, wrapping station, and so forth. We will focus on the positioning station shown in Figure P12.5(a). The positioning station is the first station that sees a paper roll. This station is responsible for receiving and weighing the roll, measuring its diameter and width, determining the desired wrap for the roll, positioning it for downstream processing, and finally ejecting it from the station.
Functionally, the RWM can be categorized as a complex operation because each functional step (e.g., measuring the width) involves a large number of field device actions and relies upon a number of accompanying sensors.
The control system for accurately positioning the width-measuring arm is shown in Figure P12.5(b). The pole \(p\) of the positioning arm is normally equal to 2 , but it is subject to change because of loading and misalignment of the machine. (a) For \(p = 2\), design a compensator so that the P.O. \(\leq 20\%\) and \(T_{s} \leq 1\text{ }s\) to a unit step input. (b) Plot \(y(t)\) for a step input \(R(s) = 1/s\). (c) Plot \(y(t)\) for a disturbance
(a)
FIGURE P12.4
Automatically guided vehicle.
(b) FIGURE P12.5
Roll-wrapping machine control.
(a)
(b)
\(T_{d}(s) = 1/s\), with \(R(s) = 0\). (d) Repeat parts (b) and (c) when \(p\) changes to 1 and \(G_{c}(s)\) remains as designed in part (a) and compare.
P12.6 The function of a steel plate mill is to roll reheated slabs into plates of scheduled thickness and dimension \(\lbrack 5,10\rbrack\). The final products are of rectangular plane view shapes having a width of up to \(3300\text{ }mm\) and a thickness of \(180\text{ }mm\).
A schematic layout of the mill is shown in Figure P12.6(a). The mill has two major rolling stands, denoted No. 1 and No. 2. These are equipped with large rolls (up to \(508\text{ }mm\) in diameter), which are driven by high-power electric motors (up to \(4470\text{ }kW\) ). Roll gaps and forces are maintained by large hydraulic cylinders.
(a)
FIGURE P12.6
Steel-rolling mill control.
(b)
Typical operation of the mill can be described as follows. Slabs coming from the reheating furnace initially go through the No. 1 stand, whose function is to reduce the slabs to the required width. The slabs proceed through the No. 2 stand, where finishing passes are carried out to produce the required slab thickness. Finally, they go through the hot plate leveller, which gives each plate a smooth finish.
One of the key systems controls the thickness of the plates by adjusting the rolls. The block diagram of this control system is shown in Figure P12.6(b).
The controller is a PID with two equal real zeros. (a) Select the PID zeros and the gains so that the closed-loop system has two pairs of equal roots.
428. FIGURE P12.7
PID controller for the motor and load system.
(b) For the design of part (a), obtain the step response without a prefilter \(\left( G_{p}(s) = 1 \right)\). (c) Repeat part (b) for an appropriate prefilter. (d) For the system, determine the effect of a unit step disturbance by evaluating \(y(t)\) with \(r(t) = 0\).
P12.7 A motor and load with negligible friction and a voltage-to-current amplifier \(K_{a}\) is used in the feedback control system, shown in Figure P12.7. A designer selects a PID controller
where \(K_{P} = 5,K_{I} = 500\), and \(K_{D} = 0.0475\).
(a) Determine the appropriate value of \(K_{a}\) so that the phase margin of the system is \(P.M. = 30^{\circ}\). (b) For the gain \(K_{a}\), plot the root locus of the system and determine the roots of the system for the \(K_{a}\) of part (a). (c) Determine the maximum value of \(y(t)\) when \(T_{d}(s) = 1/s\) and \(R(s) = 0\) for the \(K_{a}\) of part (a). (d) Determine the response to a step input \(r(t)\), with and without a prefilter.
P12.8 A unity feedback system has a nominal characteristic equation
The coefficients vary as follows:
Determine whether the system is stable for these uncertain coefficients.
P12.9 Future astronauts may drive on the Moon in a pressurized vehicle, shown in Figure P12.9(a), that would have a long range and could be used for missions of up to six months. Engineers first analyzed the Apolloera Lunar Roving Vehicle, then designed the new vehicle, incorporating improvements in radiation and thermal protection, shock and vibration control, and lubrication and sealants.
The steering control of the moon buggy is shown in Figure P12.9(b). The objective of the control design is to achieve a step response to a steering command with zero steady-state error, a percent overshoot of P.O. \(\leq 20\%\), and a peak time of \(T_{P} \leq 1\text{ }s\). It is also necessary to determine the effect of a step disturbance \(T_{d}(s) = 1/s\) when \(R(s) = 0\), in order to ensure the reduction of moon surface effects. Using (a) a PI controller and (b) a PID controller, design an acceptable controller. Record the results for each design in a table. Compare the performance of each design.
P12.10 A satellite system can be modeled as a double integrator with a plant transfer function \(G(s) = \frac{10}{s^{2}}\). We want to use a PID controller and a prefilter with unity feedback for the system to achieve the requirements of P.O. \(= 2\%\) and settling time \(T_{S} = 2sec\). The desired characteristic poles for third-order systems as per ITAE and Bessel polynomials normalized at \(\omega_{n} = 1rad/s\) are given as:
429. ITAE:
Bessel:
\((s + 0.9420)(s + 0.7455 + j0.7112)(s + 0.7455 - j0.7112)\).
Design the PID and the prefilter. Plot the step response of the system.
P12.11 Consider the three dimensional cam shown in Figure P12.11 [18]. The control of \(x\) may be achieved with a DC motor and position feedback of the form shown in Figure P12.11.
Assume \(1 \leq K \leq 5\) and \(2 \leq p \leq 5\). Normally \(K = 1\) and \(p = 3\). Design a PID controller so that the settling time response to a step input is \(T_{s} \leq 3\text{ }s\) for all \(p\) and \(K\) in the ranges given.
P12.12 Consider a control system with the plant's model
The system uses output feedback \(u(t) = r(t) - Ky(t)\), where \(r(t)\) is the reference input. Plot the root locus. Derive the transfer function of the closed-loop system and its sensitivity to variations in parameter \(K\). FIGURE P12.9
(a) A moon vehicle.
(b) Steering control for the moon vehicle.
FIGURE P12.11
An \(x\)-axis control system of a three dimensional cam.
(a)
(b)
430. ADVANCED PROBLEMS
AP12.1 To minimize vibrational effects, a telescope is magnetically levitated. This method also eliminates friction in the azimuth magnetic drive system. The photodetectors for the sensing system require electrical connections. The system block diagram is shown in Figure AP12.1. Design a PID controller so that the maximum percent overshoot for a step input is P.O. \(\leq 20\%\) and the \(T_{s} \leq 1\text{ }s\).
AP12.2 One promising solution to traffic gridlock is a magnetic levitation (maglev) system. Vehicles are suspended on a guideway above the highway and guided by magnetic forces instead of relying on wheels or aerodynamic forces. Magnets provide the propulsion for the vehicles \(\lbrack 7,12,17\rbrack\). Ideally, maglev can offer the environmental and safety advantages of a high-speed train, the speed and low friction of an airplane, and FIGURE AP12.1
Magnetically levitated telescope position control system.
the convenience of an automobile. All these shared attributes notwithstanding, the maglev system is truly a new mode of travel and will enhance the other modes of travel by relieving congestion and providing connections among them. Maglev travel would be fast, operating at 150 to 300 miles per hour.
The tilt control of a maglev vehicle is illustrated in Figures AP12.2(a) and (b). The dynamics of the plant \(G(s)\) are subject to variation so that the poles will lie within the boxes shown in Figure AP12.2(c), and \(1 \leq K \leq 2\).
The objective is to achieve a robust system with a step response possessing a percent overshoot of P.O. \(\leq 10\%\), as well as a settling time (with a \(2\%\) criterion) of \(T_{s} \leq 2\text{ }s\) when \(|u(t)| \leq 100\). Obtain a design with a PI, PD, and PID controller and compare the results. Use a prefilter \(G_{p}(s)\) if necessary.
AP12.3 Antiskid braking systems present a challenging control problem, since brake/automotive system parameter variations can vary significantly (e.g., due to the brake-pad coefficient of friction changes or road slope variations) and environmental influences (e.g., due to adverse road conditions). The objective of the antiskid system is to regulate wheel slip to maximize the coefficient of friction between the tire and road for any given road surface [8]. As we expect, the braking coefficient of friction is greatest for dry asphalt, slightly reduced for wet asphalt, and greatly reduced for ice.
A unity feedback simplified model of the braking system is represented by a plant transfer function \(G(s)\) with
where normally \(a = 1\) and \(b = 4\).
a. Using a PID controller, design a very robust system where, for a step input, the percent overshoot is \(P.O. \leq 4\%\) and the settling time (with a \(2\%\) criterion) is \(T_{s} \leq 1\text{ }s\). The steady-state error must be less than \(1\%\) for a step. We expect \(a\) and \(b\) to vary by \(\pm 50\%\).
b. Design a system to yield the specifications of part (a) using an ITAE performance index. Predict the percent overshoot and settling time for this design.
AP12.4 A robot has been designed to aid in hipreplacement surgery. The device, called RoBoDoc, is used to precisely orient and mill the femoral cavity for acceptance of the prosthetic hip implant. Clearly, we want a very robust surgical tool control, because there is no opportunity to redrill a bone \(\lbrack 21,27\rbrack\). The unity feedback control system has
where \(1 \leq a \leq 2\), and \(4 \leq b \leq 12\).
Select a PID controller so that the system is robust. Use the \(s\)-plane root locus method. Select the appropriate \(G_{p}(s)\) and plot the response to a step input.
AP12.5 The plant of a driverless car is modeled as \(G(s) = \frac{K}{s(s + 10)}\), where \(K = 1\) under nominal conditions. The system has unity feedback with a controller \(G_{c}(s)\). To increase the system's robustness, a phase margin of P.M. \(= 50^{\circ}\) is required. Design a PID controller for \(G_{c}(s)\), and determine the effect of parameter variations when the system gain \(K\) changes by \(\pm 50\%\). Plot the step response of the controlled system.
AP12.6 Consider a unity feedback system with
where \(K_{1} = 1.5\) and \(\tau = 0.001\) s. Select a PID controller so that the settling time (with a \(2\%\) criterion) for a step input is \(T_{s} \leq 1\text{ }s\) and the percent overshoot is P.O. \(\leq 10\%\). Also, the effect of a disturbance at the output must be reduced to less than \(5\%\) of the magnitude of the disturbance.
AP12.7 Consider a unity feedback system with
The goal is to select a PI controller using the ITAE design criterion while constraining the control signal as \(|u(t)| \leq 1\) for a unit step input. Determine the appropriate PI controller and the settling time (with a \(2\%\) criterion) for a step input. Use a prefilter, if necessary.
(a)
(b)
FIGURE AP12.2
(a) and (b) tilt control for a maglev vehicle. (c) plant dynamics.
(c)
FIGURE AP12.8
A machine tool control system.
AP12.8 A machine tool control system is shown in Figure AP12.8. The transfer function of the power amplifier, prime mover, moving carriage, and tool bit is
The goal is to have a percent overshoot of P.O. \(\leq 25\%\) for a step input while achieving a peak time of \(T_{p} \leq 3\text{ }s\). Determine a suitable controller using (a) PD control, (b) PI control, and (c) PID control. (d) Then select the best controller.
AP12.9 The position control of a suspension system can be represented by a unity feedback system with controller \(G_{c}(s)\). The plant has a gain \(K\) and viscous friction coefficient \(b,G(s) = \frac{K}{s^{2} + bs + K}\). The system has its gain varying in a large range, \(4 \leq K \leq 25\), with low damping, \(0.5 \leq b \leq 2\). The desired \(2\%\) criterion applies to this system with a settling time \(T_{s} \leq 0.5\text{ }s\) as per an ITAE index. Design a PID controller for \(G_{c}(s)\) so that in the worst case, the system still maintains the control performance. At the smallest damping coefficient, \(b = 0.5\), plot the root locus for the controlled system with the obtained PID controller, and comment on its performance.
AP12.10 A system of the form shown in Figure 12.1 has
where \(3 \leq p \leq 5,0 \leq q \leq 1\), and \(1 \leq r \leq 2\). We will use a compensator
with all real poles and zeros. Select an appropriate compensator to achieve robust performance.
AP12.11 A unity feedback system has a plant
We want to attain a steady-state error for a step input. Select a compensator \(G_{c}(s)\) using the pseudo-QFT method, and determine the performance of the system when all the poles of \(G(s)\) change by \(- 50\%\). Describe the robust nature of the system.
431. DESIGN PROBLEMS
CDP12.1 Design a PID controller for the capstan-slide system of Figure CDP4.1. The percent overshoot should be \(P.O. \leq 3\%\) and the settling time should be (with a \(2\%\) criterion) \(T_{s} \leq 250\text{ }ms\) for a step input \(r(t)\). Determine the response to a disturbance for the designed system.
DP12.1 A position control system for a large turntable is shown in Figure DP12.1(a), and the block diagram of the system is shown in Figure DP12.1(b) \(\lbrack 11,14\rbrack\). This system uses a large torque motor with \(K_{m} = 15\). The objective is to reduce the steady-state effect of a step change in the load disturbance to \(5\%\) of the magnitude of the step disturbance while maintaining
(a)
FIGURE DP12.1
Turntable control.
(h)
a fast response to a step input command \(R(s)\), DP12.3 Many university and government laboratories with \(P.O. \leq 5\%\). Select \(K_{1}\) and the compensator when (a) \(G_{c}(s) = K\) and (b) \(G_{c}(s) = K_{P} + K_{D}s\). Plot the step response for the disturbance and the input for both compensators. Determine whether a prefilter is required to meet the percent overshoot requirement.
DP12.2 Consider the closed-loop system depicted in Figure DP12.2. The process has a parameter \(K\) that is nominally \(K = 1\). Design a controller that results in a percent overshoot P.O. \(\leq 20\%\) for a unit step input for all \(K\) in the range \(1 \leq K \leq 4\). have constructed robot hands capable of grasping and manipulating objects. But teaching the artificial devices to perform even simple tasks required formidable computer programming. However, a special hand device can be worn over a human hand to record the side-toside and bending motions of finger joints. Each joint is fitted with a sensor that changes its signal depending on position. The signals from all the sensors are translated into computer data and used to operate robot hands [1].
The joint angle control system is shown in part Figure DP12.3. The normal value of \(K_{m}\) is 1.0. The goal is to design a PID controller so that the
FIGURE DP12.2 A unity feedback system with a process with varying parameter K.
FIGURE DP12.3 Special hand device to train robot hands.
steady-state error for a ramp input is zero. Also, the settling time (with a \(2\%\) criterion) must be \(T_{s} \leq 3\text{ }s\) for the ramp input. We want the controller to be
(a) Select \(K_{D}\) and obtain the ramp response. Plot the root locus as \(K_{D}\) varies. (b) If \(K_{m}\) changes to one-half of its normal value and \(G_{c}(s)\) remains as designed in part (a), obtain the ramp response of the system. Compare the results of parts (a) and (b) and discuss the robustness of the system.
DP12.4 Objects smaller than the wavelengths of visible light are a staple of contemporary science and technology. Biologists study single molecules of protein or DNA; materials scientists examine atomic-scale flaws in crystals; microelectronics engineers lay out circuit patterns only a few tenths of atoms thick. Until recently, this minute world could be seen only by cumbersome, often destructive methods, such as electron microscopy and X-ray diffraction. It lay beyond the reach of any instrument as simple and direct as the familiar light microscope. New microscopes, typified by the scanning tunneling microscope (STM), are now available [3].
The precision of position control required is in the order of nanometers. The STM relies on piezoelectric sensors that change size when an electric voltage across the material is changed. The "aperture" in the STM is a tiny tungsten probe, its tip ground so fine that it may consist of only a single atom and measure just 0.2 nanometer in width. Piezoelectric controls maneuver the tip to within a nanometer or two of the surface of a conducting specimen-so close that the electron clouds of the atom at the probe tip and of the nearest atom of the specimen overlap. A feedback mechanism senses the variations in tunneling current and varies the voltage applied to a third, \(z\)-axis, control. The \(z\)-axis piezoelectric moves the probe vertically to stabilize the current and to maintain a constant gap between the microscope's tip and the surface. The control system is shown in Figure DP12.4(a), and the block diagram is shown in Figure DP12.4(b).
