MATLAB: Function c2d output unstable system as Ts decrease (frequency increase)

c2ddiscretizeMATLABmimosimulink

I use the Matlab(2018b) c2d function to discretize a transfer function. The transfer function is 15th order, with two poles and two zeros very close to imaginary axis.

sysd = c2d(sysc,Ts,method)

I tried with Ts of 0.001s, 0.01s, 0.02s, and with 'tustin' and 'zoh' method. The 0.02s Ts still gives me stable discretized tf, but 0.01Ts starts to give me some unstable poles outside 1, and 0.001Ts gives me more unstable poles outside 1. I put the transfer function in LTI block in Simulink, run a step test, and the result shows below.

It made me confused. I am under the impression that the higher sampling frequency will gives you better reconstruction of the original system. But the result of the c2d function show reversed: the higher sampling frequency, the more unstable. I also make up a system with zero/poles very closed to imaginary axis, but it is only a 5th order SISO system. The c2d function seems working fine on the SISO 5th order system without problem.

The conclusion is that the c2d has some issues discretizing my 15th order transfer function. Please help me to understand what happen, and how could I discretized my original system with higher sampling frequency. Thanks a lot.

Best Answer

Just found an old post. This issue could be solved by converting the transfer function by state space format and redo the discretization. thanks

Related Solutions

MATLAB: Is the discrete approximation of a continuous system less accurate using the C2D function with a smaller sample time step

The corresponding discrete poles for Ts=0.001 are:

>> exp(.001*pole(Hc))

9.9998e-001

9.9999e-001

1.0000e+000

which is very close to a triple pole at z=1. Using the transfer function representation for discrete systems with multiple poles near z=1 will typically result in significant loss of accuracy at low frequencies.

The Control System Toolbox documentation includes the following examples that explain this in detail:

- Control System Toolbox, Using the Right Model Representation

- Control System Toolbox, Sensitivity of Multiple Roots

Such problems go away when using the state-space representation. Try the following code:

Hz = c2d(ss(Hc),.001,'t');
figure, step(Hc,Hz,1000)
figure, bode(Hc,Hz)

Note that there is a limit to how small Ts can get before all useful information about the dynamics gets vanished by the round off noise. Given the time constant of the continuous system, a sampling time of Ts = 1 or Ts = 0.1 is enough to get an accurate discretization.

MATLAB: Is the frequency response of the continuous time system different than the discretized system, when I use the ‘imp’ (Impulse Invariant method) option in the C2D function in the Control System Toolbox

This bug has been fixed in Release 14 Service Pack 3 (R14SP3). For previous product releases, read below for any possible workarounds:

The Impulse Invariant discretization method implemented in the C2D function with the 'imp' option is such that the impulse responses for the continuous system and the obtained discretized system match. For example,

n=1;d=[1 1];          % simple 1st order continuous system
sc=ss(tf(n,d));       % state space representation
sd1=c2d(sc,0.01,'imp'); % convert to discrete system via impulse invariant
figure
impulse(sc,sd1)

However the frequency responses for the systems may not match due to the scaling factor (the sampling time). For example,

figure
bode(sc,0.01*sd1) % scaled by Ts

Thus, the Impulse Invariant method is a good choice if the same impulse response is desired. It may not be a good choice for obtaining the same frequency response, since it is susceptible to aliasing. In general, a bilinear transform (such as 'tustin') is a better choice for obtaining a discretized system with matching frequency response. For example,

bode(sc,c2d(sc,0.01,'tustin'),c2d(sc,0.2,'tustin'),c2d(sc,0.5,'tustin'))

To obtain similar frequency responses between the continuous and discretized systems with the Impulse Invariant method, adjust the scaling factor in the C2D function inputs. Matching frequency responses for the continuous and discretized systems can then be obtained as shown in the following code:

Ts=[0.1 0.01 0.001];    
n=1;d=[1 1];            
sc=ss(tf(n,d));
sd1=c2d(sc*Ts(1),Ts(1),'imp');  
sd2=c2d(sc*Ts(2),Ts(2),'imp');  
sd3=c2d(sc*Ts(3),Ts(3),'imp');  
bode(sc,sd1,sd2,sd3,{0.01,100});

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