Chapter 6 Robust Design for Multivariable Systems Above, analysis for multivariable control systems with resp ect to nominal and robust st ability as well as nominal and robust performance has been assessed. It was assumed that the spec ifications for robust ness were given in terms of weight matrices Wul(s)and Wu2(s), and that the performance specifications similarly were given by weight matrices WpI(s) and wp2( s) How to derive weight matrices leading to good compensators is to some extent still an open question and possibly the most difficult in robust control. In the following, two approaches to weight matrix selection will be proposed. These approaches, though, can not be considered to be final answers to the weight selection problem in any sense 6.1 Loop Shaping The idea behind loop shap ing is to find a compensator K(s) which shapes the open loop system o(L(w)), such that cert ain requirements for robustness and performance are satisfied Natural requirements for performance would be a good dist urb ance attenuation, resulting in a small tracking error. As seen above, the tracking error can be determined as e(8s)=S(s)(T(8)-d(8s)+T(8)n(s) 6.1) Sometimes it can be reasonable to neglect the measurement noise n(s), so the most significant requirement for performance is the output sensitivity o(Solu)) to be small in the frequency range where the most dominant disturb an ces o As most disturbances are low frequent this leads to a requirement for the output sensitivity to be small at frequencies up to a cert ain an evaluation of the domina disturbances. This can be achieved by specifiy ing pl (6.3) here Wp(s)is a scalar transfer function with low pass characteristics. A requirement of this
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Page 64 of 92 ty pe is equivalent to a condition for the smallest singular value g( Lo(jw)) of the open loop transfer function at the output to be large at low frequencies g(Lo Gw))>wp(u) The uncertainties of physical Systems are often largest at high frequencies. If det ailed knowl edge of the uncert aunties of the process is not available, it would be natural to specify a multiplicative un cert ainty mo del for the output (6.5) (6.6) where Wu(s)is a scalar transfer function with high p ass characteristics, such that Wu gw) has a value corresponding to the dc gain at low frequencies and a value at high frequencies of more than unity. Requirements of this type lead to the condition 7(T0(j) Vu gw) (6.7) This is equivalent to requiring the largest singular value of the open loop transfer matrix o( Lo Gu)) to be small at high frequencies (Lo gu)) (6.8) Wuga) In this way, the specification of weight functions b ecomes a matter of trade-off between a good disturbance attenuation and robustness, and the interesting choice is the frequen cy where the curves intersect. At this frequency, it should be ensured that either transfer function is smaller than unity. Otherwise, it will be imp ossible to meet the requirements. Figure 6. 1 shows an example of how such requirement s could manifest. The method does not give an expli loop singular values should proceed close to the cross- over frequencies, i.e. where the family of curves for the open loop singular values intersect unity(0 dB). It can be seen, though, that it is convenient if the singular values can be made Other transfer functions than So(s) and To(s) could be of interest to the loop shaping ap proach. Ensuring that the control signals u(s) remain reasonably bounded, can be obtained y bounding the control sensitivity M()=(I+ k(sG(s)K(s. Moreover, in the case where the uncertainties can be described in terms of an additive uncert ainty description, this also leads to upper bounds for the control sensitivity M(s) 2 Modeling individual Channels In the loop shaping approach, the interest is mainly focused on the size of the sensitivity and the complentary sensitivity functions, which for multivariable Sy stems leads to requirements for the largest and smallest singular values of the open loop transfer matrix. This has the
