1.27 Given a closed loop block diagram, generate the closed loop Z transform and check its stability

Problem: Given the following block diagram, with sampling time \(T=0.1\ sec\), generate the closed loop transfer function, and that poles of the closed loop transfer function are inside the unit circle

System block diagram.

Mathematica

Remove["Global`*"] 
plant  = TransferFunctionModel[s/(1.0+s),s]; 
plantd = ToDiscreteTimeModel[ 
             plant, 
             1, 
             z, 
             Method->"ZeroOrderHold"]
 

d = ToDiscreteTimeModel[ 
     TransferFunctionModel[(s+1)/(s+5.0),s], 
          1, 
          z, 
          Method->"ZeroOrderHold"]
 

sys=SystemsModelSeriesConnect[d,plantd]
 

loopBack=SystemsModelFeedbackConnect[sys]
 

Now generate unit step response

ListPlot[OutputResponse[loopBack, 
      Table[1, {k, 0, 20}]], 
        Joined->True, 
        PlotRange->All, 
        InterpolationOrder->0, 
        Frame->True, 
        PlotStyle->{Thick,Red}, 
        GridLines->Automatic, 
        GridLinesStyle->Dashed, 
        FrameLabel->{{"y(k)",None}, 
          {"n","plant response"}}, 
        RotateLabel->False]
 

ss=StateSpaceModel[loopBack]
 

poles=TransferFunctionPoles[loopBack]
 

\[ {\begin {array}{c} \{0.543991\, -0.325556 i,0.543991\, +0.325556 i\} \\ \end {array}} \]

Abs[#]&/@poles
 

\[ \left ( {\begin {array}{c} \{0.633966,0.633966\} \\ \end {array}} \right ) \]

Poles are inside the unit circle, hence stable.

Matlab

clear all; close all 
 
s      = tf('s'); 
plant  = s/(s+1); 
T      = 1; %sampeling time; 
plantd = c2d(plant,T,'zoh'); 
d      = c2d( (s+1)/(s+5), T , 'zoh'); 
% obtain the open loop 
sys    = series(d,plantd); 
% obtain the closed loop 
loop   = feedback(sys,1)
 

loop = 
 
   z^2 - 1.801 z + 0.8013 
  ------------------------ 
  2 z^2 - 2.176 z + 0.8038 
 
Sample time: 1 seconds 
Discrete-time transfer function.
 

step(loop) 
grid
 

[z,p,k]=zpkdata(loop,'v'); 
fprintf('poles of closed loop discrete system as inside unit circle\n'); 
abs(p)
 

ans = 
    0.6340 
    0.6340