Chapter 8 Advanced Loopshaping In Chapter 7 we saw how to convert performance specifications on S and T into specifications on the loop transfer function L.For a stable,minimum-phase plant and L having at least the same relat ive degree as P,the controller was obtained from C=L/P.In this chapter we discuss extensions to this basic idea by doing loopshaping directly with C or other quantities and by considering plants with right half-plane(RHP)poles or zeros.Finally,we introduce several optimal control problems and explore in what sense loopshaping designs can be said to be optimal.Our aim is to extend and deepen our understanding of loopshaping,to provide an introduction to optimal design,and to establish some connections between the two approaches.Much of this chapter is closely related to what has traditionally been called classical control,particularly the work of Bode. 8.1 Optimal Controllers Recall from Section 4.5 that in general the norm (WIS2+W2T2)1/2 is a reasonable performance measure,a compromise norm for the robust performance problem. Throughout this chapter we consider problems where P,Wi,and W2 are fixed but C is variable, so it is convenient to indicate the funct ional dependence of this norm on C by defining (C):=(WiS+W2T2)2Ilo (8.1) Throughout this chapter we refer to the optimal C,the controller that minimizes for the purpose of comparing it to controllers obtained via ot her methods in order to help evaluate their effective- ness A procedure for determining the optimal C given P and weights Wi and W2 is developed in Chapter 12. We shall treat the unity-feedback loop of Figure 8.1.Suppose that we focus on the response of e and u to the two inputs d andn(r has the same effect as n),and recall that the transfer funct ions from d and n to e and u are given as follows: []-[][] If we introduce weights We on e,Wu on u,Wd on d,and Wn on n,then we could make our performance specification to keep the matrix [R][][w]-[ WeS WaP Wn] 107
Chapter Advanced Loopshaping In Chapter we saw how to convert performance specications on S and T into specications on the loop transfer function L For a stable minimumphase plant and L having at least the same relative degree as P the controller was obtained from C LP In this chapter we discuss extensions to this basic idea by doing loopshaping directly with C or other quantities and by considering plants with right halfplane RHP poles or zeros Finally we introduce several optimal control problems and explore in what sense loopshaping designs can be said to be optimal Our aim is to extend and deepen our understanding of loopshaping to provide an introduction to optimal design and to establish some connections between the two approaches Much of this chapter is closely related to what has traditionally been called classical control particularly the work of Bode Optimal Controllers Recall from Section that in general the norm kjWSj jWT j k is a reasonable performance measure a compromise norm for the robust performance problem Throughout this chapter we consider problems where P W and W are xed but C is variable so it is convenient to indicate the functional dependence of this norm on C by dening C kjWSj jWT j k Throughout this chapter we refer to the optimal C the controller that minimizes for the purpose of comparing it to controllers obtained via other methods in order to help evaluate their eective ness A procedure for determining the optimal C given P and weights W and W is developed in Chapter We shall treat the unityfeedback loop of Figure Suppose that we focus on the response of e and u to the two inputs d and n r has the same eect as n and recall that the transfer functions from d and n to e and u are given as follows e u P S S T CS d n If we introduce weights We on e Wu on u Wd on d and Wn on n then we could make our performance specication to keep the matrix We Wu P S S T CS Wd Wn WeS WuCS WdP Wn
108 CHAPTER 8.ADVANCED LOOPSHAPING Figure 8.1:Unity-feedback loop. small in some sense.A convenient specificat ion is (W.SP+WCS2)(WaPP+W)<1, which is equivalent to wsP+WT2)/。<1 where IWil=IWel(IWaP2+Wn12)12, IW2l IWul(IWn12P+Wal2)1/2. (8.2) Thus this problem fits the type of performance specification in (8.1). We will use this setup throughout this chapter,as it makes a useful "standard"problem for a number of reasons.First,it leads to some very interesting control problems,even when simple constant weights are used.Second,there is a rich theory available for this problem,although it will only be hinted at in this chapter.Third,it is easy to motivate problems in this framework that greatly stretch the loopshaping methods.Finally,despite its simplicity,the problem setup is easy to relate to what might arise in many practical situations. 8.2 Loopshaping with C The loopshaping procedure developed in Chapter 7 involved converting performance specifications on S and T into specifications on the loop transfer function L,and then constructing an L to satisfy the resulting specifications and have reasonable crossover characteristics.Assuming that the plant had neither RHP poles nor zeros,and that L had at least the same relative degree as P, the controller was obtained from C=L/P.In this section we consider a slightly different approach that focuses more directly on C.Rather than construct L without regard to P,we could begin with a very simple controller,say C=1,and compare the resulting L with the specificat ion.It is often easy then to add simple dynamics to C to meet the specification. In some instances it is more convenient to do loopshaping directly in terms of C rat her than L. This will typically occur when the weights on S and T share substantial dynamics with P,as in (8.2).If we put a constant weight of 1 on d and n and a constant weight of 0.5 on u and e,then the weights from (8.2)become We=Wu=0.5, Wa Wn =1, (8.3)
