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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Copyright (c) 2023, Amon Lahr, Simon Muntwiler, Antoine Leeman & Fabian Flürenbrock Institute for Dynamic Systems and Control, ETH Zurich.
%
% All rights reserved.
%
% Please see the LICENSE file that has been included as part of this package.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function [H, h] = lqr_maxPI(Q, R, params)
% % Define the linear system
% A = params.model.A;
% B = params.model.B;
%
% % Define the state and input constraints
% s_max=params.constraints.MaxAbsPositionXZ;
% y_max=params.constraints.MaxAbsPositionY;
% u_max = params.constraints.MaxAbsThrust;
%
% % Define the cost function
% [K, ~, ~] = dlqr(A, B, Q, R);
% P = dare(A, B, Q, R)
% F = -inv((B'*P*B+R))*B'*P*A
% A
% B
% Ax=[1,0;
% 0,1;
% -1,0;
% 0,-1];
% Ax=[1,0;
% -1,0;
% 0,1;
% 0,-1];
% Au=Ax;
% bx=[s_max;s_max;s_max;s_max];
% bu=[u_max;u_max;u_max;u_max];
% H=[Ax;Au*F]/2
% h=[bx;bu]
% end
function [H, h] = lqr_maxPI(Q, R, params)
% Define the linear system
A = params.model.A;
B = params.model.B;
nx=params.model.nx;
nu=params.model.nu;
% Define the state and input constraints
s_max= params.constraints.MaxAbsPositionXZ;
y_max= params.constraints.MaxAbsPositionY;
u_max = params.constraints.MaxAbsThrust;
% Define the cost function
K = -dlqr(A, B, Q, R);
systemLQR = LTISystem('A', A+B*K);
% absxmax = ones(nx,1)*s_max;
% absumax = ones(nu,1)*u_max;
% Xp = Polyhedron('A',[eye(nx); -eye(nx); K; -K], 'b', [absxmax;absxmax; absumax;absumax]);
Hx=params.constraints.StateMatrix;
hx=params.constraints.StateRHS;
Hu=params.constraints.InputMatrix;
hu=params.constraints.InputRHS;
Xp = Polyhedron('A',[Hx;Hu*K], 'b', [hx;hu]);
figure(1);
% Xp.plot(), alpha(0.25), title('Resulting State Constraints under LQR Control'), xlabel('x_1'), ylabel('x_2'), zlabel('x_3');
systemLQR.x.with('setConstraint');
systemLQR.x.setConstraint = Xp;
InvSetLQR = systemLQR.invariantSet();
InvSetLQR.plot(), alpha(0.25), title('Invariant Set for Triple Integrator under LQR Control'), xlabel('x_1'), ylabel('x_2'), zlabel('x_3');
H=InvSetLQR.A
h=InvSetLQR.b
end
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