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LQ Regulation via H2 Control

Suppose people were interested in quadratic optimal regulation problem, where instead of the Ricatti equation approach (which is traditionally used in such situations), it is approached using LMI's instead (thereby making it a Linear Quadratic (LQ) problem). Such an approach would be possible by converting the LQ problem into a standard $${H2}$$ problem.

The System
Consider, for example, constant linear multi-variable system in the form of:



\begin{align} \dot x&=Ax+Bu, x(0)={x_0},\\ \end{align}$$

where

\begin{align} &x\in\mathbb{R}^{n}\text{ and }u\in\mathbb{R}^{r}\text{ are the state and input vectors, respectively, and }\\ &A\text{ and }B\text{ are the system coefficient matrices of appropriate dimensions,}\\ \end{align}$$ Then the LQ optimal regulation problem for the given system is stated as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A$$, $$B$$, $$Q$$, and $$R$$ (the latter two of which are obtained as follows).

The Optimization Problem
Using the multi-variable system as described above, we can see that the optimal state feedback controller $$u=Kx$$ is obtained where

\begin{align} J(x,u) &=\int_0^\infty ({x^T}Qx+{u^T}Ru)\mathrm{d}t\\ \end{align}$$

is minimized where $$Q={Q^T}{\geq}0$$ and $$R={R^T}{\geq}0$$. However, it is to be noted that in order for the problem to have a solution, two assumptions are made - both of which must be held true at all times:
 * $$ 1)\quad (A,B)\text{ is stabilizable, and}$$
 * $$ 2)\quad (A,L)\text{ is observable, where }L={Q^{1/2}}$$.

Relating this to $${H_2}$$ performance, let us now consider the auxiliary system:

\begin{align} \begin{cases} \dot{x} &= Ax + Bu + {x_0}\omega\\ y &= Cx + Du\\ \end{cases} \end{align}$$,

where $$\omega$$ represents an impulse disturbance, and

\begin{align} C = \begin{bmatrix}{Q^{1/2}}\\0\end{bmatrix}&& D = \begin{bmatrix}0\\{R^{1/2}}\end{bmatrix}\\ \end{align}$$

Using the state feedback controller $$u=Kx$$ and applying it on the above auxiliary system results in the closed-loop system:

\begin{align} \begin{cases} \dot{x} &= (A+BK)x + {x_0}\omega\\ y &= (C+DK)x\\ \end{cases} \end{align}$$,

and the resulting transfer function from disturbance $$\omega$$ to output $$y$$ being:

\begin{align} {G_{y\omega}}=(C+DK)[sI-(A+BK)]^{-1}{x_0} \end{align}$$ thereby resulting in $$J(x,u)=||{G_{y\omega}}||^2_2$$.

The LMI: LQ Regulation via H2 Control
From the given pieces of information, and letting the 2 assumptions as described above hold, then there exist matrices $$X\in\mathbb{S}^n$$, $$Y\in\mathbb{S}^r$$ and $$W\in\mathbb{R}^{rxn}$$ satisfying:



\begin{align} (AX+BW)+{(AX+BW)^T}+{x_0}{x^T_0}&<0\\ trace({Q^{1/2}}X({Q^{1/2}})^T)+trace(Y)&<\gamma\\ \begin{bmatrix}-Y&&{R^{1/2}}W\\({R^{1/2}}W)^T&&-X\end{bmatrix}&<0 \end{align}$$

Conclusion:
From the LMI, it can be seen that the state feedback control in the form of $$u=Kx$$ (where $$K=WX^{-1}$$) exists such that $$J(x,u)<\gamma$$ if and only if the matrices $$X\text{, }Y\text{, and }W$$ are of the appropriate matrix sizes.

Implementation

 * Example Code - A GitHub link that contains code (titled "LQRegH2.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * ../LMI for System H2 Norm/ - LMI to determine the $$H_2$$-norm of a system.