(a) Use the ITAE design method to determine \(G_{c}(s)\). (b) Determine the step response of the system with and without a prefilter \(G_{p}(s)\). (c) Determine the response of the system to a disturbance when \(T_{d}(s) = 1/s\). (d) Using the prefilter and of \(G_{c}(s)\) parts (a) and (b), determine the actual response when the process changes to
DP12.5 The system described in DP12.4 is to be designed using the frequency response techniques. Select
(a)
the coefficients of \(G_{c}(s)\) so that the phase margin is \(P.M. = 45^{\circ}\). Obtain the step response of the system with and without a prefilter \(G_{p}(s)\).
DP12.6 The use of control theory to provide insight into neurophysiology has a long history. As early as the beginning of the last century, many investigators described a muscle control phenomenon caused by the feedback action of muscle spindles and by sensors based on a combination of muscle length and rate of change of muscle length.
This analysis of muscle regulation has been based on the theory of single-input, single-output control systems. An example is a proposal that the stretch reflex is an experimental observation of a motor control strategy, namely, control of individual muscle length by the spindles. Others later proposed the regulation of individual muscle stiffness (by sensors of both length and force) as the motor control strategy [30].
One model of the human standing-balance mechanism is shown in Figure DP12.6. Consider the case of a paraplegic who has lost control of his standing mechanism. We propose to add an artificial controller to enable the person to stand and move his legs. (a) Design a controller when the normal values of the parameters are \(K = 10,a = 12\), and \(b = 100\), in order to achieve a step response with percent overshoot of P.O. \(\leq 10\%\), steady-state error of \(e_{ss} \leq 5\%\), and a settling time (with a \(2\%\) criterion) of \(T_{s} \leq 2\text{ }s\). Try a controller with proportional gain, PI, PD, and PID. (b) When the person is fatigued, the parameters may change to \(K = 15,a = 8\), and \(b = 144\). Examine the performance of this system with the controllers of part (a). Prepare a table contrasting the results of parts (a) and (b).
P12.7 The goal is to design an elevator control system so that the elevator will move from floor to floor rapidly and stop accurately at the selected floor (Figure DP12.7). The elevator will contain from one to three occupants. However, the weight of the elevator should be greater than the weight of the occupants; you may assume that the elevator weighs 1000 pounds and each occupant weighs 150 pounds. Design a system to accurately control the elevator to within one centimeter. Assume that the large DC motor is field-controlled. Also, assume that the time constant of the motor and load is one second, the time constant of the power amplifier driving the motor is one-half
FIGURE DP12.6
Artificial control of standing and leg articulation.
FIGURE DP12.7
Elevator position control.
FIGURE DP12.8
Feedback control system for an electric ventricular assist device.
second, and the time constant of the field is negligible. We seek a percent overshoot of P.O. \(\leq 6\%\) and a settling time (with a \(2\%\) criterion) of \(T_{S} \leq 4\text{ }s\).
DP12.8 A model of the feedback control system is shown in Figure DP12.8 for an electric ventricular assist device. This problem was introduced in AP9.11. The motor, pump, and blood sac can be modeled by a time delay with \(T = 1\text{ }s\). The goal is to achieve a step response with less than 5% steady-state error and P.O. \(\leq 10\%\). Furthermore, to prolong the batteries, the voltage is limited to \(30\text{ }V\) [26]. Design a controller using (a) \(G_{c}(s) = K/s\), (b) a PI controller, and (c) a PID controller. In each case, also design the pre-filter. Compare the results for the three controllers by recording in a table the percent overshoot, peak time, settling time (with \(2\%\) criterion) and the maximum value of \(v(t)\).
DP12.9 One arm of a space robot is shown in Figure DP12.9(a). The block diagram for the control of the arm is shown in Figure DP12.9(b).
(a) If \(G_{c}(s) = K\), determine the gain necessary for a percent overshoot of P.O. \(= 4.5\%\), and plot the step response. (b) Design a proportional plus derivative (PD) controller using the ITAE method and \(\omega_{n} = 10\). Determine the required prefilter \(G_{p}(s)\). (c) Design a PI controller and a prefilter using the ITAE method. (d) Design a PID controller and a prefilter using the ITAE method with \(\omega_{n} = 10\). (e) Determine the effect of a unit step disturbance for each design. Record the maximum value of \(y(t)\) and the final value of \(y(t)\) for the disturbance input. (f) Determine the overshoot, peak time, and settling time (with a \(2\%\) criterion) step \(R(s)\) for each design above. ( \(g\) ) The process is subject to variation due to load changes. Find the magnitude of the sensitivity at \(\omega = 5,\left| S_{G}^{T}(j5) \right|\), where
(h) Based on the results of parts (e), (f), and (g), select the best controller.
DP12.10 A photovoltaic system is mounted on a space station in order to develop the power for the station. The photovoltaic panels should follow the Sun with
FIGURE DP12.9
Space robot control. (a)
(b) good accuracy in order to maximize the energy from the panels. The unity feedback control system uses a DC motor, so that the transfer function of the panel mount and the motor is
where \(b = 10\). Design a controller \(G_{c}(s)\) assuming that an optical sensor is available to accurately track the sun's position.
The goal is to design \(G_{c}(s)\) so that (1) the percent overshoot to a unit step is \(P.O. \leq 15\%\) and (2) the settling time is \(T_{s} \leq 0.75\text{ }s\). Examine the robustness of the system when \(b\) varies by \(\pm 10\%\).
DP12.11 Electromagnetic suspension systems for aircushioned trains are known as magnetic levitation (maglev) trains. One maglev train uses a superconducting magnet system [17]. It uses superconducting coils, and the levitation distance \(x(t)\) is inherently unstable. The model of the levitation is
FIGURE DP12.12
Two-mass cart system. where \(V(s)\) is the coil voltage; \(\tau_{1}\) is the magnet time constant; and \(\omega_{1}\) is the natural frequency. The system uses a position sensor with a negligible time constant. A train traveling at \(250\text{ }km/hr\) would have \(\tau_{1} = 0.75\text{ }s\) and \(\omega_{1} = 75rad/s\). Determine a controller in a unity feedback system that can maintain steady, accurate levitation when disturbances occur along the railway.
DP12.12 A benchmark problem consists of the massspring system shown in Figure DP12.12, which represents a flexible structure. Let \(m_{1} = m_{2} = 1\) and \(0.5 \leq k \leq 2.0\) [29]. It is possible to measure \(x_{1}(t)\) and \(x_{2}(t)\) and use a controller prior to \(u(t)\). Obtain the system description, choose a control structure, and design a robust system. Determine the response of the system to a unit step disturbance. Assume that the output \(x_{2}(t)\) is the variable to be controlled.
432. COMPUTER PROBLEMS
CP12.1 A closed-loop feedback system is shown in Figure CP12.1. Use an m-file to obtain a plot of \(\left| S_{K}^{T}(j\omega) \right|\) versus \(\omega\). Plot \(|T(j\omega)|\) versus \(\omega\), where \(T(s)\) is the closedloop transfer function.
FIGURE CP12.1 Closed-loop feedback system with gain \(K\).
CP12.2 An aircraft aileron can be modeled as a first-order system
where \(p\) depends on the aircraft. Obtain a family of step responses for the aileron system in the feedback configuration shown in Figure CP12.2.
The nominal value of \(p = 15\). Compute reasonable values of \(K_{p}\) and \(K_{I}\) so that the step response (with \(p = 15)\) has \(P.O. \leq 20\%\) and \(T_{s} \leq 0.5\) s. Then, use an m-file to obtain the step responses for \(12 < p < 18\) with the controller as determined above. Plot the settling time as a function of \(p\).
FIGURE CP12.2 Closed-loop control system for the aircraft aileron. CP12.3 Consider the control system in Figure CP12.3. The value of \(J\) is known to change slowly with time, although, for design purposes, the nominal value is chosen to be \(J = 28\).
(a) Design a PID controller (denoted by \(G_{c}(s)\) ) to achieve a phase margin \(P.M. \geq 45^{\circ}\) and a bandwidth \(\omega_{B} \leq 4rad/s\). (b) Using the PID controller designed in part (a), develop an m-file script to generate a plot of the phase margin as \(J\) varies from 10 to 40 . At what \(J\) is the closed-loop system unstable.
FIGURE CP12.3 A feedback control system with compensation.
CP12.4 Consider the feedback control system in Figure CP12.4. The exact value of parameter \(b\) is unknown; however, for design purposes, the nominal value is taken to be \(b = 4\). The value of \(a = 8\) is known very precisely.
a. Design the proportional controller \(K\) so that the closed-loop system response to a unit step input has a settling time (with a \(2\%\) criterion) of \(T_{s} \leq 5\) s and a percent overshoot of P.O. \(\leq 10\%\). Use the nominal value of \(b\) in the design.
b. Investigate the effects of variations in the parameter \(b\) on the closed-loop system unit step response. Let \(b = 0,1,4\), and 40, and co-plot the step response associated with each value of \(b\). In all cases, use the proportional controller from part (a). Discuss the results.
CP12.5 A model of a flexible structure is given by
where \(\omega_{n}\) is the natural frequency of the flexible mode, and \(\zeta\) is the corresponding damping ratio. In general, it is difficult to know the structural damping precisely, while the natural frequency can be predicted more accurately using well-established modeling techniques. Assume the nominal values of \(\omega_{n} = 2rad/s,\zeta = 0.005\), and \(k = 0.1\).
a. Design a lead compensator to meet the following specifications: (1) a closed-loop system response to a unit step input with a settling time (with a \(2\%\) criterion) \(T_{s} \leq 200\text{ }s\) and (2) a percent overshoot of P.O. \(\leq 50\%\).
b. With the controller from part (a), investigate the closed-loop system unit step response with \(\zeta = 0,0.005,0.1\), and 1. Co-plot the various unit step responses and discuss the results.
c. From a control system point of view, is it preferable to have the actual flexible structure damping less than or greater than the design value? Explain.
CP12.6 The industrial process shown in Figure CP12.6 is known to have a time delay in the loop. In practice, it is often the case that the magnitude of system time delays cannot be precisely determined. The magnitude of the time delay may change in an unpredictable manner depending on the process environment. A robust control system should be able to operate satisfactorily in the presence of the system time delays.
a. Develop an m-file script to compute and plot the phase margin for the industrial process in Figure CP12.6 when the time delay, \(T\), varies between 0 and 5 seconds. Use the pade function with a second-order approximation to approximate the time delay. Plot the phase margin as a function of the time delay.
b. Determine the maximum time delay allowable for system stability. Use the plot generated in part (a) to compute the maximum time delay approximately.
FIGURE CP12.4
A feedback control system with uncertain parameter \(b\).
FIGURE CP12.6 An industrial controlled process with a time delay in the loop.
CP12.7 A unity feedback control system has the loop transfer function
We know from the underlying physics of the problem that the parameter \(a\) can vary only between \(0 < a < 1\). Develop an m-file script to generate the following plots:
a. The unit step response for the range of \(a\) given.
b. The percent overshoot, P.O., due to the unit step input versus parameter \(a\).
c. The gain margin versus the parameter \(a\).
d. Based on the results in parts (a)-(c), comment on the robustness of the system to changes in parameter \(a\) in terms of stability and transient time response.
CP12.8 The Gamma-Ray Imaging Device (GRID) is a NASA experiment to be flown on a long-duration, high-altitude balloon during the coming solar maximum. The GRID on a balloon is an instrument that will qualitatively improve hard X-ray imaging and carry out the first gamma-ray imaging for the study of solar high-energy phenomena in the next phase of peak solar activity. From its long-duration balloon platform, GRID will observe numerous hard X-ray bursts, coronal hard X-ray sources, "superhot" thermal events, and microflares [2]. Figure CP12.8(a) depicts the GRID payload attached to the balloon. The major components of the GRID experiment consist of a 5.2-meter canister and mounting gondola, a high-altitude balloon, and a cable connecting the gondola and balloon. The instrument-sun pointing requirements of the experiment are 0.1 degree pointing accuracy and 0.2 arcsecond per \(4\text{ }ms\) pointing stability.
An optical sun sensor provides a measure of the sun-instrument angle and is modeled as a first-order system with a DC gain and a pole at \(s = - 500\). A torque motor actuates the canister/gondola assembly. The azimuth angle control system is shown in Figure CP12.8(b). The PID controller is selected by the design team so that
where \(a\) and \(b\) are to be selected. A prefilter is used as shown in Figure CP12.8(b). Determine the value of \(K_{D},a\), and \(b\) so that the dominant roots have a damping ratio \(\zeta = 0.8\) and the percent overshoot to a step input is \(P.O. \leq 3\%\). Develop a simulation to study the control system performance. Use a step response to confirm the percent overshoot meets the specification.
(a)
FIGURE CP12.8 The GRID device.
(b)
433. ANSWERS TO SKILLS CHECK
True or False: (1) True; (2) False; (3) True; (4) True; (5) Word Match (in order, top to bottom): d, g, i, b, c, k, f, h, False
Multiple Choice: (6) b; (7) b; (8) c; (9) d; (10) a; (11) d; \(a,j,e\)
(12) a; (13) c; (14) a; (15) b
434. TERMS AND CONCEPTS
Additive perturbation A system perturbation model ex- Process controller See PID controller. pressed in the additive form \(G_{a}(s) = G(s) + A(s)\), where \(G(s)\) is the nominal process function, \(A(s)\) is the perturbation that is bounded in magnitude, and \(G_{a}(s)\) is the family of perturbed process functions.
Complementary sensitivity function The function \(C(s) = \frac{G_{c}(s)G(s)}{1 + G_{c}(s)G(s)}\) that satisfies the relationship \(S(s) + C(s) = 1\), where \(S(s)\) is the sensitivity function.
Internal model principle The principle that states that if \(G_{c}(s)G(s)\) contains the input \(R(s)\), then the output \(y(t)\) will track \(R(s)\) asymptotically (in the steadystate) and the tracking is robust.
Multiplicative perturbation A system perturbation model expressed in the multiplicative form \(G_{m}(s) = G(s)(1 + M(s))\), where \(G(s)\) is the nominal process function, \(M(s)\) is the perturbation that is bounded in magnitude, and \(G_{m}(s)\) is the family of perturbed process functions.
PID controller A controller with three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term.
Prefilter A transfer function \(G_{p}(s)\) that filters the input signal \(R(s)\) prior to the calculation of the error signal.
Robust control system A system that maintains acceptable performance in the presence of significant model uncertainty, disturbances, and noise.
Robust stability criterion A test for robustness with respect to multiplicative perturbations in which stability is guaranteed if \(|M(j\omega)| < \left| 1 + \frac{1}{G(j\omega)} \right|\), for all \(\omega\), where \(M(s)\) is the multiplicative perturbation.
Root sensitivity A measure of the sensitivity of the roots (i.e., the poles and zeros) of the system to changes in a parameter defined by \(S_{\alpha}^{r_{i}} = \frac{\partial r_{i}}{\partial\alpha/\alpha}\), where \(\alpha\) is the parameter and \(r_{i}\) is the root.
Sensitivity function The function \(S(s) = \left\lbrack 1 + G_{c}(s)G(s) \right\rbrack^{- 1}\) that satisfies the relationship \(S(s) + C(s) = 1\), where \(C(s)\) is the complementary sensitivity function.
System sensitivity A measure of the system sensitivity to changes in a parameter defined by \(S_{\alpha}^{T} = \frac{\partial T/T}{\partial\alpha/\alpha}\), where \(\alpha\) is the parameter and \(T\) is the system transfer function.
435. CHAPTER
436. Digital Control Systems
13.1 Introduction 946
13.2 Digital Computer Control System Applications 946
13.3 Sampled-Data Systems 948
13.4 The \(z\)-Transform 951
13.5 Closed-Loop Feedback Sampled-Data Systems 955
13.6 Performance of a Sampled-Data, Second-Order System 959
13.7 Closed-Loop Systems with Digital Computer Compensation 961
13.8 The Root Locus of Digital Control Systems 964
13.9 Implementation of Digital Controllers 968
13.10 Design Examples 968
13.11 Digital Control Systems Using Control Design Software 977
13.12 Sequential Design Example: Disk Drive Read System 982
13.13 Summary 984
437. PREVIEW
A digital computer often hosts the controller algorithm in a feedback control system. Since the computer receives data only at specific intervals, it is necessary to develop a method for describing and analyzing the performance of computer control systems. In this chapter, we provide an introduction to the topic of digital control systems. The notion of a sampled-data system is presented followed by a discussion of the \(z\)-transform. We may use the \(z\)-transform of a transfer function to analyze the stability and transient response of a system. The basics of closedloop stability with a digital controller in the loop are covered with a short presentation on the role of root locus in the design process. This chapter concludes with the design of a digital controller for the Sequential Design Example: Disk Drive Read System.
