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

$\dot{x}(t)=Ax(t)+B_{1}u(t)+B_{2}w(t),x(0)=x_0,$

$y(t)=C_{1}x(t)+D_{1}u(t)+D_{2}w(t),$

$z(t)=C_{2}x(t),$

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

$w\in {ℝ}^p$ 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:

$\dot{\hat{x}}=(A+L{C}_1)\hat{x}-Ly+(B_1+L{D}_1)u$

where $\hat{x}$ is the state observation vector and $L\in {ℝ}^{n\times m}$ 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,

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+B_{1}u(t)+B_{2}w(t)\\ & =Ax(t)+Ly-Ly+B_{1}u(t)+B_{2}w(t)\\ & =(A+L{C}_1)x(t)-Ly(t)+(B_1+L{D}_1)u(t)+(B_2+L{D}_2)w(t)\end{array}$

Denoting

$\dot{e}=(A+L{C}_1)e+(B_2+L{D}_2)w$

$\tilde{z}(t)=C_{2}e$.

The transfer function of the system is clearly given by

$G_{\tilde{z}w}(s)=C_2(sI-A-L{C}_1{)}^{-1}(B_2+L{D}_2)$.

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

Problem 1

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

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

Problem 2

(${\mathscr{H}}_2$ 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

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

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 ${\mathscr{H}}_{\mathrm{\infty }}$ 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

$\left[\begin{array}{ccc}{A}^{T}P+C_1^T{W}^T+PA+W{C}_1& P{B}_2+W{D}_2& C_2^T\\ (P{B}_2+W{D}_2{)}^T& -\gamma I& 0\\ C_2& 0& -\gamma I\end{array}\right]<0$

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. $\begin{array}{cc}P& >0\\ \left[\begin{array}{ccc}{A}^{T}P+C_1^T{W}^T+PA+W{C}_1& P{B}_2+W{D}_2& C_2^T\\ (P{B}_2+W{D}_2{)}^T& -\gamma I& 0\\ C_2& 0& -\gamma I\end{array}\right]& <0\end{array}$

Theorem 2

The ${\mathscr{H}}_2$ 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

$\left[\begin{array}{cc}XA+W{C}_1+(XA+W{C}_1{)}^T& X{B}_2+W{D}_2\\ (X{B}_2+W{D}_2{)}^T& -I\end{array}\right]<0$,

$\left[\begin{array}{cc}-Q& C_2\\ C_2^T& -X\end{array}\right]<0$,

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

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

$A^{T}Y+C_1^T{V}^T+YA+V{C}_1+C_2^T{C}_2<0$,

$\left[\begin{array}{cc}-Z& (Y{B}_2+V{D}_2{)}^T\\ Y{B}_2+V{D}_2& -Y\end{array}\right]<0$,

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. $\left[\begin{array}{cc}XA+W{C}_1+(XA+W{C}_1{)}^T& X{B}_2+W{D}_2\\ (X{B}_2+W{D}_2{)}^T& -I\end{array}\right]<0$,

$\left[\begin{array}{cc}-Q& C_2\\ C_2^T& -X\end{array}\right]<0$,

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

or

min $\rho$

$A^{T}Y+C_1^T{V}^T+YA+V{C}_1+C_2^T{C}_2<0$

$\left[\begin{array}{cc}-Z& (Y{B}_2+V{D}_2{)}^T\\ Y{B}_2+V{D}_2& -Y\end{array}\right]<0$

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

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

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.