机器翻译-Data-driven Design of Fault Diagnosis and Fault-tolerant Control Systems 8.2.1.2-8.4节

英文原文

8.2.1.2 Coprime Factorization Techniques

Coprime factorization of a transfer function (matrix) gives a further system
rep-resentation form which will be intensively used in our subsequent study.
Roughly speaking, a coprime factorization over

\[\mathcal{R}\mathcal{H}_{\infty}$$ is to factorize a transfer matrix into two stable and coprime transfer matrices. \]

Definition 8.1 Two stable transfer matrices $$\hat{M}(z)$$,

\[\hat{N}(z)$$ are called left coprime if \]

there exist two stable transfer matrices $$\hat{X}(z)$$ and $$\hat{Y}(z)$$ such
that

\[\begin{bmatrix} \hat{M}\left( z \right) & \hat{N}\left( z \right) \\ \end{bmatrix}\begin{bmatrix} \hat{X}\left( z \right) \\ \hat{Y}\left( z \right) \\ \end{bmatrix} = \text{I.} \]

138 8 Introduction, Preliminaries and I/O Data Set Models

Similarly, two stable transfer matrices M(z), N(z) are right coprime if there
exist two stable matrices Y(z), X(z) such that

\[\begin{bmatrix} X\left( z \right) & Y\left( z \right) \\ \end{bmatrix}\begin{bmatrix} M\left( z \right) \\ N\left( z \right) \\ \end{bmatrix} = I. \]

Let G(z) be a proper real-rational transfer matrix. The left coprime
factorization (LCF) of G(z) is a factorization of G(z) into two stable and
coprime matrices which will play a key role in designing the so-called residual
generator. To complete the notation, we also introduce the right coprime
factorization (RCF), which is however only occasionally applied in our study.

Definition 8.2 $$G\left( z \right) = {\hat{M}}^{- 1}\left( z \right)\hat{N}(z)$$
with the left coprime pair $$\left( \hat{M}\left( z \right),\hat{N}\left( z
\right) \right)$$ is

called LCF of G(z). Similarly, RCF of G(z) is defined by $$G\left( z \right)
= N\left( z \right)M\left( z \right)^{- 1}$$ with the right coprime pair

\[\left( M\left( z \right),N\left( z \right) \right)$$*.* Below, we present a lemma that provides us with a state space computation algorithm of $$\left( M\left( z \right),N\left( z \right) \right)$$*, (M(z), N(z))* and the associated pairs *X(z), Y(z)* and *(X(z), Y(z)).* **Lemma 8.1** *Suppose G(z) is a proper real-rational transfer matrix with a state space realization ( A, B, C, D), and it is stabilizable and detectable. Let F and L be so that A* + *B F and A* − *LC are Schur matrices (that is, their eigenvalues are inside the unit circle on the complex plane), and define* \]

\hat{M}\left( z \right) = \left( A - LC, - L,C,I \right),\hat{N}\left( z \right) = \left( A - LC,B - LD,C,D \right)

\[ \]

M\left( z \right) = \left( A + BF,B,F,I \right),N\left( z \right) = \left( A + BF,B,C,C + DF,D \right)

\[ \]

\hat{X}\left( z \right) = \left( A - LC, - \left( B - LD \right),F,I \right),Y\left( z \right) = \left( A - LC, - L,F,0 \right)

\[ \]

X\left( z \right) = \left( A - LC, - \left( B - LD \right),F,I \right),Y\left( z \right) = \left( A - LC, - L,F,0 \right)

\[ *Then* \]

G\left( z \right) = {\hat{M}}^{- 1}\left( z \right)\hat{N}\left( z \right) = N\left( z \right)M^{- 1}\left( z \right)

\[ *are the LCF and RCF of G(z), respectively. Moreover, the so-called Bezout identity* *holds* \]

\begin{bmatrix}
X\left( z \right) & Y\left( z \right) \

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  M\left( z \right) & - \hat{Y}\left( z \right) \
  N\left( z \right) & X\left( z \right) \
  \end{bmatrix} = \begin{bmatrix}
  I & 0 \
  0 & I \
  \end{bmatrix}\

\[ Note that \]

r\left( z \right) = \begin{bmatrix}

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  u\left( z \right) \
  y\left( z \right) \
  \end

\[ is a dynamic system with the process input and output vectors *u, y* as its inputs and the residual vector *r* as its output. It is called residual generator. In some literature, $$\begin{bmatrix} - \hat{N}\left( z \right) & \hat{M}\left( z \right) \\ \end{bmatrix}$$is also called kernel representation of system (8.2), (8.3). | 8.2 Preliminaries and Review of Model-Based FDI Schemes | 139 | |---------------------------------------------------------|-----| > **8.2.1.3 Representations of Systems With Disturbances** > Disturbances around the process under consideration, unexpected changes > within the technical process as well as measurement and process noises are > often modeled as unknown input vectors. We denote them by *d, v* or *η* and > integrate them into the state space model (8.2), (8.3) or input–output model > (8.1) as follows: - state space representation \]

x\left( k + 1 \right) = Ax\left( k \right) + Bu\left( k \right) + E_{d}d\left( k \right) + \eta\left( k \right)

\[ \]

y\left( k \right) = C_{x}\left( k \right) + D_{u}\left( k \right) + F_{d}d\left( k \right) + v\left( k \right)

\[ > with *Ed , Fd* being constant matrices of compatible dimensions, $$d \in > \mathcal{R}^{k_{d}}$$ is a deterministic unknown input vector, $$\eta \in > \mathcal{R}^{k_{\eta}},v \in \mathcal{R}^{k_{v}}$$ are, if no additional > remark is made, white, normal distributed noise vectors with $$\eta\sim > N\left( 0,\Sigma_{\eta} \right),\ v\sim N\left( 0,\Sigma_{v} \right)\]

• input–output model

\[y\left( z \right) = G_{\text{yu}}\left( z \right)u\left( z \right) + G_{\text{yd}}\left( z \right)d\left( z \right) + G_{\text{yf}}\left( z \right)f\left( z \right) \]

where Gyd(z), Gyν(z) are known.

8.2.1.4 Modeling of Faults

There exists a number of ways of modeling faults. Extending model (8.16) to

\[y\left( z \right) = G_{\text{yu}}\left( z \right)u\left( z \right) + G_{\text{yd}}\left( z \right)d\left( z \right) + G_{\text{yf}}\left( z \right)f\left( z \right) \]

is a widely adopted one, where $$f \in \mathcal{R}^{k_{f}}$$ is a unknown
vector that represents all possible faults and will be zero in the
fault-free case, $$G_{\text{yf}}(z\mathcal{) \in L}\mathcal{H}_{\infty}$$ is
a known transfer matrix. Throughout this book, f is assumed to be a
deterministic time func-tion. No further assumption on it is made, provided
that the type of the fault is not specified.

Suppose that a minimal state space realization of (8.17) is given by

\[x\left( k + 1 \right) = Ax\left( k \right) + Bu\left( k \right) + E_{d}d\left( k \right) + E_{f}f\left( k \right) \]

\[y\left( k \right) = Cx\left( k \right) + Du\left( k \right) + F_{d}d\left( k \right) + F_{f}f\left( k \right) \]

With know matrixs Ef , Ff .Then we have

\[G_{\text{yf}}\left( z \right) = F_{f} + C\left( zI - A \right)^{- 1}E_{f} \]

140 8 Introduction, Preliminaries and I/O Data Set Models

It becomes evident that Ef , Ff indicate the place where a fault occurs
and its influ-ence on the system dynamics. It is the state of the art that
faults are divided into three categories:

• sensor faults fS : these are faults that directly act on the process
  measurement

• actuator faults fA: these faults cause changes in the actuator

• process faults fP : they are used to indicate malfunctions within the
  process.

A sensor fault is often modeled by setting Ff = I, that is,

\[y\left( k \right) = C_{x}\left( k \right) + D_{u}\left( k \right) + F_{d}d\left( k \right) + fs\left( k \right) \]

while an actuator fault by setting Ef = B, Ff = D, which leads to

\[x\left( k + 1 \right) = Ax\left( k \right) + B\left( u\left( k \right) + f_{A} \right) + E_{d}d(k) \]

\[y\left( k \right) = Cx\left( k \right) + D\left( u\left( k \right) + f_{A} \right) + F_{d}d\left( k \right) \]

Depending on their type and location, process faults can be modeled by Ef
= EP and Ff = FP for some EP, FP. For a system with sensor, actuator
and process faults, we define

\[f = \begin{bmatrix} f_{A} \\ f_{P} \\ f_{S} \\ \end{bmatrix},E_{f} = \begin{bmatrix} B & E_{P} & 0 \\ \end{bmatrix},F_{f} = \begin{bmatrix} D & F_{P} & I \\ \end{bmatrix} \]

and apply (8.18), (8.19) to represent the system dynamics.

Due to the way how they affect the system dynamics, the faults described by
(8.18), (8.19) are called additive faults. It is very important to note that
the occurrence of an additive fault will not affect the system stability,
independent of the system configuration. Typical additive faults met in
practice are, for instance, an offset in sensors and actuators or a drift in
sensors. The former can be described by a constant, while the latter by a
ramp.

