考虑一般的光滑线性时变系统
\begin{equation}
\dot{x}=A(t)x+b(t)u
\end{equation}
其中控制输入选取为\(u=K(t)x+u_{d}(t)\)。
找到一个压缩变化\(\delta z=\Theta(t)\delta x\)得到一般的雅克比矩阵\(F\)为
\[F=(\dot{\Theta}+\Theta(A+bK))\Theta^{-1}
\]
\[=
\begin{gathered}
\begin{bmatrix}
0 & 1 & 0 &\cdots& 0 \\
0 & 0 & 1 &\cdots& 0 \\
\vdots & \vdots & \vdots &\ddots & 0 \\
0 & 0 & 0 &\cdots & 1 \\
-a_{0} & -a_{1}& -a_{2} &\cdots & -a_{n-1} \\
\end{bmatrix}
\end{gathered}
\]
上式的书写中省去时间\(t\)，矩阵\(\Theta\)的行向量\(\theta_{j}(j=1,2,...,n)\)可以表示为
\[\dot{\theta}_{j}+\theta_{j}(A+bK)=\theta_{j+1}
\]
\[\dot{\theta}_{n}+\theta_{n}(A+bK)=[-a_{0},-a_{0},\cdots,-a_{n-1}]
\begin{gathered}
\begin{bmatrix}
\theta_{1} \\
\vdots \\
\theta_{n} \\
\end{bmatrix}
\end{gathered}
\]
为了使得坐标变换\(\Theta\)独立于系统的输入信号，我们给\(\theta_{j}\)施加如下的限制
\[0=\theta_{1}b=\theta_{1}L^{0}b,\\
0=\theta_{2}b=(\dot{\theta}_{1}+\theta_{1}A)b-\frac{d}{dt}(\theta_{1}b)=\theta_{1}L^{1}b,\\
0=\theta_{3}b=(\dot{\theta}_{2}+\theta_{2}A)b-\frac{d}{dt}(\theta_{2}b)=\theta_{2}L^{1}b=(\dot{\theta}_{1}+\theta_{1}A)L^{1}b-\frac{d}{dt}(\theta_{1}L^{1}b)=\theta_{1}L^{2}b,\\
\vdots\\
D= \theta_{n}b = \theta_{1}L^{n-1}b ,
\]
其中\(L^{j}b\)为广义李导数（Lovelock, D. and H. Rund (1989). Tensors, Differential Forms,and variational Principles, Dover, New York）
\[L^{0}b=b,\\
L^{j+1}b=AL^{j}b-\frac{d}{dt}L^{j}b,j=0,...,n-1
\]
若选取\(D=det|L^{0}b,...,L^{n-1}b|\),那么总可以找到一个光滑的解\(\theta_{1}\),然后通过迭代可以得到所有的\(\theta_{j}\),
\[\theta_{j+1}=\dot{\theta}_{j}+\theta_{j}A, j=1,...,n-1
\]
这样可以得到光滑的有界矩阵，反馈系数\(K(t)\)可以由下面的方程得到
\[DK(t)=[-a_{0},-a_{0},\cdots,-a_{n-1}]
\begin{gathered}
\begin{bmatrix}
\theta_{1} \\
\vdots \\
\theta_{n} \\
\end{bmatrix}
\end{gathered}-\dot{\theta}_{n}-\theta_{n}A
\]
正定矩阵\(M=\Theta^{T}\Theta\)可以确保\(\delta x\)为一致指数型收敛。\(M\)表示充分的压缩条件。
