MATLAB: Discrete time optimization with non-linear constraints

Question

Hello,

I have an optimization problem to solve with non-linear constraints. It is a control theory based discrete time model (which i feel fules out using fmincon) over a time horizon say N seconds.

I found a few old posts similar to this, but none I felt had clear answers.

I am going nuts trying to find a way to implement it in matlab. Which Matlab tool would be best suited in this case?

Thanks,

italic EDIT: the system is continuous, but we analyse it in a discrete time domain. Thus, variables have discrete values. There are n_v entities and each of the entity has each parameter described below:

variables: p,v,u; size(p)=size(v)=size(u)=(1,N) vectors

Obj. fn: minimize sum(u(1,:))

constraints:-

Upper and lower bounds like:

p_min<= p(1,:)<= p_max

v_min<= v(1,:)<= v_max

u_min<= u(1,:)<= u_max

linear equalities:

p(n+1)=p(n)+ v(n)t + 0.5u(n)t^2;

v(n+1)=v(n)+u(n)t;

u(n+1)= (s_star / (p1-p2))^2 ; p1 and p2 correspond to entity 1 and 2's p parameter

non linear equalities:

s_star= v1(n) * (v1(n)-v2(n)) ; where v1 and v2 are parameters of entities 1 and 2 respectively at nth instant. * this is the non-linear equality/constraint.

s_star2 = v1(n) * (v1(n)-v1(n-1));

Note: we use either s_star or s_star2 in our code

initial conditions:

p(1)=p_init;

v(1)=vel_init;

end conditions:

p(N)<= p_max;

v(N)=0;

I can give more information if required.

thanks,

Answer 1

I'm not sure what you want to get, but I tried to create an example. Hope this helps.

function [x,f,eflag,outpt] = myModelParamsSolver
xLast = []; % Last place computeall was called
myf = []; % Use for objective at xLast
myc = []; % Use for nonlinear inequality constraint
myceq = []; % Use for nonlinear equality constraint
N = 10;
x0 = rand(1,2+2*N);
fun = @objfun; % the objective function, nested below
cfun = @constr; % the constraint function, nested below
options = optimoptions('fmincon');
lb = zeros(1,2+2*N);
ub = ones(1,2+2*N);
% Call fmincon
[x,f,eflag,outpt] = fmincon(fun,x0,[],[],[],[],lb,ub,cfun,options);
      function y = objfun(x)
          if ~isequal(x,xLast) % Check if computation is necessary

              [myf,myc,myceq] = myDiscreteTimeModel(x);
              xLast = x;
          end
          % Now compute objective function
          y = myf;
      end
      function [c,ceq] = constr(x)
          if ~isequal(x,xLast) % Check if computation is necessary
              [myf,myc,myceq] = myDiscreteTimeModel(x);
              xLast = x;
          end
          % Now compute constraint functions
          c = myc; % In this case, the computation is trivial
          ceq = myceq;
      end
      function [f,c,ceq] = myDiscreteTimeModel(x)
          p = zeros(1,N);
          v = p;
          u = p;
          Ts = 0.01;
          p_max = 100;
          p_init = 10;
          vel_init = 1;
          % initial conditions:
          p(1)=p_init;
          v(1)=vel_init;
          p1 = x(1);
          p2 = x(2);
          v1 = x(3:3+N-1);
          v2 = x(3+N:end);
          for n = 1:N
              t = N*Ts;
              % non linear equalities:
              s_star= v1(n) * (v1(n)-v2(n));
              % where v1 and v2 are parameters of entities 1 and 2 respectively at nth instant.
              % * this is the non-linear equality/constraint.
              %s_star2 = v1(n) * (v1(n)-v1(n-1));
              % Note: we use either s_star or s_star2 in our code
              % linear equalities:
              p(n+1)= p(n)+ v(n)*t + 0.5*u(n)*t^2;
              v(n+1)= v(n)+ u(n)*t;
              u(n+1)= (s_star / (p1-p2))^2 ; % p1 and p2 correspond to entity 1 and 2's p parameter
          end
          % end conditions:
          ceq = v(N);
          c = p(N)-p_max;
          f = sum(u(:));
      end
  end