LMIs in Control/pages/Switched systems H2 Optimization

Switched Systems $H_{2}$ Optimization

The Optimization Problem

This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix $A$ . which ever matrix is switching states, there must be an optimization for both cases using the same variables for both.

This is first done by defining the 9-matrix plant as such: $A\in \mathbb {R} ^{m\times m}$ , $B_{1}\in \mathbb {R} ^{m\times n}$ , $B_{2}\in \mathbb {R} ^{m\times p}$ , $C_{1}\in \mathbb {R} ^{n\times m}$ , $D_{11}\in \mathbb {R} ^{n\times n}$ , $D_{12}\in \mathbb {R} ^{n\times p}$ , $C_{2}=I$ , $D_{21}=0$ , and $D_{22}=0$ . Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for $H_{\infty }$ .

The Data

The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $A\in \mathbb {R} ^{m\times m}$ , $B_{1}\in \mathbb {R} ^{m\times n}$ , $B_{2}\in \mathbb {R} ^{m\times p}$ , $C_{1}\in \mathbb {R} ^{n\times m}$ , $D_{11}\in \mathbb {R} ^{n\times n}$ , $D_{12}\in \mathbb {R} ^{n\times p}$ , $C_{2}=I$ , $D_{21}=0$ , and $D_{22}=0$ . What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as $A(\delta )$ .

The LMI: Switched Systems $H_{2}$ Optimization

There exists a scalar $\gamma$ , along with the matrices $X>0$ , $W$ , and $Z$ where:

{\begin{aligned}||S(P,K(0,0,0,F))||_{H_{2}}^{2}\leq \gamma \\\\traceW<\gamma \\\\AX+XA^{T}+B_{2}Z+Z'B_{2}'+B_{1}B_{1}^{T}<0\\\\{\begin{bmatrix}X&(C_{1}X+D_{12}Z)^{T}\\C_{1}X+D_{12}Z&W\\\end{bmatrix}}&>0\\\end{aligned}}

Where $F=ZX^{-1}$ is the controller matrix. This also assumes that the only switching matrix is $A$ ; however, other matrices can be switched in states in order for more robustness in the controller.

Conclusion:

The results from this LMI gives a controller gain that is an optimization of $H_{2}$ for a switched system optimization.

Implementation

H2 switched state-feedback optimization example

% Switched System H2 example
% -- EXAMPLE --

clear; clc; close all;

%Given
A  = [ 1  1  0  1  0  1;
      -1 -1 -1  0  0  1;
       1  0  1 -1  1  1;
      -1  1 -1 -1  0  0;
      -1 -1  1  1  1 -1;
       0 -1  0  0 -1 -1];
  
B1 = [ 0 -1 -1;
       0  0  0;
      -1  1  1;
      -1  0  0;
       0  0  1;
      -1  1  1];

B21= [ 0  0  0;
      -1  0  1;
      -1  1  0;
       1 -1  0;
      -1  0 -1;
       0  1  1];

B22= [ 0   0   0;
      -1   0   1;
      -1   1   0;
       1   1   0;
       1   0   1;
       0  -3  -1];

C1 = [ 0  1  0 -1 -1 -1;
       0  0  0 -1  0  0;
       1  0  0  0 -1  0];

D12= [ 1    1    1;
       0    0    0;
       0.1  0.2  0.4];

D11= [ 1  2  3;
       0  0  0;
       0  0  0];
   
%Error
eta = 1E-4;

%sizes of matrices
numa  = size(A,1);    %states
numb2 = size(B21,2);  %actuators
numb1 = size(B1,2);   %external inputs
numc1 = size(C1,1);   %regulated outputs

%variables
gam = sdpvar(1);
Y   = sdpvar(numa);
Z   = sdpvar(numb2,numa,'full');
W   = sdpvar(numc1);

%Matrix for LMI optimization
M11 = Y*A'+A*Y+B21*Z+Z'*B21'+B1*B1';
M12 = Y*A'+A*Y+B22*Z+Z'*B22'+B1*B1';
M2  = [Y            (C1*Y+D12*Z)'  ;
       C1*Y+D12*Z   W              ];

%Constraints
Fc = (M11 <= 0);
Fc = [Fc; M12 <= 0];
Fc = [Fc; trace(W) <= gam];
Fc = [Fc; M2 >= zeros(numa+numc1)];

opt = sdpsettings('solver','sedumi');

%Objective function
obj = gam;

%Optimizing given constraints
optimize(Fc,obj,opt);

%Displays output Hinf gain
fprintf('\n\nHinf for H2 optimal state-feedback problem is: ')
display(value(gam))

F = value(Z)*inv(value(Y)); %#ok<MINV>

fprintf('\n\nState-Feedback controller F matrix')
display(F)

External Links

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

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