Maintaining C and D matrices when converting state space to discrete-time

Hi,

I am encountering some confusion when it comes to comparing different types of controllers, as my C and D matrices keep changing with different methods. I know that these are all valid realizations of the transfer functions involved, but I'm unsure about how the outputs compare to each other.

I have a Laplace transfer function G and a pid controller (that I synthesized), connected with negative feedback:

G = 1/(s^2*(0.1*s + 1));
Cpid = 180*( ( s+0.1 )*( s+0.04 ) )/(s*(s+60));
H = feedback(G*Cpid,1);

With the state-space of G being:

sys = ss(G);
A = sys.A;
B = sys.B;
C = sys.C;
D = sys.D;

I am trying to convert the system into discrete-time. I learned that the Tustin method is the best approximation for this. However, I am confused about the physical meaning of the output of the state-space system after converting to discrete-time. With the default zoh method, I can keep the C and D matrices the same by just converting the A and B matrices to Φ and Γ:

Ts = 0.05;
[Phi, Gamma] = c2d(sys.A,sys.B, Ts);  % zoh

And then finalize the state-space by including C and D:

ss(Phi, Gamma, C, D, Ts);

The problem I encounter is that this does not work when using the Tustin method, as this results in an error:

[Phi, Gamma] = c2d(sys.A,sys.B, Ts, 'tustin');

Later on, I need to use pole placement to come up with a discrete controller as well, and it would be much easier to compare the controllers when I know the C and D matrices are the same.

Is what I'm trying to do completely wrong or am I on the right track? Do the different C and D matrices sort of 'cancel out' because of the feedback connection with the PID controller or is there something else going on?

I would really appreciate some help.

Thank you in advance!

Answer 1

[Michael Hubatka]

I suggest that you use

sysd1 = c2d(sys, Ts); % ZOH
sysd2 = c2d(sys, Ts, 'tustin'); % tustin

instead of extracting A, B, C, D and recombining them after conversion.

The C, D and the state vector are preserved for ZOH, but not for Tustin and other methods, see for example Continuous-Discrete Conversion Methods - MATLAB & Simulink - MathWorks.

If you want to compare the controllers I suggest not to use numerical comparisons of the matrices. Use bode, pzmap or compute Eigenvalues.