Determine the response of a system to step input with nonzero initial conditions

1.30 Determine the response of a system to step input with nonzero initial conditions

Problem: Given a system represented by state space, how to determine the response with nonzero initial conditions in the states and when the input is a step input?

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}; 
y  = OutputResponse[{sys,x0}, 
         UnitStep[t],{t,0,20}]; 
 
Plot[y,{t,0,20}, 
  PlotRange->{{0,20},{0,13}}, 
  GridLines->Automatic, 
  GridLinesStyle->Dashed, 
  Frame->True, 
  ImageSize->300, 
  AspectRatio->1, 
  FrameLabel->{{"y(t)",None}, 
  {"t", 
  "system response to initial\ 
      conditions and step input"}}, 
  RotateLabel->False, 
  PlotStyle->Red]

pict

Matlab

A = [-0.5572   -0.7814;0.7814  0]; 
B = [1;0]; 
C = [1.9691  6.4493]; 
x0 = [1 ; 0]; 
 
sys = ss(A,B,C,[]); 
 
[y1,t] = initial(sys,x0,20); 
y2    = step(sys,t); 
 
plot(t,y1+y2,'r'); 
 
title(['initial conditions and step'... 
          'input response']); 
xlabel('t *sec)'); ylabel('y(t)'); 
grid 
set(gcf,'Position',[10,10,320,320]);

pict

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