In practice, malfunctions in the process or in the sensors and actuators
often cause changes in the model parameters. They are called multiplicative
faults and generally modeled in terms of parameter changes.

8.2.2 Model-Based Residual Generation Schemes

Next, we introduce some standard model- and observer-based residual
generation schemes.

8.2 Preliminaries and Review of Model-Based FDI Schemes	141

8.2.2.1 Fault Detection Filter

Fault detection filter (FDF) is the first type of observer-based residual
generators proposed by Beard and Jones in the early 1970s. Their work marked the
beginning of a stormy development of model-based FDI techniques.

Core of an FDF is a full-order state observer

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + L\left( y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \right) \]

which is constructed on the basis of the nominal system model Gyu(z) =
C(zI − A)−1B + D. Built upon (8.25), the residual is simply defined
by

\[r\left( k \right) = y\left( k \right) - \hat{y}\left( k \right) = y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \]

Introducing variable

\[e\left( k \right) = x\left( k \right) - \hat{x}\left( k \right) \]

yields, on the assumption of process model (8.18), (8.19),

\[e\left( k + 1 \right) = \left( A - LC \right)e\left( k \right) + \left( E_{d} - LF_{d} \right)d\left( k \right) + \left( E_{f} - LF_{f} \right) \]

\[r\left( k \right) = Ce\left( k \right) + F_{d}d\left( k \right) + F_{f}f\left( k \right) \]

It is evident that r(k) has the characteristic features of a residual when the
observer gain matrix L is chosen so that A − LC is stable.

The advantages of an FDF lie in its simple construction form and, for the reader
who is familiar with the modern control theory, in its intimate relationship
with the state observer design and especially with the well-established robust
control theory by designing robust residual generators.

We see that the design of an FDF is in fact the determination of the observer
gain matrix L. To increase the degree of design freedom, we can switch a
matrix to the output estimation error $$y\left( z \right) - \hat{y}(z)$$, that
is

\[r\left( z \right) = V\left( y\left( z \right) - \hat{y}\left( z \right) \right) \]

A disadvantage of FDF scheme is the online implementation of the full-order
state observer, since in many practical cases a reduced order observer can
provide us with the same or similar performance but with less online
computation. This is one of the motivations for the development of Luenberger
type residual generators, also called diagnostic observers.

142 8 Introduction, Preliminaries and I/O Data Set Models

8.2.2.2 Diagnostic Observer Scheme

The diagnostic observer (DO) is, thanks to its flexible structure and
similarity to the Luenberger type observer, one of the mostly investigated
model-based residual generator forms.

The core of a DO is a Luenberger type (output) observer described by

\[z\left( k + 1 \right) = Gz\left( k \right) + Hu\left( k \right) + Ly\left( k \right) \]

\[r\left( k \right) = Vy\left( k \right) - Wz\left( k \right) - Qu\left( k \right) \]

where $$z \in \mathcal{R}^{s}$$, s denotes the observer order and can be
equal to or lower or higher than the system order n. Although most
contributions to the Luenberger type observer are focused on the first case
aiming at getting a reduced order observer, higher order observers will play
an important role in the optimization of FDI systems.

Assume Gyu(z) = C(zI − A)−1B + D, then matrices G, H, L , Q, V
and W together with a matrix $$T \in \mathcal{R}^{s \times n}$$ have to
satisfy the so-called Luenberger conditions,

\[{\text{I.\ \ G\ is\ stable}\backslash n}{II.\ \ TA - GT = LC,H = TB - LD\backslash n}{III.\ \ VC - WT = 0,Q = VD} \]

under which system (8.30), (8.31) delivers a residual vector, that is

\[\forall u,x\left( 0 \right),\operatorname{}{r\left( k \right) = 0} \]

Let e(k) = Tx(k) − z(k), it is straightforward to show that the system
dynamics of DO is, on the assumption of process model (8.18), (8.19),
governed by

\[e\left( k + 1 \right) = Ge\left( k \right) + \left( TEd - LFd \right)d\left( k \right) + \left( TE_{f} - LF_{f} \right)f\left( k \right) \]

\[r\left( k \right) = Ve\left( k \right) + VF_{d}d\left( k \right) + VF_{f}f(k) \]

Remember that in the last section it has been claimed all residual generator
design schemes can be formulated as the search for an observer gain matrix
and a post-filter. It is therefore of practical and theoretical interest to
reveal the relationships between matrices G, L , T , V and W solving
Luenberger equations (8.32)–(8.34) and observer gain matrix as well as
post-filter.

A comparison with the FDF scheme makes it clear that

• the diagnostic observer scheme may lead to a reduced order residual
  generator, which is desirable and useful for online implementation,

• we have more degree of design freedom but, on the other hand,

• more involved desig

8.2 Preliminaries and Review of Model-Based FDI Schemes

8.2.2.3 Kalman Filter Based Residual Generation

Consider (8.14), (8.15). Assume that η(k), ν(k) are white Gaussian
processes and independent of initial state vector x(0), u(k) with

\[\varepsilon\begin{bmatrix} \eta\left( i \right)\eta^{T}\left( j \right) & \eta\left( i \right)v^{T}\left( j \right) \\ v\left( i \right)\eta^{T}\left( j \right) & v\left( i \right)v^{T}\left( j \right) \\ \end{bmatrix} = \begin{bmatrix} \Sigma_{\eta} & S_{\text{ηv}} \\ S_{\text{vη}} & \Sigma_{v} \\ \end{bmatrix}\delta_{\text{ij}},\delta_{\text{ij}} = \left\{ \begin{matrix} 1,\ \ i = j \\ 0,\ \ i \neq j \\ \end{matrix} \right.\ \]

\[\Sigma_{v} > 0,\Sigma_{\eta} \geq 0,\varepsilon\left\lbrack \eta\left( k \right) \right\rbrack = 0,\varepsilon\left\lbrack v\left( k \right) \right\rbrack = 0 \]

\[\varepsilon\left\lbrack x\left( 0 \right) \right\rbrack = \overline{x},\varepsilon\left\lbrack \left( x\left( 0 \right) - \overline{x} \right)\left( x\left( 0 \right) - \overline{x} \right)^{T} \right\rbrack = P_{o} \]

A Kalman filter is, although structured similar to an observer of the
full-order, a time-varying system given by the following recursions:

recursive computation for optimal state estimation:

\[\ \hat{x}\left( 0 \right) = \overline{x} \]

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + K\left( k \right)\left( y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \right) \]

recursive computation for Kalman filter gain:

\[P\left( 0 \right) = P_{0} \]

\[P\left( K + 1 \right) = AP\left( k \right)A^{T} - K\left( k \right)R_{e}\left( k \right)K^{T}\left( k \right) + \Sigma_{\eta} \]

\[K\left( k \right) = \left( \text{AP}\left( k \right)C^{T} + S_{\text{ηv}} \right)R_{e}^{- 1}\left( k \right) \]

\[R_{e}\left( k \right) = \Sigma_{v} + CP\left( k \right)C^{T} \]

where xˆ(k) denotes the estimation of x(k) and

\[P\left( k \right) = \varepsilon\left\lbrack \left( x\left( k \right) - \hat{x}\left( k \right) \right)\left( x\left( k \right) - \hat{x}\left( k \right) \right)^{T} \right\rbrack \]

is the associated estimation error covariance.

The significant characteristics of Kalman filter is

• the state estimation is optimal in the sense of

\[P\left( k \right) = \varepsilon\left\lbrack \left( x\left( k \right) - \hat{x}\left( k \right) \right)\left( x\left( k \right) - \hat{x}\left( k \right) \right)^{T} \right\rbrack \Longrightarrow \min \]

• the so-called innovation process e(k) = y(k) −C xˆ (k)− Du(k) is a
  white Gaussian process with covariance

\[\varepsilon\left( e\left( k \right)e^{T}\left( k \right) \right) = R_{e}\left( k \right) = \Sigma_{v} + CP\left( k \right)C^{T} \]

144 8 Introduction, Preliminaries and I/O Data Set Models

Below is an algorithm for the online implementation of the Kalman filter
algorithm given by (8.40)–(8.45).

Algorithm 8.1 On-line implementation of Kalman filter

S0: Set xˆ(0), P(0) as given in (8.40) and (8.42)

S1: Calculate Re(k), K (k), xˆ(k), according to (8.45), (8.44) and (8.41)

S2: Increase k and calculate P(k + 1) according to (8.43)

S3: Go S1.

Suppose the process under consideration is stationary, then

\[\operatorname{}{K\left( k \right) =}K = constant\ matrix \]

which is subject to

\[K = \left( \text{AP}C^{T} + S_{\eta v^{T}} \right)R_{e}^{- 1} \]

With

\[P = \lim_{k \rightarrow \infty}P\left( k \right),R_{e} = \Sigma_{v} + CPC^{T} \]

It holds

\[P = \text{AP}A^{T} - KR_{e}K^{T} + \Sigma_{\eta} \]

Equation (8.49) is an algebraic Riccati equation whose solution P is positive
definite under certain conditions. It thus becomes evident that given system
model the gain matrix K can be calculated offline by solving Riccati equation
(8.49). The corre-sponding residual generator is then given by

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + K\left( y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \right) \]

\[r(k) = y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \]

Note that we now have in fact an observer of the full-order.

