# ch9soln - Chapter 9 Control-System Design: Principles and...

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Chapter 9 Control-System Design: Principles and Case Studies Problems and Solutions for Chapter 9 1. Of the three types of PID control (proportional, integral, or derivative), which one is the most e f ective in reducing the error resulting from a constant disturbance? Explain. Solution: Integral control is the most e f ective in reducing the error due to constant disturbances. Block diagram for showing integral control is the most e f ective means of reducing steady-state errors. Using the above block diagram, Y = G ( W + ED c ) , E = R Y = R G ( W + c ) , E = 1 1+ D c G R G D c G W, e = lim t →∞ e ( t ) = lim s 0 sE ( s )=lim s 0 s & 1 D c G R G D c G W . Writing G ( s )= n G ( s ) d G ( s ) , and using a step input R ( s k r s , and a step disturbance W ( s k w s ,we can show that integral control leads to zero steady-state error, while proportional and derivative 617

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618 CHAPTER 9. CONTROL-SYSTEM DESIGN: PRINCIPLES AND CASE STUDIES control, in general, do not. Integral control ,D c ( s )= 1 s ,e = lim s 0 sE ( s ) = lim s 0 & d G sk r d G s + n G + n G sk w d G s + n G =0 , if n G (0) 6 , Proportional control c ( s K p 6 , Derivative control c ( s s, e = k r G (0) k w 6 , if d G (0) 6 . This analysis assumes that there are no pole-zero cancellations between the plant, G, and the compensator, D c . In general, proportional or derivative control will not have zero steady-state error. 2. Is there a greater chance of instability when the sensor in a feedback control system for a mechanical structure is not collocated with the actuator? Explain. Solution: Yes. For comparison, see the following two root loci which were taken from the discussion in the text on satellite attitude control. In Fig. 9.26, the sensor and the actuator are collocated, resulting in a stable closed-loop system with PD control. In Fig. 9.5, the sensor and the actuator are not collocated creating an unstable system with the same PD control. Root Locus Real Axis I m a g A x i s -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 [Text Fig. 9.26] PD control of satellite: collocated.
619 Root Locus Real Axis I m a g A x i s -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 [Text Fig. 9.5] PD control of satellite: non-collocated. 3. Consider the plant G ( s )=1 /s 3 . Determine whether or not it is possible to stabilize this plant by adding the lead compensator D c ( s )= K s + a s + b , ( a<b ) . (a) What is the maximum phase margin of the resulting feedback system? (b) Can a system with this plant, together with any number of lead compensators, be made unconditionally stable? Explain why or why not. Solution: (a) G ( s /s 3 has phase angle of -270 for all frequencies. The maximum phase lead from a compensator D c ( s K s + a s + b is 90 with b a = . In practice a lead compensator with b a = 100 contributes phase lead of approximately 80 . Hence the closed-loop system will be unstable with PM = 10 .T o h a v eP M 70 we need, for example, a double lead compensator D c ( s ( s + a ) 2 ( s + b ) 2 with b a = 100.

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## This note was uploaded on 03/17/2009 for the course MEEM 4700 taught by Professor Staff during the Spring '08 term at Michigan Technological University.

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ch9soln - Chapter 9 Control-System Design: Principles and...

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