438. DESIRED OUTCOMES
Upon completion of Chapter 13, students should be able to:
$\square\ $ Explain the role of digital computers in control system design and application.
$\square\ $ Describe the \(z\)-transform and sampled-data systems.
$\square\ $ Design digital controllers using root locus methods.
$\square\ $ Identify the potential issues of implementing digital controllers.
438.1. INTRODUCTION
The use of digital computer compensator (controller) devices continues to increase as the price and reliability of digital computers improves [1,2]. A block diagram of a single-loop digital control system is shown in Figure 13.1. The digital computer in this system configuration receives the error in digital form and performs calculations in order to provide an output in digital form. The computer may be programmed to provide an output so that the performance of the process is near or equal to the desired performance. Many computers are able to receive and manipulate several inputs, so a digital computer control system can often be a multivariable system.
A digital computer receives and operates on signals in digital (numerical) form, as contrasted to continuous signals [3]. A digital control system uses digital signals and a digital computer to control a process. The measurement data are converted from analog form to digital form by means of the analog-to-digital converter shown in Figure 13.1. After processing the inputs, the digital computer provides an output in digital form. This output is then converted to analog form by the digital-to-analog converter shown in Figure 13.1.
438.2. DIGITAL COMPUTER CONTROL SYSTEM APPLICATIONS
A digital computer consists of a central processing unit (CPU), input-output units, and a memory unit. The size and power of a computer will vary according to the size, speed, and power of the CPU, as well as the size, speed, and organization of the memory unit. Powerful but inexpensive computers, called microcomputers are everywhere. These systems use a microprocessor as a CPU. Therefore, the nature of the control task, the extent of the data required in memory, and the speed of calculation required will dictate the selection of the computer within the range of available computers.
The size of computers and the cost for the active logic devices used to construct them have both declined exponentially. The active components per cubic centimeter have increased so that the actual computer can be reduced in size to the point where relatively inexpensive, powerful laptop computers are providing mobile high-performance computational capability to students and professionals alike, and are, in many instances, replacing traditional desktop microcomputers. The speed of computers has also increased exponentially. The transistor density (a measure of computational performance) on microprocessor integrated circuits has increased exponentially over the last 40 years, as illustrated in Figure 13.2. In fact, according to "Moore's law," the transistor density doubles every year, and will probably
FIGURE 13.1
A block diagram of a computer control system, including the signal converters. The signal is indicated as digital or analog.
FIGURE 13.2
The development of microprocessors measured in millions of transistors.
FIGURE 13.3
The flight deck of the Boeing 787 Dreamliner features digital control electronics. The aircraft is equipped with a complete suite of navigation and communication avionics. (Courtesy of Craig F. Walker/ Getty Images.)
continue to do so. Significant progress in computation capability has been and will continue to be made. Clearly, improvements in computational capability have revolutionized the application of control theory and design in the modern era.
Digital control systems are used in many applications: for machine tools, metal-working processes, biomedical, environmental, chemical processes, aircraft control, and automobile traffic control, and many others [4-8]. An example of a computer control system used in the aircraft industry is shown in Figure 13.3. Automatic computer-controlled systems are used for purposes as diverse as measuring the objective refraction of the human eye and controlling the engine spark timing or air-fuel ratio of automobile engines.
The advantages of using digital control include improved measurement sensitivity; the use of digitally coded signals, digital sensors and transducers, and microprocessors; reduced sensitivity to signal noise; and the capability to easily reconfigure FIGURE 13.4
A digital control system.
the control algorithm in software. Improved sensitivity results from the low-energy signals required by digital sensors and devices. The use of digitally coded signals permits the wide application of digital devices and communications. Digital sensors and transducers can effectively measure, transmit, and couple signals and devices. In addition, many systems are inherently digital because they send out pulse signals.
438.3. SAMPLED-DATA SYSTEMS
Computers used in control systems are interconnected to the actuator and the process by means of signal converters. The output of the computer is processed by a digital-to-analog converter. We will assume that all the numbers that enter or leave the computer do so at the same fixed period \(T\), called the sampling period. Thus, for example, the reference input shown in Figure 13.4 is a sequence of sample values \(r(kT)\). The variables \(r(kT),m(kT)\), and \(u(kT)\) are discrete signals in contrast to \(m(t)\) and \(y(t)\), which are continuous functions of time.
439. Sampled data (or a discrete signal) are data obtained for the system variables only at discrete intervals and are denoted as \(x(kT)\).
A system where part of the system acts on sampled data is called a sampled-data system. A sampler is basically a switch that closes every \(T\) seconds for one instant of time. Consider an ideal sampler, as shown in Figure 13.5. The input is \(r(t)\), and the output is \(r^{*}(t)\), where \(nT\) is the current sample time, and the current value of \(r^{*}(t)\) is \(r(nT)\). We then have \(r^{*}(t) = r(nT)\delta(t - nT)\), where \(\delta\) is the impulse function.
Let us assume that we sample a signal \(r(t)\), as shown in Figure 13.5, and obtain \(r^{*}(t)\). Then, we portray the series for \(r^{*}(t)\) as a string of impulses starting at \(t = 0\), spaced at \(T\) seconds, and of amplitude \(r(kT)\). For example, consider the input signal \(r(t)\) shown in Figure 13.6(a). The sampled signal is shown in Figure 13.6(b) with an impulse represented by a vertical arrow of magnitude \(r(kT)\).
A digital-to-analog converter serves as a device that converts the sampled signal \(r^{*}(t)\) to a continuous signal \(p(t)\). The digital-to-analog converter can usually be
FIGURE 13.5
An ideal sampler with an input \(r(t)\).
FIGURE 13.6
(a) An input signal \(r(t)\).
(b) The sampled signal \(r^{*}(t) =\) \(\Sigma_{k = 0}^{x}r(kT)\delta(t - kT)\).
The vertical arrow represents an impulse.
FIGURE 13.7 A sampler and zero-order hold circuit.
FIGURE 13.8
The response of a zero-order hold to an impulse input \(r(kT)\), which equals unity when \(k = 0\) and equals zero when \(k \neq 0\), so that \(r^{*}(t) = r(0)\delta(t)\).
(a)
(b)
represented by a zero-order hold circuit, as shown in Figure 13.7. The zero-order hold takes the value \(r(kT)\) and holds it constant for \(kT \leq t < (k + 1)T\), as shown in Figure 13.8 for \(k = 0\). Thus, we use \(r(kT)\) during the sampling period.
A sampler and zero-order hold can accurately follow the input signal if \(T\) is small compared to the transient changes in the signal. The response of a sampler and zero-order hold for a ramp input is shown in Figure 13.9. Finally, the response of a sampler and zero-order hold for an exponentially decaying signal is shown in Figure 13.10 for two values of the sampling period. Clearly, the output \(p(t)\) will approach the input \(r(t)\) as \(T\) approaches zero, meaning that we sample frequently. FIGURE 13.9
The response of a sampler and zero-order hold for a ramp input \(r(t) = t\).
(a) \(T = 0.5\text{ }s\)
(b) \(T = 0.2\text{ }s\)
FIGURE 13.10 The response of a sampler and zero-order hold to an input \(r(t) = e^{- t}\) for two values of sampling period \(T\). The precision of the digital computer and the associated signal converters is limited. Precision is the degree of exactness or discrimination with which a quantity is stated. The precision of the computer is limited by a finite word length. The precision of the analog-to-digital converter is limited by an ability to store its output only in digital logic composed of a finite number of binary digits. The converted signal \(m(kT)\) is then said to include an amplitude quantization error. When the quantization error and the error due to the computer finite word size are small relative to the amplitude of the signal \(\lbrack 13,16\rbrack\), the system is sufficiently precise, and the precision limitations can be neglected.
439.1. THE \(z\)-TRANSFORM
Because the output of the ideal sampler, \(r^{*}(t)\), is a series of impulses with values \(r(kT)\), we have
for a signal for \(t > 0\). Using the Laplace transform, we have
We now have an infinite series that involves multiples of \(e^{sT}\) and its powers. We define
where this relationship involves a conformal mapping from the \(s\)-plane to the \(z\)-plane. We then define a new transform, called the \(z\)-transform, so that
As an example, let us determine the \(z\)-transform of the unit step function \(u(t)\) (not to be confused with the control signal \(u(t))\). We obtain
since \(u(kT) = 1\) for \(k \geq 0\). This series can be written in closed form as \(\ ^{1}\)
In general, we will define the \(z\)-transform of a function \(f(t)\) as
\(\ ^{1}\) Recall that the infinite geometric series may be written \((1 - bx)^{- 1} = 1 + bx + (bx)^{2} + (bx)^{3} + \ldots\), if \(|bx| < 1\).
440. EXAMPLE 13.1 Transform of an exponential
Let us determine the \(z\)-transform of \(f(t) = e^{- at}\) for \(t \geq 0\). Then
Again, this series can be written in closed form as
In general, we may show that
441. EXAMPLE 13.2 Transform of a sinusoid
Let us determine the \(z\)-transform of \(f(t) = sin(\omega t)\) for \(t \geq 0\). We can write \(sin(\omega t)\) as
Then, it follows that
A table of \(z\)-transforms is given in Table 13.1 and at the MCS website. Note that we use the same letter to denote both the Laplace and \(z\)-transforms, distinguishing them by the argument \(s\) or \(z\). A table of properties of the \(z\)-transform is given in Table 13.2. As in the case of Laplace transforms, we are ultimately interested in the output \(y(t)\) of the system. Therefore, we must use an inverse transform to obtain \(y(t)\) from \(Y(z)\). We may obtain the output by (1) expanding \(Y(z)\) in a power series, (2) expanding \(Y(z)\) into partial fractions and using Table 13.1 to obtain the inverse of each term, or (3) obtaining the inverse \(z\)-transform by an inversion integral. We will limit our methods to (1) and (2) in this limited discussion.
442. EXAMPLE 13.3 Transfer function of an open-loop system
Consider the system shown in Figure 13.11 for \(T = 1\). The transfer function of the zero-order hold is
Therefore, the transfer function \(Y(s)/R^{*}(s)\) is
443. Table \(13.1\ z\)-Transforms
Expanding into partial fractions, we have
and the \(z\)-transform is
FIGURE 13.11
An open-loop, sampled-data system (without feedback).
Table 13.2 Properties of the z-Transform
| $$\mathbf{x}(\mathbf{t})$$ | $$\mathbf{X}(\mathbf{z})$$ |
| 1. \(kx(t)\) | $$kX(z)$$ |
| 2. \(x_{1}(t) + x_{2}(t)\) | $$X_{1}(z) + X_{2}(z)$$ |
| 3. \(x(t + T)\) | $$zX(z) - zx(0)$$ |
| 4. \(tx(t)\) | $$- Tz\frac{dX(z)}{dz}$$ |
-
\(e^{- at}x(t)\) \(X\left( ze^{aT} \right)\)
-
\(x(0)\), initial value \(\lim_{z \rightarrow \infty}\mspace{2mu} X(z)\) if the limit exists
-
\(x(\infty)\), final value \(\lim_{z \rightarrow 1}\mspace{2mu}(z - 1)X(z)\) if the limit exists and the system is stable; that is, if all poles of \((z - 1)X(z)\) are inside the unit circle \(|z| = 1\) on \(z\)-plane.
Using the entries of Table 13.1 to convert from the Laplace transform to the corresponding \(z\)-transform of each term, we have
When \(T = 1\), we obtain
The response of this system to a unit impulse is obtained for \(R(z) = 1\) so that \(Y(z) = G(z) \cdot 1\). We obtain \(Y(z)\) by dividing the denominator into the numerator:
FIGURE 13.12
System with sampled output.
FIGURE 13.13
The \(z\)-transform transfer function in block diagram form.
This calculation yields the response at the sampling instants and can be carried as far as is needed for \(Y(z)\). From Equation (13.5), we have
In this case, we have obtained \(y(kT)\) as follows: \(y(0) = 0,y(T) = 0.3678,y(2T) =\) 0.7675 , and \(y(3T) = 0.9145\). Note that \(y(kT)\) provides the values of \(y(t)\) at \(t = kT\).
We have determined \(Y(z)\), the \(z\)-transform of the output sampled signal. The \(z\)-transform of the input sampled signal is \(R(z)\). The transfer function in the \(z\)-domain is
Since we determined the sampled output, we can use an output sampler to depict this condition, as shown in Figure 13.12; this represents the system of Figure 13.11 with the sampled input passing to the process. We assume that both samplers have the same sampling period and operate synchronously. Then
as required. We may represent Equation (13.19), which is a \(z\)-transform equation, by the block diagram of Figure 13.13.
443.1. CLOSED-LOOP FEEDBACK SAMPLED-DATA SYSTEMS
In this section, we consider closed-loop, sampled-data control systems. Consider the system shown in Figure 13.14(a). The sampled-data \(z\)-transform model of this figure with a sampled-output signal \(Y(z)\) is shown in Figure 13.14(b). The closed-loop transfer function (using block diagram reduction) is
Here, we assume that the \(G(z)\) is the \(z\)-transform of \(G(s) = G_{0}(s)G_{p}(s)\), where \(G_{0}(s)\) is the zero-order hold, and \(G_{p}(s)\) is the process transfer function. FIGURE 13.14
Feedback control system with unity feedback. \(G(z)\) is the \(z\)-transform corresponding to \(G(s)\), which represents the process and the zero-order hold.
FIGURE 13.15
(a) Feedback control system with a digital controller. (b) Block diagram model. Note that \(G(z) =\) \(Z\left\{ G_{0}(s)G_{p}(s) \right\}\).
(a)
(b)
(a)
(b)
A digital control system with a digital controller is shown in Figure 13.15(a). The \(z\)-transform block diagram model is shown in Figure 13.15(b). The closed-loop transfer function is
444. EXAMPLE 13.4 Response of a closed-loop system
Consider the closed-loop system shown in Figure 13.16. We have obtained the \(z\)-transform model of this system, as shown in Figure 13.14. Therefore, we have
FIGURE 13.16 A closed-loop, sampled-data system.
In Example 13.3, we obtained \(G(z)\) as Equation (13.16) when \(T = 1\text{ }s\). Substituting \(G(z)\) into Equation (13.22), we obtain
Since the input is a unit step,
it follows that
Completing the division, we have
The values of \(y(kT)\) are shown in Figure 13.17, using the symbol \(\square\). The complete response of the sampled-data, closed-loop system is shown and contrasted to the response of a continuous system (when \(T = 0\) ). The overshoot of the sampled system is \(45\%\), in contrast to \(17\%\) for the continuous system. Furthermore, the settling time of the sampled system is twice as long as that of the continuous system.
of the sampled system is twice as long as that of the continuous system.\(\text{~}\text{Time (s)}\text{~}\)
FIGURE 13.17
The response of a secondorder system: (a) continuous \((T = 0)\), not sampled; (b) sampled system, \(T = 1\text{ }s\). A linear continuous feedback control system is stable if all poles of the closed-loop transfer function \(T(s)\) lie in the left half of the \(s\)-plane. The \(z\)-plane is related to the \(s\)-plane by the transformation
We may also write this relationship as
and
In the left-hand \(s\)-plane, \(\sigma < 0\); therefore, the related magnitude of \(z\) varies between 0 and 1 . Thus, the imaginary axis of the \(s\)-plane corresponds to the unit circle in the \(z\)-plane, and the inside of the unit circle corresponds to the left half of the \(s\)-plane [14].
Therefore, we can state that the stability of a sampled-data system exists if all the poles of the closed-loop transfer function \(T(z)\) lie within the unit circle of the \(z\)-plane.
445. EXAMPLE 13.5 Stability of a closed-loop system
Let us consider the system shown in Figure 13.18 when \(T = 1\) and
Recalling Equation (13.16), we note that
where \(a = 0.3678\) and \(b = 0.2644\).