Remark 8.1 The offline set up (S0) in the above algorithm is needed only for
one time, but S1–S3 have to be repeated at each time instant. Thus, the online
implemen-tation, compared with the steady-state Kalman filter, is
computationally consuming. For the FDI purpose, we can generally assume that the
system under consideration is operating in its steady state before a fault
occurs. Therefore, the use of the steady-state type residual generator (8.50),
(8.51) is advantageous. In this case, the most involved computation is finding a
solution for Riccati equation (8.49), which, nevertheless, is carried out
offline, and moreover for which there exists a number of numerically reliable
methods and CAD programs.

8.2 Preliminaries and Review of Model-Based FDI Schemes	145

8.2.2.4 Parity Space Approach

The parity space approach was initiated by Chow and Willsky in their pioneering
work in the early 1980s. Although a state space model is used for the purpose of
residual generation, the so-called parity relation, instead of an observer,
builds the core of this approach. The parity space approach is generally
recognized as one of the important model-based residual generation approaches,
parallel to the observer-based and the parameter estimation schemes.

We consider in the following the state space model (8.18), (8.19) and, without
loss of generality, assume that r ank(C) = m. For the purpose of
constructing a residual generator, we first suppose f (k) = 0, d(k) = 0.
Following (8.18), (8.19), y(k − s), s > 0, can be expressed in terms of
x(k − s), u(k − s), and y(k − s + 1) in terms of x(k − s), u(k −
s + 1), u(k − s) as follows

\[y\left( k - s \right) = Cx\left( k - s \right) + Du\left( k - s \right) \]

\[{y\left( k - s + 1 \right) = Cx\left( k - s + 1 \right) + Du\left( k - s + 1 \right)\backslash n}{= CAx\left( k - s \right) + CBu\left( k - s \right) + Du(k - s + 1)} \]

Repeating this procedure yields

\[y\left( k - s + 2 \right) = CA^{2}x\left( k - 2 \right) + CABu\left( k - s \right) + CBu\left( k - s + 1 \right) + Du\left( k - x + 2 \right),\ldots, \]

\[y\left( k \right) = CA^{s}x\left( k - s \right) + CA^{s - 1}\text{Bu}\left( k - s \right) + \cdots + CBu\left( k + 1 \right) + Du\left( k \right) \]

Introducing the notations

\[y_{s}\left( k \right) = \begin{bmatrix} y\left( k - s \right) \\ y\left( k - s + 1 \right) \\ \vdots \\ y\left( k \right) \\ \end{bmatrix},u_{s}\left( k \right) = \begin{bmatrix} u\left( k - s \right) \\ u\left( k - s + 1 \right) \\ \vdots \\ u\left( k \right) \\ \end{bmatrix} \]

\[H_{o,s} = \begin{bmatrix} C \\ \text{CA} \\ \vdots \\ CA^{s} \\ \end{bmatrix},H_{u,s} = \begin{bmatrix} D & 0 & \cdots & 0 \\ \text{CB} & D & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ CA^{s - 1}B & \cdots & \text{CB} & D \\ \end{bmatrix} \]

leads to the following compact model form

\[{y_{s}(k) = H}_{o,s}x\left( k - s \right) + H_{u,s}u_{s}\left( k - s \right) \]

Note that (8.54) describes the input and output relationship in dependence on
the past state vector x(k − s). It is expressed in an explicit form, in
which

146 8 Introduction, Preliminaries and I/O Data Set Models

• ys (k) and us (k) consist of the temporal and past outputs and inputs
  respectively and are known

• matrices Ho,s and Hu,s are composite of system matrices A, B, C, D and
  also known

• the only unknown variable is x(k − s).

The underlying idea of the parity relation based residual generation lies in
the utilization of the fact, known from the linear control theory, that for
s ≥ n the following rank condition holds:

r ank Ho,s ≤ n < the row number of matrix Ho,s = (s + 1)m.

This ensures that for s ≥ n there exists at least a (row) vector

\[v_{s}\left( \neq 0 \right) \in \mathcal{R}^{\left( s + 1 \right)m}$$ such that \]

\[u_{s}H_{o,s} = 0 \]

Hence, a parity relation based residual generator is constructed by

\[r\left( k \right) = v_{s}\left( y_{s}\left( k - s \right) - H_{u,s}u_{s}\left( k - s \right) \right) \]

whose dynamics is governed by, in case of f (k) = 0, d(k) = 0,

\[r\left( k \right) = v_{s}\left( y_{s}\left( k \right) - H_{u,s}u_{s}\left( k \right) \right) = v_{s}H_{o,s}\left( k - s \right) = 0 \]

Vectors satisfying (8.55) are called parity vectors, the set of which,

\[P_{s} = \left\{ v_{s} \middle| v_{s}H_{o,s} = 0 \right\} \]

is called the parity space of the sth order.

In order to study the influence of f, d on residual generator (8.56), let
f(k) ≠ 0, d(k) ≠ 0. It is straightforward that

\[y_{s}\left( k \right) = H_{o,s}x\left( k - s \right) + H_{u,s}u_{s}\left( k \right) + H_{f,s}f_{s}\left( k \right) + H_{d,s}d_{s}\left( k \right) \]

where

\[f_{s}\left( k \right) = \begin{bmatrix} f\left( k - s \right) \\ f\left( k - s + 1 \right) \\ \vdots \\ f\left( k \right) \\ \end{bmatrix},\ H_{f,s} = \begin{bmatrix} F_{f} & 0 & \cdots & 0 \\ CE_{f} & F_{f} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ CA^{s - 1}E_{f} & \cdots & CE_{f} & F_{f} \\ \end{bmatrix} \]

8.2 Preliminaries and Review of Model-Based FDI Schemes

\[d_{s}\left( k \right) = \begin{bmatrix} d\left( k - s \right) \\ d\left( k - s + 1 \right) \\ \vdots \\ d\left( k \right) \\ \end{bmatrix},\ H_{d,s} = \begin{bmatrix} F_{d} & 0 & \cdots & 0 \\ CE_{d} & F_{d} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ CA^{s - 1}E_{d} & \cdots & CE_{d} & F_{d} \\ \end{bmatrix} \]

Constructing a residual generator according to (8.56) finally results in

\[r\left( k \right) = v_{s}(H_{f,s}f_{s}\left( k \right) + H_{d,s}d_{s}\left( k \right),v_{s} \in P_{s} \]

We see that the design parameter of the parity relation based residual generator
is the parity vector whose selection has decisive impact on the performance of
the residual generator.

Remark 8.2 One of the significant properties of parity relation based residual
gener-ators, also widely viewed as the main advantage over the observer-based
approaches, is that the design can be carried out in a straightforward manner.
In fact, it only deals with solutions of linear equations or linear optimization
problems. In against, the implementation form (8.56) is surely not ideal for an
online realization, since it is presented in an explicit form, and thus not only
the temporal but also the past measurement and input data are needed and have to
be recorded.

8.2.2.5 Kernel Representation and Parameterization of Residual Generators

In the model-based FD study, the FDF, DO and Kalman filter based residual
gener-ators are called closed-loop configurated, since a feedback of the
residual signal is embedded in the system configuration and the computation is
realized in a recursive form. Differently, the parity space residual generator
is open-loop structured. In fact, it is an FIR (finite impulse response) filter.
Below, we introduce a general form for all types of LTI residual generators,
which is also called parameterization of residual generators.

A fundamental property of the LCF is that in the fault- and noise-free case

\[\forall u,\begin{bmatrix} - \hat{N}\left( z \right) & \hat{M}\left( z \right) \\ \end{bmatrix}\begin{bmatrix} u\left( z \right) \\ y\left( z \right) \\ \end{bmatrix} = 0 \]

Equation (8.61) is called kernel representation (KR) of the system under
consid-eration and useful in parameterizing all residual generators. For our
purpose, we introduce below a more general definition of kernel representation.

Definition 8.3 Given system (8.2), (8.3), a stable linear system K driven
by u(z), y(z) and satisfying

\[\forall u\left( z \right),r\left( z \right) = \kappa\begin{bmatrix} u\left( z \right) \\ y\left( z \right) \\ \end{bmatrix} = 0 \]

148 8 Introduction, Preliminaries and I/O Data Set Models

is called stable kernel representation (SKR) of (8.2), (8.3).

model (8.18), (8.19) with unknown input vectors. The parameterization forms
of all LTI residual generators and their dynamics are described by

\[{r\left( z \right) = R\left( z \right)\begin{bmatrix} - \hat{N}\left( z \right) & \hat{M}\left( z \right) \\ \end{bmatrix}\begin{bmatrix} u\left( z \right) \\ y\left( z \right) \\ \end{bmatrix}\backslash n}{= R\left( z \right)\left( {\hat{N}}_{d}\left( z \right)d\left( z \right) + {\hat{N}}_{f}\left( z \right)f\left( z \right) \right)} \]

where

\[{\hat{N}}_{d}\left( z \right) = \left( A - \text{LC},\text{Ed} - \text{LFd},C,\text{Fd} \right),{\hat{N}}_{f}\left( z \right) = \left( A - \text{LC},E_{f} - LF_{f},C,F_{f} \right) \]

R(z)(≠0) is a stable parameterization matrix and called post-filter.
Moreover, in order to avoid loss of information about faults to be detected,
the condition r ank(R(z)) = m is to be satisfied.

