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find_metric_Allgower.m
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find_metric_Allgower.m
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clear all; close all; clc;
yalmip('clear');
%% Allgower
lambda_range = 1.742857142857143;
% cond =1.000207519531250;
% lambda_range = linspace(0.8,2,15);
bounds = nan(15,1);
eps = 1e-3;
for ll = 1:length(lambda_range)
lambda = lambda_range(ll);
fprintf('lambda: %f ', lambda);
cond_l = 1;
cond_u = 5;
cond_best = cond_u;
while (cond_u-cond_l>eps)
cond = (cond_l+cond_u)/2;
fprintf(' cond: %f ', cond);
yalmip('clear');
w_upper = sdpvar(1); w_lower = sdpvar(1);
x1 = sdpvar(1);
x2 = sdpvar(1);
x = [x1 x2]';
f = [-1*x1 + 2*x2;
-3*x1 + 4*x2 - 0.25*(x2^3)];
B = [0.5;-2];
A = jacobian(f,x);
A0 = replace(A,x,zeros(2,1));
Q_LQR = 10*eye(2); R_LQR = 2;
[Klqr, P] = lqr(A0,B,Q_LQR,R_LQR);
Wlqr = inv(P);
delta = sdpvar(2,1);
[W_11,c_11]= polynomial(x,4,0);
[W_12,c_12]= polynomial(x,4,0);
[W_22,c_22]= polynomial(x,4,0);
[rho,c_rho] = polynomial(x,4,0);
W = [W_11,W_12;
W_12,W_22];
dW_f = jacobian(W(:),x)*f;
dW_f = reshape(dW_f,2,2);
% Enforce LQR at origin
LQR_constr_W = [replace(W,x,0) == Wlqr];
LQR_constr_rho = [replace(rho,x,0) == 2*inv(R_LQR)];
%Box constraints
g = [x1+5; 5-x1; x2+5; 5-x2];
%Box Lagrange functions
%Upper and Lower definiteness
[su_1,c_us_1] = polynomial([x;delta],4,0);
[su_2,c_us_2] = polynomial([x;delta],4,0);
[su_3,c_us_3] = polynomial([x;delta],4,0);
[su_4,c_us_4] = polynomial([x;delta],4,0);
[sl_1,c_ls_1] = polynomial([x;delta],4,0);
[sl_2,c_ls_2] = polynomial([x;delta],4,0);
[sl_3,c_ls_3] = polynomial([x;delta],4,0);
[sl_4,c_ls_4] = polynomial([x;delta],4,0);
%Killing Field
[seq_11_1,ceq_11_1] = polynomial(x,4,0);
[seq_11_2,ceq_11_2] = polynomial(x,4,0);
[seq_11_3,ceq_11_3] = polynomial(x,4,0);
[seq_11_4,ceq_11_4] = polynomial(x,4,0);
[seq_11_5,ceq_11_5] = polynomial(x,4,0);
[seq_11_6,ceq_11_6] = polynomial(x,4,0);
[seq_11_7,ceq_11_7] = polynomial(x,4,0);
[seq_11_8,ceq_11_8] = polynomial(x,4,0);
[seq_12_1,ceq_12_1] = polynomial(x,4,0);
[seq_12_2,ceq_12_2] = polynomial(x,4,0);
[seq_12_3,ceq_12_3] = polynomial(x,4,0);
[seq_12_4,ceq_12_4] = polynomial(x,4,0);
[seq_12_5,ceq_12_5] = polynomial(x,4,0);
[seq_12_6,ceq_12_6] = polynomial(x,4,0);
[seq_12_7,ceq_12_7] = polynomial(x,4,0);
[seq_12_8,ceq_12_8] = polynomial(x,4,0);
[seq_22_1,ceq_22_1] = polynomial(x,4,0);
