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

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

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_1x(t) +T_2y(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{align} \Gamma_{i,j}^{11} &= (A_i-K_iC)^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{align} $$

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_max $$ then

\lim_{t \rightarrow \infty} \sup \|\tilde{x}\| \leq \epsilon x_max \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.