It has been demonstrated that all LTI residual generators can be expressed
by (8.63), while their dynamics with respect to the unknown inputs and
faults are para-meterized by (8.64). Moreover, it holds

\[\left\lbrack - \hat{N}\left( z \right)\hat{M}\left( z \right) \right\rbrack\begin{bmatrix} u\left( z \right) \\ y\left( z \right) \\ \end{bmatrix} = y\left( z \right) - \hat{y}\left( z \right) \]

with yˆ delivered by a full order observer as an estimate of y, we can
apply an FDF,

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + L\left( y\left( k \right) - \hat{y}\left( k \right) \right),\hat{y}\left( k \right) = C\hat{x}\left( x \right) + Du\left( k \right) \]

for the online realization of (8.63).

As a summary, we present a theorem which immediately follows from Definition
8.3 and the residual generator parametrization.

Theorem 8.1 Given process model (8.18), (8.19), an LTI dynamic system
is a resid-ual generator if and only if it is an SKR of (8.2), (8.3).

8.3 I/O Data Models

In order to connect analytical models and process data, we now introduce
different I/O data models. They are essential in our subsequent study and
build the link between the model-based FD schemes introduced in the previous
sections and the data-driven design methods to be introduced below. For our
purpose, the following LTI process model is assumed to be the underlying
model form adopted in our study

8.3 I/O Data Models 149

where u √ Rl , y √ Rm and x √ Rn , w √ Rn and v √ Rm denote
noise sequences that are normally distributed and statistically independent
of u and x(0).

Let ω(k) √ Rξ be a data vector. We introduce the following notations

\[{w_{s}\left( k \right) = \begin{bmatrix} w\left( k - 1 \right) \\ \vdots \\ w\left( k \right) \\ \end{bmatrix} \in \mathcal{R}^{\left( s + 1 \right)\xi}\backslash n}{\ \Omega_{k} = \begin{bmatrix} w\left( k \right) & \cdots & w\left( k + N - 1 \right) \\ \end{bmatrix} \in \mathcal{R}^{\xi \times N}} \]

\[\Omega_{k,s} = \begin{bmatrix} w_{s}\left( k \right)\cdots w_{s}\left( k + N - 1 \right) \\ \end{bmatrix} = \begin{bmatrix} \Omega_{k - s} \\ \vdots \\ \Omega_{k} \\ \end{bmatrix} \in \mathcal{R}^{\xi \times N} \]

where s, N are some (large) integers. In our study, ω(k) can be y(k),
u(k), x(k), and

represents m or l or n given in (8.65), (8.66). The first I/O data
model described by

\[Y_{k,s} = \Gamma_{s}X_{k - s} + H_{u,s}U_{k,s} + H_{w,s}W_{k,s} + V_{k,s} \in \mathcal{R}^{\left( s + 1 \right)m \times N} \]

\[Y_{s} = \begin{bmatrix} C \\ \text{CA} \\ \vdots \\ CA^{s} \\ \end{bmatrix} \in \mathcal{R}^{\left( s + 1 \right)m \times n},H_{u,s} = \begin{bmatrix} D & 0 & \ & \ \\ \text{CB} & \ddots & \ddots & \ \\ \vdots & \ddots & \ddots & 0 \\ CA^{s - 1}B & \cdots & \text{CB} & D \\ \end{bmatrix} \]

follows directly from the I/O model (8.54) introduced in the study on parity
space scheme, where Hu,s √ R(s+1)m ×(s+1)l , Hw,s Wk,s + Vk,s
represents the influence of the noise vectors on the process output with
Hw,s having the same structure like Hu,s and Wk,s , Vk,s as defined in
(8.67), (8.68).

In the SIM framework, the so-called innovation form, instead of (8.69), is
often applied to build an I/O model. The core of the innovation form is a
(steady) Kalman filter, which is written as

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + K\left( y\left( k \right) - \hat{y}\left( k \right) \right),\hat{y}\left( k \right) = C\hat{x}\left( k \right) + Du\left( k \right) \]

with the innovation sequence y(k) − yˆ(k) := e(k) being a white
noise sequence and

1. the Kalman filter gain matrix. Based on (8.70), the I/O relation of the
   process can be alternatively written into

\[\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + Ke\left( k \right) = A_{K}\hat{x}\left( k \right) + B_{K}u\left( k \right) + Ky\left( k \right) \]

\[y\left( k \right) = C\hat{x}\left( k \right) + Du\left( k \right) + e\left( k \right),A_{K} = A - KC,B_{K} = B - KD \]

150 8 Introduction, Preliminaries and I/O Data Set Models

The following two I/O data models follow from (8.71), (8.72):

\[Y_{k,s} = \Gamma_{s}{\hat{X}}_{k - s} + H_{u,s}U_{k,s} + H_{e,s}E_{k,s},H_{e,s} = \begin{bmatrix} I & 0 & \ & \ \\ \text{CK} & \ddots & \ddots & \ \\ \vdots & \ddots & \ddots & 0 \\ CA^{\left( s - 1 \right)}K & \cdots & \text{CK} & I \\ \end{bmatrix} \]

\[\left( I - H_{y,s}^{K} \right)Y_{k,s} = \Gamma_{s}^{K}{\hat{X}}_{k - s} + H_{u,s}^{K}U_{k,s},\Gamma_{s}^{K} = \begin{bmatrix} C \\ CA_{K} \\ \vdots \\ CA_{K}^{s} \\ \end{bmatrix} \]

\[H_{y,s}^{K} = \begin{bmatrix} 0 & 0 & \ & \ \\ \text{CK} & \ddots & \ddots & \ \\ \vdots & \ddots & \ddots & 0 \\ CA_{K}^{s - 1}K & \cdots & \text{CK} & 0 \\ \end{bmatrix},H_{u,s}^{K} = \begin{bmatrix} 0 & 0 & \ & \ \\ CB_{K} & \ddots & \ddots & \ \\ \vdots & \ddots & \ddots & 0 \\ CA_{K}^{s - 1}K & \cdots & CK_{B} & 0 \\ \end{bmatrix} \]

8.4 Notes and References

In order to apply the MVA technique to solving FD problems in dynamic
processes, numerous methods have been developed in the last two decades.
Among them, dynamic PCA/PLS [1–3], recursive implementation of PCA/PLS [4,
5], fast moving window PCA [6] and multiple-mode PCA [7] are widely used in
the research and applications in recent years.

SIM is a well-established technique and widely applied in process
identification [8–11]. The application of the SIM technique to FDI study is
new and has been first proposed by [12–15].

In Sect. 8.2, the basics of the model-based FDI framework have been
reviewed. They can be found in the monographs [1, 16–25] and in the survey
papers [26–29]. The first work on FDF has been reported by Beard and Jones
in [30, 31], and Chow and Willsky have proposed the first optimal FDI
solution using the parity space scheme [32]. The reader is referred to
Chaps. 3 and 5 in [25] for a systematic handling of the issues on process
and fault modeling and model-based residual generation schemes.

The concept SKR will play an important role in our subsequent studies. In
fact, residual generator design is to find an SKR, as given in Theorem 8.1.
The SKR definition given in Definition 8.3 is similar to the one introduced
in [33] for nonlinear systems.

初步译文

8.2.1.2 互质分解技术

传递函数(矩阵)的互质分解给出了一种更进一步的系统表示形式，这种表示形式将在后续的研究中得到广泛的应用。粗略地说，$$\mathcal{R}\mathcal{H}_{\infty}$$的上的互质互质分解就是把一个转移矩阵分解为两个稳定的互质转移矩阵。

定义8.1
如果存在两个稳定转移矩阵$$\hat{X}(z)$$和$$\hat{Y}(z)$$满足(8.5)，则两个稳定转移矩阵

\[\hat{M}(z)$$*，*$$\hat{N}(z)$$称之为左互质。 \]

\begin{bmatrix}
\hat{M}\left( z \right) & \hat{N}\left( z \right) \
\end{bmatrix}\begin{bmatrix}
\hat{X}\left( z \right) \
\hat{Y}\left( z \right) \
\end{bmatrix} = I

\[ 类似地，如果存在两个稳定矩阵*Y*(*z*)，*X*(*z*)满足(8.6)，则两个稳定传递矩阵*M*(*z*)，*N*(*z*)是右互质的。 \]

\begin{bmatrix}
X\left( z \right) & Y\left( z \right) \
\end{bmatrix}\begin{bmatrix}
M\left( z \right) \
N\left( z \right) \
\end{bmatrix} = I.