[seq_22_2,ceq_22_2] = polynomial(x,4,0);
[seq_22_3,ceq_22_3] = polynomial(x,4,0);
[seq_22_4,ceq_22_4] = polynomial(x,4,0);
[seq_22_5,ceq_22_5] = polynomial(x,4,0);
[seq_22_6,ceq_22_6] = polynomial(x,4,0);
[seq_22_7,ceq_22_7] = polynomial(x,4,0);
[seq_22_8,ceq_22_8] = polynomial(x,4,0);
%CCM condition
[s_ccm_1,c_ccm_1] = polynomial([x;delta],4,0);
[s_ccm_2,c_ccm_2] = polynomial([x;delta],4,0);
[s_ccm_3,c_ccm_3] = polynomial([x;delta],4,0);
[s_ccm_4,c_ccm_4] = polynomial([x;delta],4,0);
%CCM condition
R_CCM = -(-dW_f + A*W + W*A' - rho*(B*B') + 2*lambda*W);
p_CCM = delta'*R_CCM*delta - [s_ccm_1,s_ccm_2,s_ccm_3,s_ccm_4]*g;
%Killing field condition
dW_B = jacobian(W(:),x)*B;
dW_B = reshape(dW_B,2,2);
killing_11 = dW_B(1,1) - [seq_11_1,seq_11_2,seq_11_3,seq_11_4]*g;
killing_11_2 = (-dW_B(1,1)) - [seq_11_5,seq_11_6,seq_11_7,seq_11_8]*g;
killing_12 = dW_B(1,2) - [seq_12_1,seq_12_2,seq_12_3,seq_12_4]*g;
killing_12_2 = (-dW_B(1,2)) - [seq_12_5,seq_12_6,seq_12_7,seq_12_8]*g;
killing_22 = dW_B(2,2) - [seq_22_1,seq_22_2,seq_22_3,seq_22_4]*g;
killing_22_2 = (-dW_B(2,2)) - [seq_22_5,seq_22_6,seq_22_7,seq_22_8]*g;
%W uniform bounds
W_bounds = [w_lower>=0.0035; w_upper>=w_lower];
%Condition bound
W_cond = [w_upper-cond*w_lower <= 0];
%W pos def
p_W_up = (w_upper*(delta'*delta)-delta'*W*delta) - ...
[su_1,su_2,su_3,su_4]*g;
p_W_low = (delta'*W*delta - w_lower*(delta'*delta))-...
[sl_1,sl_2,sl_3,sl_4]*g;
coeff_List = [c_rho;c_11;c_12;c_22;
c_us_1;c_us_2;c_us_3;c_us_4;c_ls_1;c_ls_2;c_ls_3;c_ls_4;
ceq_11_1;ceq_11_2;ceq_11_3;ceq_11_4;ceq_11_5;ceq_11_6;ceq_11_7;ceq_11_8;
ceq_12_1;ceq_12_2;ceq_12_3;ceq_12_4;ceq_12_5;ceq_12_6;ceq_12_7;ceq_12_8;
ceq_22_1;ceq_22_2;ceq_22_3;ceq_22_4;ceq_22_5;ceq_22_6;ceq_22_7;ceq_22_8;
c_ccm_1;c_ccm_2;c_ccm_3;c_ccm_4];
constr_List = [ sos(p_W_low);sos(p_W_up);
sos(p_CCM);
sos(su_1); sos(su_2);
sos(su_3); sos(su_4);
sos(sl_1); sos(sl_2);
sos(sl_3); sos(sl_4);
W_bounds;W_cond;
sos(seq_11_1);sos(seq_11_2);sos(seq_11_3);sos(seq_11_4);
sos(seq_11_5);sos(seq_11_6);sos(seq_11_7);sos(seq_11_8);
sos(seq_12_1);sos(seq_12_2);sos(seq_12_3);sos(seq_12_4);
sos(seq_12_5);sos(seq_12_6);sos(seq_12_7);sos(seq_12_8);
sos(seq_22_1);sos(seq_22_2);sos(seq_22_3);sos(seq_22_4);
sos(seq_22_5);sos(seq_22_6);sos(seq_22_7);sos(seq_22_8);
sos(killing_11); sos(killing_11_2);
sos(killing_12); sos(killing_12_2);
sos(killing_22); sos(killing_22_2);