CHAPTER ADVANCED LOOPSHAPING C P r u y d n e Figure Unityfeedback loop small in some sense A convenient specication is jWeSj jWuCSj jWdP j jWnj which is equivalent to jWSj jWT j where jWj jWejjWdP j jWnj jWj jWujjWnj jP j jWdj Thus this problem ts the type of performance specication in We will use this setup throughout this chapter as it makes a useful standard problem for a number of reasons First it leads to some very interesting control problems even when simple constant weights are used Second there is a rich theory available for this problem although it will only be hinted at in this chapter Third it is easy to motivate problems in this framework that greatly stretch the loopshaping methods Finally despite its simplicity the problem setup is easy to relate to what might arise in many practical situations Loopshaping with C The loopshaping procedure developed in Chapter involved converting performance specications on S and T into specications on the loop transfer function L and then constructing an L to satisfy the resulting specications and have reasonable crossover characteristics Assuming that the plant had neither RHP poles nor zeros and that L had at least the same relative degree as P the controller was obtained from C LP In this section we consider a slightly dierent approach that focuses more directly on C Rather than construct L without regard to P we could begin with a very simple controller say C and compare the resulting L with the specication It is often easy then to add simple dynamics to C to meet the specication In some instances it is more convenient to do loopshaping directly in terms of C rather than L This will typically occur when the weights on S and T share substantial dynamics with P as in If we put a constant weight of on d and n and a constant weight of on u and e then the weights from become We Wu Wd Wn
8.2.LOOPSHAPING WITH C 109 Iwl=0.50P2+1)/2, 1W2=0.5(P2+1)1/2. (8.4) With Wi and W2 so defined,for the performance specification to be met the loopshape will usually be very similar to P,so that the controller C will have C1.We can get some insight into why this is so by interpreting the weights in terms of the requirements they place on C as follows For L=PC 1WS=0.25PP+1 Z+12 IWTP=0.251P2+1 |L-1+12 and W2Tl W S =C1. Assuming that (C)<1,at frequencies for which P>1 we have that lW|≈0.5P, lW2l≈0.5， w,S≈0.5O |W2Tl≈0.51Tl, and when|P≤l, 1 lW|≈0.5， 1W2l≈0.5， PI IW1S|≈0.5lS1, 1W2T|≈0.5lCS1. The crossover region can occur wherever P1.When P=1,we have wl=w= v② Viewing this as a standard loopshaping problem,we would expect thatLP and C=1. Example 1 Consider P given by P(s)=0.5 + 0.2s s2+2Cw8+w7 i=1 where wI=0.2,w2 =0.5,w3 =2,w4 =10,Si =0.02.The Bode magnit ude plot of P is shown in Figure 8.2,and the result ing weights Wi and W2 on S and T are shown in Figure 8.3. The complicated weights would appear to make this a tricky loopshaping problem,but in fact the controller C=1 meets the specification.The quant ity (WS2+IW2T2)1/2 (8.5) for C=1 is plotted in Figure 8.4.The optimal for this problem is approximately 0.69,so C=1 is very close to optimal. This example illustratesthe point that loopshaping directly using C can often be much simpler than loopshaping with L and solving for C.While the example is somewhat contrived,similar things can happen quite naturally.In particular,this example exhibits some characteristics of plants that arise in the control of flexible structures. Example 2 We will now use the same setup but with a slightly simpler plant P which we will motivate with a simple mechanical analog of a two-mode flexible structure (Figure 8.5).Shown is a one-dimensional rigid beam of mass M,length 21,and moment of inertia I connected to a
LOOPSHAPING WITH C jWj jP j jWj jP j With W and W so dened for the performance specication to be met the loopshape will usually be very similar to P so that the controller C will have jCj We can get some insight into why this is so by interpreting the weights in terms of the requirements they place on jCj as follows For L P C jWSj jP j jL j jWT j jP j jL j and jWT j jWSj jCj Assuming that C at frequencies for which jP j we have that jWj jP j jWj jWSj jT j jCj jWT j jT j and when jP j jWj jWj jP j jWSj jSj jWT j jCSj The crossover region can occur wherever jP j When jP j we have jWj jWj p Viewing this as a standard loopshaping problem we would expect that jLjjP j and jCj Example Consider P given by P s s X i s s iis i where i The Bode magnitude plot of P is shown in Figure and the resulting weights W and W on S and T are shown in Figure The complicated weights would appear to make this a tricky loopshaping problem but in fact the controller C meets the specication The quantity jWSj jWT j for C is plotted in Figure The optimal for this problem is approximately so C is very close to optimal This example illustrates the point that loopshaping directly using C can often be much simpler than loopshaping with L and solving for C While the example is somewhat contrived similar things can happen quite naturally In particular this example exhibits some characteristics of plants that arise in the control of exible structures Example We will now use the same setup but with a slightly simpler plant P which we will motivate with a simple mechanical analog of a twomode exible structure Figure Shown is a onedimensional rigid beam of mass M length l and moment of inertia I connected to a
110 CHAPTER 8 ADVANCED LOOPSHAPING 么 101 100 101 10 10 100 101 102 Figure 82:Bode plot of P2 102 101 100 10 10- 101 100 101 102 Figure 823:Bode plots of W(solid)and W2(dash)2
CHAPTER ADVANCED LOOPSHAPING 10-2 10-1 100 101 102 10-2 10-1 100 101 102 Figure Bode plot of jP j 10-1 100 101 102 10-2 10-1 100 101 102 Figure Bode plots of jWj solid and jWj dash
8.2.LOOPSHAPING WITH C 111 0.75 0.7 0.65 0.6 0.55 L1LL 101 100 101 102 frequency Figure 8.4:(WiS+W2T2)/2 for C=1 (solid)and optimal C (dash). M.I 。。 Figure 8.5:Mechanical structure,Example 2
LOOPSHAPING WITH C 0.5 0.55 0.6 0.65 0.7 0.75 10-2 10-1 100 101 102 frequency magnitude Figure jWSj jWT j for C solid and optimal C dash M I lu l k k x m U Figure Mechanical structure Example