Unable to meet integration tolerances in fmincon to solve ODE optimization problem - matlab

Question

I have constrained ODE optimization problem to be solved using Matlab, I started by solving the ODE using ode15s which working will this initial values, also I had very good results without constraints using fminsearch, the problem started when I used fmincon it gave me these warnings:
Warning: Length of lower bounds is < length(x); filling in missing lower bounds with -Inf.
> In checkbounds (line 33)
In fmincon (line 318)
In Optimization (line 64)
Warning: Length of upper bounds is < length(x); filling in missing upper bounds with +Inf.
> In checkbounds (line 47)
In fmincon (line 318)
In Optimization (line 64)
Warning: Failure at t=2.340250e+01. Unable to meet integration tolerances without reducing the step size below the smallest value
allowed (5.684342e-14) at time t.
> In ode15s (line 668)
In ObjFun (line 6)
In barrier
In fmincon (line 813)
In Optimization (line 64)
I tried to remove constraints, but nothing changed...
y1_max = 5; y2_max = 2.3;
y01 = 1; y02 = 1.6;
%Control function parameter
Umin = -0.3; Umax = -0.1;
%Creation the structure of model parameters
params.T0=T0; params.Tf=Tf;
params.y1d=y1d; params.y2d=y2d;
params.y01=y01; params.y02=y02;
params.y2_max=y2_max;
n=2;
U0=-0.2*ones(1,n);
params.n=n;
params.U=U0; params.Umax=Umax; params.Umin=Umin;
params.dt=(Tf-T0)/n;
%for initial value of optimization parameters
options = odeset('RelTol',1e-7,'AbsTol',1e-7);
U=U0;
[t,y]=ode15s(#ODEsolver,[T0,Tf],[y01 y02],options,params);
%adding the ode solution as input prameters
params.t=t; params.y1= y(:,1); params.y2= y(:,2);
U0=-0.2*ones(1,n);
params.n=n; params.U=U0;
params.dt=(Tf-T0)/n;
A = [];
B = [];
Aq = []; Bq = [];
options1 = optimset('MaxIter',5000);
[x,fval,exitflag,output] = fmincon(#ObjFun,U0,A,B,Aq,Bq,Umin,Umax,IneqConst(U0,params),options1,params)
function y = ODEsolver(t,y,params)
dt = params.dt;
nu = floor(t/dt)+1;
nu = min(nu,params.n-1);
t1 = (nu-1)*dt;
Ut = params.U(nu) + ((t-t1)/dt)*(params.U(nu+1) - params.U(nu));
dy1a = ((0.63*y(1))/(1 + y(1)));
dy1b = (Ut*y(1)*y(2))/(1+0.6*y(1));
dy1 = dy1a + dy1b;
dy2a = (0.15*y(2))/(0.05-0.2*y(2));
dy2 = -0.2*y(2) + dy1b * dy2a;
y = [dy1; dy2];
end
function ObjFun1 = ObjFun(u,params)
% Calculating the value of the optimization criteria
params.U=u;
options = odeset('RelTol',1e-8,'AbsTol',1e-8);
[t,y]=ode15s(#ODEsolver,[0,100],[params.y01,
params.y02],options,params);
ObjFun1 =trapz(t,(y(:,1)-params.y1d).^2);
end
function [c,Const] = IneqConst(u, params)
params.U=u;
options = odeset('RelTol',1e-8,'AbsTol',1e-8);
[t,y]=ode15s(#ODEsolver,[0,100],[params.y01, params.y02],options,params);
c =[];
yCon = y(:,2)-params.y2_max;
Const = trapz(t,(abs(yCon)+yCon).^2);
end

Answer 1

The bug in this code is that the dimension of the bounds (Umin, Umax) should be an array with the same size of the control function U
so if the control function parameters n = 2 then Umin = [-0.3 -0.3] & Umax = [-0.1 -0.1]