\[ 设*G(z)*是一个适当的有理传递矩阵。G(z)的左互素分解(LCF)是将G(z)分解为两个稳定并且互质的矩阵，该矩阵将在设计所谓的残差生成器中起着关键的作用。为了完善这一概念，我们还介绍了右互质分解(RCF)，但是在我们的研究中仅偶尔使用它。 **定义8.2** 有左互质对$$\left( \hat{M}\left( z \right),\hat{N}\left( z \right) \right)$$的$$G\left( z \right) = {\hat{M}}^{- 1}\left( z \right)\hat{N}(z)$$称为*G(z)*的LCF*。*类似地，*G*(*z*)的RCF定义为有右互质对$$\left( M\left( z \right),N\left( z \right) \right)$$的$$G\left( z \right) = N\left( z \right)M\left( z \right)^{- 1}$$。 下面，我们提出一个引理，该引理为我们提供了$$\left( \hat{M}\left( z \right),\hat{N}\left( z \right) \right)$$、(*M*(*z*)*,N*(*z*))和关联对$$\left( \hat{X}\left( z \right),\hat{Y}\left( z \right) \right)(X(z),Y(z))$$的状态空间计算算法。 **引理8.1** 假设*G*(*z*)是一个具有状态空间实现(*A*，*B*，*C*，*D*)的适当的实际有理传递矩阵，并且它是可稳定的和可检测的。令*F*和*L*为*A+BF*和*A−LC*为Schur矩阵（即它们的特征值在复平面上的单位圆内），并定义 \]

\hat{M}\left( z \right) = \left( A - LC, - L,C,I \right),\hat{N}\left( z \right) = \left( A - LC,B - LD,C,D \right)

\[ \]

M\left( z \right) = \left( A + BF,B,F,I \right),N\left( z \right) = \left( A + BF,B,C,C + DF,D \right)

\[ \]

\hat{X}\left( z \right) = \left( A - LC, - \left( B - LD \right),F,I \right),Y\left( z \right) = \left( A - LC, - L,F,0 \right)

\[ \]

X\left( z \right) = \left( A - LC, - \left( B - LD \right),F,I \right),Y\left( z \right) = \left( A - LC, - L,F,0 \right)

\[ 然后 \]

G\left( z \right) = {\hat{M}}^{- 1}\left( z \right)\hat{N}\left( z \right) = N\left( z \right)M^{- 1}\left( z \right)

\[ 分别是*G*(*z*)的LCF和RCF*。*而且，所谓的Bezout恒等式成立 \]

\begin{bmatrix}
X\left( z \right) & Y\left( z \right) \

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  M\left( z \right) & - \hat{Y}\left( z \right) \
  N\left( z \right) & X\left( z \right) \
  \end{bmatrix} = \begin{bmatrix}
  I & 0 \
  0 & I \
  \end{bmatrix}\

\[ 注意 \]

r\left( z \right) = \begin{bmatrix}

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  u\left( z \right) \
  y\left( z \right) \
  \end

\[ 是一个动态系统，其过程输入和输出向量*u，y为*该系统的输入，而残差向量*r*作为该系统的输出。它称为残差生成器。在一些文献中$$\begin{bmatrix} - \hat{N}\left( z \right) & \hat{M}\left( z \right) \\ \end{bmatrix}$$是被称为系统(8.2)与(8.3)的核心表。 8.2.1.3 带有扰动的系统的表示 ---------------------------- 工艺过程中已考虑的干扰变动、工艺过程中的意外更改以及测量和过程噪声通常被建模为未知的输入向量。我们用*d，v*或*η*表示它们，并将它们整合到状态空间模型(8.2)，(8.3)或输入输出模型(8.1)中，如下所示： - 状态空间表示 \]

x\left( k + 1 \right) = Ax\left( k \right) + Bu\left( k \right) + E_{d}d\left( k \right) + \eta\left( k \right)

\[ \]

y\left( k \right) = C_{x}\left( k \right) + D_{u}\left( k \right) + F_{d}d\left( k \right) + v\left( k \right)

\[ 其中*Ed*，*Fd*是相容维数的常数矩阵，$$d \in \mathcal{R}^{k_{d}}$$是确定性未知输入向量，如果不作其他说明，$$\eta \in \mathcal{R}^{k_{\eta}},v \in \mathcal{R}^{k_{v}}$$是服从$$\eta\sim N\left( 0,\Sigma_{\eta} \right),\ v\sim N\left( 0,\Sigma_{v} \right)$$分布的白噪声向量。 - 输入-输出模型 \]

y\left( z \right) = G_{\text{yu}}\left( z \right)u\left( z \right) + G_{\text{yd}}\left( z \right)d\left( z \right) + G_{\text{yf}}\left( z \right)f\left( z \right)

\[ 其中*Gyd*(*z*)，*Gyv*(*z*)是已知的。 8.2.1.4 故障建模 ---------------- 有许多对故障进行建模的方法。将模型(8.16)扩展成 \]

y\left( z \right) = G_{\text{yu}}\left( z \right)u\left( z \right) + G_{\text{yd}}\left( z \right)d\left( z \right) + G_{\text{yf}}\left( z \right)f\left( z \right)

\[ 是一个被广泛采用的方法，其中$$f \in \mathcal{R}^{k_{f}}$$是一个代表所有可能故障的未知向量，并且在无故障情况下值为0。$$G_{\text{yf}}(z\mathcal{) \in L}\mathcal{H}_{\infty}$$是一个已知的传递矩阵。在这本书中，*f*被认为是一个确定性的时间函数。如果没有指定故障的类型，则不对其作进一步的假设。 假设(8.17)的最小状态空间实现由下式得出 \]

x\left( k + 1 \right) = Ax\left( k \right) + Bu\left( k \right) + E_{d}d\left( k \right) + E_{f}f\left( k \right)

\[ \]

y\left( k \right) = Cx\left( k \right) + Du\left( k \right) + F_{d}d\left( k \right) + F_{f}f\left( k \right)

\[ 如果已知矩阵*Ef*，*Ff*那么我们有 \]

G_{\text{yf}}\left( z \right) = F_{f} + C\left( zI - A \right)^{- 1}E_{f}

\[ 显然，*Ef，Ff*指示了故障发生的位置及其对系统动态的影响。现有技术将故障分为三类： - 传感器故障*fS*：这些是直接作用于过程测量的故障。 - 执行器故障*fA*：这些故障导致执行器发生变化。 - 过程故障*fP*：它们用于指示过程中的故障。 传感器故障通常通过设置$$F_{f} = I$$来建模，即 \]

y\left( k \right) = C_{x}\left( k \right) + D_{u}\left( k \right) + F_{d}d\left( k \right) + fs\left( k \right)

\[ 而执行器故障是通过设置*Ef = B，Ff = D*，这会导致 \]

x\left( k + 1 \right) = Ax\left( k \right) + B\left( u\left( k \right) + f_{A} \right) + E_{d}d(k)

\[ \]

y\left( k \right) = Cx\left( k \right) + D\left( u\left( k \right) + f_{A} \right) + F_{d}d\left( k \right)

\[ 根据故障的类型和位置，可以用*Ef* = *EP*和*Ff* = *FP*对某些*EP*，*FP*进行过程故障建模。针对具有传感器，执行器和过程故障的系统，我们定义 \]

f = \begin{bmatrix}
f_{A} \
f_{P} \
f_{S} \
\end{bmatrix},E_{f} = \begin{bmatrix}
B & E_{P} & 0 \
\end{bmatrix},F_{f} = \begin{bmatrix}
D & F_{P} & I \
\end{bmatrix}

\[ 并应用(8.18)，(8.19)表示系统动态。 由它们影响系统动态的方式，(8.18)，(8.19)所描述的故障称为附加性故障。值得注意的是，附加故障的发生不会影响系统稳定性，与系统配置无关。在实践中遇到的典型附加故障是，例如，传感器和执行器的偏移或传感器的漂移。前者可以用一个常数来描述，而后者可以用一个斜坡来描述。 实际上，过程中或传感器和执行器中的故障通常会导致模型参数发生变化。它们被称为乘法故障，通常根据参数变化来进行建模。 8.2.2 基于模型的残差生成方案 ============================ 接下来，我们介绍一些基于模型和观测器的标准残差生成方法。 8.2.2.1 故障检测滤波器 ---------------------- 故障检测滤波器(FDF)是Beard和Jones在1970年代初提出的第一类基于观测器的残差生成器。他们的工作标志着基于模型的FDI技术迅猛发展的开始。 FDF的核心是全序状态观测器 \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + L\left( y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \right)

\[ 该观测器是在标称系统模型$$G_{\text{yu}}\left( z \right) = C\left( zI - A \right)^{- 1}B + D$$的基础上构造的。 建立在（8.25）之上，残差可以简单地定义为 \]

r\left( k \right) = y\left( k \right) - \hat{y}\left( k \right) = y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right)

\[ 引入变量 \]

e\left( k \right) = x\left( k \right) - \hat{x}\left( k \right)

\[ 根据过程模型(8.18)，(8.19)的假设 \]

