# 可观测性

## 定义

${\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix}}}$

${\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}$

${\displaystyle G:R^{n}\rightarrow {\mathcal {C}}(t_{0},t_{1};R^{n})}$
${\displaystyle x_{0}\mapsto C\Phi (t_{0},t_{1})x_{0}}$ ,

${\displaystyle N=\bigcap _{k=0}^{n-1}\ker(CA^{k})=\ker {\mathcal {O}}}$

${\displaystyle A={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$ , ${\displaystyle C={\begin{bmatrix}0&1\\\end{bmatrix}}}$ .

${\displaystyle {\mathcal {O}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}$

${\displaystyle {\mathcal {O}}v=0}$

${\displaystyle {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}v1\\v2\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\to v={\begin{bmatrix}v1\\0\end{bmatrix}}\to v=v1{\begin{bmatrix}1\\0\end{bmatrix}}}$

${\displaystyle Ker({\mathcal {O}})=N=span\{{\begin{bmatrix}1\\0\end{bmatrix}}\}}$

• ${\displaystyle N\subset Ke(C)}$
• ${\displaystyle A(N)\subset N}$
• ${\displaystyle N=\bigcup {\{S\subset R^{n}\mid S\subset Ke(C),A(S)\subset N\}}}$

## 线性时变系统

${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}$
${\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\,}$

${\displaystyle t\in [t_{0},t_{1}];}$ 的时间内，${\displaystyle A,B}$ ${\displaystyle C}$ 矩阵都已知，而输入及输出${\displaystyle u}$ ${\displaystyle y}$ 也都已知，可以透过一个额外在${\displaystyle M(t_{0},t_{1})}$ 之内的向量来确认${\displaystyle x(t_{0})}$ ${\displaystyle M(t_{0},t_{1})}$ 定义如下

${\displaystyle M(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\phi (t,t_{0})^{T}C(t)^{T}C(t)\phi (t,t_{0})dt}$

${\displaystyle M(t_{0},t_{1})}$ 非奇异方阵，可以找到一个唯一的${\displaystyle x(t_{0})}$ 。而且若${\displaystyle x_{1}-x_{2}}$ 是在${\displaystyle M(t_{0},t_{1})}$ 的核内，不可能由${\displaystyle x_{2}}$ 找到对应的启始状态${\displaystyle x_{1}}$

• ${\displaystyle M(t_{0},t_{1})}$ 对称矩阵
• ${\displaystyle M(t_{0},t_{1})}$ ${\displaystyle t_{1}\geq t_{0}}$ 时，为半正定矩阵
• ${\displaystyle M(t_{0},t_{1})}$ 满足线性
${\displaystyle {\frac {d}{dt}}M(t,t_{1})=-A(t)^{T}M(t,t_{1})-M(t,t_{1})A(t)-C(t)^{T}C(t),\;M(t_{1},t_{1})=0}$
• ${\displaystyle M(t_{0},t_{1})}$ 满足以下方程
${\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\phi (t,t_{0})^{T}M(t,t_{1})\phi (t,t_{0})}$ [5]

### 可观测性

${\displaystyle A(t),C(t)}$ 可解析，则系统在[${\displaystyle t_{0}}$ ,${\displaystyle t_{1}}$ ]可观测的条件是存在${\displaystyle {\bar {t}}\in [t_{0},t_{1}]}$ 以及正数k使得[6]

${\displaystyle rank{\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&:&\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}$

${\displaystyle N_{i+1}(t):=-N_{i}(t)A(t)-{\frac {\mathrm {d} }{\mathrm {d} t}}N_{i}(t),\ i=0,\ldots ,k-1}$

### 例子

${\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}}}$ , ${\displaystyle C(t)={\begin{bmatrix}1&0&1\end{bmatrix}}.}$ ${\displaystyle {\begin{bmatrix}N_{0}(0)\\N_{1}(0)\\N_{2}(0)\end{bmatrix}}={\begin{bmatrix}1&0&1\\0&-1&0\\1&0&0\end{bmatrix}}}$ ，因为矩阵的秩为3，因此在 ${\displaystyle \mathbb {R} }$ 内所有非平凡区间内都是可控制的。

## 非线性系统

${\displaystyle d{\mathcal {O}}_{s}(x_{0})=\mathrm {span} (dh_{1}(x_{0}),\ldots ,dh_{p}(x_{0}),dL_{v_{i}}L_{v_{i-1}},\ldots ,L_{v_{1}}h_{j}(x_{0})),\ j\in p,k=1,2,\ldots .}$ [7]

Griffith及Kumar,[8]、Kou、Elliot及Tarn[9]及Singh[10]是早期发展非线性动态系统的可观测性准则的先驱。

## 参考资料

1. ^ Kalman R. E., "On the General Theory of Control Systems", Proc. 1st Int. Cong. of IFAC, Moscow 1960 1481, Butterworth, London 1961.
2. ^ Kalman R. E., "Mathematical Description of Linear Dynamical Systems", SIAM J. Contr. 1963 1 152
3. Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
4. ^ 存档副本 (PDF). [2017-10-21]. （原始内容存档 (PDF)于2019-06-10）.
5. ^ Brockett, Roger W. Finite Dimensional Linear Systems. John Wiley & Sons. 1970. ISBN 978-0-471-10585-5.
6. ^ Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
7. ^ Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, prof dr. J.M.A.Scherpen and prof dr. A.J.van der Schaft.
8. ^ Griffith E. W. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971 35 135
9. ^ Kou S. R., Elliott D. L. and Tarn T. J. Observability of nonlinear systems. Information and Control, 22:89–99, 1973
10. ^ Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975
11. ^ Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981) (PDF). [2017-10-25]. （原始内容 (PDF)存档于2020-01-26）.
12. ^ Stanley G.M., and Mah R.S.H., "Observability and Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981) (PDF). [2017-10-25]. （原始内容存档 (PDF)于2017-08-10）.