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Picard iteration convergence for solving non-linear state space system

Nasser M. Abbasi

January 7, 2015 page compiled on June 30, 2015 at 7:25pm

1 introduction

Picard iteration for the solution of non-linear system x′(t) = f (x (t) ,t)  is given by

                 t
 k+1        0  ∫   (  k      )
x    (t) = x  +    f x  (τ),τ  dτ
                0

The above iteration was implemented numerically for a two state system with the forcing function      (           )
         cosx1
f =   tx  + e−tx
        1       2

The initial guess used is the same as the initial conditions which is given by      (    )
 0      2
x  =
       − 1 . Matlab was used for the implementation. The source code is in one m file and can be downloaded from the link above. A movie showing the convergence is given below as well.

2 Observation

Numerical solution was required since there is no closed form solution using symbolic integration after the second iteration for the first state.  The Picard solution is compared to the final numerical solution obtained from ODE45. Time span of 30 seconds was used. For numerical integration, different sampling times were tried. The results shows that the smaller the time span, the less Picard iterations are needed to converge.

Reference: Lecture notes, ECE 717, Fall 2014, University of Wisconsin, Madison, by Professor B. Ross Barmish

2.1 Source code

 
function   nma_picard() 
%version oct 17, 2014 
%study of picard iteration method for non-linear state space system 
%Matlab 2013a 
%The following parameters can be changed: max simulation time, 
%time spacing between each sample, initial conditions, and 
%initial guess. 
%by Nasser M. Abbasi 
 
close all 
 
number_of_iterations = 100;    %How many Picard iterations to do? 
initial_conditions   = [2 -1]; %initial conditions for x_1 and x_2 
initial_guess        = [2 -1]; %initial guess 
max_time             = 50;     %simulation time in seconds 
delT                 = 0.02;   %time spacing for sampling, numerical integration 
 
nSamples             = round(max_time/delT); 
first_K              = bsxfun(@times, initial_guess,ones(nSamples,2)); 
next_K               = zeros(nSamples,2); 
t                    = (0:delT:max_time-delT)'; 
 
%obtain numerical ODE45 solution to use to compare with 
odeTime                    = 0:delT:max_time-delT; 
[odeTime,ODE45_solution]   = ode45(@rhs,odeTime,initial_conditions); 
 
for n = 1:number_of_iterations 
    for i = 1:nSamples %numerical integration as time is increased 
                       %t vector above hold incremental time values. 
        next_K(i,1) = initial_conditions(1) + delT*trapz(cos(first_K(1:i,1))); 
        z = t(1:i).*first_K(1:i,1)+exp(-t(1:i)).*first_K(1:i,2); 
        next_K(i,2) = initial_conditions(2) + delT*trapz(z); 
    end 
 
    makePlot(first_K,t,n,delT,initial_guess,max_time,ODE45_solution); 
    first_K  = next_K; 
end 
    function dxdt=rhs(t,x) 
        dxdt = [cos(x(1));t*x(1)+exp(-t)*x(2)]; 
    end 
end 
 
%------------------------------- 
function makePlot(x,t,n,delT,guess,max_time,ODE45_solution) 
if n==1 
    scrsz = get(groot,'ScreenSize'); 
    figure('Position',[.25*scrsz(3) .35*scrsz(4) .5*scrsz(3) .5*scrsz(4)]); 
    set(0,'DefaultAxesFontName', 'Times New Roman'); 
    set(0,'DefaultAxesFontSize',10); 
    set(0,'DefaultTextFontname', 'Times New Roman'); 
    set(0,'DefaultTextFontSize', 12); 
end 
 
%fix the Y scale, to make the plots easier to see as simulation runs 
minY1   = -1; 
maxY1   = 2.5; 
minY2   = -10; 
maxY2   = 1000; 
 
subplot(1,2,1); 
hold off; 
plot(t,x(:,1)); 
hold on; 
plot(t,ODE45_solution(:,1),'r:'); 
title(sprintf('$$x_1^{%d}, \\Delta t=%3.3f, guess=[%d,%d]$$',n,delT,guess(1),guess(2)),'FontSize', 12,'interpreter','latex'); 
xlabel('time(sec)'); 
ylim([minY1,maxY1]); 
xlim([0,max_time]); 
subplot(1,2,2); 
hold off; 
plot(t,x(:,2)); 
hold on; 
plot(t,ODE45_solution(:,2),'r:'); 
title(sprintf('$$x_2^{%d}$$',n),'FontSize', 14,'interpreter','latex'); 
xlabel('time(sec)'); 
ylim([minY2,maxY2]); 
xlim([0,max_time]); 
drawnow 
end