Solving Riccati equation using LMI algorithm

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Gilad Shaul
Gilad Shaul 2023-4-29
回答： Samyuktha 2023-5-24
I am tring to solve Riccati equation using LMI algorithm, but I can't find a solver that can support quadratic semidefinite constraints. The goal is to minimize P then calculate the optimal gain K.
I tried: 'mosek', 'scs', 'sdpt3'
I run the following code and get an error:
Warning: Solver not applicable (scs does not support quadratic semidefinite constraints)
LMI problem infeasible or failed
%% Vehicle Parameters
m=1573;
Iz=2873;
l1=1.58;
l2=1.1;
lh=7;
Ca1=80000;
Ca2=80000;
vx_max=25;
vx_min=15;
%% Matrix Variables
A_22=-2*(Ca2+Ca1)/(m*vx_max);
A_23= 2*(Ca2+Ca1)/m;
A_24=-2*(Ca2*l2-Ca1*l1)/(m*vx_max);
A_42=-2*(Ca2*l2-Ca1*l1)/(Iz*vx_max);
A_43= 2*(Ca2*l2-Ca1*l1)/Iz;
A_44=-2*(-Ca2*((l2)^2)+Ca1*((l1)^2))/(Iz*vx_max);
A_1 = [0 1 0 0; 0 A_22 A_23 A_24; 0 0 0 1; 0 A_42 A_43 A_44];
%% B&Q&R
B = [0; 2*Ca2/m; 0; 2*Ca2*l2/Iz];
q11=1;
q12=0;
q21=0;
q22=0;
Q=diag([q11,q12,q21,q22]);
R = 1;
%% Riccati Equations
% Care_max = A_1'*P + P*A_1 - P*B*inv(R)*B'*P + Q <= 0;
% Care_min = A_2'*P + P*A_2 - P*B*inv(R)*B'*P + Q <= 0;
%% Define LMI variables
P = sdpvar(4);
L = sdpvar(1, 4);
%% Define LMI constraints
% F = [A_1'*P + P*A_1 - P*B*inv(R)*B'*P + Q <= 0; P >= 0; [1; 0; 0; 0]*L*B == 1;
% L >= 0;
% (A_1 + B*L)'*P + P*(A_1 + B*L) <= 0];
F = [[A_1'*P+P*A_1+Q P*B; B'*P -eye(1,1)] <= 0; P >= 0; [1; 0; 0; 0]*L*B == 1;
L >= 0;
(A_1 + B*L)'*P + P*(A_1 + B*L) <= 0];
%% Define objective function
obj = trace(Q*P);
%% Define LMI options
opts = sdpsettings('solver', 'scs', 'verbose', 0);
%% Solve LMI
sol = optimize(F, obj, opts);
%% Check feasibility and extract solution
if sol.problem ~= 0
error('LMI problem infeasible or failed');
else
P_sol = value(P);
L_sol = value(L);
end
%% Compute optimal gain
K = inv(R)*B'*P_sol;
%% Display results
disp('Optimal Gain =');
disp(K);

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回答（1 个）

Samyuktha
Samyuktha 2023-5-24
Hi Gilad,
You can choose any of the three algorithms that Quadratic Programming Algorithms ('quadprog') supports:
  • 'interior-point-convex' (default)
  • 'trust-region-reflective'
  • 'active-set'
Since you have a positive semidefinite problem with a large number of linear constraints and not a large number of variables, try 'active-set'. Please refer to the following documentation links for more information:
Choosing the Algorithm, When the Solver Fails
Hope it helps!

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