1. Discrete Controller Design (Deadbeat & Dahlin Controllers)
Chapter 9:
Discrete Controller Design
(Deadbeat & Dahlin Controllers)

2. Discrete-time controller design
The design procedure is outlined as follows:
Derive the transfer function of the plant (process).
Transform the system transfer function into the z-plane.
Design a suitable digital controller in the z-plane.
Implement the controller algorithm on a digital computer.

3. Discrete-time controller design
Today, we will study two methods for discrete controller design:
Deadbeat controller
a technique to achieve an ideal response but require excessive control signals.
Dahlin controller
a technique to approximate the deadbeat controller but does not require excessive control signals.

4. Digital controllers
The closed-loop transfer function of the system shown is
Suppose that the closed-loop transfer function is chosen to be:
Then, the controller required to achieve T(z) is given by:

5. Digital controllers The equation,
states that the required controller D(z) can be designed if we know the model of the process GH(z).
The controller D(z) must be realizable. That is, D(z) must not have a numerator whose order exceeds that of the numerator.

6. Deadbeat Controller
Deadbeat controller is one in which a step input is followed by the system but delayed by one or more sampling periods, i.e. the system response is required to be equal to unity at every sampling instant after the application of a unit step input.

7. Deadbeat Controller
Therefore, the desired closed-loop transfer function is
and the controller achieving this is

8. Example 9.1
The open-loop transfer function of a plant is given by
Design a dead-beat digital controller for the system. Assume that T = 1 s.
Answer:
The transfer function of the system with a ZOHis given by

9. From the z-transform tables
So,
Hence, the controller is given by

10. For realizability, we must choose k ≥ 3.
Choosing k = 3, we obtain the controller
With this controller, the block diagram of the closed-loop is
To analyze the designed system performance, let us simulate the step response of the closed-loop and the control signal.

11. As designed, the step response is unity after 3 secconds and stays at this value. Note: It is important to realize that the response is correct only at the sampling instants and the response can have an oscillatory behavior between the sampling instants.

12. For the control signal, we can realize that the maximum magnitude of the control signal is very high at the beginning (≈11). Usually, this is not acceptable in practice.

13. Drawbacks of deadbeat control
It requires large control signals which may not be acceptable.
It is very sensitive to plant characteristics and a small change in the plant may lead to ringing or oscillatory response.

14. MATLAB code for Deadbeat control
% Deadbeat control, Dogan Ibrahim, Chapter 9, Example 9.1 Gp = tf(1,[10 1],'iodelay',2); Gpd = c2d(Gp,1); Gc = tf([ ],[ ],1); Gcl=Gc*Gpd/(1+Gc*Gpd); t=0:1:10; y=step(Gcl,t) figure(1) plot(t,y,'o','MarkerSize',6) xlabel('time, sec') ylabel('output, y') xlabel('time, t') axis([ ]) title('Step response') Gru=Gc/(1+Gc*Gpd); u=step(Gru,t) figure(2) plot(t,u,'o','MarkerSize',6) ylabel('control signal, u') axis([ ]) title('Control signal')

15. Dahlin Controller
Dahlin controller is a modification of the deadbeat controller which produces an exponential response that is smoother than deadbeat control.
The desired closed-loop response for step input looks like:

16. Dahlin Controller Hence, the desired closed-loop transfer function is:
As step input is assumed (which is constant between samples), the desired closed-loop transfer function in the z-domain will be

17. Example The open-loop transfer function of a plant is given by
Design a Dahlin digital controller for the system to achieve a closed-loop time constant of 5 sec. Assume that the sampling period, T = 1 s.

18. Answer
First, we need to find the z-transform of the process (preceded by a ZOH). From the previous example, this is found to be:
Second, we need to find the z-transform of the desired closed-loop transfer function, T(z).
As the desired closed-loop time constant, τ, is 5 sec,

19. Therefore,

20. The Dahlin controller is thus given by

21. For the controller to be realizable, the degree of the numerator must be less than or equal to the degree of the denominator, i.e.
Choosing k=2, the controller is, then, given by:
Using the designed controller, the closed-loop step response and control signal are shown next.

22. It is clear that the response is exponential as designed but slower than deadbeat control.
What is response time delay? time constant?

23. It is clear that the maximum magnitude of the control signal (≈1
It is clear that the maximum magnitude of the control signal (≈1.9) is much smaller than the control signal obtained using a deadbeat controller (≈11), which is more acceptable in practice.

24. MATLAB code for Dahlin controller
% Dahlin control, Dogan Ibrahim, Chapter 9, Example 9.1 Gp = tf(1,[10 1],'iodelay',2); Gpd = c2d(Gp,1); Gc = tf([ ],[ ],1); Gcl=Gc*Gpd/(1+Gc*Gpd); t=0:1:30; y=step(Gcl,t) figure(1) plot(t,y,'o','MarkerSize',6) xlabel('time, sec') ylabel('output, y') xlabel('time, t') axis([ ]) title('Step response') Gru=Gc/(1+Gc*Gpd); u=step(Gru,t) figure(2) plot(t,u,'o','MarkerSize',6) ylabel('control signal, u') axis([ ]) title('Control signal')