LMIs in Control/Click here to continue/Observer synthesis/Switched Observer with State Jumps

Switched Systems

Observer synthesis for switched linear systems results in switched observers with state jumps.

The System

${\begin{aligned}{\dot {x}}(t)&=A_{\sigma (t)}x(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $u(t)\in \mathbb {R} ^{m}$, $y(t)\in \mathbb {R} ^{p}$ and $\sigma (t)$ is the index function in discrete state given by $\sigma :[0,\infty )\rightarrow I_{N}=\{1,\cdots ,N\}$ deciding which one of the linear vector fields is active at a certain time instant.

The Data

• The matrices $A_{\sigma },B,C$ are system matrices of appropriate dimensions and are known.

• The unknown variables of the observer synthesis LMI are $P_{i},d_{i},j,\mu _{i,j}$ and $K_{i,j}$.

The Problem Formulation

Given a State-space representation of a system given as above. The dynamics of the continuous time observer is defined as:

${\begin{aligned}{\dot {\hat {x}}}&=A_{{\hat {\sigma }}(t){\hat {x}}}+Bu+K_{\hat {\sigma (t)}}(y-{\hat {y}})\\{\hat {y}}&=C{\hat {x}}\end{aligned}}$

where ${\hat {x(t)}}\in \mathbb {R} ^{n}$ is the state estimate of the vector field $x$ , $K_{j}\in \mathbb {R} ^{n\times p},j\in I_{N}$ is the observer gains, ${\hat {q}}(t):[0,\infty )\rightarrow I_{N}=\{1,\cdots ,N\}$ is the index function, and ${\hat {y(t)}}$ is the output of the mode location observer.

The observer is divided into two parts, the mode location observer estimating the active dynamics and the continuous-time observer estimating the continuous state of the switched system.

The estimated state jumps will be updated according to

${\hat {x}}^{+}=T_{1}x(t)+T_{2}y(t)\quad t\in \mathbb {T}$

where $\mathbb {T}$ is the set of times when the mode location observer switches mode, which are the times when ${\hat {\sigma }}$ changes value.

The LMI:

The following are equivalent:

(a)There exists $P_{i}>0,d_{i,j}\geq 0,\alpha >0,\mu _{i,j}\geq 0,\gamma \geq 0$ and $K_{i}$ such that

$\alpha I\leq P_{i}\leq \beta I,\qquad i\in I_{N}$

$\Gamma _{i,j}={\begin{bmatrix}\Gamma _{i,j}^{11}&\Gamma _{i,j}^{12}\\(\Gamma _{i,j}^{12})^{T}&\Gamma _{ij}^{22}\\\end{bmatrix}}\leq 0,\qquad (i,j)\in I_{s}$

$P_{j}=P_{i}+d_{i,j}^{T}C+C^{T}d_{i,j},\qquad (i,j)\in I_{s}$

${\begin{bmatrix}\lambda _{i}^{2}I_{p\times p}&W_{i}^{T}\\W_{i}&I_{n\times n}\\\end{bmatrix}}\geq 0,\qquad i\in I_{N}$

where

${\begin{aligned}\Gamma _{i,j}^{11}&=(A_{i}-K_{i}C)^{T}P_{i}+P_{i}(A_{i}-K_{i}C)+\gamma I\\\Gamma _{i,j}^{12}&=P_{i}(A_{j}-A_{i})\\\Gamma _{i,j}^{22}&=\mu _{i,j}Q_{j}-\gamma \epsilon ^{2}I\\\epsilon \geq 0,\alpha >0\end{aligned}}$

and the states of the hybrid observer is updated according to

${\hat {x}}^{+}=(I-R_{i}^{-1}(CR_{i}^{-1})^{\dagger }C){\hat {x}}+R_{i}^{-1}(CR_{i}^{-1})^{\dagger }y$

(b) If for some $T_{0}>0$

$\sup _{t>T_{0}}\|x(t)\|\leq x_{m}ax$

then

$\lim _{t\rightarrow \infty }\sup \|{\tilde {x}}\|\leq \epsilon x_{m}ax{\sqrt {\frac {\beta }{\alpha }}}$

Conclusion:

Using multiple Lyapunov functions and properly updating the continuous estimated states when the mode changes occur, an observer can be synthesized by solving a linear matrix inequality problem above.

External Links

A list of references documenting and validating the LMI.

• S. Pettersson and B. Lennartson. Hybrid system stability and robustness verification using linear matrix inequalities. International Journal of Control, 75(16-17):1335–1355, 2002.
• Stefan Pettersson. Designing switched observers for switched systems using multiple Lyapunov functions and dwell-time switching. IFAC Proceedings Volumes, 39(5):18–23, 2006. 2nd IFAC Conference on Analysis and Design of Hybrid Systems.
• Stefan Pettersson. Switched state jump observers for switched systems. IFAC Proceedings Volumes, 38(1):127–132, 2005. 16th IFAC World Congress.