e\left( k + 1 \right) = \left( A - LC \right)e\left( k \right) + \left( E_{d} - LF_{d} \right)d\left( k \right) + \left( E_{f} - LF_{f} \right)

\[ \]

r\left( k \right) = Ce\left( k \right) + F_{d}d\left( k \right) + F_{f}f\left( k \right)

\[ 显然，当选择观测器增益矩阵*L*使得*A*-*LC*稳定时，*r*(*k*)具有残差特性。 FDF的优点在于它的结构简单，并且对于熟悉现代控制理论的读者而言，FDF还与状态观测器设计（尤其是与设计鲁棒残差生成器成熟的鲁棒控制理论）有着密切的关系。 我们看到，一个FDF的设计实际上是由观测器增益矩阵*L*决定的。为了提高设计的自由程度，我们可以将一个矩阵转换为输出估计误差$$y\left( z \right) - \hat{y}(z)$$，即 \]

r\left( z \right) = V\left( y\left( z \right) - \hat{y}\left( z \right) \right)

\[ FDF法的一个缺点是全阶状态观测器的在线实现，因为在许多实际情况下，降阶观测器可以为我们提供相同或相似的性能，但只需要较少的在线计算量。这是开发Luenberger型残差发生器（也称为诊断观测器）的动机之一。 8.2.2.2 诊断观测器方案 ---------------------- 诊断观测器（DO）由于其灵活的结构和与Luenberger型观察器的相似性而成为研究最多的基于模型的残差生成器形式之一。 DO的核心是Luenberger类型（输出）的观测器，描述为 \]

z\left( k + 1 \right) = Gz\left( k \right) + Hu\left( k \right) + Ly\left( k \right)

\[ \]

r\left( k \right) = Vy\left( k \right) - Wz\left( k \right) - Qu\left( k \right)

\[ 其中$$z \in \mathcal{R}^{s}$$，s表示观测器的阶数，并且可以等于、低于或高于系统的阶数。 尽管对Luenberger型观测器起主要作用的是第一种情况（目的是得到一个降阶观测器），但是高阶观测器在FDI系统的优化中会发挥重要作用。 假定，$$G_{\text{yz}}\left( z \right) = C\left( zI - A \right)^{- 1}B + D$$,那么矩阵*G*，*H*，*L*，*Q*，*V*和*W*和矩阵$$T \in \mathcal{R}^{s \times n}$$必须满足所谓的Luenberger 条件 \]

{\text{I.\ \ G\ is\ stable}\backslash n}{II.\ \ TA - GT = LC,H = TB - LD\backslash n}{III.\ \ VC - WT = 0,Q = VD}

\[ 在该条件下，系统(8.30)，(8.31)传递残差矢量，即 \]

\forall u,x\left( 0 \right),\operatorname{}{r\left( k \right) = 0}

\[ 令*e*(*k*)=*Tx*(k)-*z*(*k*)，直接表明了在过程模型(8.18)，(8.19)的前提下，DO的系统动态性取决于 \]

e\left( k + 1 \right) = Ge\left( k \right) + \left( TEd - LFd \right)d\left( k \right) + \left( TE_{f} - LF_{f} \right)f\left( k \right)

\[ \]

r\left( k \right) = Ve\left( k \right) + VF_{d}d\left( k \right) + VF_{f}f(k)

\[ 请记住，在上一节中我们已经指出可以将所有残差发生器设计方案表述为对观测器增益矩阵和后置滤波器的寻求。因此，在揭示求解Luenberger方程(8.32)-(8.34)的矩阵*G*，*L*，*T*，*V*与*W*和观测器增益矩阵以及后置滤波器 之间的关系上具有实际意义和理论意义。 与FDF方案的比较能够清楚地表明 - 诊断观测器方案可能会导致残差生成器的阶数降低，这很适合且有利于在线实现。 - 会让我们拥有更大程度的设计自由。 - 以及更多的复杂设计。 8.2.2.3 基于卡尔曼滤波器的残差生成 ---------------------------------- 根据(8.14)，(8.15)，假设*η*(*k*),*ν*(*k*)是白高斯过程并且与初始状态矢量*x*(0)，*u*(*k*)无关 \]

\varepsilon\begin{bmatrix}
\eta\left( i \right)\eta^{T}\left( j \right) & \eta\left( i \right)v^{T}\left( j \right) \
v\left( i \right)\eta^{T}\left( j \right) & v\left( i \right)v^{T}\left( j \right) \
\end{bmatrix} = \begin{bmatrix}
\Sigma_{\eta} & S_{\text{ηv}} \
S_{\text{vη}} & \Sigma_{v} \
\end{bmatrix}\delta_{\text{ij}},\delta_{\text{ij}} = \left{ \begin{matrix}
1,\ \ i = j \
0,\ \ i \neq j \
\end{matrix} \right.\

\[ \]

\Sigma_{v} > 0,\Sigma_{\eta} \geq 0,\varepsilon\left\lbrack \eta\left( k \right) \right\rbrack = 0,\varepsilon\left\lbrack v\left( k \right) \right\rbrack = 0

\[ \]

\varepsilon\left\lbrack x\left( 0 \right) \right\rbrack = \overline{x},\varepsilon\left\lbrack \left( x\left( 0 \right) - \overline{x} \right)\left( x\left( 0 \right) - \overline{x} \right)^{T} \right\rbrack = P_{o}

\[ 虽然卡尔曼滤波器的结构相似全阶观测器，但它是一个时变系统，由以下 递归形式给出： 最佳状态估计的递归计算： \]

\ \hat{x}\left( 0 \right) = \overline{x}

\[ \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + \text{Bu}\left( k \right) + K\left( k \right)\left( y\left( k \right) - C\hat{x}\left( k \right) - \text{Du}\left( k \right) \right)

\[ 卡尔曼滤波器增益的递归计算： \]

P\left( 0 \right) = P_{0}

\[ \]

P\left( K + 1 \right) = \text{AP}\left( k \right)A^{T} - K\left( k \right)R_{e}\left( k \right)K^{T}\left( k \right) + \Sigma_{\eta}

\[ \]

K\left( k \right) = \left( \text{AP}\left( k \right)C^{T} + S_{\text{ηv}} \right)R_{e}^{- 1}\left( k \right)

\[ \]

R_{e}\left( k \right) = \Sigma_{v} + \text{CP}\left( k \right)C^{T}

\[ 其中$$\hat{x}(k)$$表示x(k)的估计并且 \]

P\left( k \right) = \varepsilon\left\lbrack \left( x\left( k \right) - \hat{x}\left( k \right) \right)\left( x\left( k \right) - \hat{x}\left( k \right) \right)^{T} \right\rbrack

\[ 是联合估计误差协方差 **卡尔曼滤波器的重要特征是** - 在这种情况下，状态估计是最佳的 \]

P\left( k \right) = \varepsilon\left\lbrack \left( x\left( k \right) - \hat{x}\left( k \right) \right)\left( x\left( k \right) - \hat{x}\left( k \right) \right)^{T} \right\rbrack \Longrightarrow min

\[ - 所谓的新息过程$$e\left( k \right) = y\left( k \right) - C\hat{x}\left( k \right) - \text{Du}\left( k \right)$$是一个具有协方差的白高斯过程 \]

\varepsilon\left( e\left( k \right)e^{T}\left( k \right) \right) = R_{e}\left( k \right) = \Sigma_{v} + \text{CP}\left( k \right)C^{T}

\[ 下面是由(8.40)，(8.45)给出的一种算法，可以在线实现卡尔曼滤波算法。 **算法8.1** 卡尔曼滤波器的在线实现 *S0*：设$$\hat{x}(0)$$，P(0)值为如(8.40)和(8.42)所示 *S1*：根据(8.45)、(8.44)及(8.41)计算*Re*(*k*)、*k*(*k*)、*x*(*k*) *S2*：增加*k*并根据(8.43)计算*p*(*k*+1) *S3*：转到*S*1。 假设所考虑的过程是稳定的，则 \]

\operatorname{}{K\left( k \right) =}K = constant\ matrix

\[ 这取决于 \]

K = \left( \text{AP}C^{T} + S_{\eta v^{T}} \right)R_{e}^{- 1}

\[ 让 \]

P = \lim_{k \rightarrow \infty}P\left( k \right),R_{e} = \Sigma_{v} + CPC^{T}

\[ 则下式成立 \]

