DIGITAL CONTROLLERS
In general, we can make use of the block diagram shown in Figure 9.3 when designing a digital controller. In this figure, R(z) is the reference input, E (z) is the error signal, U (z) is the output of the controller, and Y (z) is the output of the system. HG(z) represents the digitized plant transfer function together with the zero-order hold.
The closed-loop transfer function of the system in Figure 9.3 can be written as
Equation (9.3) states that the required controller D(z) can be designed if we know the model of the process. The controller D(z) must be chosen so that it is stable and can be realized. One of the restrictions affecting realizability is that D(z) must not have a numerator whose order exceeds that of the numerator. Some common controllers based on (9.3) are described below.
Dead-Beat Controller
The dead-beat 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.
The required closed-loop transfer function is then
Figure 9.4 shows the system block diagram with the controller, while Figure 9.5 shows the step response of the system. The output response is unity after 3 s (third sample) and stays at this value. It is important to realize that the response is correct only at the sampling instants and the response can have an oscillatory behaviour between the sampling instants.
The control signal applied to the plant is shown in Figure 9.6. Although the dead-beat controller has provided an excellent response, the magnitude of the control signal may not be acceptable, and it may even saturate in practice.
The dead-beat controller is very sensitive to plant characteristics and a small change in the plant may lead to ringing or oscillatory response.
Dahlin Controller
The Dahlin controller is a modification of the dead-beat controller and produces an exponential response which is smoother than that of the dead-beat controller.
The required response of the system in the s-plane can be shown to be
Figure 9.8 shows the step response of the system. It is clear that the response is exponential as expected.
The response of the controller is shown in Figure 9.9. Although the system response is slower, the controller signal is more acceptable.
Pole-Placement Control – Analytical
The response of a system is determined by the positions of its closed-loop poles. Thus, by placing the poles at the required points we should be able to control the response of a system.
Pole-Placement Control – Graphical
In the previous subsection we saw how the response of a closed-loop system can be shaped by placing its poles at required points in the z-plane. In this subsection we will be looking at some examples of pole placement using the root-locus graphical approach.
When it is required to place the poles of a system at required points in the z-plane we can either modify the gain of the system or use a dynamic compensator (such as a phase lead or a phase lag). Given a first-order system, we can modify only the d.c. gain to achieve the required time constant. For a second-order system we can generally modify the d.c. gain to achieve a constant damping ratio greater than or less than a required value, and, depending on the system, we may also be able to design for a required natural frequency by simply varying the
d.c. gain. For more complex requirements, such as placing the system poles at specific points in the z-plane, we will need to use dynamic compensators, and a simple gain adjustment alone
will not be adequate. Some example pole-placement techniques are given below using the root locus approach.
Example 9.4
The block diagram of a sampled data control system is shown in Figure 9.11. Find the value of d.c. gain K which yields a damping ratio of ζ = 0.7.
Solution
In this example, we will draw the root locus of the system as the gain K is varied, and then we will superimpose the lines of constant damping ratio on the locus. The value of K for the required damping ratio can then be read from the locus.
The root locus of the system is shown in Figure 9.12. The locus has been expanded for clarity between the real axis points (−1, 1) and the imaginary axis points (−1, 1), and the lines of constant damping ratio are shown in Figure 9.13. A vertical and a horizontal line are
drawn from the point where the damping factor is 0.7. At the required point the roots are z1,2 = 0.7191 ± j 0.2114. The value of K at this point is calculated to be K = 0.0618.
In this example, the required specification was obtained by simply modifying the d.c. gain of the system. A more complex example is given below where it is required to place the poles at specific points in the z-plane.
Example 9.5
The block diagram of a digital control system is given in Figure 9.14. It is required to design a controller for this system such that the system poles are at the points z1,2 = 0.3 ± j 0.3.
Solution
In this example, we will draw the root locus of the system and then use a dynamic compensator to modify the shape of the locus so that it passes through the required points in the z-plane.
The root locus of the system without the compensator is shown in Figure 9.15. The point where we want the roots to be is marked with a × and clearly the locus will not pass through this point by simply modifying the d.c. gain K of the system.
The angle of G(z) at the required point is
There are many combinations of p and n which will give the required angle. For example, if we choose n = 0.5, then,
The compensator introduces a zero at z = 0.5 and a pole at z = 0.242. The root locus of the compensated system is shown in Figure 9.16. Clearly the new locus passes through the required points z1,2 = 0.3 ± j 0.3, and it will be at these points that the d.c. gain is K = 0.185. The step response of the system with the compensator is shown in Figure 9.17. It is clear from this diagram that the system has a steady-state error.
The block diagram of the controller and the system is given in Figure 9.18.
Example 9.6
The block diagram of a system is as shown in Figure 9.19. It is required to design a controller for this system with percent overshoot (PO) less than 17 % and settling time ts ≤ 10 s. Assume that T = 0.1 s.
The root locus of the uncompensated system and the required root position is shown in Figure 9.20.
It is clear from the figure that the root locus will not pass through the marked point by simply changing the d.c. gain. We can design a compensator as in Example 9.5 such that the locus passes through the required point, i.e.
Since the sum of the angles at a point in root locus must be a multiple of −180◦, the compensator must introduce an angle of −180 − (−259) = 79◦. The required angle can be obtained using a compensator with a transfer function, and the angle introduced by the compensator is
Figure 9.21 shows the root locus of the compensated system. Clearly the locus passes through the required point. The d.c. gain at this point is K = 123.9.
The time response of the compensated system is shown in Figure 9.22.