The poles of the closed-loop transfer function \(T(z)\) are the roots of the equation \(1 + G(z) = 0\). We call \(q(z) = 1 + G(z) = 0\) the characteristic equation. Therefore, we obtain
When \(K = 1\), we have
Therefore, the system is stable because the roots lie within the unit circle. When \(K = 10\), we have
and the system is unstable because both roots lie outside the unit circle. This system is stable for \(0 < K < 2.39\). The locus of the roots as \(K\) varies is discussed in Section 13.8.
We notice that a second-order sampled system can be unstable with increasing gain where a second-order continuous system is stable for all values of gain (assuming both the poles of the open-loop system lie in the left half \(s\)-plane).
445.1. PERFORMANCE OF A SAMPLED-DATA, SECOND-ORDER SYSTEM
Consider the performance of a sampled second-order system with a zero-order hold, as shown in Figure 13.18, when the process is
We then obtain \(G(z)\) for the sampling period \(T\) as
where \(E = e^{- T/\tau}\). The stability of the system is analyzed by considering the characteristic equation
Because the polynomial \(q(z)\) is a quadratic and has real coefficients, the necessary and sufficient conditions for \(q(z)\) to have all its roots within the unit circle are
These stability conditions for a second-order system can be established by mapping the \(z\)-plane characteristic equation into the \(s\)-plane and checking for positive coefficients of \(q(s)\). Using these conditions, we establish the necessary conditions from Equation (13.35) as
FIGURE 13.19
The maximum percent overshoot for a second-order sampled system for a unit step input.
FIGURE 13.20 The loci of integral squared error for a second-order sampled system for constant values of \(I\).
Table 13.3 Maximum Gain for a Second-Order Sampled System
| $$T/\tau$$ | 0 | 0.1 | 0.5 | 1 | 2 | |
| Maximum | $$K\tau$$ | $$\infty$$ | 20.4 | 4.0 | 2.32 | 1.45 |
and \(K > 0,T > 0\). For this system, we can calculate the maximum gain permissible for a stable system. The maximum gain allowable is given in Table 13.3 for several values of \(T/\tau\). It is possible to set \(T/\tau = 0.1\) and vary \(K\) to obtain system characteristics approaching those of a continuous (nonsampled) system. The maximum percent overshoot of the second-order system for a unit step input is shown in Figure 13.19.
The integral squared error performance criterion can be written as
The loci of this criterion are given in Figure 13.20 for constant values of \(I\). For a given value of \(T/\tau\), we can determine the minimum value of \(I\) and the required
FIGURE 13.21
The steadystate error of a second-order sampled system for a unit ramp input \(r(t) = t,t > 0\).
value of \(K\tau\). The optimal curve shown in Figure 13.20 indicates the required \(K\tau\) for a specified \(T/\tau\) that minimizes \(I\). For example, when \(T/\tau = 0.75\), we require \(K\tau = 1\) in order to minimize the performance criterion \(I\).
The steady-state error for a unit ramp input \(r(t) = t\) is shown in Figure 13.21. For a given \(T/\tau\), we can reduce the steady-state error, but then the system yields a greater overshoot and settling time for a step input.
446. EXAMPLE 13.6 Design of a sampled system
Consider a closed-loop sampled system as shown in Figure 13.18 when
We seek to select \(T\) and \(K\) for suitable performance. We use Figures 13.19-13.21 to select \(K\) and \(T\) for \(\tau = 0.1\). Limiting the percent overshoot to P.O. \(= 30\%\) for the step input, we select \(T/\tau = 0.25\), yielding \(K\tau = 1.4\). For these values, the steadystate error for a unit ramp input is approximately \(e_{ss} = 0.6\) (see Figure 13.21).
Because \(\tau = 0.1\), we then set \(T = 0.025\text{ }s\) and \(K = 14\). The sampling rate is 40 samples per second. The percent overshoot to the step input and the steady-state error for a ramp input may be reduced if we set \(T/\tau\) to 0.1 . The percent overshoot to a step input will be P.O. \(= 25\%\) for \(K\tau = 1.6\). Using Figure 13.21, we estimate that the steady-state error for a unit ramp input is \(e_{ss} = 0.55\) for \(K\tau = 1.6\).
446.1. CLOSED-LOOP SYSTEMS WITH DIGITAL COMPUTER COMPENSATION
A closed-loop, sampled system with a digital computer used to improve the performance is shown in Figure 13.15. The closed-loop transfer function is
The transfer function of the computer is represented by
In our prior examples, \(D(z)\) was represented by a gain \(K\). As an illustration of the power of the computer as a compensator, we consider again the second-order system with a zero-order hold and process
Then (see Equation 13.16)
If we select
we cancel the pole of \(G(z)\) at \(z = 0.3678\) and have to set two parameters, \(r\) and \(K\). If we select
we have
If we calculate the response of the system to a unit step, we find that the output is equal to the input at the fourth sampling instant and thereafter. The responses for both the uncompensated and the compensated system are shown in Figure 13.22. The overshoot of the compensated system is \(4\%\), whereas the percent overshoot of the uncompensated system is \(P.O. = 45\%\). It is beyond the objective of this book
FIGURE 13.22
The response of a sampled-data second-order system to a unit step input.
FIGURE 13.23
The continuous system model of a sampled system.
to discuss all the extensive methods for the analytical selection of the parameters of \(D(z)\); other texts [2-4] can provide further information. However, we will consider two methods of compensator design: (1) the \(G_{c}(s)\)-to- \(D(z)\) conversion method (in the following paragraphs) and (2) the root locus \(z\)-plane method (in Section 13.8).
One method for determining \(D(z)\) first determines a controller \(G_{c}(s)\) for a given process \(G_{p}(s)\) for the system shown in Figure 13.23. Then, the controller is converted to \(D(z)\) for the given sampling period \(T\). This design method is called the \(G_{c}(s)\)-to- \(D(z)\) conversion method. It converts the \(G_{c}(s)\) of Figure 13.23 to \(D(z)\) of Figure 13.15 [7].
We consider a first-order compensator
and a digital controller
We determine the \(z\)-transform corresponding to \(G_{c}(s)\) and set it equal to \(D(z)\) as
Then the relationship between the two transfer functions is \(A = e^{- aT},B = e^{- bT}\), and when \(s = 0\), we require that
447. EXAMPLE 13.7 Design to meet a phase margin specification
Consider a system with a process
We will design \(G_{c}(s)\) so that we achieve a phase margin of \(P.M. = 45^{\circ}\) with a crossover frequency \(\omega_{c} = 125rad/s\). Using the Bode plot of \(G_{p}(s)\), we find that the phase margin is \(P.M. = 2^{\circ}\). Consider the phase-lead compensator
We select \(K\) in order to yield \(20\log_{10}\left| G_{c}(j\omega)G(j\omega) \right| = 0\) when \(\omega = \omega_{c} = 125rad/s\) yielding \(K = 5.0\). The compensator \(G_{c}(s)\) is to be realized by \(D(z)\), so we solve the relationships with a selected sampling period. Setting \(T = 0.003\text{ }s\), we have
Then we have
Of course, if we select another value for the sampling period, then the coefficients of \(D(z)\) would differ.
In general, we select a small sampling period so that the design based on the continuous system will accurately carry over to the \(z\)-plane. However, we should not select too small a \(T\), or the computation requirements may be more than necessary. In general, we use a sampling period \(T \approx 1/\left( 10f_{B} \right)\), where \(f_{B} = \omega_{B}/(2\pi)\), and \(\omega_{B}\) is the bandwidth of the closed-loop continuous system. The bandwidth of the system designed in Example 13.7 is \(\omega_{B} = 208rad/s\) or \(f_{B} = 33.2\text{ }Hz\). Thus, we select a period \(T = 0.003\text{ }s\).
447.1. THE ROOT LOCUS OF DIGITAL CONTROL SYSTEMS
Consider the transfer function of the system shown in Figure 13.24. Recall that \(G(s) = G_{0}(s)G_{p}(s)\). The closed-loop transfer function is
The characteristic equation is
Thus, we can plot the root locus for the characteristic equation of the sampled system as \(K\) varies. The rules for obtaining the root locus are summarized in Table 13.4.
448. EXAMPLE 13.8 Root locus of a second-order system
Consider the system shown in Figure 13.24 with \(D(z) = 1\) and \(G_{p}(s) = 1/s^{2}\). Then we obtain
FIGURE 13.24
Closed-loop system with a digital controller.
449. Table 13.4 Root Locus in the z-Plane
-
The root locus starts at the poles and progresses to the zeros.
-
The root locus lies on a section of the real axis to the left of an odd number of poles and zeros.
-
The root locus is symmetrical with respect to the horizontal real axis.
-
The root locus may break away from the real axis and may reenter the real axis. The breakaway and entry points are determined from the equation
with \(z = \sigma\). Then obtain the solution of \(\frac{dF(\sigma)}{d\sigma} = 0\).
- Plot the locus of roots that satisfy
or
and
Let \(T = \sqrt{2}\) and plot the root locus. We now have
and the poles and zeros are shown on the \(z\)-plane in Figure 13.25. The characteristic equation is
Let \(z = \sigma\) and solve for \(K\) to obtain
Then obtain the derivative \(dF(\sigma)/d\sigma = 0\) and calculate the roots as \(\sigma_{1} = - 3\) and \(\sigma_{2} = 1\). The locus leaves the two poles at \(\sigma_{2} = 1\) and reenters at \(\sigma_{1} = - 3\), as shown in Figure 13.25. The unit circle is also shown in Figure 13.25. The system always has two roots outside the unit circle and is always unstable for all \(K > 0\).
We now turn to the design of a digital controller \(D(z)\) to achieve a specified response utilizing a root locus method. We will select a controller
FIGURE 13.25
Root locus for Example 13.8.
We then use \(z - a\) to cancel one pole at \(G(z)\) that lies on the positive real axis of the \(z\)-plane. Then we select \(z - b\) so that the locus of the compensated system will give a set of complex roots at a desired point within the unit circle on the \(z\)-plane.
450. EXAMPLE 13.9 Design of a digital compensator
Let us design a compensator \(D(z)\) that will result in a stable system when \(G_{p}(s)\) is as described in Example 13.8. With \(D(z) = 1\), we have an unstable system. Select
so that
If we set \(a = 1\) and \(b = 0.2\), we have
Using the equation for \(F(\sigma)\), we obtain the entry point as \(z = - 2.56\), as shown in Figure 13.26. The root locus is on the unit circle at \(K = 0.8\). Thus, the system is stable for \(K < 0.8\). If we select \(K = 0.25\), we find that the step response has a percent overshoot of \(P.O. = 20\%\) and a settling time (with a \(2\%\) criterion) \(T_{s} = 8.5\text{ }s\).
We can draw lines of constant \(\zeta\) on the \(z\)-plane. The mapping between the \(s\)-plane and the \(z\)-plane is obtained by the relation \(z = e^{sT}\). The lines of constant \(\zeta\) on the \(s\)-plane are radial lines with
FIGURE 13.26
Root locus for Example 13.9.
FIGURE 13.27 Curves of constant \(\zeta\) on the \(z\)-plane.
Since \(s = \sigma + j\omega\), we have
where
The plot of these lines for constant \(\zeta\) is shown in Figure 13.27 for a range of T. A common value of \(\zeta\) for many design specifications is \(\zeta = 1/\sqrt{2}\). Then we have \(\sigma = - \omega\) and
where \(\theta = \omega T\).
450.1. IMPLEMENTATION OF DIGITAL CONTROLLERS
Consider the PID controller with an \(s\)-domain transfer function
We can determine a digital implementation of this controller using a discrete approximation for the derivative and integration. For the time derivative, we use the backward difference rule
The \(z\)-transform of Equation (13.55) is then
The integration of \(x(t)\) can be represented by the forward rectangular integration at \(t = kT\) as
where \(u(kT)\) is the output of the integrator at \(t = kT\). The \(z\)-transform of Equation (13.56) is
and the transfer function is then
Hence, the \(z\)-domain transfer function of the PID controller is
The complete difference equation algorithm that provides the PID controller is obtained by adding the three terms to obtain [we use \(x(kT) = x(k)\) ]
Equation (13.58) can be implemented using a digital computer or microprocessor. Of course, we can obtain a PI or PD controller by setting an appropriate gain equal to zero.
450.2. DESIGN EXAMPLES
In this section we present two illustrative examples. In the first example, two controllers are designed to control the motor and lead screw of a movable worktable. Using a zero-order hold formulation, a proportional controller and a lead compensator FIGURE 13.28
A table motion control system: (a) actuator and table; (b) block diagram.
(a)
(b)
are obtained and their performance compared. In the second example, a control system is designed to control an aircraft control surface as part of a fly-by-wire system. Using root locus methods, the design process focuses on the design of a digital controller to meet settling time and percent overshoot performance specifications.
451. EXAMPLE 13.10 Worktable motion control system
An important positioning system in manufacturing systems is a worktable motion control system. The system controls the motion of a worktable at a certain location [18]. We assume that the table is activated in each axis by a motor and lead screw, as shown in Figure 13.28(a). We consider the \(x\)-axis and examine the motion control for a feedback system, as shown in Figure 13.28(b). The goal is to obtain a fast response with a rapid rise time and settling time to a step command while not exceeding a percent overshoot of P.O. \(= 5\%\).
The specifications are then (1) a percent overshoot equal to P.O. \(= 5\%\) and (2) a minimum settling time (with a \(2\%\) criterion) and rise time.
To configure the system, we choose a power amplifier and motor so that the system is described by Figure 13.29. Obtaining the transfer function of the motor and power amplifier, we have
We will initially use a continuous system and design \(G_{c}(s)\) as described in Section 13.8. We then obtain \(D(z)\) from \(G_{c}(s)\). Consider the controller
FIGURE 13.29 Model of the wheel control for a work table.
FIGURE 13.30
Root locus for \(L(s) =\) \(KG_{c}(s)G_{P}(s)\) where \(G_{c}(s) = K(s + a)/\) \((s + b),a = 30\) and \(b = 25\).
The root locus is shown in Figure 13.30 when \(a = 30\) and \(b = 25\). In Figure 13.30, the desired region for the pole placement is shown consistent with a targeted percent overshoot P.O. \(\leq 5\%\) (corresponding to \(\zeta \geq 0.69\) ). The selected point corresponds to \(K = 545\). The actual percent overshoot is \(P.O. = 5\%\), the settling time is \(T_{s} = 1.18\text{ }s\), and the rise time \(T_{r} = 0.4\text{ }s\), therefore, the performance specifications are satisfied. The final controller design is
The closed-loop system bandwidth is \(\omega_{B} = 5.3rad/s\) ( or \(f_{B} = 0.85\text{ }Hz\) ). Hence, the sampling frequency is selected to be \(T = 1/\left( 10f_{B} \right) = 0.12\text{ }s\). Following the design strategy in Section 13.7, we determine that
The digital controller is then given by
Using this \(D(z)\), we expect a response very similar to that obtained for the continuous system model.
452. EXAMPLE 13.11 Fly-by-wire aircraft control surface
Increasing constraints on weight, performance, fuel consumption, and reliability created a need for the flight control system known as fly-by-wire. This approach implies that particular system components are interconnected electrically rather than mechanically and that they operate under the supervision of a computer responsible for monitoring, controlling, and coordinating the tasks. The fly-by-wire principle allows for the implementation of totally digital and highly redundant control systems reaching a remarkable level of reliability and performance [19].
Operational characteristics of a flight control system depend on the dynamic stiffness of an actuator, which represents its ability to maintain the position of the control surface in spite of the disturbing effects of random external forces. One flight actuator system consists of a special type of DC motor, driven by a power amplifier, which drives a hydraulic pump that is connected to either side of a hydraulic cylinder. The piston of the hydraulic cylinder is directly connected to a control surface of an aircraft through some appropriate mechanical linkage, as shown in Figure 13.31. The elements of the design process emphasized in this example are highlighted in Figure 13.32.
The process model is given by
The zero-order hold is modeled by
Combining the process and the zero-order hold in series yields
The control goal is to design a compensator, \(D(z)\), so that the control surface angle \(Y(s) = \theta(s)\) tracks the desired angle, denoted by \(R(s)\). We state the control goal as
453. Control Goal
Design a controller \(D(z)\) so that the control surface angle tracks the desired angle.