P = \text{AP}A^{T} - KR_{e}K^{T} + \Sigma_{\eta}

\[ 方程(8.49)是代数Riccati方程，其解*P*在某些条件下为正定。因此很明显，给定的系统模型可以通过求解Riccati方程(8.49)离线计算出增益矩阵*K。*相应的残差发生器则由下式给出 \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + Bu\left( k \right) + K\left( y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right) \right)

\[ \]

r(k) = y\left( k \right) - C\hat{x}\left( k \right) - Du\left( k \right)

\[ 请注意，实际上，我们现在已经有了一个全阶观测器。 **备注8.1** 上述算法中的离线设置(*S0*)仅需要一次，但每次必须重复S1-S3。因此，与稳态卡尔曼滤波器相比，在线实现的计算量很大。出于FDI的目的，我们通常可以假定所考虑的系统在故障发生之前就处于稳定状态。因此，使用稳态型残差产生器(8.50)、(8.51)是有利的。在这种情况下，最复杂的计算是求解Riccati方程(8.49)，然而该求解方法是离线执行的，而且需要大量可靠的数值方法和计算机辅助程序。 8.2.2.4 奇偶空间法 ------------------ 奇偶空间法是Chow和Willsky在十九世纪八十年代初期的开拓性工作中提出的。该方法虽然采用状态空间模型来生成残差，但其是核心是所谓的奇偶关系，而不是观测器。奇偶空间法是一种重要的基于模型的残差生成方法，与基于观测器的残差生成方法和参数估计方法并行。 我们考虑下面的状态空间模型(8.18)，(8.19)，并且在不失一般性的前提下，假设*rank*(*C*)=*m*。为了构造一个残差生成器，我们首先假设*f*(*k*)=0*，d*(*k*)=0。由于(8.18)，(8.19)，*y*(*k*-*s*)，*s\>*0，可以用*x(k-s)*，*u(k-s)*，*y(k-s+1)*表示*x*(*k-s*)，*u*(*k-s+*1)，*u(k-s)*，如下 \]

y\left( k - s \right) = Cx\left( k - s \right) + Du\left( k - s \right)

\[ \]

{y\left( k - s + 1 \right) = Cx\left( k - s + 1 \right) + Du\left( k - s + 1 \right)\backslash n}{= CAx\left( k - s \right) + CBu\left( k - s \right) + Du(k - s + 1)}

\[ 重复此过程将产生 \]

y\left( k - s + 2 \right) = CA^{2}x\left( k - 2 \right) + CABu\left( k - s \right) + CBu\left( k - s + 1 \right) + Du\left( k - x + 2 \right),\ldots,

\[ \]

y\left( k \right) = CA^{s}x\left( k - s \right) + CA^{s - 1}\text{Bu}\left( k - s \right) + \cdots + CBu\left( k + 1 \right) + Du\left( k \right)

\[ 引入下面的记法 \]

y_{s}\left( k \right) = \begin{bmatrix}
y\left( k - s \right) \
y\left( k - s + 1 \right) \
\vdots \
y\left( k \right) \
\end{bmatrix},u_{s}\left( k \right) = \begin{bmatrix}
u\left( k - s \right) \
u\left( k - s + 1 \right) \
\vdots \
u\left( k \right) \
\end{bmatrix}

\[ \]

H_{o,s} = \begin{bmatrix}
C \
\text{CA} \
\vdots \
CA^{s} \
\end{bmatrix},H_{u,s} = \begin{bmatrix}
D & 0 & \cdots & 0 \
\text{CB} & D & \ddots & \vdots \
\vdots & \ddots & \ddots & 0 \
CA^{s - 1}B & \cdots & \text{CB} & D \
\end{bmatrix}

\[ 然后得出简洁的模型形式如下 \]

{y_{s}(k) = H}{o,s}x\left( k - s \right) + Hu_{s}\left( k - s \right)

\[ 注意(8.54)描述了依赖于过去状态向量*x*(*k*-*s*)的输入和输出关系。 它表达为一个明确的形式，其中 - *Ys*(*k*)*和us*(*k*)分别由当前及过去的输出和输入组成，并且是已知的*。* - 矩阵*Ho,s*和*Hu,s*是系统矩阵*A*，*B*，*C*，*D*的组合*，*并且也是已知的。 - 唯一的未知变量是*x*(*k-s*)。 基于奇偶关系的残差生成，其基本思想在于利用线性控制理论中得知的事实，即对于*s*≥*n*，以下秩条件成立： \]

\text{rank}\left( H_{o,s} \right) \leq n < the\ row\ number\ of\ matrix\ H_{o,s} = \left( s + 1 \right)m

\[ 这保证了对于*s* ≥ *n*，至少存在一个(行)向量$$v_{s}\left( \neq 0 \right) \in \mathcal{R}^{\left( s + 1 \right)m}$$使得 \]

u_{s}H_{o,s} = 0

\[ 因此，通过以下方式构造了一个基于奇偶关系的残差生成器 \]

r\left( k \right) = v_{s}\left( y_{s}\left( k - s \right) - H_{u,s}u_{s}\left( k - s \right) \right)

\[ 在*f*(*k*)=0，*d*(*k*)= 0的情况下，其动态变化受如下控制 \]

r\left( k \right) = v_{s}\left( y_{s}\left( k \right) - H_{u,s}u_{s}\left( k \right) \right) = v_{s}H_{o,s}\left( k - s \right) = 0

\[ 满足(8.55)的向量称为奇偶向量，其集合 \]

P_{s} = \left{ v_{s} \middle| v_{s}H_{o,s} = 0 \right}

\[ 被称为某阶次的奇偶空间。 为了研究*f,d对*残差产生器(8.56)的影响，设*f*(*k*)≠0,*d*(*k*)≠0。很明显 \]

y_{s}\left( k \right) = H_{o,s}x\left( k - s \right) + H_{u,s}u_{s}\left( k \right) + H_{f,s}f_{s}\left( k \right) + H_{d,s}d_{s}\left( k \right)

\[ 其中 \]

f_{s}\left( k \right) = \begin{bmatrix}
f\left( k - s \right) \
f\left( k - s + 1 \right) \
\vdots \
f\left( k \right) \
\end{bmatrix},\ H_{f,s} = \begin{bmatrix}
F_{f} & 0 & \cdots & 0 \
CE_{f} & F_{f} & \ddots & \vdots \
\vdots & \ddots & \ddots & 0 \
CA^{s - 1}E_{f} & \cdots & CE_{f} & F_{f} \
\end{bmatrix}

\[ \]

d_{s}\left( k \right) = \begin{bmatrix}
d\left( k - s \right) \
d\left( k - s + 1 \right) \
\vdots \
d\left( k \right) \
\end{bmatrix},\ H_{d,s} = \begin{bmatrix}
F_{d} & 0 & \cdots & 0 \
CE_{d} & F_{d} & \ddots & \vdots \
\vdots & \ddots & \ddots & 0 \
CA^{s - 1}E_{d} & \cdots & CE_{d} & F_{d} \
\end{bmatrix}

\[ 根据(8.56)构造残差产生器，最终得到 \]

r\left( k \right) = v_{s}(H_{f,s}f_{s}\left( k \right) + H_{d,s}d_{s}\left( k \right),v_{s} \in P_{s}

\[ 我们看到，基于奇偶关系的残差生成器的设计参数是奇偶向量，其选择对残差生成器的性能具有决定性的影响。 **注释8.2**基于奇偶关系的残差生成器的重要特性之一（也被广泛认为是其优于基于观测器的方法的主要优点）在于，可以以简单的方式进行设计。实际上，它仅处理线性方程组或线性最优化问题的解。相反，实现形式(8.56)肯定不是在线实现的理想形式，因为它是以一种明确的形式呈现的，因此不仅需要当前的数据，而且还需要过去的测量和输入数据，并且必须将其记录下来。 8.2.2.5 残差生成器的核表示和参数化 ---------------------------------- 在基于模型的FD研究中，由于将残差信号的反馈嵌入到系统配置中，并以递归形式实现计算，因此将基于模型的FDF，DO和Kalman滤波器的残差生成器称为闭环配置器。不同的是，奇偶空间残差生成器是开环结构。实际上，它是一个FIR（有限脉冲响应）滤波器。下面，我们介绍所有类型的LTI残差生成器的一般形式，也称为残差生成器的参数化。 LCF（低周波滤波器）的基本特性是在无故障和无噪声的情况下 \]

\forall u,\begin{bmatrix}

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  u\left( z \right) \
  y\left( z \right) \
  \end{bmatrix} = 0

\[ 方程(8.61)被称为这个系统的核表示(KR)，可用于参数化所有残差生成器。为此，我们在下面介绍一个更一般的核表示定义。 **定义8.3**假设有系统(8.2),(8.3)和一个由*u*(*z*)及*y*(*z*)驱动的稳定线性系统$$\kappa$$，并满足 \]

\forall u\left( z \right),r\left( z \right) = \kappa\begin{bmatrix}
u\left( z \right) \
y\left( z \right) \
\end{bmatrix} = 0

\[ 那么这样的系统被称为(8.2)、(8.3)的稳定核表示（SKR）。 显然，系统$$\left\lbrack - \hat{N}\left( z \right)\hat{M}\left( z \right) \right\rbrack$$是一个SKR.现在，考虑带有未知输入向量 的模型(8.18)，(8.19)。所有LTI残差生成器的参数化形式及其动态描述如下： \]

{r\left( z \right) = R\left( z \right)\begin{bmatrix}

• \hat{N}\left( z \right) & \hat{M}\left( z \right) \
  \end{bmatrix}\begin{bmatrix}
  u\left( z \right) \
  y\left( z \right) \
  \end{bmatrix}\backslash n}{= R\left( z \right)\left( {\hat{N}}{d}\left( z \right)d\left( z \right) + {\hat{N}}\left( z \right)f\left( z \right) \right)}

\[ 其中 \]

