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 $x\in \mathbb {R} ^{n},\ y\in \mathbb {R} ^{l},\ z\in \mathbb {R} ^{m}$ 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,

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}u(t)+B_{2}w(t)\\&=Ax(t)+Ly-Ly+B_{1}u(t)+B_{2}w(t)\\&=(A+LC_{1})x(t)-Ly(t)+(B_{1}+LD_{1})u(t)+(B_{2}+LD_{2})w(t)\end{aligned}}$

Denoting

${\dot {e}}=(A+LC_{1})e+(B_{2}+LD_{2})w$

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

The transfer function of the system is clearly given by

$G_{{\tilde {z}}w}(s)=C_{2}(sI-A-LC_{1})^{-1}(B_{2}+LD_{2})$.

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

${\begin{bmatrix}A^{T}P+C_{1}^{T}W^{T}+PA+WC_{1}&PB_{2}+WD_{2}&C_{2}^{T}\\(PB_{2}+WD_{2})^{T}&-\gamma I&0\\C_{2}&0&-\gamma I\end{bmatrix}}<0$

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. ${\begin{aligned}P&>0\\{\begin{bmatrix}A^{T}P+C_{1}^{T}W^{T}+PA+WC_{1}&PB_{2}+WD_{2}&C_{2}^{T}\\(PB_{2}+WD_{2})^{T}&-\gamma I&0\\C_{2}&0&-\gamma I\end{bmatrix}}&<0\end{aligned}}$

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

$A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0$,

${\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0$,

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 $\rho$

$A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0$

${\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0$

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

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

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.