The variable to be controlled is the control surface angle \(\theta(t)\) :
454. Variable to Be Controlled
Control surface angle \(\theta(t)\). FIGURE 13.31
(a) Fly-by-wire aircraft control surface system and (b) block diagram. The sampling period is 0.1 second.
(a)
(b)
The design specifications are as follows:
455. Design Specifications
DS1 Percent overshoot of P.O. \(\leq 5\%\) to a unit step input.
DS2 Settling time of \(T_{S} \leq 1\text{ }s\) to a unit step input.
We begin the design process by determining \(G(z)\) from \(G(s)\). Expanding \(G(s)\) in Equation (13.63) in partial fractions yields
and
where \(Z\{ \cdot \}\) represents the \(z\)-transform. Choosing \(T = 0.1\), we have
FIGURE 13.32 Elements of the control system design process emphasized in this fly-by-wire aircraft control surface example.
Topics emphasized in this example
For a simple compensator, \(D(z) = K\), the root locus is shown in Figure 13.33. For stability we require \(K < 21\). Using an iterative approach we discover that as \(K \rightarrow 21\), the step response is very oscillatory, and the percent overshoot is too large; conversely, as \(K\) gets smaller, the settling time gets too long, although the percent overshoot decreases. In any case the design specifications cannot be satisfied with a simple proportional controller, \(D(z) = K\). We need to utilize a more sophisticated controller.
We have the freedom to select the controller type. As with control design for continuous-time systems, the choice of compensator is always a challenge and problem-dependent. Here we choose a compensator with the general structure
Therefore, the key tuning parameters are the compensation parameters:
456. Select Key Tuning Parameters
\(K,a\), and \(b\). FIGURE 13.33
Root locus for \(D(z) = K\).
For continuous systems we know that a design rule-of-thumb formula for the settling time is
where we use a \(2\%\) bound to define settling. This design rule-of-thumb is valid for second-order systems with no zeros. So to meet the \(T_{s}\) requirement, we want
where \(s_{i},i = 1,2\) are the dominant complex-conjugate poles. In the definition of the desired region of the \(z\)-plane for placing the dominant poles, we use the transform
Computing the magnitude of \(z\) yields
To meet the settling time specification, we need the \(z\)-plane poles to be inside the circle defined by
where we have used the result in Equation (13.66).
Consider the settling time requirement \(T_{s} < 1\text{ }s\). In our case \(T = 0.1\text{ }s\). From Equation (13.67) we determine that the dominant \(z\)-plane poles should lie inside the circle defined by
FIGURE 13.34
Root locus for \(D(z) = K\) with the stability and performance regions shown.
As shown previously we can draw lines of constant \(\zeta\) on the \(z\)-plane. The lines of constant \(\zeta\) on the \(s\)-plane are radial lines with
Then, with \(s = \sigma + j\omega\) and using the transform \(z = e^{sT}\), we have
For a given \(\zeta\), we can plot \(Re(z)\) vs \(Im(z)\) for \(z\) given in Equation (13.68).
If we were working with a second-order transfer function in the \(s\)-domain, we would need to have the damping ratio associated with the dominant roots be greater than \(\zeta \geq 0.69\). When \(\zeta \geq 0.69\), the percent overshoot for a second-order system (with no zeros) will be P.O. \(\leq 5\%\). The curves of constant \(\zeta\) on the \(z\)-plane will define the region in the \(z\)-plane where we need to place the dominant \(z\)-plane poles to meet the percent overshoot specification.
The root locus in Figure 13.33 is repeated in Figure 13.34 with the stability and desired performance regions included. We can see that the root locus does not lie in the intersection of the stability and performance regions. The question is how to select the controller parameters \(K,a\), and \(b\) so that the root locus lies in the desired regions.
One approach to the design is to choose \(a\) such that the pole of \(G(z)\) at \(z = 0.9048\) is cancelled. Then we must select \(b\) so that the root locus lies in the desired region. For example, when \(a = - 0.9048\) and \(b = 0.25\), the compensated root locus appears as shown in Figure 13.35. The root locus lies inside the performance region, as desired. FIGURE 13.35
Compensated root locus.
FIGURE 13.36
Closed-loop system step response.
A valid value of \(K\) is \(K = 70\). Thus the compensator is
The closed-loop step response is shown in Figure 13.36. Notice that the percent overshoot specification \((P.O. \leq 5\%)\) is satisfied, and the system response settles in less than 10 samples \((10\) samples \(= 1\) second because the sampling time is \(T = 0.1\text{ }s\) ).
456.1. DIGITAL CONTROL SYSTEMS USING CONTROL DESIGN SOFTWARE
The process of designing and analyzing sampled-data systems is enhanced with the use of interactive computer tools. Many of the control design functions for continuous-time control design have equivalent counterparts for sampled-data systems. Discrete-time transfer function model objects are obtained with the tf function. Figure 13.37 illustrates the use of tf. Model conversion can be accomplished with the functions c2d and d2c, shown in Figure 13.37. The function c2d converts continuous-time systems to discrete-time systems; the function d2c converts discrete-time systems to continuous-time systems. For example, consider the process transfer function
For a sampling period of \(T = 1\text{ }s\), we have
We can use an m-file script to obtain the \(G(z)\), as shown in Figure 13.38.
(a)
(b)
FIGURE 13.37
(a) The tf function.
(b) The c2d
function. (c) The
d2c function.
FIGURE 13.38
Using the c2d function to convert \(G(s) = G_{0}(s)G_{p}(s)\) to \(G(z)\).
Transfer function:
Sampling time: 1
The functions step, impulse, and Isim are used for simulation of sampled-data systems. The unit step response is generated by step. The step function format is shown in Figure 13.39. The unit impulse response is generated by the function impulse, and the response to an arbitrary input is obtained by the Isim function. The impulse and Isim functions are shown in Figures 13.40 and 13.41, respectively. These sampled-data system simulation functions operate in essentially the same manner as their counterparts for continuous-time (unsampled) systems. The output is \(y(kT)\) and is shown as \(y(kT)\) held constant for the period \(T\).
457. EXAMPLE 13.12 Unit step response
In Example 13.4, we considered the problem of computing the step response of a closed-loop sampled-data system. In that example, the response, \(y(kT)\), was
FIGURE 13.39 The step function generates the output \(y(kT)\) for a step input. FIGURE 13.40
The impulse function generates the output \(y(t)\) for an impulse input.
\(y =\) output response
\(T =\) simulation time vector
Gys
\(T\) should be in the form \(0:T_{s}:T_{f}\), where \(T_{s}\) is the sample time.
FIGURE 13.41
The Isim function generates the output \(y(kT)\) for an arbitrary input.
\(y =\) output response
\(T =\) simulation time vector
\(u\) : input should be sampled at the same rate as sys
computed using long division. We can compute the response \(y(kT)\) using the step function, shown in Figure 13.39. With the closed-loop transfer function given by
the associated closed-loop step response is shown in Figure 13.42. The discrete step response shown in this figure is also shown in Figure 13.17. To determine the actual continuous response \(y(t)\), we use the m-file script as shown in Figure 13.43. The zero-order hold is modeled by the transfer function
In the m-file script in Figure 13.43, we approximate the \(e^{- sT}\) term using the pade function with a second-order approximation and a sampling time of \(T = 1\text{ }s\).
FIGURE 13.42 The discrete response, \(y(kT)\), of a sampled second-order system to a unit step.
We then compute an approximation for \(G_{0}(s)\) based on the Padé approximation of \(e^{- sT}\).
The subject of digital computer compensation was discussed in Section 13.7. In the next example, we consider again the subject utilizing control design software.
458. EXAMPLE 13.13 Root locus of a digital control system
Consider
and the compensator
FIGURE 13.43 The continuous response \(y(t)\) to a unit step for the system of Figure 13.16.
with the parameter \(K\) as a variable yet to be determined. The sampling time is \(T = 1\text{ }s\). When
we have the problem in a form for which the root locus method is directly applicable. The rlocus function works for discrete-time systems in the same way as for continuous-time systems. Using a m-file script, the root locus associated with Equation (13.70) is easily generated, as shown in Figure 13.44. Remember that the stability region is defined by the unit circle in the complex plane. The function rlocfind can be used with the discrete-time system root locus in exactly the same way as for continuous-time systems to determine the value of the system gain associated with any point on the locus. Using rlocfind, we determine that \(K = 4.639\) places the roots on the unit circle. FIGURE 13.44
The rlocus function for sampled-data systems.
rlocfind(sys)
Select a point in the graphics window
Determine \(K\) at the unit circle
selected_point \(=\) boundary
458.1. SEQUENTIAL DESIGN EXAMPLE: DISK DRIVE READ SYSTEM
In this chapter, we design a digital controller for the disk drive system. As the disk rotates, the sensor head reads the patterns used to provide the reference error information. This error information pattern is read intermittently as the head reads the stored data, and then the pattern in turn. Because the disk is rotating at a constant speed, the time \(T\) between position-error readings is a constant. This sampling period is typically \(100\mu s\) to \(1\text{ }ms\) [20]. Thus, we have sampled error information. We may also use a digital controller, as shown in Figure 13.45, to achieve a satisfactory system response. In this section, we will design \(D(z)\).
First, we determine
Since
FIGURE 13.45
Feedback control system with a digital controller. Note that \(G(z) =\) \(Z\left\lbrack G_{0}(s)G_{p}(s) \right\rbrack\).
we have
We note that for \(s = 20\) and \(T = 1\text{ }ms,e^{- sT} = 0.98\). Then we see that the pole at \(s = - 20\) in Equation (13.71) has an insignificant effect. Therefore, we could approximate
Then we need
We need to select the digital controller \(D(z)\) so that the desired response is achieved for a step input. If we set \(D(z) = K\), then we have
The root locus for this system is shown in Figure 13.46. When \(K = 4000\),
Therefore, the closed-loop transfer function is
We expect a rapid response for the system. The percent overshoot to a step input is P.O. \(= 0\%\), and the settling time is \(T_{s} = 2\text{ }ms\).
458.2. SUMMARY
The use of a digital computer as the compensation device for a closed-loop control system has grown during the past two decades as the price and reliability of computers have improved dramatically. A computer can be used to complete many calculations during the sampling interval \(T\) and to provide an output signal that is used to drive an actuator of a process. Computer control is used today for chemical processes, aircraft control, machine tools, and many common processes.
The \(z\)-transform can be used to analyze the stability and response of a sampled system and to design appropriate systems incorporating a computer. Computer control systems have become increasingly common as low-cost computers have become readily available.
459. SKILLS CHECK
In this section, we provide three sets of problems to test your knowledge: True or False, Multiple Choice, and Word Match. To obtain direct feedback, check your answers with the answer key provided at the conclusion of the end-of-chapter problems. Use the block diagram in Figure 13.47 as specified in the various problem statements.
FIGURE 13.47 Block diagram for the Skills Check.
In the following True or False and Multiple Choice problems, circle the correct answer.
- A digital control system uses digital signals and a digital computer to control a process.
True or False
- The sampled signal is available only with limited precision.
True or False
- Root locus methods are not applicable to digital control system design and analysis.
True or False 4. A sampled system is stable if all the poles of the closed-loop transfer function lie outside the unit circle of the \(z\)-plane.
- The \(z\)-transform is a conformal mapping from the \(s\)-plane to the \(z\)-plane by the relation \(z = e^{sT}\).
True or False
- Consider the function in the \(s\)-domain
Let \(T\) be the sampling time. Then, in the \(z\)-domain the function \(Y(s)\) is
a. \(Y(z) = \frac{5}{6}\frac{z}{z - 1} - \frac{5}{4}\frac{z}{z - e^{- 2T}} + \frac{5}{12}\frac{z}{z - e^{- 6T}}\)
b. \(Y(z) = \frac{5}{6}\frac{z}{z - 1} - \frac{5}{4}\frac{z}{z - e^{- 6T}} + \frac{5}{12}\frac{z}{z - e^{- T}}\)
c. \(Y(z) = \frac{5}{6}\frac{z}{z - 1} - \frac{z}{z - e^{- 6T}} + \frac{5}{12}\frac{z}{z - e^{- 2T}}\)
d. \(Y(z) = \frac{1}{6}\frac{z}{z - 1} - \frac{z}{1 - e^{- 2T}} + \frac{5}{6}\frac{z}{1 - e^{- 6T}}\)
- The impulse response of a system is given by
Determine the values of \(y(nT)\) at the first four sampling instants.
a. \(y(0) = 1,y(T) = 27,y(2T) = 647,y(3T) = 660.05\)
b. \(y(0) = 0,y(T) = 27,y(2T) = 47,y(3T) = 60.05\)
c. \(y(0) = 1,y(T) = 27,y(2T) = 674.4,y(3T) = 16845.8\)
d. \(y(0) = 1,y(T) = 647,y(2T) = 47,y(3T) = 27\)
- Consider a sampled-data system with the closed-loop system transfer function
This system is:
a. Stable for all finite \(K\).
b. Stable for \(- 0.5 < K < \infty\).
c. Unstable for all finite \(K\).
d. Unstable for \(- 0.5 < K < \infty\).
- The characteristic equation of a sampled system is
where \(K > 0\). The range of \(K\) for a stable system is:
a. \(0 < K < 2.63\)
b. \(K \geq 2.63\)
c. The system is stable for all \(K > 0\).
d. The system is unstable for all \(K > 0\). 10. Consider the unity feedback system in Figure 13.47, where
with the sampling time \(T = 0.4\text{ }s\). The maximum value for \(K\) for a stable closed-loop system is which of the following:
a. \(K = 7.27\)
b. \(K = 10.5\)
c. Closed-loop system is stable for all finite \(K\).
d. Closed-loop system is unstable for all \(K > 0\).
In Problems 11 and 12, consider the sampled data system in Figure 13.47 where
-
The closed-loop transfer function \(T(z)\) of this system with sampling at \(T = 1\text{ }s\) is
a. \(T(z) = \frac{1.76z + 1.76}{z^{2} + 3.279z + 2.76}\)
b. \(T(z) = \frac{z + 1.76}{z^{2} + 2.76}\)
c. \(T(z) = \frac{1.76z + 1.76}{z^{2} + 1.519z + 1}\)
d. \(T(z) = \frac{z}{z^{2} + 1}\) -
The unit step response of the closed-loop system is:
a. \(Y(z) = \frac{1.76z + 1.76}{z^{2} + 3.279z + 2.76}\)
b. \(Y(z) = \frac{1.76z + 1.76}{z^{3} + 2.279z^{2} - 0.5194z - 2.76}\)
c. \(Y(z) = \frac{1.76z^{2} + 1.76z}{z^{3} + 2.279z^{2} - 0.5194z - 2.76}\)
d. \(Y(z) = \frac{1.76z^{2} + 1.76z}{2.279z^{2} - 0.5194z - 2.76}\)
In Problems 13 and 14, consider the sampled data system with a zero-order hold where
-
The closed-loop transfer function \(T(z)\) of this system using a sampling period of \(T = 0.5\text{ }s\) is which of the following:
a. \(T(z) = \frac{1.76z + 1.76}{z^{2} + 2.76}\)
b. \(T(z) = \frac{0.87z + 0.23}{z^{2} - 0.14z + 0.24}\)
c. \(T(z) = \frac{0.87z + 0.23}{z^{2} - 1.01z + 0.011}\)
d. \(T(z) = \frac{0.92z + 0.46}{z^{2} - 1.01z}\) 14. The range of the sampling period \(T\) for which the closed-loop system is stable is:
a. \(T \leq 1.12\)
b. The system is stable for all \(T > 0\).
c. \(1.12 \leq T \leq 10\)
d. \(T \leq 4.23\) -
Consider a continuous-time system with the closed-loop transfer function
Using a zero-order hold on the inputs and a sampling period of \(T = 0.02\text{ }s\), determine which of the following is the equivalent discrete-time closed-loop transfer function representation:
a. \(T(z) = \frac{0.019z - 0.019}{z^{2} + 2.76}\)
b. \(T(z) = \frac{0.87z + 0.23}{z^{2} - 0.14z + 0.24}\)
c. \(T(z) = \frac{0.019z - 0.019}{z^{2} - 1.9z + 0.9}\)
d. \(T(z) = \frac{0.043z - 0.02}{z^{2} + 1.9231}\)
In the following Word Match problems, match the term with the definition by writing the correct letter in the space provided.
a. Precision
b. Digital computer compensator
c. \(z\)-plane
d. Backward difference rule
e. Minicomputer
f. Sampled-data system
g. Sampled data
h. Digital control system
i. Microcomputer
j. Forward rectangular integration
k. Stability of a sampled-data system
A system where part of the system acts on sampled data (sampled variables).
