%Generate the data for m=2, c=2, and k=1 by solving the 
%differential equation at t=0, 0.5, 1, 1.5, ..., 10.
m=2;
c=2;
k=1;
x0=[1; 0];
tspan=0.:.5:10;
options=odeset('RelTol',1e-5,'AbsTol',1.e-5);
[tdata,xdata]=ode45('fcnspring',tspan,x0,options,m,c,k);

plot(tdata,xdata(:,1),'b-*')
grid on
xlabel('Time')
ylabel('Displacement')
title('Simulated Data')

%Inverse problem: Now given that we know the data at 
%t=0, 0.5, 1, 1.5, ..., 10, we want to determine the 
%parameters m, c, and k so that the solution of the 
%spring-mass-dashpot model will best fit the data at 
%t=0, 0.5, 1, 1.5, ..., 10

options=optimset('MaxFunEvals',2000,'MaxIter',1000,'TolFun',1.e-12,'TolX',1.e-10);
parguess=[3 1 5];
[par,fval]=fminsearch('lscost',parguess,options,tdata,xdata)

%Plot the solution of the model against the data

m=par(1);
c=par(2);
k=par(3);
x0=[1; 0];
options=odeset('RelTol',1e-5,'AbsTol',1.e-5);
[tmodel,xmodel]=ode45('fcnspring',tdata,x0,options,m,c,k);

plot(tdata,xdata(:,1),'b-*',tmodel,xmodel(:,1),'r-v')
grid on
xlabel('Time')
ylabel('Displacement')
title('Model Solution versus Simulated Data')
legend('Data',['Model, 'num2str(par)],0)