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yov es anly Open loop bounds for large Wp and Wu for large Wp ound for lo Angular frequency (rad/secl igur6 6.1: Oen 0)glin requirementfrm u 2 Ind u 1 in frequency regi)n=wtere either Gimitotigf thot prga 6ms wh6r6 diff6r 6f t r6quir6m6f ts or6 r6%ot 6d tg if dieiduo9 chof f 69 cof f gt a 6 6osi 2 oddr 6ss6d. Tg icustrot 6 this4 ossum 6 thot th6 uf cbrt oif t2 hos a 66f fguf d tg a 6 △=IAa1(p (6.9) Tg 6f sur 6 rga ust stoaisit24 th6 requir 6m6ft o(u.< 1 suffic6s. Hgw6e6r 4 if gf 2 th6 Sorg6st sif gugor eo916 is aguf d6d4 th6 requir 6m 6f t a 6cgm 6s (6.10) This cgf ditigf cof a 6 for mgr6 r6strictie64 6sp cio 2 if th 6r6 or6 org 6 diffEr 6f c6s if th6 uf cbrtoif t2 fgr th6 if dieiduo9 chof f 69 gf th6 trof sfGr motrix Thus4it wguSd a 6 much a 6tt 6r tg 6eoquot6 th6 uf cErt oif t 2 fgr 6och if dieiduogtrof sf6r fuf ctigf if th6 trof sf Gr motrix 6xpsicit S2 Th6 6eoQuotigf cof a 6 hof d6d46 g a 2 cgf sid Grif g p orom6tric uf cort oif t2 d6scriptigf s gr a 2 6eoquotif g o fr 6qu6f c2 aguf d fgr th6 eofidit2 gf th6 mg d69 of d if trgduc6 muStipSicotie 6 uf cortoif ti6s if 6och chof f 694 imp 6m 6f tif g this aguf d If th6 som foshigf 4it wiC9 a 6 gf sigf ificof c6 fgr the of c6s fgr 6och chof f 69is 6stimot 6d4 of d wh6th6r th6 imp grt of c6 gf th6 6rrgrs or6 6eoluot 6d fgr 6och gutput. Esp 6cioCc2 if th6 if dieiduo9 sigf o9 differ if mogf itud64 it is impgrtoft tg sco 6 6och gutput 4 such thot o cgmmgf aguf d fgr th6 f grm gf th6 gutput e6ctgr a 6cgm 6s m 6of if gfug
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Ws F ILwe sr I CcSnrsa In this sectio n, a st ate space Solutio n to the p)Control prop lem in the l s I plo ck formulation will pe presented. This solution was derived py Doyle rt F6 in 1988. The structure of this p Solution will pe comp ared to the well-known LQG structure, where it will pe app arent that the two structures have several similarities. u ctually, the LQG Solution can pe interpreted as a sp ecial case o al Hence, Co nceptually, a Solutio n will pe given to the prop lem KAebv ysi f m hFIANAb KAbbh+ (6.11) The prop lem of finding a Solution to(6. 11) was prop ap ly the most important research area within Control theory during the 1980s. Initially, only algorithms that provided p) optimal Controllers of a very high order-see e g. [Fra87I-were known, or algorithms that were only siple for Siso systems-see e g. Gri86. In 1988, ho wever, Doyle, Glo ver, Khargo nekar Lnd Francis anno unced a st ate sp ace Solutio n, involving only two algepraic Riccati equatio ns, and yielding a Compensator of the same order as the augmented system NAb, just as for the well-known LQG Solution. This was a major p reak-thro ugh for p)theo ry. It no w pecame evident that the LQG and the p)prop lems and their solutions were related in many ways Bo th Co mp ensator types have a st ate estimatio n-st ate feed ack structure, and two algepraic Riccati equations pro vide the state feed ack matrix Kc and the op server gain matrix Kf d' Asb u.Asb KAb Ab 2: Tanl s 1 b6ck 1g 66m Given a I s I p lo ck matrix NAsb, see Figure 6. 2, and a desired upper pound y for the p norm hFiAAb, KAsbbh+, the p)Solutio n returns a Co mpensator parametriz atio n, often referred to as the dkgf parameterization KAbV FlAAb 6.12) f all stapilizing Compensators for which hF AAsb, KAstbh-t <?Y, see Figure 6.3. u ny stap transfer matrix Q Ab for which hQab-to y will stapilize the dlo sed loop system and make hFiANAsh KAsbbh+o < y. u ny QAb that is unst aple or has p)norm larger than y would either make the clo sed loo or imply that hFiANAb, KAbbhta The p)Solution is given py Definitio n 6.1 and py Theorem 6.1 this xist for technical reasons. To be more precise, this section will be present a method for obtaining near optimal solu tions
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N(8) u(8) y(s) J(8) Q(8) Figure 6.3: The DGKF parameterization of all stabilizing compensators satisfying F(N(s),K(s)川ls< Definition 6.1 (Riccati Solution)Ass ume that the algebraic Riccati equation A X+XA-XRX +Q=0 has a unique s tabil izing solution X, i.e. a solution for which the eigenvalues of a-rx are all negative. This solution will be denoted by X= Ric(H) where h is the associated Hamiltonian matri T R (6.14) Theorem 6.1(The Ho Subopt imal Control Problem) This formulation cf the solu tion has been taken from/Dai90/. Let N(s be given by its state space realization A, B, C, D and introduce the notation A B1 B N(s)= C1 Du D12 15) where b, C, and d are partitioned consistenty with d, e, u, and y. Now, make the following assumptions 1.(A, B1) and(A, B2)are stabilizable(controllable 2.(C1, A)and(C2, A)are detectable (observable) D12=I and D21 Da1=1. D D21 (6.16) and soloe the two Riccati equation A-B2DICU B,Bt-B2BA CTDED D2C1)2 (6.17)
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