LMIs in Control/pages/full order Hinf H2 state observers

WIP, Description in progress

In this section, we treat the problem of designing a full-order state observer for system such that the effect of the disturbance $$w(t)$$ to the estimate error is prohibited to a desired level.

System Setting
The system is following

$$\dot{x}(t) = Ax(t) + B_1u(t) + B_2w(t), x(0) = x_0, $$

$$y(t) = C_1x(t) + D_1u(t) + D_2w(t), $$

$$z(t) = C_2x(t), $$

where $$x\in \mathbb{R}^n, \ y\in \mathbb{R}^l, \ z\in \mathbb{R}^m$$ are respectively the state vector, the measured output vector, and the output vector of interests.

$$ w \in \mathbb{R}^p$$ are the disturbance vector and control vector, respectively.

$$ A, \ B_1, \ B_2, \ C_1, \ C_2, \ D_1, \text{ and } D_2$$ are the system coefficient matrices of appropriate dimensions.

Problem Formulation
For the system, we introduce a full-order state observer in the following form:

$$\dot{\hat{x}}=(A+LC_1)\hat{x}-Ly+(B_1+LD_1)u$$

where $$\hat{x}$$ is the state observation vector and $$L\in \mathbb{R}^{n\times m}$$ is the observer gain. Obviously, the estimate of the interested output is given by

$$\hat{z}(t) = C_2\hat{x}(t)$$

which is desired to have as little affection as possible from the disturbance $$w(t)$$.

Using system dynamics,

$$\begin{align}\dot{x}(t) &= Ax(t) + B_1u(t) + B_2w(t) \\ &=Ax(t)+Ly-Ly+B_1u(t)+B_2w(t) \\ &=(A+LC_1)x(t)-Ly(t)+(B_1+LD_1)u(t)+(B_2+LD_2)w(t)\end{align}$$

Denoting

$$\dot{e} = (A+LC_1)e+(B_2+LD_2)w$$

$$\tilde{z}(t)=C_2e$$.

The transfer function of the system is clearly given by

$$ G_{\tilde{z}w}(s) = C_2(sI-A-LC_1)^{-1}(B_2+LD_2)$$.

With the aforementioned preparation, the problems of $$\mathcal{H}_\infty \text{ and } \mathcal{H}_2$$ state observer designs can be stated as follows.

Problem 1
($$\mathcal{H}_\infty$$ state observers) Given system (9.22) and a positive scalar $$\gamma$$ , find a matrix $$L$$ such that

$$||G_{\tilde{z}w}(s)||_\infty < \gamma$$.

Problem 2
($$\mathcal{H}_2$$ state observers) Given system (9.22) and a positive scalar $$\gamma$$ , find a matrix $$L$$ such that

$$||G_{\tilde{z}w}(s)||_2 < \gamma$$

As a consequence of the requirements in the previous problems, the error system is asymptotically stable, and hence we have

$$e(t) = x(t) - \hat{x}(t) \rightarrow 0, \text{ as } t\rightarrow \ \infty$$

This states that $$\hat{x}(t)$$ is an asymptotic estimate of $$x (t)$$.

Solution/Theorem
Regarding the solution to the problem of H∞ state observers design, we have the following theorem.

Theorem 1
The $$\mathcal{H}_\infty$$ state observers problem 1 has a solution if and only if there exist a matrix $$W$$ and a symmetric positive definite matrix $$P$$ such that

$$ \begin{bmatrix} A^TP+C_1^TW^T+PA+WC_1 & PB_2+WD_2 & C_2^T \\ (PB_2+WD_2)^T & -\gamma I & 0 \\ C_2 & 0 & -\gamma I \end{bmatrix}<0$$

When such a pair of matrices W and P are found, a solution to the problem is given as

$$L=P^{-1}W$$

With a prescribed attenuation level, the problem of H∞ state observers design is turned into an LMI feasibility problem in the form problem stated before. The problem with a minimal attenuation level $$\gamma$$ can be sought via the following optimization problem:

min $$\gamma$$

s.t. $$ \begin{align}P&>0 \\ \begin{bmatrix} A^TP+C_1^TW^T+PA+WC_1 & PB_2+WD_2 & C_2^T \\ (PB_2+WD_2)^T & -\gamma I & 0 \\ C_2 & 0 & -\gamma I \end{bmatrix}&<0 \end{align}$$

Theorem 2
The $$\mathcal{H}_2$$ state observers problem 2 has a solution the following 2 conclusions hold.

1.It has a solution if and only if there exists a matrix W, a symmetric matrix Q, and a symmetric matrix X such that

$$ \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2 \\ (XB_2+WD_2)^T & -I \end{bmatrix}<0$$,

$$\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0$$,

$$ \text{trace}(Q) < \gamma^2 $$.

When such a triple of matrices are obtained, a solution to the problem is given as

$$L=X^{-1}W$$.

2. It has a solution if and only if there exists a matrix V, a symmetric matrix Z, and a symmetric matrix Y such that

$$A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0$$,

$$\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0$$,

trace$$(Z)<\gamma^2$$.

When such a triple of matrices are obtained, a solution to the problem is given as

$$L=Y^{-1}V$$.

In applications, we are often concerned with the problem of finding the minimal attenuation level $$\gamma$$. This problem can be solved via the optimization

min $$\rho$$

s.t. $$ \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2 \\ (XB_2+WD_2)^T & -I \end{bmatrix}<0$$,

$$\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0$$,

$$ \text{trace}(Q) < \gamma^2 $$,

or

min $$\rho$$

$$A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0$$

$$\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0$$

trace$$(Z)<\gamma^2$$

When a minimal ρ is obtained, the minimal attenuation level is $$\gamma=\sqrt{\rho}$$.

WIP, additional references to be added