Infeasible models occurs after applying robust MPC frame

[peige guo]

Dear all,

I'm new to Yalmip. When I try to use MPC robust frame, I find "Model is infeasible or unbounded" errors will come. However, compared with the model without robust no parameters are modified. I have attached my codes. Could you please tell me what is going wrong for the robust frame? I appreciate your time so much!

Best regards

Peige

[Johan Löfberg]

Interesting catch. It is an unexpected combination of nonconvex use of norm operator which is MILP-represented, and uncertain expression containing same norm ooerator, where YALMIP assumes you are in a convex world. At the moment, you will have to do some manual coding to get around it

% Nominal case without any uncertainty works

x = sdpvar(2,1);

Model = [-3 <= x <= 3, norm(x - [0.1;0.2],1) >= 1];

optimize(Model, x'*x)

clf

plot(Model);hold on;plot(value(x(1)),value(x(2)),'*')

% Add uncertainty, fails due to infeasibility

x = sdpvar(2,1);

w = sdpvar(2,1);

Model = [-3 <= x <= 3, norm(x + w - [0.1;0.2],1) >= 1];

Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]

optimize(Model, x'*x)

% Manual derivation of nonconvex constraint 

% and it works as expected

x = sdpvar(2,1);

w = sdpvar(2,1);

d = binvar(4,1);

q = x + w - [0.1;0.2];

M = 10;

Model = [-3 <= x <= 3, q(1) + q(2) >= 1-M*(1-d(1)),

                       q(1) - q(2) >= 1-M*(1-d(2)),

                      -q(1) + q(2) >= 1-M*(1-d(3)),

                      -q(1) - q(2) >= 1-M*(1-d(4)),

                      sum(d)==1]

Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]

optimize(Model, x'*x)

clf

plot(Model,x);hold on;plot(value(x(1)),value(x(2)),'*')

% Semimanual, represent the polytope manually but use implies
% for the combinatorial stuff

A = [1 1;-1 1;1 -1;-1 -1];

b = [1;1;1;1];

x = sdpvar(2,1);

w = sdpvar(2,1);

d = binvar(4,1);

q = A*(x + w - [0.1;0.2]) - 1;

Model = [-3 <= x <= 3, implies(d,q>=0),sum(d)==1];

Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]

optimize(Model, x'*x)

clf

plot(Model,x);hold on;plot(value(x(1)),value(x(2)),'*')

[peige guo]

Dear Dr.Johan,

After reviewing some math knowledge, now I get the reason why I should derive the nonconvex constraint manually. I also find the methods to transfer it into a mixed-integer form by introducing binary variables. The YALMIP robust frame works well now.  I appreciate your time and detailed enlighten so much.

Best regards,

Peige

[peige guo]

Dear Dr.Johan,

After reviewing some math knowledge, now I get the reason why I should derive the nonconvex constraint manually. I also find the methods to transfer it into a mixed-integer form by introducing binary variables. The YALMIP robust frame works well now.  I appreciate your time and detailed enlighten so much.

Best regards,

Peige Guo