MATLAB: C2d with ‘impulse’ method

c2dcontrol

According to the z-transform theory, the z-transform of the step unit u(t) is z/(z-1), regardless the sampling time. Aldo the s-transform of u(t) is 1/s. I want to obtain in Matlab z/(z-1) using the fucntion c2d for a sampling time T=0.1. This is my code:

T=0.1.

g_s = tf([1],[1 0]);

g_z = c2d (g_s, T, 'impulse');

However, the result for g_z is: 0.1*z/(z-1). May anyone explain why the factor 0.1 (the sampling time) appears in g_z? Which command should I use to obtain z/(z-1)? I would appreciate any help.

PS: I also observed that when I plotted the impulse response of g_z using the command impulse(g_z), the plot was right.

Best Answer

I don't think you can use c2d() and get z/(z-1) with a sample time T = 0.1.

In, short, the factor of 0.1 is needed to have the gain of g_s and the gain of g_z match in the frequency domain. You can see this with:

bode(g_s,g_z)

impulse(g_z) gives the "right" plot because the Control System Toolbox command impulse(g_z) does not return the inverse z-transform of g_z, or the unit pulse response. Rather, it returns the ouput in response to a unit pulse scaled by 1/T. So the output of impulse(g_z) is really the inverse z-transform of 1/T*g_z = 1/T * T * z / (z-1) = z/(z-1) which are the samples of the continuous time unit step.

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MATLAB: How i can solve this problem? Pleas!

Your ‘T’ matrix must become a cell array. See my sample of ‘T’ for an example.

Once you do that, I would do something like this:

Y = [12.004
    13.1573 
    11.1665  
    11.8082    
    12.2129   
    12.8075   
    12.4167  
    11.5708   
    12.5798    
    13.7489];
T={'cluster2'; 'cluster1'; 'cluster3'; 'cluster9'; 'cluster0'; 'cluster7'; 'cluster4'}; % Sample Of Cell Array
[UnqClust,~,idx] = unique(T);                                                           % Unique Values & Locations
Out = Y(idx);                                                                           % Mapped Output
for k1 = 1:length(idx)
    fprintf(1, '\t%s\t%.4f\n', T{k1}, Out(k1))              % Demonstration Output (Delete Loop)
end
  cluster2  11.1665
  cluster1  13.1573
  cluster3  11.8082
  cluster9  12.4167
  cluster0  12.0040
  cluster7  12.8075
  cluster4  12.2129

MATLAB: Effect of sampling time on the time-domain response of discrete systems

There is a difference between the results from R2011a and R2014b. It is however expected.

In the release notes of Controls Systems Toolbox R2012a, you can find the following:

Rescaled Impulse Response and Impulse-Invariant Time Domain Conversion

For discrete-time dynamic system models, the input signal applied by impulse is now a unit area pulse of length Ts and height 1/Ts. Ts is the sampling time of the discrete time system. Previously, impulse applied a pulse of length Ts and unit height.

( http://www.mathworks.com/help/control/release-notes.html )

That is to say, before R2012a, the magnitude of the impulse response is always 1, but after R2012a, the magnitude is changed to Ts. Therefore, by specifying different sampling times, you end up with different results. What you can do in your use case is to multiply the results with 1/Ts. Then, you will have consistency in the results with different sample times.

For example, instead of

>> y = impulse(G, 3*0.05)

Please do

y = impulse(G, 3*0.05)*(1/0.05)

Hope this helps!

-Yu

Best Answer

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MATLAB: How i can solve this problem? Pleas!

MATLAB: Effect of sampling time on the time-domain response of discrete systems

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