The stable condition exists when all the poles of the closed-loop transfer function \(T(z)\) are within the unit circle on the \(z\)-plane.
The plane with the vertical axis equal to the imaginary part of \(z\) and the horizontal axis equal to the real part of \(z\).
A control system using digital signals and a digital computer to control a process.
Data obtained for the system variables only at discrete intervals.
The period when all the numbers leave or enter the computer.
A conformal mapping from the \(s\)-plane to the \(z\)-plane by the relation \(z = e^{sT}\).
The sampled signal available only with a limited precision.
A system that uses a digital computer as the compensator element.
A computational method of approximating the time derivative of a function.
A computational method of approximating the integration of a function.
- Amplitude quantization error
m. PID controller
n. \(z\)-transform
o. Sampling period
p. Zero-order hold
A small personal computer (PC) based on a microprocessor.
A stand-alone computer with size and performance between a microcomputer and a large mainframe.
A controller with three terms in which the output is the sum of a proportional term, an integral term, and a differentiating term.
The degree of exactness or discrimination with which a quantity is stated.
A mathematical model of a sample and data hold operation.
460. EXERCISES
E13.1 State whether the following signals are discrete or continuous:
(a) Elevation contours on a map.
(b) Temperature in a room.
(c) Digital clock display.
(d) The score of a soccer game.
(e) The output of a loudspeaker.
E13.2 (a) Find the values \(y(kT)\) when
for \(k = 0\) to 3 .
(b) Obtain a closed form of solution for \(y(kT)\) as a function of \(k\).
Answer: \(y(0) = 0,y(T) = 2,y(2T) = 8,y(3T) = 26\); \(y(kT) = e^{kIn3} - 1\).
E13.3 Obtain the \(z\)-transform \(Y(z)\) for the response \(y(kT) = kT,k \geq 0\), where \(T\) is the sampling time,
(a) by using the definition \(X(z) = \sum_{k = 0}^{\infty}\mspace{2mu} x(kT)z^{- k}\),
(b) by applying the property of differentiation in \(z\)-domain, \(Z\{ tx(t)\} = - zT\frac{dX(z)}{dz}\), given that \(Z\{ u(t)\} = \frac{z}{z - 1}\).
E13.4 We have a function
Using a partial fraction expansion of \(Y(s)\) and a table of \(z\)-transforms, find \(Y(z)\) when \(T = 0.2\text{ }s\).
E13.5 The space shuttle, with its robotic arm, is shown in Figure E13.5(a). An astronaut controls the robotic arm and gripper by using a window and the TV cameras [9]. Discuss the use of digital control for this system and sketch a block diagram for the system, including a computer for display generation and control.
E13.6 Computer control of a robot to spraypaint an automobile is shown by the system in Figure E13.6(a) [1]. The system is of the type shown in Figure E13.6(b), where
and we want a phase margin of \(P.M. = 45^{\circ}\). Using frequency response methods, a compensator was developed, given by \(G_{c}(s) = \frac{0.508(s + 0.15)}{s + 0.015}\). Obtain the \(D(z)\) required when \(T = 0.05\text{ }s\),
E13.7 Find the response for the first four sampling instants for
Then, find \(y(0),y(1),y(2)\), and \(y(3)\).
E13.8 Determine whether the closed-loop system with \(T(z)\) is stable when
Answer: unstable
E13.9 (a) Determine \(y(kT)\) for \(k = 0\) to 3 when
(b) Obtain a closed form solution for \(y(kT)\) as a function of \(k\).
E13.10 A system has
with \(T = 0.01\text{ }s\) and \(\tau = 0.008\text{ }s\). (a) Find \(K\) so that the percent overshoot is P.O. \(\leq 40\%\). (b) Determine the steady-state error in response to a unit ramp input. (c) Determine \(K\) to minimize the integral squared error.
(a)
FIGURE E13.5
(a) Space shuttle and robotic arm.
(b) Astronaut
control of the arm.
FIGURE E13.6
(a) Automobile spraypaint system. (b) Closed-loop system with digital controller.
(b)
(a)
(b) FIGURE E13.10
A closed-loop sampled system.
E13.11 A system has a process transfer function
(a) Determine \(G(z)\) for \(G_{p}(s)\) preceded by a zeroorder hold with \(T = 0.15\text{ }s\). (b) Determine whether the digital system is stable. (c) Plot the impulse response of \(G(z)\) for the first 15 samples. (d) Plot the first 30 samples of the output response of \(G(z)\) when the input is a step of 0.5 unit.
E13.12 Find the \(z\)-transform of
when the sampling period is \(T = 1\text{ }s\).
E13.13 The characteristic equation of a sampled system is
Find the range of \(K\) so that the system is stable.
Answer: \(1.3 < K < 4.7\)
E13.14 A unity feedback system, as shown in Figure E13.10, has a plant
with \(T = 0.5\text{ }s\). Determine whether the system is stable when \(K = 4\). Determine the maximum value of \(K\) for stability.
E13.15 Consider the sampled-data system shown in Figure E13.15. Determine the transfer function \(G(z)\) when the sampling time \(T = 1\text{ }s\).
E13.16 Consider the sampled-data system shown in Figure E13.16. Determine the transfer function \(G(z)\) and when the sampling time \(T = 0.5\text{ }s\).
FIGURE E13.15
An open-loop sampled-data system with sampling time \(T = 1\text{ }s\).
FIGURE E13.16
An open-loop sampled-data system with sampling time \(T = 0.5\text{ }s\).
461. PROBLEMS
P13.1 The input to a sampler is \(r(t) = sin(\omega t)\), where \(\omega = 1.5\pi\). Plot the input to the sampler and the output \(r^{*}(t)\) for the first 2 seconds when \(T = 0.25\text{ }s\).
P13.2 The input to a sampler is \(r(t) = sin(\omega t)\), where \(\omega = 2\pi\). The output of the sampler enters a zeroorder hold. Plot the output of the zero-order hold \(p(t)\) for the first 2 seconds when \(T = 0.125\text{ }s\).
P13.3 A unit ramp \(r(t) = t,t > 0\), is used as an input to a process where \(G(s) = 1/(s + 1)\), as shown in Figure
P13.3. Determine the output \(y(kT)\) for the first four sampling instants.
FIGURE P13.3 Sampling system. P13.4 A closed-loop system has a hold circuit and process as shown in Figure E13.10. Determine \(G(z)\) when \(T = 0.5\text{ }s\) and
P13.5 For the system in Problem P13.4, let \(r(t)\) be a unit step input and calculate the response of the system by synthetic division for five time steps.
P13.6 Consider the closed-loop system shown in Figure E13.10, where \(G_{p}(s) = \frac{1}{(0.2s + 1)}\). Given the sampling period \(T = 0.05\text{ }s\), find the output \(Y(z)\) to a unit step input. Find the initial and final value directly from \(Y(z)\), and plot the unit step response.
P13.7 A closed-loop system is shown in Figure E13.10. This system represents the pitch control of an aircraft. The process transfer function is \(G_{p}(s) =\) \(K/\lbrack s(0.5s + 1)\rbrack\). Select a gain \(K\) and sampling period \(T\) so that the percent overshoot is limited to 0.3 for a unit step input and the steady-state error for a unit ramp input is less than 1.0.
P13.8 Consider the computer-compensated system shown in Figure E13.6(b) when \(T = 1\text{ }s\) and
Select the parameters \(K\) and \(r\) of \(D(z)\) when
Select within the range \(1 < K < 2\) and \(0 < r < 1\).
Determine the response of the compensated system and compare it with the uncompensated system.
P13.9 A suspended, mobile, remote-controlled system to bring three-dimensional mobility to professional NFL football is shown in Figure P13.9. The camera can be moved over the field as well as up and down. The motor control on each pulley is represented by Figure E13.10 with
FIGURE P13.10 Feedback control system with a digital controller.
We wish to achieve a phase margin of \(P.M. = 45^{\circ}\) using \(G_{c}(s)\). Select a suitable crossover frequency and sampling period to obtain \(D(z)\). Use the \(G_{c}(s)\)-to- \(D(z)\) conversion method.
P13.10 Consider a system as shown in Figure P13.10 with a zero-order hold, a process
and \(T = 0.1\) s. Note that \(G(z) = Z\left\{ G_{0}(s)G_{p}(s) \right\}\).
(a) Let \(D(z) = K\) and determine the transfer function \(G(z)D(z)\). (b) Determine the characteristic equation of the closed-loop system. (c) Calculate the maximum value of \(K\) for a stable system. (d) Determine \(K\) such that the percent overshoot is P.O. \(\leq 30\%\). (e) Calculate the closed-loop transfer function \(T(z)\) for \(K\) of part (d) and plot the step response. (f) Determine the location of the closedloop roots and the percent overshoot if \(K\) is one-half of the value determined in part (c). (g) Plot the step response for the \(K\) of part (f).
P13.11 (a) For the system described in Problem P13.10, design a lag compensator \(G_{c}(s)\) to achieve a percent overshoot P.O. \(\leq 30\%\) and a steady-state error of \(e_{ss} = 0.01\) for a ramp input. Assume a continuous nonsampled system with \(G_{p}(s)\).
(b) Determine a suitable \(D(z)\) to satisfy the requirements of part (a) with a sampling period \(T = 0.1\text{ }s\). Assume a zero-order hold and sampler, and use the \(G_{c}(s)\)-to- \(D(z)\) conversion method.
(c) Plot the step response of the system with the continuous-time compensator \(G_{c}(s)\) of part (a) and of the digital system with the \(D(z)\) of part (b). Compare the results.
(d) Repeat part (b) for \(T = 0.01\text{ }s\) and then repeat part (c).
(e) Plot the ramp response for \(D(z)\) with \(T = 0.1\text{ }s\) and compare it with the continuous-system response.
P13.12 The transfer function of a plant and a zero-order hold is
(a) Plot the root locus. (b) Determine the range of gain \(K\) for a stable system.
FIGURE P13.9 Mobile camera for football field. P13.13 The azimuth control system of an aircraft has a transfer function \(G_{p}(s) = \frac{(s + 3)}{s(s + 1)^{2}}\). It is implemented with a sampler and hold as shown in Figure E13.10.
(a) Find the transfer function of the plant and zero-order hold at a sampling rate \(1\text{ }Hz\).
(b) Plot the root locus, and determine the value of \(K\) so that the system is stable.
(c) Determine the value of \(K\) so that the system has two equal roots, and calculate all the roots in this case.
P13.14 A sampled-data system with a sampling period \(T = 0.05\text{ }s\) is
\(G(z) = \frac{K\left( z^{3} + 10.3614z^{2} + 9.758z + 0.8353 \right)}{z^{4} - 3.7123z^{3} + 5.1644z^{2} - 3.195z + 0.7408}\).
(a) Plot the root locus. (b) Determine \(K\) when the two real poles break away from the real axis. (c) Calculate the maximum \(K\) for stability.
P13.15 A closed-loop system with a sampler and hold, as shown in Figure E13.10, has a process transfer function
Determine the first 6 samples of \(y(kT)\) when \(T = 0.1\text{ }s\). The input signal is a unit step.
P13.16 A closed-loop system as shown in Figure E13.10 has
Calculate and plot \(y(kT)\) for \(0 \leq k \leq 10\) when \(T = 1\text{ }s\), and the input is a unit step.
P13.17 A closed-loop system, as shown in Figure E13.10, has
and \(T = 1.5\text{ }s\). Plot the root locus for \(K \geq 0\), and determine the gain \(K\) that results in the two roots of the characteristic equation on the \(z\)-circle (at the stability limit).
P13.18 A unity feedback system, as shown in Figure E13.10, has
If the system is continuous \((T = 0)\), then \(K = 1\) yields a step response with a percent overshoot of P.O. \(= 16\%\) and a settling time (with a \(2\%\) criterion) of \(T_{S} = 8\text{ }s\). Plot the response for \(0 \leq T \leq 1.2\), varying \(T\) by increments of 0.2 when \(K = 1\). Complete a table recording the percent overshoot and the settling time versus \(T\).
462. ADVANCED PROBLEMS
AP13.1 A closed-loop system, as shown in Figure E13.16, has a process
where \(a\) is adjustable to achieve a suitable response. Plot the root locus when \(a = 6\). Determine the range of \(K\) for stability when \(T = 0.5\text{ }s\).
AP13.2 A manufacturer uses an adhesive to form a seam along the edge of the material, as shown in Figure AP13.2. It is critical that the glue be applied evenly to avoid flaws; however, the speed at which the material passes beneath the dispensing head is not constant. The glue needs to be dispensed at a rate proportional to the varying speed of the material. The controller adjusts the valve that dispenses the glue [12].
The system can be represented by the block diagram shown in Figure P13.10, where \(G_{p}(s) = 5/(0.04s + 1)\) with a zero-order hold \(G_{0}(s)\). Use a controller
FIGURE AP13.2 A glue control system. that represents an integral controller. Determine \(G(z)\) \(D(z)\) for \(T = 40\text{ }ms\), and plot the root locus. Select an appropriate gain \(K\), and plot the step response.
AP13.3 A system of the form shown in Figure P13.10 has \(D(z) = K\) and
When \(T = 0.05\text{ }s\), find a suitable \(K\) for a rapid step response with a percent overshoot of P.O. \(\leq 10\%\).
AP13.4 A system of the form shown in Figure E13.10 has
Determine the range of sampling period \(T\) for which the system is stable. Select a sampling period \(T\) so that the system is stable and provides a rapid response.
AP13.5 Consider the closed-loop sampled-data system shown in Figure AP13.5. Determine the acceptable range of the parameter \(K\) for closed-loop stability.
FIGURE AP13.5
A closed-loop sampled-data system with sampling time \(T = 0.1\text{ }s\).
463. DESIGN PROBLEMS
CDP13.1 Design a digital controller for the system using the second-order model of the motor-capstan-slide as described in CDP2.1 and CDP4.1. Use a sampling period of \(T = 1\text{ }ms\) and select a suitable \(D(z)\) for the system shown in Figure P13.10. Determine the response of the designed system to a step input \(r(t)\).
DP13.1 A temperature system, as shown in Figure \(P13.10\), has a process transfer function
and a sampling period \(T\) of \(0.5\text{ }s\).
(a) Using \(D(z) = K\), select a gain \(K\) so that the system is stable. (b) The system may be slow and overdamped, and thus we seek to design a phase-lead compensator. Determine a suitable compensator \(G_{c}(s)\) and then calculate \(D(z)\). (c) Verify the design obtained in part (b) by plotting the step response of the system for the selected \(D(z)\).
DP13.2 A disk drive read-write head-positioning system has a system as shown in Figure P13.10 [11]. The process transfer function is
Accurate control using a digital compensator is required. Let \(T = 10\text{ }ms\) and design a compensator, \(D(z)\), using (a) the \(G_{c}(s) -\) to \(- D(z)\) conversion method and (b) the root locus method.
DP13.3 Vehicle traction control, which includes antiskid braking and antispin acceleration, can enhance vehicle performance and handling. The objective of this control is to maximize tire traction by preventing the wheels from locking during braking and from spinning during acceleration.
Wheel slip, the difference between the vehicle speed and the wheel speed (normalized by the vehicle speed for braking and the wheel speed for acceleration), is chosen as the controlled variable for most of the traction-control algorithm because of its strong influence on the tractive force between the tire and the road [17].
A model for one wheel is shown in Figure DP13.3 when \(y\) is the wheel slip. The goal is to minimize the slip when a disturbance occurs due to road
FIGURE DP13.3
Vehicle fraction control system.
conditions. Design a controller \(D(z)\) so that the damping ratio of the system \(\zeta = 1/\sqrt{2}\), and determine the resulting \(K\). Assume \(T = 0.1\text{ }s\). Plot the resulting step response, and find the percent overshoot and settling time (with a \(2\%\) criterion).
