1. Discrete Controller Design
Lecture 7:
Discrete Controller Design
(Pole Placement)

2. Objectives Outline the procedure for discrete-time controller design.
Review the step response characteristics (rise, peak, and settling times, and percent overshoot) and relate them to the locations of the poles of the system.
Introduce the pole placement method for controller design.

3. Procedure for discrete-time controller design
Derive the transfer function of the plant (process).
Transform the process transfer function into z-plane.
Design a suitable digital controller in z-plane.
Implement the controller algorithm on a digital computer.

4. Digital controllers
The closed-loop transfer function of the system is:
Suppose it is desired that the closed-loop transfer function be:
Then, the controller required to achieve T(z) is given by:
Note that the controller D(z) must be realizable, that is, the degree of the numerator must not exceeds that of the denominator.

5. Time domain specifications
Most commonly used time domain performance measures refer to a 2nd order system:
where ωn is the undamped natural frequency and ζ is the damping ratio of the system.
The performance of a control system is usually measured in terms of its response to a step input.
The step input is easy to generate and gives the system a nonzero steady-state condition, which can be measured.

6. The typical response of a 2nd order system when excited with a unit step input is shown. In this response, the following performance indices are usually defined: rise time; peak time; settling time; maximum overshoot; and steady-state error.

7. Rise time (Tr) Peak time (Tp) Settling time (Ts)
The time required for the response to go from 0% to 100% of its final value.
It is a measure of the speed of response. The smaller is the rise time the faster is the response.
The time required for the response to reach its peak value.
The response is faster when the peak time is smaller, but this gives rise to a higher overshoot.
The time required for the response to reach and stay within a range (2 or 5%) about the final value.

8. The percent overshoot (Mp)
The percent overshoot is defined as
Where y0 is the initial, ym is the maximum, and yss is the steady state (final) values of the step response, respectively.

9. The percent overshoot (Mp)
For 2nd order system, the percent overshoot is calculated as
The amount of overshoot depends on the damping ratio (ζ) and directly indicates the relative stability of the system. The lower is the damping ratio, the higher the is maximum overshoot.

10. The steady-state error (ess)
ess is the difference between the reference input and the output response at steady-state.
Small ess is required in most control systems. However, in some systems such as position control, it is important to have zero ess.
ess can be found using the final value theorem. If the Laplace transform of the output is Y(s), then the final (steady-state) value is given by
Hence, for a unit step input, ess is given as

11. Example
Determine the step response performance indices of the system with the following closed-loop transfer function Answer: Comparing the given system with the standard 2nd order transfer function We find that ζ = 0.5 and ωn = 1 rad/s. Thus, the damped natural frequency is

12. The percent overshoot is
The peak time is

13. Pole placement approach
It is seen that the performance indices of the step response are functions of (ζ,wn). But ζ,wn determine the poles of the system:
Therefore, the response of a system is determined by the position of its poles. By placing the closed-loop poles at good locations, we can shape the response of the system and achieve desired time response characteristics.
Although the previous analysis is conducted for 2nd order continuous-time system, the pole placement approach is also applicable for discrete-time systems thanks to the mapping between s- and z-plane poles (z = esT) provided that the sampling interval is chosen sufficiently small.

14. ζ and wn in the z-plane

15. Pole placement design procedure
Given some specifications of
maximum overshoot, rise time or settling time ↓
Find the damping ratio (ζ) and undamped natural frequency (ωn) of the closed-loop system
Locate the desired closed-loop poles in the z-plane
Design a controller

16. Example
The open-loop transfer function of a system together with a ZOH is given by
Design a digital controller so that for the closed-loop system:
settling time (2%) is sec,
percent overshoot is 9.5%,
steady-state error to a step input is zero.
steady-state error to a ramp input is 0.2.
Assume that T = 0.2 sec.

17. Solution From the required percent overshoot and settling time,
The damped natural frequency
Hence, the required pole positions in the z-plane are

18. The desired closed-loop transfer function is thus given by:
Before we proceed, we must determine the degree of the numerator N(z). This degree must be ≤ 2 so that T(z) is realizable.
If we choose N(z) to be exactly of order 2, this means that we ask the closed-loop to start its response instantaneously without delay.

19. However, by recalling the transfer function of the process,
We can realize that the process itself has a delay of 1 sample (the order of the denominator exceeds that of the numerator by 1), so the closed-loop must have, at least, the same delay.
With this reasoning, the desired transfer function T(z) is chosen as
Note that this choice insures the realizability of the controller we are designing.

20. Now, let us find b1 and b0. These parameters can be determined from the steady-state requirements.
For input R(z), the error is given by
E(z) = R(z) – Y(z)
= R(z) – T(z)R(z)
= R(z) [1– T(z)]
So, the steady-state error is given by

21. From the given specifications, it is required that, for a unit-step input, ess = 0, so
Hence,
b1 + b0 =

22. Also, from the given specifications, it is required that, for a unit-ramp input, ess = 0.2
Hence, we need to apply L’Hospital rule !!!
This gives:
From which we can find that
b1 = 0.595
b0 =

23. Hence, the desired closed-loop transfer function is
The controller that achieves this closed-loop transfer function is given by:

24. The step response of the closed-loop system with the designed controller is shown below.