96 PERFORMANCE SPECIFICATIONS AND LIMITATIONS 6.7 Notes and References The loop shaping design is well known for SISO systems in the classical control theory. The idea was extended to MIMO systems by Doyle and Stein 1981]using LQG design technique.The limitations of the loop shaping design are discussed in detail in Stein and Doyle [1991].Chapter 16 presents another loop shaping method using Hoo control theory which has the potential to overcome the limitations of the LQG/LTR method. The design tradeoffs and limitations for SISO systems are discussed in detail in Bode 1945,Horowitz 1963,and Doyle,Francis,and Tannenbaum [1992.The monograph by Freudenberg and Looze [1988 contains many multivariable generalizations.The multivariable generalization of Bode's integral relation can be found in Chen 1995]. Some related results can be found in Boyd and Desoer 1985.Additional related results can be found in a recent book by Seron,Braslavsky and Goodwin [1997. 6.8 Problems Problem 6.1 Let P be an open loop plant.It is desired to design a controller so that the overshoot 10%and settling time <10sec.Estimate the allowable peak sensitivity Ms and the closed-loop bandwidth. Problem 6.2 LetL be an open loop transfer function of a unity feedback system.Find the phase margin,overshoot,settling time,and the corresponding Ms. Problem 6.3 Repeated the last problem with 100(s+10) L2=8+1)(8+2)8+20 Problem6.4LletP=.tseclasicalopshapingmdhodtodesigmaomntrolilg so that the system has at least 300 phase margin and as large crossover frequency as possible. Problem 6.5 Use root locus method to show that a nonminimum phase system cannot be stabilized by a high gain controller. Problem 6.6 Let P-Design a controller so that the system has at least 5 300 phase margin and the smallest possible bandwith (or crossover frequency). Problem 6.7 Use root locus method to show that a unstable system cannot be stabilized by a low gain controller. Problem 6.8 Consider the unity-feedback loop with proper controller K(s)and strictly proper plant P(s),both assumed square.Assume internal stability.PERFORMANCE SPECIFICATIONS AND LIMITATIONS ￾ Notes and References The loop shaping design is well known for SISO systems in the classical control theory The idea was extended to MIMO systems by Doyle and Stein ￾ using LQG design technique The limitations of the loop shaping design are discussed in detail in Stein and Doyle  Chapter  presents another loop shaping method using H￾ control theory which has the potential to overcome the limitations of the LQGLTR method The design tradeos and limitations for SISO systems are discussed in detail in Bode   Horowitz  and Doyle Francis and Tannenbaum  The monograph by Freudenberg and Looze ￾￾ contains many multivariable generalizations The multivariable generalization of Bodes integral relation can be found in Chen  Some related results can be found in Boyd and Desoer ￾ Additional related results can be found in a recent book by Seron Braslavsky and Goodwin  ￾ Problems Problem  Let P be an open loop plant It is desired to design a control ler so that the overshoot   and settling time  sec Estimate the al lowable peak sensitivity Ms and the closedloop bandwidth Problem  Let L   ss￾ be an open loop transfer function of a unity feedback system Find the phase margin overshoot settling time and the corresponding Ms Problem Repeated the last problem with L   s   s   s   s    Problem  Let P  ￾s ss￾ Use classical loop shaping method to design a control ler so that the system has at least ￾ phase margin and as large crossover frequency as possible Problem  Use root locus method to show that a nonminimum phase system cannot be stabilized by a high gain control ler Problem Let P   ss Design a control ler so that the system has at least ￾ phase margin and the smal lest possible bandwith or crossover frequency Problem Use root locus method to show that a unstable system cannot be stabilized by a low gain control ler Problem  Consider the unityfeedback loop with proper control ler K s and strictly proper plant P s both assumed square Assume internal stability