考慮一般的光滑線性時變系統
\begin{equation}
\dot{x}=A(t)x+b(t)u
\end{equation}
其中控制輸入選取為\(u=K(t)x+u_{d}(t)\)。
找到一個壓縮變化\(\delta z=\Theta(t)\delta x\)得到一般的雅克比矩陣\(F\)為

\[F=(\dot{\Theta}+\Theta(A+bK))\Theta^{-1} \] \[= \begin{gathered} \begin{bmatrix} 0 & 1 & 0 &\cdots& 0 \\ 0 & 0 & 1 &\cdots& 0 \\ \vdots & \vdots & \vdots &\ddots & 0 \\ 0 & 0 & 0 &\cdots & 1 \\ -a_{0} & -a_{1}& -a_{2} &\cdots & -a_{n-1} \\ \end{bmatrix} \end{gathered} \]

上式的書寫中省去時間\(t\)

，矩陣\(\Theta\)的行向量\(\theta_{j}(j=1,2,...,n)\)可以表示為

\[\dot{\theta}_{j}+\theta_{j}(A+bK)=\theta_{j+1} \] \[\dot{\theta}_{n}+\theta_{n}(A+bK)=[-a_{0},-a_{0},\cdots,-a_{n-1}] \begin{gathered} \begin{bmatrix} \theta_{1} \\ \vdots \\ \theta_{n} \\ \end{bmatrix} \end{gathered} \]

為了使得座標變換\(\Theta\)

獨立於系統的輸入訊號，我們給\(\theta_{j}\)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\(L^{j}b\)

為廣義李導數（Lovelock, D. and H. Rund (1989). Tensors, Differential Forms,and variational Principles, Dover, New York）

\[L^{0}b=b,\\ L^{j+1}b=AL^{j}b-\frac{d}{dt}L^{j}b,j=0,...,n-1 \]

若選取\(D=det|L^{0}b,...,L^{n-1}b|\),那麼總可以找到一個光滑的解\(\theta_{1}\),然後通過迭代可以得到所有的\(\theta_{j}\),

\[\theta_{j+1}=\dot{\theta}_{j}+\theta_{j}A, j=1,...,n-1 \]

這樣可以得到光滑的有界矩陣，反饋係數\(K(t)\)可以由下面的方程得到

\[DK(t)=[-a_{0},-a_{0},\cdots,-a_{n-1}] \begin{gathered} \begin{bmatrix} \theta_{1} \\ \vdots \\ \theta_{n} \\ \end{bmatrix} \end{gathered}-\dot{\theta}_{n}-\theta_{n}A \]

正定矩陣\(M=\Theta^{T}\Theta\)可以確保\(\delta x\)為一致指數型收斂。\(M\)表示充分的壓縮條件。