On the loop transfer recovery - Berkeley Robotics & Human ...

INT. J. CONTROL,1986, YOLo43, No.3, 981-996

On the loop transfer recovery H. KAZEROONI t and P. K. HOUPTt One method otmodel-based compensatordesign for linear multivariable systems consists of state-feedbackdesignand observerdesign(Athans 1971).A key step in recentwork in multi variable synthesisinvolvesselectingan observergain so the final loop-transfer function is the sameas the state-feedbackloop transfer function (Doyle and Stein 1979,Goodman 1984).This is called loop transfer recovery(LTR). This paper shows how identification of the eigenstructure of the compensators that achieveLTRmakes possiblea designprocedurefor observergain (Luenberger1971). This procedure is based on the eigenstructure assignment of the observers.The sufficient condition for LTR and the stability of the closed-loop systemis that the plant be minimum-phase.The limitation of this method might arise when the plant has multiple transmissionzeros.

Nomenclature A, B, C plant parameters dj(t), do(t) input and output disturbances x(t), u(t), y(t) states,input and output of the system x(t), Y(t) statesand output of the observers e(t) error signal of the observer Gp(s) transfer-function matrix of the plant Ai eigenvaluesof A -BG .LLi eigenvaluesof A -HC Sj transmissionzeros of A, B, C Ui transmissionzeros of A, B, G G state-feedbackgain K(s) transfer-function matrix of the compensator p positive scalar vi left-eigenvectorof A-HC Ui right-eigenvectorof A -BG W square non-singular m x m matrix zi zero direction of the transmissionzero wi input direction of the transmissionzero j maximum number of the finite transmissionzeros xi left-eigenvectorof A -BG -HC $ol(S) open-loop characteristic equation of the plant $cl(S) closed-loopcharacteristic equation of the observer n order of the system m rank of matrices Band C P(s) precompensator Received5 June 1985.

t Massachusetts Institute of Technology,MechanicalEngineeringDepartment,Cambridge,MA 02139,U.S.A. t GeneralElectric Company,Automationand Control Laboratory, Schenectady, NY 12345,U.S.A.

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H. Kazerooni and P. K. Houpt

1. Introduction Historically, the LTR method is the consequenceof attempts by Doyle and Stein (1979,1981)to improve the robustnessof linear quadratic gaussian(LQG) regulators. In their seminalwork, Doyle and Steinaddressthe problem of finding the steady-state observergain that assuresthe recoveryof the loop transfer function resulting from full state feedback.First, they demonstratea key lemma that gives a sufficient condition for the steady-stateobservergain such that LTR takes place. To compute the gain, they show that the infinite time-horizon Kalman-filter formalism with 'small' white measurement-noisecovariance yields an observer gain that satisfies the sufficient condition for loop-transferrecovery.In this paperwe presenta method for computing obseryer gain that obviates the need for Kalman-filter formalism. The goal of this paper is to analysethe eigenstructureproperties of the LTR method for the general classof feedbackcontrol systemsthat usemodel-basedcompensators.After examining the eigenstructure of LTR, a design methodology for LTR via eigenstructure assignmentwill be given.

2. Background We will deal with the standard feedbackconfiguration shown in Fig. 1, which consistsof: plant model Gp(s);compensatorK(s), forced by command r(t); measurement noise n(t);and the disturbancesdj(t) and do(t).The precompensatorP(s),is used to filter the input for command-following. do(t)

Idj(t)

r~p(S

)

V(tI

K( S)

nIt) Figure

Standardclosed-loop system.

Throughout this paper,we assumethat the plant canbe describedby the following equations: x(t) = Ax(t) + Bu(t)+ Bdj(t) (1) y(t) = Cx(t)+ do(t)+ n(t) (2) where x(t) E Rft, u(t), y(t), dj(t), do(t),n(t)ERm [A, B] is a stabilizable (controllable) pair [A, C] is a detectable(observable)pair rank (B) = rank (C) = m Once we specifythe plant model Gp(s),we must find K(s) so that: (i) the nominal feedbackdesign y(s)= Gp(s)[Imm + K(s)Gp(s)]-ldj(s) is stable; (ii) the perturbed systemin the presenceof bounded unstructured uncertaintiesis stable; (iii) application-dependentdesignspecificationsare achieved.

..(~ 'T

Loop transfer recovery

983

The designspecificationscan be expressedas frequency-dependentconstraints on the loop transfer function K(s)Gp(s).The standardpractice is to shapethe loop transfer function, K(s)Gp(s)so it does not violate the frequency-dependentconstraints (Doyle and Stein 1981).The loop-shaping problem canbe consideredto be a designtrade-off among performance objectives, stability in the face of unstructured uncertainties (Lehtomaki et at. 1981, Safonov and Athans 1977) and performance limitations imposedby the gain/phaserelationship. Here we assumethat n(t)is a noise signal that operatesover a frequencyrange beyond the frequencyrange of r(t), dj(t) and do(t).We also usea precompensatorP(s)to shapethe input for command-following. Therefore the performanceobjectivesare consideredas only input-disturbance rejection over a bounded frequency range. The design specifications may be frequency-dependent constraints on Gp(s)K(s),which is the loop transfer function broken at the output of the plant, rather than on K(s)Gp(s),which is the loop transferfunction broken at the input to the plant. Since Doyle and Stein first applied LTR to the loop transfer function K(s)Gp(s),for consistencyand continuity, we will also assumethroughout this article that all designspecificationsapply to K(s)Gp(s). One method of designing K(s) consistof two stages.The first stageconcernsstatefeedbackdesign.A state-feedbackgain G is designedso that the loop transfer function (;(sl nn-A) -1 B, which is shown in Fig. 2, meets the frequency-dependentdesign specificationsand satisfiesthe following to guaranteestability:

Ilo-

xlf)

B A

-

Figure 2. State-feedbackconfiguration.