{\hat{N}}{d}\left( z \right) = \left( A - \text{LC},\text{Ed} - \text{LFd},C,\text{Fd} \right),{\hat{N}}\left( z \right) = \left( A - \text{LC},E_{f} - LF_{f},C,F_{f} \right)

\[ R(z)(≠0)是一种稳定的参数化矩阵，称为后滤波器。此外，为了避免故障检测信息的丢失，所述条件$$\text{rank}\left( R\left( z \right) \right) = m$$要被满足。 已经证明，所有LTI残差生成器都可以用(8.63)表示，而它们相对于未知输入和故障的动态特性可以用(8.64)进行参数化。而且，下式的成立 \]

\left\lbrack - \hat{N}\left( z \right)\hat{M}\left( z \right) \right\rbrack\begin{bmatrix}
u\left( z \right) \
y\left( z \right) \
\end{bmatrix} = y\left( z \right) - \hat{y}\left( z \right)

\[ 是由一个全阶观测器推得的$$\hat{y}$$作为y的估计，我们可以将一个FDF \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + \text{Bu}\left( k \right) + L\left( y\left( k \right) - \hat{y}\left( k \right) \right),\hat{y}\left( k \right) = C\hat{x}\left( x \right) + \text{Du}\left( k \right)

\[ 应用于(8.63)的在线实现。 总而言之，我们提出一个定理，它紧接着定义8.3和残差生成器参数化。 定理8.1在给定过程模型(8.18)，(8.19)的情况下，当且仅当LTI动态系统是(8.2)，(8.3)的SKR时，LTI动态系统才是残差生成器。 8.3 I/O数据模型 =============== 为了连接分析模型和过程数据，我们现在介绍不同的I/O数据模型。它们在我们的后续研究中至关重要，并在前面各节中介绍的基于模型的FD方案与下面将介绍的数据驱动设计方法之间建立了联系。为此，假定以下LTI过程模型是我们研究中采用的基础模型形式 \]

x\left( k + 1 \right) = \text{Ax}\left( k \right) + \text{Bu}\left( k \right) + w\left( k \right)

\[ \]

y\left( k \right) = \text{Cx}\left( k \right) + \text{Du}\left( k \right) + v\left( k \right)

\[ 其中$$u \in \mathcal{R}^{l}$$，$$y \in \mathcal{R}^{m}$$与$$x \in \mathcal{R}^{n}$$,$$\ w \in \mathcal{R}^{n}$$与$$v \in \mathcal{R}^{m}$$表示噪声序列，其符合正态分布并且统计上独立于u和x(0)。 令$$w\left( k \right) \in \mathcal{R}^{\xi}$$是数据矢量。我们引入下面的记法 \]

{w_{s}\left( k \right) = \begin{bmatrix}
w\left( k - 1 \right) \
\vdots \
w\left( k \right) \
\end{bmatrix} \in \mathcal{R}^{\left( s + 1 \right)\xi}\backslash n}{\ \Omega_{k} = \begin{bmatrix}
w\left( k \right) & \cdots & w\left( k + N - 1 \right) \
\end{bmatrix} \in \mathcal{R}^{\xi \times N}}

\[ \]

\Omega_{k,s} = \begin{bmatrix}
w_{s}\left( k \right)\cdots w_{s}\left( k + N - 1 \right) \
\end{bmatrix} = \begin{bmatrix}
\Omega_{k - s} \
\vdots \
\Omega_{k} \
\end{bmatrix} \in \mathcal{R}^{\xi \times N}

\[ 其中*s*，*N*是一些（大的）整数。在我们的研究中，*ω*(*k*)可以是*y*(*k*)，*u*(*k*) *x*(*k*)，并且$$\xi$$表示由(8.65)，(8.66)中给出的*m*、*l*或者*n* 第一个*I*/*O*数据模型描述如下 \]

Y_{k,s} = \Gamma_{s}X_{k - s} + H_{u,s}U_{k,s} + H_{w,s}W_{k,s} + V_{k,s} \in \mathcal{R}^{\left( s + 1 \right)m \times N}

\[ \]

Y_{s} = \begin{bmatrix}
C \
\text{CA} \
\vdots \
CA^{s} \
\end{bmatrix} \in \mathcal{R}^{\left( s + 1 \right)m \times n},H_{u,s} = \begin{bmatrix}
D & 0 & \ & \ \
\text{CB} & \ddots & \ddots & \ \
\vdots & \ddots & \ddots & 0 \
CA^{s - 1}B & \cdots & \text{CB} & D \
\end{bmatrix}

\[ 上式是直接根据在对奇偶空间法的研究中介绍的I/O模型(8.54)得出来的 其中 表示噪声矢量对过程输出的影响， 其中$$H_{u,s} \in \mathcal{R}^{\left( s + 1 \right)m \times (s + 1)l}$$，$$H_{w,s}W_{k,s} + V_{k,s}$$代表噪声矢量对具有相同结构的$$H_{w,s}$$(如$$H_{u,s}\left( 8.67 \right),(8.68)W_{k,s}V_{k,s}$$)的过程输出影响。 在SIM框架中，通常使用所谓的改进形式代替(8.69)来构建I/O模型。改进形式的核心是一个（稳定的）卡尔曼滤波器，其写为 \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + \text{Bu}\left( k \right) + K\left( y\left( k \right) - \hat{y}\left( k \right) \right),\hat{y}\left( k \right) = C\hat{x}\left( k \right) + \text{Du}\left( k \right)

\[ 新息序列$$y\left( k \right) - \hat{y}\left( k \right) e(k)$$是一个白噪声序列并且K是卡尔曼滤波器增益矩阵。基于(8.70)，过程的I/O关系可以重新写为 \]

\hat{x}\left( k + 1 \right) = A\hat{x}\left( k \right) + \text{Bu}\left( k \right) + \text{Ke}\left( k \right) = A_{K}\hat{x}\left( k \right) + B_{K}u\left( k \right) + \text{Ky}\left( k \right)

\[ \]

y\left( k \right) = C\hat{x}\left( k \right) + \text{Du}\left( k \right) + e\left( k \right),A_{K} = A - \text{KC},B_{K} = B - \text{KD}

\[ 以下两个I/O数据模型分别来自(8.71)，(8.72)： \]

Y_{k,s} = \Gamma_{s}{\hat{X}}{k - s} + HU_{k,s} + H_{e,s}E_{k,s},H_{e,s} = \begin{bmatrix}
I & 0 & \ & \ \
\text{CK} & \ddots & \ddots & \ \
\vdots & \ddots & \ddots & 0 \
CA^{\left( s - 1 \right)}K & \cdots & \text{CK} & I \
\end{bmatrix}

\[ \]

\left( I - H_{y,s}^{K} \right)Y_{k,s} = \Gamma_{s}^{K}{\hat{X}}{k - s} + H^{{K}U_{k,s},\Gamma_{s}} = \begin{bmatrix}
C \
CA_{K} \
\vdots \
CA_{K}^{s} \
\end{bmatrix}

\[ \]

H_{y,s}^{K} = \begin{bmatrix}
0 & 0 & \ & \ \
\text{CK} & \ddots & \ddots & \ \
\vdots & \ddots & \ddots & 0 \
CA_{K}^{s - 1}K & \cdots & \text{CK} & 0 \
\end{bmatrix},H_{u,s}^{K} = \begin{bmatrix}
0 & 0 & \ & \ \
CB_{K} & \ddots & \ddots & \ \
\vdots & \ddots & \ddots & 0 \
CA_{K}^{s - 1}K & \cdots & CK_{B} & 0 \
\end{bmatrix}

\[ 其中$${H\hat{}K\ }_{y,s} \in \mathcal{R}^{\left( s + 1 \right)m \times (s + 1)m}$$,$$\ H_{e,s} \in \mathcal{R}^{\left( s + 1 \right)m \times (s + 1)m}$$,$$\ E_{k,s} \in \mathcal{R}^{\left( s + 1 \right)m \times N}\]

与(8.67)，(8.68)中给出的结构相同。

8.4 注释和参考

为了将MVA技术应用于解决动态过程中的FD问题，最近二十年来已开发了出很多方法。其中，动态PCA/PLS[1-3]，PCA/PLS[4,5]的递归实现，快速移动窗体PCA[6]和多模式PCA[7]在近年来得到了广泛的应用和研究。

SIM是一个成熟的技术，在过程识别中得到了广泛的应用[8-11]。SIM技术在FDI中的应用，是[12-15]首次提出的。

第8.2节回顾了基于模型的FDI框架的基本内容。它们可以在专著[1,16-
25]和调查文件[26-29]中找到。

Beard 和 Jones 在[30,31]中报道了关于FDF的第一项工作，Chow
和Willsky提出了第一个利用奇偶空间法的最优FDI解决方法[32]。读者可以参考[25]中的第三章和第五章来系统地处理过程及故障建模以及基于模型的残差生成法等问题。

SKR的概念将在我们的后续研究中发挥重要作用。实际上，残差生成器的设计就是要找到定理8.1中给出的SKR。定义8.3中给出的SKR定义类似于[33]中针对非线性系统引入的定义。