sos(s_ccm_1); sos(s_ccm_2); sos(s_ccm_3); sos(s_ccm_4)];
% LQR_constr_W;LQR_constr_rho];
options = sdpsettings('solver','mosek','verbose',0);
SOS_soln = solvesos(constr_List,[],options,coeff_List);
if (SOS_soln.problem == 0)
fprintf('feasible\n');
cond_u = cond;
cond_best = cond;
coeff_sol = clean(double(coeff_List),1e-3);
W_sol = replace(W,coeff_List,coeff_sol);
dW_f = jacobian(W(:),x)*f; dW_f = reshape(dW_f,2,2);
dW_f_sol = replace(dW_f,coeff_List,coeff_sol);
rho_sol = replace(rho,coeff_List,coeff_sol);
else
cond_l = cond;
end
end
fprintf('\n');
bounds(ll) = (sqrt(cond))/lambda;
end
pause;
clc;
figure()
plot(lambda_range, bounds,'ro','markerfacecolor','g','markersize',20);
grid on
xlabel('\lambda');
ylabel('$\|x^{*}-x\|/\bar{w}$','Interpreter','Latex');
% title('Robustness optimization');
set(findall(gcf,'type','text'),'FontSize',32);set(gca,'FontSize',32)
% Display
disp('W')
sdisplay(W_sol)
disp('rho')
sdisplay(rho_sol)
disp('Bound achieved');
disp((1/lambda)*sqrt(cond));
%% Convert into functions
W_cell = cell(2,2);
dW_f_cell = cell(2,2);
for i = 1:2
for j = 1:2
W_ij = sdisplay(W_sol(i,j));
dW_ij = sdisplay(dW_f_sol(i,j));
W_ij = strcat('@(x1,x2)',W_ij{1});
dW_ij = strcat('@(x1,x2)',dW_ij{1});
W_cell{i,j} = str2func(W_ij);
dW_f_cell{i,j} = str2func(dW_ij);
end
end
W_mat = @(x1,x2) [W_cell{1,1}(x1,x2), W_cell{1,2}(x1,x2);
W_cell{2,1}(x1,x2), W_cell{2,2}(x1,x2)];
dW_f_mat = @(x1,x2) [dW_f_cell{1,1}(x1,x2), dW_f_cell{1,2}(x1,x2);
dW_f_cell{2,1}(x1,x2), dW_f_cell{2,2}(x1,x2)];
rho = sdisplay(rho_sol);
rho = strcat('@(x1,x2)',rho{1});
rho = str2func(rho);
x1_range = -5:0.2:5;
x2_range = -5:0.2:5;
[X,Y] = meshgrid(x1_range,x2_range);
W_max = zeros(size(X));
W_min = zeros(size(X));
for i = 1:length(x2_range)
for j = 1:length(x1_range)
l = eig(W_mat(x1_range(j),x2_range(i)));
W_max(i,j) = max(l);
W_min(i,j) = min(l);
end
end
figure()
surface(X,Y,W_min);
hold on
surface(X,Y,W_max);
grid on
% Check Contraction condition
f = @(x1,x2) [-1*x1 + 2*x2;
-3*x1 + 4*x2 - 0.25*(x2^3)];
B = [0.5;-2];
A = @(x1,x2) [-1, 2;
-3, 4-0.75*(x2^2)];
R_val = zeros(size(X));
% (-dW_f + A*W + W*A' - rho*(B*B') + 2*lambda*W)
for i = 1:length(x2_range)
for j = 1:length(x1_range)
x1 = x1_range(j); x2 = x2_range(i);
L = (-dW_f_mat(x1,x2) + ...
A(x1,x2)*W_mat(x1,x2) + W_mat(x1,x2)*A(x1,x2)' -...
rho(x1,x2)*(B*B') + 2*lambda*W_mat(x1,2));
R_val(i,j) = max(eig(L));
end
end
figure()
surface(X,Y,R_val);
grid on
hold on