(A.ilnn-A+BG)ui=On,

i=1,2,...,n

(3)

real (A.J< 0, Ui# 0. '~iis the closed-loopstate-feedbackeigenvalue,while Uiis the n x 1 right closed-loop ~~igenvector of the system.Controllability of [A. B] guaranteesthe existenceof G in (3). At this stage,one can determine whether or not state-feedbackdesign can meet the designspecifications.In this paperwe assumethat G is selectedso that (3)is satisfied and the loop transfer function G(sInn-A)-l B, which is shown in Fig. 2, meets the ,desiredfrequency-domain design specification. In the second-stageof the compensatordesign,an observeris designedto make the first stagerealizable (Luenberger 1966,Yuksel and Bongiorno 1971).Figure 3 shows the structure of the closed-loop observer.The observerdesignis not involved in meetingthe specificationsfor the loop transfer function, since all designspecificationshave been met by the state feedback gain G. For stability of the observer,the following must also be satisfied: vJ(Jlilnn-A+HC)=O~, real (JlJ< 0, vJ # O~

i=1,2,...,n

(4)

Loop transfer recovery

then K(s), as given by (5), approaches pointwise (non-uniformly) towards (7): G(sInn-A)-1

B[C(sInn -A)-1

B]-1

(7)

and since Gp(s)= C(sInn-A)-l

then K(s)Gp(s)will approach G(sInn-A)-l

B

(8)

B pointwise.t

The procedure requires only that H be stabilizing and have the asymptotic characteristicof (6). Doyle and Stein suggestedone way to meet this requirement: a steady-state Kalman-filter gain (Kwakernak and Sivan 1972) with very small measurement-noisecovariance. Now suppose we choose H with the following

structure: H = pBW

(9)

where W is any non-singular m x m matrix and p is a scalar. It can be shown (by the definition of the limit) that the structure of H chosenin (9)satisfiesthe limit in (6)as p approachesinfinity. In other words, as p approachesinfinity, H -+pBW results inH/ p -+ BW. (The reverseis not true.) Sincethe structure of H given in (9)satisfiesthe limit in (6), then if H is chosen to be pBW, K(s)Gp(s)will approach G(sInn-A) -1 B pointwise,as p approachesinfinity. Note that the structure of H given by (9)does not necessarilyyield a stable observer.We chooseH to be pBW throughout this paper. The asymptotic finite eigenstructuresof both forms given by (9)and (6) are the same, while the asymptotic infinite eigenstructuresare usually different. The form in (9) usually yields an unstable infinite eigenstructure. Although this paperis not an exposition of the properties of the transmissionzeros of a plant, beforestating the theorem,we will remind readersof somedefinitions and conceptsabout this matter. (For more information and properties of the transmission zeros, see Rosenbrock 1973, Desoer and Schulman 1974, Kouvaritakis and MacFarlane 1976.)The transmissionzeros of a square plant are defined to be the set of complexnumbers Si that satisfy the following inequality:

[

rank

SJnn -A

B

C

OYLE, J. C., and STEIN,G., 1979.,I.E.E.E. Trans. autom. Control, 24, 607; 1981, Ibid, 26, 1. (,OODMAN, G. C., 1984,The LQG/LTR method and discrete-time control systems.Master's thesis,MIT, Mechanical EngineeringDepartment. HARVEY, C. A., and STEIN,G., 1978,I.E.E.E. Trans.autom. Control, 23, 378. KAILATH,T., 1980,Linear Systems(Englewood Cliffs, N.J.: Prentice-Hall). KLEIN,G., and MOORE, B. C., 1977,I.E.E.E. Trans. autom. Control, 22, 140.

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Loop transfer recovery

KOUVARITAKIS, B., and MACFARLANE, A.G. J., 1976,Int. J. Control, 23, 149. KOUVARITAKIS, B., and SHAKED, U., 1976,Int. J. Control, 23, 297. KWAKERNAK,H., and SIVAN,R., 1972,Linear Optimal Control Systems (New York: Wiley-

Interscience). LEHTOMAKI,N. A., SANDELL,N. R., and ATHANS, M., 1981, I.E.E.E. Trans. autom. Control, 26,75.

LUENBERGER, D. G., 1966,I.E.E.E. Trans. autom.Control, 11, 190; 1971,Ibid., 16,596. MACFARLANE, A. G. J., 1972,Automatica,8, 455. MACFARLANE, A. G. J., and KARCANIAS, N., 1976,Int. J. Control, 24,33. OWENS,D. H., 1977, Feedback and Multivariable Systems (Stevenage, England: Peter

Peregrinus).

POR1'ER, B., and D' Azzo, J. J., 1978a, Int. J. Control, 27, 487; 1978 b, Ibid., 27, 943. ROSENBROCK, H. H., 1973,Int. J. Control, 18,297; 1974,Computer-AidedControl SystemDesign (London: Academic Press). SAFO:NOV, M. G., and ATHANS,M., 1977,I.E.E.E. Trans. autom. Control, 22,173. SHAKED, U., and KOUVARITAKIS, B., 1977,I.E.E.E. Trans. autom. Control, 22, 547. YUKSEL, Y. 0., and BONGIORNO, J. J., 1971,I.E.E.E. Trans. autom. Control, 16, 603.