12.Ad hoc methods II.Simple frequency response methods for controller design

Exercises

The next three exercises have to do with designing circuits to implement various controllers. Although we look at only a three specific controller transfer functions, it is possible, in principle, to design a circuit using only passive resistors, capacitors, and inductors, to realise any transfer function in RH+∞. A means of doing this was first pointed out in the famous paper of Bott and Du n [1949].

E12.1 Consider the circuit of Figure E12.1.

Figure E12.1 Circuit for lead compensation

(a)Determine the di erential equation governing the output voltage V2 given the input voltage V1.

(b)Convert this di erential equation to a transfer function, and show that the resulting transfer function is that of a lead compensator.

(c)Determine expressions for K, α, and τ in the standard form for a lead compensator in terms of R1, R2, and C.

E12.2 Consider the circuit of Figure E12.2.

Figure E12.2 Circuit for lag compensation

(a)Determine the di erential equation governing the output voltage V2 given the input voltage V1.

(b)Convert this di erential equation to a transfer function, and show that the resulting transfer function is that of a lag compensator.

(c)Determine expressions for K, α, and τ in the standard form for a lag compensator in terms of R1, R2, and C.

E12.3 Consider the circuit of Figure E12.3.

490 12 Ad hoc methods II: Simple frequency response methods for controller design 22/10/2004

Figure E12.3 Circuit for notch filter

(a)Determine the di erential equation governing the output voltage V2 given the input voltage V1.

(b)Convert this di erential equation to a transfer function, and show that the resulting transfer function is that of a notch filter.

(c)Determine expressions for K, α, and τ in the standard form for a lag compensator in terms of Rand C.

Figure E12.4 A closed-loop system with an unstable, nonminimum phase plant

(a)Why is the plant unstable? nonminimum phase?

(b)Produce the Nyquist and Bode plots for the plant transfer function. Is the closed-loop system stable with RC(s) = 1?

(c)Is it possible to design a proportional controller RC(s) = K which renders the closed-loop system IBIBO stable?

(d)We will design a stabilising controller by manipulating the Nyquist contour to look like the cartoon in Figure E12.5. Sketch the Bode plot for such a Nyquist

Figure E12.5 The desired Nyquist contour

contour, being particularly concerned with the phase of the Bode plot. Assume that the loop gain RL = RCRP is strictly proper and that its maximum magnitude occurs at ω = 0.

(e)Design a phase lead controller RC with the property that the Bode plot for RL = RCRP has the Bode plot phase which looks qualitatively like that you drew in part (d). For the moment, design RC to that RC(0) = 1. Plot the Nyquist contour for RL to verify that it has the shape shown in Figure E12.5.

(f)With the lead compensator you have designed, is your closed-loop system IBIBO stable? By adding a gain K to the lead compensator from part (e), it can be ensured that the system will be IBIBO stable for some values of K. Using your Nyquist contour from part (e), for which values of K will the closed-loop system be IBIBO stable? Pick one such value of K and produce the Nyquist contour to verify that the controller you have designed does indeed render the closed-loop system IBIBO stable.

Congratulations, you have just designed a controller for an unstable, nonminimum phase plant, albeit a contrived one. Let us see how good this controller is.

(g)Comment on the gain and phase margins for your system.

(h)Produce the step response for the closed-loop system, and comment on its performance.

E12.7 In this exercise you will be given a plant transfer function for which it is not possible to achieve arbitrary design objectives. The transfer function is

˜ s − z RP (s) = RP (s)s − p

for s, p R positive. Thus, we know that our plant has an unstable real pole p, and a nonminimum phase real zero z. All other poles and zeros are assumed to be stable and minimum phase, respectively. Let RC be a stable, minimum phase controller transfer function and define RL = RCRP .

492 12 Ad hoc methods II: Simple frequency response methods for controller design 22/10/2004

and the lead/lag controller transfer function

RC(s) = K(1 + ατs).

1 + τs

Answer the following two questions.

(a)Show that it is not possible to design a lead/lag controller RC for which RC S(RP ).

(b)Show that for any proper, second-order plant transfer function RP R(s) there exists a lead/lag controller RC for which RC S(RP ).

E12.9 Consider the plant transfer function

Answer the following two questions.

(a)Show that it is not possible to design a PID controller RC for which RC S(RP ).

(b)Show that for any proper, third-order plant transfer function RP R(s) there exists a PID controller RC for which RC S(RP ).

E12.10 Consider the coupled tank system of Exercises E1.11, E2.6, and E3.17. Take as

system parameters α = 13 , δ1 = 1, A1 = 1, A2 = 12 , a1 = 101 , a2 = 201 , and g = 9.81. As output, take the level in tank 2.

(a)Design a PID controller for the system that achieves a phase margin of at least 75◦ with as large a gain crossover frequency as possible.

We now wish to simulate the behaviour of the system with the controller that you have designed. The states of the system are nominally h1 and h2. However, since the controller involves an integration, an additional state will need to be defined. With this in mind, answer the following question.

(b)Suppose that a reference output h2,ref has been specified, and that the control u is specified by the PID controller you designed in part (a). Develop the nonlinear state di erential equations for the system, starting with the equations you linearised in Exercise E2.6. As indicated in the preamble to this part of the problem, you will have three state equations.

(c)With the controller that you have designed, simulate the di erential equations from part (b) with initial conditions equal to the equilibrium initial conditions, and subject to a reference step input of size 14 .

(d)Plot the height in both tanks, and comment on the behaviour of your controller.