DP13.4 A machine-tool system has the form shown in Figure E13.6(b) with [10]
The sampling rate is chosen as \(T = 1\text{ }s\). We desire the step response to have a percent overshoot of P.O. \(\leq 20\%\) and a settling time (with a \(2\%\) criterion) of \(T_{s} \leq 10\text{ }s\). Design a \(D(z)\) to achieve these specifications.
DP13.5 Plastic extrusion is a well-established method widely used in the polymer processing industry [12]. Such extruders typically consist of a large barrel divided into several temperature zones, with a hopper at one end and a die at the other. Polymer is fed into the barrel in raw and solid form from the hopper and is pushed forward by a powerful screw. Simultaneously, it is gradually heated while passing through the various temperature zones set in gradually increasing temperatures. The heat produced by the heaters in the barrel, together with the heat released from the friction between the raw polymer and the surfaces of the barrel and the screw, eventually causes the melting of the polymer, which is then pushed by the screw out from the die, to be processed further for various purposes.
The output variables are the outflow from the die and the polymer temperature. The main controlling variable is the screw speed, since the response of the process to it is rapid.
The control system for the output polymer temperature is shown in Figure DP13.5. Select a gain \(K\) and a sampling period \(T\) to obtain a percent overshoot of P.O. \(\leq 20\%\) and \(T_{s} \leq 10\text{ }s\) for a unit step input.
DP13.6 A sampled-data system closed-loop block diagram is shown in Figure DP13.6. Design \(D(z)\) to such that the closed-loop system response to a unit step response has a percent overshoot P.O. \(\leq 12\%\) and a settling time \(T_{s} \leq 20\text{ }s\).
FIGURE DP13.5
Control system for an extruder.
FIGURE DP13.6
A closed-loop sampled-data system with sampling time \(T = 1\text{ }s\).
(a)
(b)
464. COMPUTER PROBLEMS
CP13.1 Develop an \(m\)-file to plot the unit step response of the system
Verify graphically that the steady-state value of the output is 1 .
CP13.2 Convert the following continuous-time transfer functions to sampled-data systems using the c2d function. Assume a sample period of 1 second and a zero-order hold \(G_{0}(s)\).
(a) \(G_{p}(s) = \frac{1}{s}\)
(b) \(G_{p}(s) = \frac{s}{s^{2} + 2}\)
(c) \(G_{p}(s) = \frac{s + 4}{s + 3}\)
(d) \(G_{p}(s) = \frac{1}{s(s + 8)}\)
CP13.3 The closed-loop transfer function of a sampleddata system is given by
(a) Compute the unit step response of the system using the dstep function, and assume a sampling period. of \(T = 0.1\text{ }s\). (b) Determine the continuous-time transfer function equivalent of \(T(z)\) using the \(d2c\) function, and assume a sampling period of \(T = 0.1\text{ }s\). (c) Compute the unit step response of the continuous (nonsampled) system using the step function, and compare the plot with part (a).
CP13.4 Plot the root locus for the system
Find the range of \(K\) for stability.
FIGURE CP13.5
Control system with a digital controller.
CP13.5 Consider the feedback system in Figure CP13.5. Obtain the root locus, and determine the range of \(K\) for stability.
CP13.6 Consider the sampled-data system with the loop transfer function
(a) Plot the root locus using the rlocus function.
(b) From the root locus, determine the range of \(K\) for stability.
CP13.7 An industrial grinding process is given by the transfer function [15]
The objective is to use a digital computer to improve the performance, where the transfer function of the computer is represented by \(D(z)\). The design specifications are (1) phase margin of \(P.M. \geq 45^{\circ}\), and (2) settling time (with a \(2\%\) criterion) of \(T_{s} \leq 1\text{ }s\).
(a) Design a controller
to meet the design specifications. (b) Assuming a sampling time of \(T = 0.02\text{ }s\), convert \(G_{c}(s)\) to \(D(z)\). (c) Simulate the continuous-time, closed-loop system with a unit step input. (d) Simulate the sampled-data, closed-loop system with a unit step input. (e) Compare the results in parts (c) and (d) and comment.
465. ANSWERS TO SKILLS CHECK
True or False: (1) True; (2) True; (3) False; (4) False; (5) True
Multiple Choice: (6) a; (7) c; (8) a; (9) d; (10) a; (11) a; (12) c; (13) b; (14) a; (15) c
Word Match (in order, top to bottom): f, k, c, h, g, o, n, \(l,b,d,j,i,e,m,a,p\)
466. TERMS AND CONCEPTS
Amplitude quantization error The sampled signal available only with a limited precision. The error between the actual signal and the sampled signal.
Backward difference rule A computational method of approximating the time derivative of a function given by \(\overset{˙}{x}(kT) \approx \frac{x(kT) - x((k - 1)T)}{T}\), where \(t = kT,T\) is the sample time, and \(k = 1,2,\ldots\)
Digital computer compensator A system that uses a digital computer as the compensator element.
Digital control system A control system using digital signals and a digital computer to control a process.
Forward rectangular integration A computational method of approximating the integration of a function given by $\ x(kT) \approx x((k - 1)T) + T\overset{˙}{x}((k - 1)T),\ $ where \(t = kT,T\) is the sample time, and \(k = 1,2,\ldots\).
Microcomputer A small personal computer (PC) based on a microprocessor.
PID controller A controller with three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term, given by
Precision The degree of exactness or discrimination with which a quantity is stated.
Sampled data Data obtained for the system variables only at discrete intervals. Data obtained once every sampling period.
Sampled-data system A system where part of the system acts on sampled data (sampled variables).
Sampling period The period when all the numbers leave or enter the computer. The period for which the sampled variable is held constant.
Stability of a sampled-data system The stable condition exists when all the poles of the closed-loop transfer function \(T(z)\) are within the unit circle on the \(z\)-plane.
\(z\)-plane The plane with the vertical axis equal to the imaginary part of \(z\) and the horizontal axis equal to the real part of \(z\).
\(z\)-transform A conformal mapping from the s-plane to the \(z\)-plane by the relation \(z = e^{sT}\). A transform from the \(s\)-domain to the \(z\)-domain.
Zero-order hold A mathematical model of a sample and data hold operation whose input-output transfer function is represented by \(G_{o}(s) = \frac{1 - e^{- sT}}{s}\).
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Absolute stability, 395, 445
Acceleration error constant, 339, 393
Acceleration input, steady-state error, 339
Accelerometer, 107
Ackermann's formula, 812, 823-824, 828, 833-834, \(859 - 860,870\)
Across-variable, 81,83
Active noise control system, 76
Actuator, 30, 100, 182
Additive perturbation, 888, 944
Advanced driver-assistance (ADAS) systems, 73
Agricultural systems, 45
AGV. See Automated guided vehicle (AGV)
Aircraft, 49
unmanned, 44-45
Aircraft attitude control, 355-356
Airplane control, 309
All-pass network, 565-566, 620
Alternative signal-flow graph, and block diagram models, 205-208
Ambler, 577
Amplidyne, 166
Amplifier, feedback, 263-264
Amplitude decay, 480,544
quantization error, 951,996
Analogous variables, 85
Analog-to-digital converter, 946, 948
Analysis of robustness, 888-890
Analytical methods, 758-759
Anesthesia, blood pressure control during, 277-285
Angle of departure, 461-462, 465, 476, 543
Angle of the asymptotes, 454, 457, 543
Antiskid braking systems, 934
Arc welding, 434
Armature-controlled motor, 102, 103, 105, 117,133, 166,178
Artificial hand, 41
Artificial intelligence (AI), 38, 45, 49
Assumptions, 80, 122-123, 182
Asymptote, 454, 543
centroid, 455,543
of root locus, 454
Asymptotic approximation, 556
for a Bode diagram, 556
Automated guided vehicle (AGV), 798-799
Automated vehicles, 39-40
Automatic control, history of, 33-39
Automatic fluid dispenser, 251
Automation, 35, 78
Automobiles
hybrid fuel vehicles, 51,78
steering control system, 39
velocity control, 496-502
Auxiliary polynomial, 403,445
Avemar ferry hydrofoil, 794
Axis shift, 408
B
Backward difference rule, 968, 996
Bandwidth, 571, 620, 650, 727
Bellman, R., 36
Biological control system, 42
Biomedical engineering, 42-43
Black, H. S., 35, 169, 884
Block diagram
models, 107-112, 182, 194-204
alternative signal-flow graph, 205-208
signal-flow graphs, 194-204
transformations, 107-112
Blood pressure control and anesthesia, 277-285
Bode, H. W., 552, 884
Bode plot, 552-553, 591-592, 620, 622
asymptotic approximation, 556
boring machine system, 275-277
Boring machine system, 275-277
Bounded response, 395
Branch, 112
Breakaway point, 457-461
Break frequency, 557, 620
C
CAE. See Computer-aided engineering (CAE)
Camera control, 379
Canonical form, 196, 254
Capek, Karel, 41
Cascade compensators, 729, 731-735, 811 Cauchy's theorem, 623, 626-630, 727
CDP. See Continuous design problem (CDP)
Centroid, asymptote, 455, 543
Characteristic equation, 90, 182, 424
Circles, constant, 650
Closed epidemic system, 410-411
Closed-loop feedback control system, 31, 78
Closed-loop feedback sampled-data system, 955-959
Closed-loop frequency response, 648, 727
Closed-loop system, 258, 320
Closed-loop transfer function, 110, 122, 182, 385,423
Command following, 835, 881
Compensation, 774
using analytical methods, 758-759
using a phase-lag network on the \(s\)-plane, 750
using a phase-lead network on the Bode diagram, 733-734
using a phase-lead network on the \(s\)-plane, 731
using integration networks, 788
using state-variable feedback, 812
Compensators, 527, 729, 811, 813
cascade, 731-735, 811
design, full-state feedback and observer, 831
Complementary sensitivity function, 888, 944
in cost of feedback, 274
Complexity of design, 46, 78
Components, 320
in cost of feedback, 274
Computer-aided engineering (CAE), 51
Computer control systems, 945, 946
for electric power plant, 41
Conditionally stable system, 707
Conformal mapping, 625, 727
Congress, 43
Constant \(M\) circles, 651
Constant \(N\) circles, 651
Continuous design problem (CDP), 75, 178, 252,
313, 387, 441, 535, 615, 720, 804, 875, 936, 993
Contour map, 624-630
Control design software digital control systems using, 977-982
state variable models using, 228-232
system performance using, 364-369
Control engineering, 30, 36-37, 39
Controllability, 813-819
matrix, 814,881
Controllable system, 814,881
Control system, 30, 78, 257
characteristics using m-files, 288
description of, 29-33
design, 47-50
future evolution of, 55-56
historical developments of, 38-39
modern examples, 39-45
Control system engineering, 30
Conv function, 139
Convolution signal, 324
Corner frequency. See Break frequency
Cost of feedback, 274
Coulomb damper, 83
Critical damping, 92, 157, 182
D
Damped oscillation, 94, 182
Dampers, 83
Damping ratio, 92, 182, 325-326, 328
dB. See Decibel (dB)
DC amplifier, 106
DC motor, 100
armature-controlled, 102, 117, 178
field controlled, 102
Deadbeat response, 762-764, 811
Decade, 554, 621
of frequencies, 554
Decibel (dB), 552, 621
Decoupled state variable model, 206
Design, 46-47, 78
Design gap, 46, 78
Design of control system, 729, 811
robot control, 441
in time-domain, 813
using a phase-lag network on the Bode diagram, \(753 - 758\)
using a phase-lag network on the \(s\)-plane, 750
using a phase-lead network on the Bode diagram, 733-734
using a phase-lead network on the \(s\)-plane, 741
using integration networks, 788
using state-feedback, 812
Design specifications, 322, 393
Detectable, 817,881
Diagonal canonical form, 206, 254
Diesel electric locomotive control, 848-854
Differential equations, \(80,97,182\)
Differential equations of physical systems, 80-85
Differential operator, 90 Differentiating circuit, 104
Digital audio tape controller, 906-914
Digital computer compensator, 961-964, 996
Digital controllers, implementation of, 968
Digital control system, 945-996 using control design software, 977-982
Digital-to-analog converter, 948
Direct-drive arm, 707
Disk drive read system, 62-63, 232-235. See also Sequential design example
Disturbance, 32,78 rejection property, 265-269
signal, 264-269, 320
Disturbance signals in feedback, 264-269
Dominant roots, 330, 393, 466, 543, 572, 587
Drebbel, Cornelis, 33
Drones, 44-45, 75
Dynamics of physical systems, 79
E
Economic systems, 43-44
Electric power industry, 41-42
Electric traction motor, 119, 132-134, 149-150 control, 132-134
Electric ventricular assist device (EVAD), 719-720
Electrohydraulic actuator, 105, 167, 722-723
Electrohydraulic servomechanisms, 708
Embedded control, 53 systems, 53
Energy storage systems (green engineering), 55
Engineering design, 46-47, 78
English channel tunnel boring system, 275-277, 288-291
Engraving machine, 587-589, 590
Environmental monitoring (green engineering), 55
Error
amplitude quantization, 951, 996
estimation, 825,881
integral of absolute magnitude of the, 344
integral of square, 344
steady-state, 272-274, 339
tracking. See Error signal
Error constants
acceleration input, 339
position, 338
ramp, 338
velocity, 338
Error signal, 144, 182, 238, 258, 320
analysis, 259-260
Error-squared performance indices, 837
Error, steady-state, 272-274
Estimation error, 825, 881
EVAD. See Electric ventricular assist device (EVAD)
Evans, R., 447
Examples of control systems, 39-45
Exponential matrix function, 190
Extender, 247-248, 440-441, 800
F
Feedback, 32
amplifier, 263-264
control system, 32, 39, 774-781
cost of, 274
disturbance signals in, 264-269
full-state control design, 819-824
negative, 32,35
positive, 69
of state variables, \(837,839,881\)
Feedback amplifier, 263-264
Feedback control system, and disturbance signals, 264-269
Feedback function, 144-147, 254
with unity feedback, 144
Feedback signal, 31, 78, 144
Feedback systems, history of, 31
Field current controlled motor, 101
Fifth-order system, 405
Final value, 92,182
of response, 92
theorem, 92, 182
Flow graph. See Signal-flow graph
Fluid flow modeling, 122-132
Flyball governor, 34, 78
Fly-by-wire aircraft control surface, 971-976
Forward rectangular integration, 968, 996
Fourier transform, 548, 621
pair, 547-548, 621
Frequency response, 546, 621
closed-loop, 648
measurements, 569-571
plots, \(548 - 569\)
using control design software, 584-589
Full-state feedback control law, 813, 881
Fundamental matrix. See Transition matrix
Future evolution of control systems, 55-56 G
Gain margin, 642, 678-679, 686, 727
Gamma-Ray Imaging Device (GRID), 943
Gear train, 106
Generative design process, 49
Global navigation satellite services, 37
Global Positioning System (GPS), 36
GPS. See Global Positioning System (GPS)
Graphical evaluation of residues, 91
Gravity gradient torque, 216
Green engineering, 54-55
applications of, 54-55
principles of, 54
GRID. See Gamma-Ray Imaging Device (GRID)
Gun controllers, 36
Gyroscope, 247
481. H
Halo orbit, 879-880
Hand, robotic, 41
Helicopter control, 522, 530
High-fidelity simulations, 129
History of automatic control, 33-39
Home appliances, 53
Homogeneity, 85-86, 183
Hot ingot robot control, 667-676
Hot ingot robot mechanism, 667
Hubble telescope, 352-354
Human-in-the-loop control, 40
Hybrid fuel automobile, 51, 78
Hybrid fuel vehicles, 51-52
Hydraulic actuator, 105, 167, 866
482. I
IAE, 344
