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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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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o ohai P ag A- B1 D21C2 nci B1D21D21B1-A-B1D21C2 6.18) Form the state feedback matrit b c, the observer gain matric b f, and the matrin Zs as D12C1+B2 9 19 B,DA +Yso (6.20) Z Ys X (6.21) If Xs I 0 and Ys I 0 exist, and if the spectral radius p(Xs ys)<y, then the DGKF parameterization is given by A Zs bf Zs B2+r-2Ys CD J(o) (6.22) C2+-2D21B1X J1{o)J12(o) J21{o)J2(o) (6.23) where As is given by As=A-B2bc+-2B1BIX8 -Zs D21B1 (6.24) Stabilizing compensators b(o) satisfying FI(No),b(o))Ho <y can then be constructed by combining Jo)with any stable Q(o)for which Q(o)Ho <Y b(o)=F(J(o),Q(o)=J1(o)+J2(o)Qo)(I-J2(o)Q(o)-1J21(o) (6.25) Then, the Hs norm of the closed loop system FI(N(o), FiJ(o),Q(o))satisfies FI(N(o), F(J(o),Q(o)))8<y (6.26) The compensator J11(o obtained for Qlo)=0 is called the central Hs compensator 3.1 Remarks to the Hs solution Note, that Theorem 6. 1 does not provide the optimal Hg compensator, but rather a compen- sator, satisfying FI(N(o), b(o))lu <?, after having specified y if a compensator achieving this exist s. Thus, the designer might have to iterate on y to approximate the optimal Hs norm ?e. Therefore, the solution in Theorem 6. 1 is called the subop timal Hs compensator This is in contrast to the lQG solution, where the optimal comp ensator can be found without iteration The central sator(Q(o)=0) is never the sator which obt ains the smallest closed loop Hg norm. For a desired value of ?, however, it is al ways an admissible compensator and, hence, it is common to choose this particular compensator for implement a tion. Specifically, this compensator is returned for the Matlab algorithms implemented the Robust Control Toolbox and in the u Analysis and Synthesis Further note, that even though the Hs control problem has been formulated in frequen domain, the solution is given in st ate space form, i.e. in time domain. This combination of frequency domain specifications and state space computations is very symptomatic for mo dern control theory
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Now, consider the four assumptions in Theorem 6. 1. Assumptions 1 and 2 are just require ments of stabilizability and detect ability of the generalized system N(s). Note, that if (A, B2) is st abiliz able or if(C2, A)is not detectable, no st abilizing compensator(Hoo, PID, or other) exist s! The ot her two conditions of Assumption 1 and 2 are included for technical reasons, but they can be alleviated. The main purp ose of Assumption 3 is to ensure that the number of columns of D12 does not exceed the number of rows, i. e. that D12 is a tall'matrix. In the same fashion, D21 is not allowed to have more rows than columns, i.e. D21 must be a'flat matrix. This implies the following conditions on the dimension of the signals e(s),d'(s), u(s dime(s)≥dim(s),dimd(s)≥dimy(s) (6.27) where dim a denotes the dimension of the vector a. Hence, the number of exogenous output (error signals)[el,.,e has to be larger than or equal to the number of controllable inputs he number of actuators)[ul Similarly, the number of exogenous inputs(distur- bances)[di,., dd] has to be larger than or equal to the number of measured outputs(the number of sensors)Iy1,., y]. Especially for multivariable sy stems, it can easily happen if care is not taken, that the design problem is formulated in such a way that(6.27) is not satisfied. In such case, fictitious exogenous inputs or fictitious exogenous outputs have to be roduced, possibly with a small weight. Since the original result by Doyle, Glover, Khar also include versions that allow for instance more control inputs than exogenous outputs The reader should be warned, however, that in general a violation of(6. 27) indicates ill-posed er than demonstrating a flaw of the theory. Thus usually be advisable to reconsider the specifications than to look for an alternative optimiz ation proce- dure that could handle a violation of(6.27) However, even if a control problem would satisfy(6.27) directly from the design specifications it would not be anticipated to satisfy the seemingly rather restrictive algebraic condition Di D12=I=D21D2I. Obviously, a general D matrix, motivated from physics would not satisfy this condition automatically. It can be shown, though, that a design problem with a general D matrix satisfying cert ain rank conditions can be transformed into a new design problem which is