LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

Full State Feedback Optimal $$H_{\infty}$$ Control
Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given. $$H_{\infty}$$ methods formulate this task as an optimization problem and attempt to minimize the $$H_{\infty}$$ norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the $$H_{\infty}$$ we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The lower linear fractional transformation (LFT) is used to implement a controller $$K$$ into the system. The lower LFT is denoted as $$\underline{S}(P,K)$$ and is formed by $$\underline{S}(P,K)=P_{11}+P_{12}(I-KP_{22})^{-1}KP_{21}$$ with $$\begin{bmatrix} z\\y \end{bmatrix}=\begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w\\u \end{bmatrix} $$. For full-state feedback we consider a controller of the form $$u(t)=Fx(t)$$. This is a special case where $$y(t)=x(t)$$ and results in a controller of the form $$ K= \begin{bmatrix} 0 & 0 \\ 0 & F \end{bmatrix} $$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI:Full State Feedback Optimal $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$F$$ such that $$||\underline{S}(P,K(0,0,0,F)||_{H_{\infty}}\leq \gamma$$

2) There exists $$Y>0$$ and $$Z$$ such that


 * $$\begin{bmatrix} YA^T+AY+Z^TB_2^{T}+B_2Z & B_1 & YC^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -\gamma I & D^{T}_{11} \\ C_1 Y + D_{12} Z & D_{11} & -\gamma I \end{bmatrix}<0

$$.

Then $$F=ZY^{-1}$$.

Conclusion:
The above LMI, if feasible, will determine the bound $$\gamma$$ on the $$H_{\infty}$$ norm of the system. In addition to this $$F$$ is also determined allowing the closed loop system to be determined using the controller $$\hat{K}(0,0,0,F)$$ found during the optimization.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_Hinf.m

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Full State Feedback Optimal H2 LMI