Impulse signal, 323
Index of performance, 344-349, 393
Industrial control systems, 45
Input feedforward canonical form, 201-202, 254
Input signals, 322-324
Instability, 320
in cost of feedback, 274
Insulin
delivery control system, 57, 60-61
injections, 377-378
Integral of absolute magnitude of the error, 344
Integral of square of error, 344
Integral of time multiplied by absolute error, 344
multiplied by error squared, 344
optimum coefficient of \(T(s),347 - 348\)
Integral operator, 90
Integrating filter, 104
Integration networks, \(734,788,811\)
Integration-type compensator, 747
Intelligent vehicle/highway systems (IVHS), 497
Internal model design, 837, 845-848, 881
Internal model principle, 847, 900, 944
Internal Revenue Service, 43
The International Data Corporation, 37
Internet of Things (IoT), 37
Inverse Laplace transform, 88, 90, 92-93, 183
Inverted pendulum, 207-208, 822-824, 831-834, 873,874
IoT. See Internet of Things (IoT)
ISE, 344
ITAE, 344, 347-348
ITSE, 344
IVHS. See Intelligent vehicle/highway systems (IVHS)
J
Jordan canonical form, 206, 254
K
Kalman state-space decomposition, 814, 817, 881
Kirchhoff voltage laws, 187
Laboratory robot, 45
Lag compensator, 734
Lag network. See Phase-lag network
Laplace transform, 80, 88-95, 183, 185, 324
Laplace transform pair, 89, 547-548, 621
Lead compensator, 734 for second-order system, 738-741
for type-one system, 745-747
for type-two system, 736-738
using root locus, \(742 - 745\)
Lead-lag network, 757-758, 811
Lead network. See Phase-lead network
LEM. See Lunar excursion module (LEM)
Linear approximation, 87,183
Linear approximations of physical systems, 85-88
Linearized, 80, 183
Linear quadratic regulator, parameters, 845,881 Linear system, 85-86, 183 simplification of, 349-352, 367-368 transfer function of, 95-107
Liquid level control system, 683
Locus, 447, 543
Logarithmic magnitude, 558, 574, 621
Logarithmic (decibel) measure, 643, 727
Logarithmic plot. See Bode plot
Logarithmic sensitivity, 472, 543-544
Log-magnitude-phase diagram, 644
Loop, 113 gain, 259 on signal-flow graph, 113
Loss of gain, 320 in cost of feedback, 274
Low fidelity simulations, 129
Low-pass filter, 119, 134-136
lsim function, 231, 232
Lunar excursion module (LEM), 792
M
Machine, human versus automatic, 42
Magnetic levitation, 169, 875
Magnetic tape transport, 524
Manual control system, 41
Manual PID tuning, 479, 544
MAP. See Mean arterial pressure (MAP)
Mapping of contours in the \(s\)-plane, 624-630
Marginally stable, 397, 445
margin function, 678, 809
Margin, gain, 642, 678-679, 686, 727
phase, 647, 678-679, 686, 727, 963-964
Mars rover vehicle, 441-442, 536
Mason, 112
Mason loop rule, 158, 183
Mason's signal-flow gain formula, 112, 114, 117, 153, \(167,196,198,210 - 212,224,235\)
Mathematical models, 79-80, 183 of systems, 79
MATLAB Bode plot, 585 control system characteristics, 285 simulation of systems, 136-150 state variables and, 228-232 system performance and, 364-369
Matrix exponential function, 190, 255
Maximum overshoot, 328
Maximum power point tracking (MPPT), 119
Maximum value of the frequency response, 559, 571,621
Maxwell, J. C., 34, 38
\(M\) circles, 651
Mean arterial pressure (MAP), 281, 284
Measurement noise, 32, 78 attenuation, 267-269
Mechatronics, 50-53, 78
MEMS. See Microelectromechanical systems (MEMS)
Metallurgical industry, 45
Microcomputer, 946, 996
Microelectromechanical systems (MEMS), 51
Milling machine control system, 768-774
Minimum phase transfer function, 564, 621
Minorsky, N., 159
minreal function, 148-149
Mobile robot, 339-342
Model of, DC motor, 100 hydraulic actuator, 105, 164, 866
inverted pendulum and cart, 207-208, 822-824, 831-834, 873, 874
MPPT. See Maximum power point tracking (MPPT)
Multiloop feedback control system, 32, 78
Multiloop reduction, 147-148
Multiple-loop feedback system, 111
Multiplicative perturbation, 888, 944
Multivariable control system, 32-33, 78
Natural frequency, 92, 183, 589, 621
\(N\) circles, 651
Necessary condition, 85,183
Negative feedback, 32, 78, 431
Negative gain root locus, 488-493, 544
ngrid function, 678, 681
Nichols chart, 651-654, 678, 681-682, 687, 727, 914
nichols function, 678,679
Nodes, 113
of signal flow graph, 113
Noise, 259, 264-265, 267-269, 274, 279, 280, 296, 312-313
Nomenclature, 83
Nonminimum phase transfer functions, 562, \(565 - 566,621\)
Nontouching, 113
loops, 113-114
Nonunity feedback systems, 342-343 Nuclear reactor controls, 68-69, 793
Number of separate loci, 454,544
Numerical experiments, 129
Nyquist, H., 623
contour, 632
criterion, 630-641, 655, 686
function, 678-679
stability criterion, 622, 623, 630-641, 655, 688, 727
0
Observability, 813-819
matrix, 817, 881
Observable system, 817,881
Observer, 813, 881
design, 825-828
Octave, 555, 621
of frequencies, 555,572
Op-amp circuit, transfer function of, 97-98
Open-loop control system, 31, 78
Open-loop system, 261-263, 320
Operational amplifier, 783, 784, 921
Operators, differential and integral, 90
Optimal control system, 393, 837-845, 881
Optimization, 47, 78
Optimize parameters, 47
Optimum coefficient of for ITAE, 347-349
Optimum control system, 344
Output equation, 189, 255
Overdamped, 137, 183
Overshoot, 278-279, 288-289
483. \(\mathbf{P}\)
Padé approximation of a time delay, 657-659, 678 pade function, 678, 683, 942, 979
Papin, Dennis, 33
Parabolic input signal, 323
parallel function, 144
Parameter design, 467, 544
Parameter variations and system sensitivity, 261-264
Parkinson, D. B., 36
Path, 113
PD controller. See Proportional plus derivative (PD) controller
Peak time, 326, 393
Pendulum oscillator, 87-88
Percent overshoot, 327,393
Performance
of control system, 321 index, 344-349, 393
of sampled second-order system, 959-961
specifications in the frequency domain, 571-574
Phase-lag compensation, 734, 811
Phase-lag compensator, design of, 752-753, 754-758
Phase-lag network, 734-735, 811
on Bode diagram, 734
on the \(s\)-plane, 750
Phase-lead compensation, 734, 783, 811
Phase-lead compensator, 732
Phase-lead network, 732, 811
on Bode diagram, 733, 735
on the \(s\)-plane, 741
Phase-lock loop (detector), 435
Phase margin, 643, 647, 678-679, 686, 727
Phase variable canonical form, 198
Phase variables, 198, 255 canonical form, 198, 255
Photovoltaic generators, 119-122, 575-577
Physical state variables, 186-187
Physical systems differential equations of, \(80 - 85\)
dynamics of, 79
linear approximations of, 85-88
Physical variables, 205-206, 255
PI controller. See Proportional plus integral (PI controller)
PID controller, 281, 282, 283, 284, 285, 477-488, 544
design of robust, 896-900, 944
in discrete-time, 977,996
in frequency domain, 677
of wind turbines for clean energy, 660-663
PID tuning, 479, 544
Plant. See Process
Plants, power, 41
Plastic extrusion, 994
Plotting using MATLAB, 136-137
Pneumatica, 33
Polar plot, 549, 621
Poles, 90-91, 183
placement, 817,881
Pole-zero map, 142
Political feedback model, 44
Political systems, 43-44
poly function, \(139,424,444,879\)
polyval function, 137, 140
Polzunov, I., 34
Pontryagin, L. S., 36 Position error constant, 338, 393
Positive feedback, 69, 78
loop, 111
Potentiometer, 106
Power flow, 58
Power plants, 41
Power quality monitoring (green engineering), 55
Precision, 951, 996
speed control system, 525
Prefilter, 759-762, 811, 899-900, 944
Principle of superposition, 85, 183
Principle of the argument. See Cauchy's theorem
Printer belt drive modeling, 222-228
Process, 31, 78
controller. See PID controller
Productivity, 35, 78
Proportional plus derivative (PD) controller, 477, 544,811
Proportional plus integral (PI controller), 477, 544, \(747 - 748,811\)
Prosthetic arm, 43-44
Pseudo-quantitative feedback system, 914-916 pzmap function, 140-141, 181
484. Q
QFT. See Quantitative feedback theory (QFT)
Quantitative feedback theory (QFT), 914
Quarter amplitude decay, 480, 544
Rack and pinion, 103, 107
Radio-based navigation system, 37
Ramp input, 348 steady-state error, 338-339
test signal equation, 323
Reaction curve, 483
Reduced sensitivity, 261-262
Reference input, 144, 149, 183, 835-837
Regulator problem, 144, 820, 835, 837, 881, 948
Regulatory bodies, 43
Relative stability, 395, 407, 445
by the Nyquist criterion, 641-648
by the Routh-Hurwitz criterion, 407
Remotely controlled vehicle, 664-667, 683-686
Remote manipulators, 797
Residues, 91, 93, 94, 183
Resonant frequency, 559-560, 571-572, 621
Rise time, 326, 393
Risk, 46, 78
Robot, 41, 78
controlled motorcycle, 413-418
control system, 536
mobile, steering control, 365-368
Robot-controlled motorcycle, 413-418
Robust control system, 882-944
using control design software, 916-919
Robust PID control, 896-900
Robust stability criterion, 888-889, 944
Roll-wrapping machine (RWM), 930, 931
Root contours, 471, 544
Root locus, 447-451, 544, 689, 964-967
angle of departure, 461
asymptote, 454
breakaway point, 457
concept, 447-451
of digital control systems, 964-967
plot, obtaining, 503-508
segments on the real axis, \(453,455,544\)
sensitivity and, 472-477, 508
steps in sketching, 463
using control design software, 502-508
in the \(z\)-plane, 965-966
Root locus method, 446-466, 544
parameter design, 466-471
Root sensitivity, 472, 544
to parameters, 884,944
Roots function, 139, 142, 420, 424
Rotating disk speed control, 59-60
Rotor winder control system, 765-767, 774-781
Routh-Hurwitz criterion, 399-407, 411, 419, 434-435, 445
Routh-Hurwitz stability, 394
R.U.R. (play), 40-41
RWM. See Roll-wrapping machine (RWM)
S
Sampled data, 948, 996
Sampled-data system, 948-951, 996
Sampling period, 948, 996
Scanning tunneling microscope (STM), 938
Second-order system, 330, 824
performance of, 325-330
response, 330-335
response, effect of third pole and zero, 330-335
Self-healing process, 58
Sensitivity. See also System sensitivity
of control systems to parameter variations, 261-264 of root control systems, 473
root locus and, 472-477, 508
Sensitivity function, 260, 265, 268, 283, 296, 888,944
Sensor, 30
Separation principle, \(819,830,881\)
Sequential design example, 62-63, 150-153, 232-235, 291-295, 370-372, 425-427, 508510, 589-591, 686-689, 781-782, 860-862, 919-921, 982-984
Series connection, 143-146
series function, 143, 144, 146, 147
Settling time, 327, 393
Ship stabilization, 304
Signal-flow graph, 112-119, 183
block diagram models and, 194-204
models, 112-119
Simplification of linear systems, 349-352
Simplified model, 351-352
Simulation, 129, 183
Smart grid
control systems, 57-59
definition, 54, 57
Smart meters, 57
Social feedback model, 44
Social systems, 43-44
Solar cells, 119
Solar energy (green engineering), 55
Spacecraft, 161, 180, 214-221
Space shuttle, 608, 708-709, 988
Space station, 214-221
Space telescope, 345-349
Specifications, 46, 78
Speed control system, 265-267, 269-271, 285-288, 309, 313, 314
for automobiles, 306
for power generator, 523
s-plane, 90, 183
for steel rolling mill, 265
Spring-mass-damper system, 83, 92, 94-95, 186
Stability, 395, 445
absolute, 395, 445
concept of, 395-399
in frequency domain, 622-727
of linear feedback systems, 394-445
of a sampled-data system, 996
of second-order system, 408-410
of state variable systems, 408-411
for unstable process, 395-397 using Nyquist criterion, 630-631
using Routh-Hurwitz criterion, 399-407, 411, 419
Stabilizable, 817,881
Stabilizing controller, 831, 881
Stable system, 395, 445
State differential equation, 188-194, 255
State equation, transfer function from, 209-210
State-feedback, 812
State of a system, 185-188, 255
State-space representation, 189, 228-232, 255
State transition matrix, 191
time response and, 210-214
State variable models, 184 of dynamic system, 185-188
State variables, 185-188, 255
of dynamic system, 185-188
feedback, 248, 255, 837, 839-842, 881
system design using control design software, \(855 - 860\)
State variable systems, 408-411
stability of, 424-425
State vector, 189,255
Steady-state, 92, 183
of response of, 92, 321, 322, 393
Steady-state error, 272-274, 320, 342
of feedback control system, 337-343
Steel rolling mill, 265, 655-656, 723, 724, 931
Steering control system
of automobile, 39, 615
of mobile robot, 339-342
of ship, 711-712
Step input, 337-338
optimum coefficient of \(T(s),347\)
steady-state error, 337-338
test signal equation, 323
STM. See Scanning tunneling microscope (STM)
Submarine control system, 243, 245
Superposition, principle of, 85
Symbols, in MATLAB
used in book, 83
Synthesis, 46-47, 78
sys function, 140, 141
System design, approaches to, 730-731
Systems, 30, 78
bandwidth, 655
performance, 364-369
with uncertain parameters, 890-892
System sensitivity, 262, 320
to parameters, 884,944
485. T
Tables, 82
of Laplace transform pairs, 89
through- and across-variables for physical systems, 81
of transfer function plots, 690-697
of transfer functions, 104-107
Tachometer, 106
Taylor series, 86, 183
Temperature control system, 748-750
Test input signal, 322-324, 393
Thermal heating system, 107
Third-order system, 401-403
Through-variable, 80,81
Time constant, 96, 183
Time delay, 655-659, 727
Time domain, 185, 255
design, 813
Time-domain specifications, 364-367
Time response
by a discrete-time evaluation, 210
and state transition matrix, 210-214
and state transition matrix, 210-214
Time-varying system, 95, 255
Tracked vehicle turning control, 411-413, 420-423
Tracking error. See Error signal
Trade-off, 46, 78
Transfer function, 95, 141-143, 183
of complex system, 118-119
of DC motor, 100-107
of dynamic elements and networks, 104-107
in frequency domain, 552, 621
of interacting system, 115-117
of linear systems, 95-107
in m-file script, 141,144
minimum phase and nonminimum phase, 562,564
of multi-loop system, 117-118
of op-amp circuit, 97-98
from the state equation, 209-210
of system, 98-100
table of dynamic elements and networks, 104-107
Transient response, 320, 322, 393
control of, 269-272
relationship to root location, 335-337
of second-order system, 326
Transition matrix, 191, 255
Twin lift, 72
Twin-T network, 562
Two state variable models, 202-204
Type number, 338, 393
UAVs. See Unmanned aerial vehicles (UAVs)
Ubiquitous computing, 37,78
Ubiquitous positioning, 37,78
Ultimate gain, 483
Ultimate period, 483
Ultra-precision diamond turning machine, 903-906
Uncertain parameters, 890-892
Underdamped, 85, 92
Unit impulse, 323, 393
Unity feedback, 144-146, 175, 177, 183, 518
Unmanned aerial vehicles (UAVs), 44-45
Unstable system, 395, 397, 403-404
V
Variables
models, two state, 202-204
for physical systems, 81
Vehicle traction control, 993
Velocity error constant, 338, 393
Velocity input, 308
Vertical takeoff and landing (VTOL) aircraft, 437, 707,879
Viscous damper, 83
VTOL aircraft. See Vertical takeoff and landing (VTOL) aircraft
Vyshnegradskii, I. A., 35
W
Water clock, 33
Water level control, 33-34, 70, 125-132, 174-175
Water-level float regulator, 34
Watt, James, 34, 38
Wearable computers, 53
Welding control, 405-407
Wind energy (green engineering), 55
Wind power, 51-53
Wind turbines, 659-663
rotor speed control, 493-496
Worktable motion control,969-971
Zero-order hold, 950, 952, 996
Zeros, 90-91, 183
Zettabytes (ZB), 37
Ziegler-Nichols PID tuning method, 479, 483-487, 544
\(z\)-plane, 996
root locus, 965-966
z-transform, 951-955, 996 This page is intentionally left blank
- \(\ ^{\dagger}\) Matrix operations are discussed on the MCS website.
