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1.28 Determine the state response of a system to only initial conditions in state space

Problem: Given a system with 2 states, represented in state space, how to determine the state change due some existing initial conditions, when there is no input forces?

Mathematica 

Remove["Global`*"]; 
a = {{-0.5572,-0.7814},{0.7814, 0}}; 
c = {{1.9691, 6.4493}}; 
sys=StateSpaceModel[ 
       {a,{{1},{0}},c,{{0}}}]

pict 

x0 = {1,0}; 
{x1,x2} = StateResponse[ 
            {sys,x0},0,{t,0,20}]; 
 
Plot[{x1,x2},{t,0,20}, 
  PlotRange->All, 
  GridLines->Automatic, 
  GridLinesStyle->Dashed, 
  Frame->True, 
  ImageSize->350, 
  AspectRatio->1, 
  FrameLabel->{{"y(t)",None}, 
   {"t", 
   "first and second state\ 
    change with time" 
   }}, 
    RotateLabel->False, 
    PlotStyle->{Red,Blue}, 
    BaseStyle -> 12]

pict

 

Matlab 

clear; 
A = [-0.5572   -0.7814;0.7814  0]; 
B = [1;0]; 
C = [1.9691  6.4493]; 
x0 = [1 ; 0]; 
 
sys = ss(A,B,C,[]); 
 
[y,t,x]=initial(sys,x0,20); 
plot(t,x(:,1),'r',t,x(:,2),'b'); 
title('x1 and x2 state change with time'); 
xlabel('t (sec)'); 
ylabel('x(t)'); 
grid 
set(gcf,'Position',[10,10,320,320]);

pict

 

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