solvable for the same y value and by the same controller, but such that the new design problem satisfies the following conditions DDDD 21 6666 implying that both Assumption 3 and 4 are satisfied. This reformulation is obt ained by a cert ain scaling and 'loop-shifting' procedure. It is beyond the scope of these lecture notes, however, to describe the det ails of this. A reference is TC96, App. B. The only conditions for the reformulation to be possible are that D12 has full column rank, and that D2 has full row rank i.e. that rank D12= dim u, rank D21 dim y (6.32) The rank constraint for D12 can be interpreted as the condition that all control signals must be penalized in the optimization. This is a natural constraint, since otherwise some of th feedback gains would tend to infinity as y would tend to its optimal value. Similarly, the
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Psge 71 oe 92 rank constraint for DbAcan 2e interpreted as the condition that all measurement s have some noise contri] ution. This prevents the optimiz ation from creating very large o2 server gains y would tend to its optimal value. In the Ro2 ust Control Tool2 Ox, and in the u 1 naly sis and Synthesis Tool2 ox in M. H. B,(6.32) must 2e satisfied in order to compute the( su2- opUn H) Hence, when setting up a design pro2 lem,(6.32)must 2e taken into consideration. In short this amounts to making sure that for each control input of u(a), there is at least one of the transfer functions to the error signals of f'(a) having equally many poles and zeros, this corresponds to having a feed-through term(a ' term). Similarly, for each of the measurement signals of y(a) there has to 2 e at least one of the transfer functions from the distur ances of d(a) having equally many poles and zeros. Often, in the process of making 2 oth rank constraint s satisfied, the designer would 2 e lead to intro ducing additional(fictitious) In the sequel, the structure of the H)compensator will 2e compared to the well-known L structure. To that end, Figure 6.4 illustrates the structure of the LQG compensator, Figure 6.5 similarly illustrates the H)compensator d BA Bb CA DAb Bb OH Figure 6.4: LQG compensator structure as a 2 X 2 block problem. N(a is given by the upper bor and the log compensator as the lower box
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for the LQG Riccati equations g-B)DOTO B T2B2B (6.33) qY D B(B(B B -(g-B and for the Hoo(see(6. 17)-(6. 18))Riccati equations are comp ared, it is seen that the only difference is the additional terms I) B(Bf and I/t( in the upper right comer. Th LQG state feedback o o does not depend on B( i.e. how the disturb ances s'N. enter the system. Likewise, the Kalman gain o f in the LQG st ate estimator does not depend on T( i. e. all states are equally weighted. In contrast, o f in the Hoo st ate estimator depend on T( i.e. on the particular linear combination of st ates which corresponds to the output nNI One of the problems with the LQG compensator is, that even though LQ (full state informa- tion) has excellent guaranteed st ability margins (infinite gain margin and a phase margin of 00), it turns out that the observer based LQG compensator often is not very robust. The problem is, that some states often contribute more to the gain than others, but this can not compensated in the lQg Kalman filter as all st ates are weighted equally. In the Hoo st ate estimator, however, the designer can perform such weighting by virtue of the T( matrix and, hence, make the design more robust. Likewise, the dist urb ances s'N. can be weighted differently when computing the state feedb ack, and thereby the robustness of the sy stem can e improved Commercially available software now exists, which support the Hoo design problem, such as the MATLAB toolboxes [CS92, BDG 93]. Both toolboxes perform an iteration for I in order to find a near optimal Hoo compensator. 6.3.2 The Matlabtm toolbox es In addition to the Control Toolbox in MATLABTM, two toolboxes are available specifically for robust control design, i.e. robust control toolbox 6 Analy sis and Synthesis Toolbox. Both toolboxes are valuable pieces of software that provide a significant help in designing robust compensators in a smooth way, and they can both be recommended. Some of the advant ages of the 6 toolb ox are lent utility for converting a 2 x 2 block strukture into a st ate sp ace description Very natural functions for the ms, e.g. for interconnecting sy ster or closing loops(st arp. m) Good approach to 6, see Chapter 7
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