LMIs in Control/pages/H2-OSE

The H2 norm of a stable system H is the root-mean-square of the impulse response of the system. The H2 norm measures the steady-state covariance (or power) of the output response to unit noise input. In this module, the goal of H2 optimal state estimation is to design an observer that minimizes the H2 norm of the closed loop transfer matrix

The System
Consider the continuous-time generalized plant P with state-space realization



\begin{align} \dot x(t)&=Ax(t)+B_1u(t)+B_2w(t) & x(0)=x_0\\ y(t)&=C_1x(t)+D_{1}u(t)+D_2w(t)\\ &z(t) = C_2x(t) \end{align}$$

where it is assumed that (A,C2) is detectable. An observer of the form

The Data
output vector, and the output vector of interests
 * $$x$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$n, $$y$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$l , $$z$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$m are respectively the state vector, the measured

respectively
 * $$w$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$p and $$u$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$r are the disturbance vector and the control vector,

appropriate dimensions
 * A, B1, B2, C1, C2, D1, and D2 are the system coefficient matrices of

The Optimization Problem
Given the system and a positive scalar $$\gamma$$ we have to find the matrix L such that

|| $$G_{zw}(s)$$ || 2 < $$\gamma$$

An observer of the form

\begin{align} \dot x(t) = Ax(t) + (Ly - Ly) + B_1u(t) + B_2w(t)\\ \dot x(t)=(A+LC_1)x(t) - Ly + (B_1+LD_1)u(t) \\+ (B_2+LD_2)w(t) \end{align} $$ is to be designed, where L is the observer gain. Defining the error state as  $$e = x - \hat x$$ The break dynamics are found to be
 * $$\begin{align}

\dot e = (A+LC_1)e + (B_2+LD_2)w\\ \bar z(t) = C_2e \end{align} $$

For the system we introduce a full state observer in the following form: $$ \dot \hat x = (A+LC_1)\hat x - Ly + (B_1+LD_1)u$$ $$ \hat x, L$$ are the observation vector and the observer gain. The transfer function for this case is $$ G_{zw}(s) = C_2(sI-A-LC_1)^{-1}(B_2+LD_2)$$ and thus the problem of $$H_2$$ state observer design is to find L such that ||$$G_{zs}(s)||_2 < \gamma $$

The LMI: LMI for H2 Observer estimation
The H2 state observer problem 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}$$$$\begin{align}< 0\end{align}$$

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



\begin{align} trace(Q)< \gamma^2 \end{align} $$

and from the solution of the above LMIs we can obtain the observer matrix as



L = X^{-1}W $$

Conclusion
Thus by formulation, we have converted the problem of H2 state observer design into an LMI feasibility problem by optimizing the above LMIs. In application we are often concerned with the problem of finding the minimal attenuation level  $$\gamma$$  On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 3 matrices as output, $$X,W and Q$$ and also $$\rho$$ which is used to calculate $$\gamma$$ which is the H2 norm of the system.

There exists another set of LMIs which holds true for the same optimization problem as above.

\begin{align} A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2 < 0\\ \end{align}$$



\begin{bmatrix} -Z & YB_2 +VD_2 \\ (YB_2 +VD_2)^T & -Y \end{bmatrix}$$$$\begin{align}< 0\end{align}$$ '''$$ \begin{align} trace(Z)< \gamma^2 \end{align} $$'''

When a minimal $$\rho$$ is obtained, the minimal attenuation level is

$$\gamma = \sqrt\rho$$

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs
H$\infty$ State Observer Design

Discrete time H2 State Observer Design