CTMS: PID controller with Bilinear Approximation

PID Compensator with Bilinear Approximation

Bilinear Transformation

There are several ways for mapping from the s-plane to z-plane. One of these, the bilinear transformation, has the advantage that it maps the entire left half of the s-plane into the unit circle in the z-plane. The bilinear transformation maps from s-plane to z-plane using the following relation:

where Ts is the sampling time of discrete-time system.

PID Compensator with Bilinear Transformation

In the continuous-time system, a transfer function for a PID controller is described as follows:

where K_p,K_i and K_d are the proportional, integral and derivative gains respectively. To map to the z-plane we substitute the bilinear relation into the above PID transfer function, so the discrete-time PID controller becomes.

You can add the following MATLAB commands into your m-file for discrete-time PID controller with appropriate gains, K_p,K_i and K_d.

% Discrete PID Controller
% Ts = sampling time

num = [Kp+(Ki*Ts/2)+((2*Kd)/Ts) 
         (Ki*Ts)-(4*(Kd/Ts))
         -Kp+(Ki*Ts/2)+((2*Kd)/Ts)]';
den = [1 0 -1];
pid = tf(num,den,Ts);

In addition, the c2d command in MATLAB can convert continuous-time to discrete-time PID controller without going through an algebraic substitution or mapping. This command requires a system representation of a continuous system, sampling time and the method that will be used. For the bilinear approximation, the "tustin" method should be specified. For example, the c2d command can be used as follows.

% Discrete PID Controller using c2dm command
% Ts = sampling time

pid_c = tf([Kd Kp Ki],[1 0]);
pid   = c2d(pid_c,Ts,'tustin');

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