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## Material Information- Title:
- Canonical forms for linear-quadratic optimal control problems
- Creator:
- Khargonekar, Pramod Prabhakar, 1956-
- Publication Date:
- 1981
- Language:
- English
- Physical Description:
- v, 91 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Algebra ( jstor )
Canonical forms ( jstor ) Differentials ( jstor ) Eigenvalues ( jstor ) Half planes ( jstor ) Mathematics ( jstor ) Matrices ( jstor ) Optimal control ( jstor ) Polynomials ( jstor ) Riccati equation ( jstor ) Control theory ( lcsh ) Riccati equation ( lcsh )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1981.
- Bibliography:
- Includes bibliographical references (leaves 86-90).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Pramod Prabhakar Khargonekar.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 000295846 ( ALEPH )
ABS2198 ( NOTIS ) 07971946 ( OCLC )
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CANONICAL FORMS FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS By PRAMOD PRABHAKAR KHARGONEKAR A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1981 ACKNOWLEDGEMENTS rish express sincere gratitude to all those who contributed I w towards making this work possible. been Professor R a constant KALMAN, source the chairman encouragement of my super during my entire committee, graduate work. The main motivation several control invaluable problems. for this work was discu am also ssions provided his research with him on linear-quadratic grateful for his early proposal optimal intuition the relevance of the Riccati transformation group ctral theory of linear-quadratic optimal control problems. Without the financial support which he arranged for me for the last three years, this work would not exist toda would like my supervisory express committee: sincere Professor apprec s T. E nation to the other members BULLOCK LONG, M. POPOV, V. SHAFFER. Discu ssions with Professor BULLOCK on the linear-quadr atic optimal control and algebraic problem geometry and with helped Profe a great ssor deal LONG gaining on invariant an under ending problems considered in this sser station. A doctorate more than a dissertation--it a slow educational process. KAMEN, grat A. fully acknowledge TMTJENBAUI4, Profe BA SIL;E, ssors and others EMRE SONTAG, for their guidance help. would also like express my deep appreciation the untiring help and unending pati ence of my fellow students and fri ends Tryphon, Bulent, Ja ime, Satis am also teful to Ms. Eleanor Onoda for doing an excellent typing job. course, no research would reported here were it not for the constant love encouragement of my parents. To them, am eternally This research was supported in part US Army Research Grant DAAG29-80-C0050 and US Air Force Grant AFOSR76-3034D through the Center for Mathematical System Theory, University of Florida, Gainesville, 32611, USA. TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT CHAPTER INTRODUCTION * S S S S S S 1 RICCATI TRANSFORMATION GROUP The Linear-Quadratic Optimizati Riccati Transformation Group . Invariants, Group Actions, and Problem Canonical F S S S TII. STATE SPACE AND FEEDBACK GROUPS Invariants Action of for the State Space the Feedback Group . Group Action ACTION OF THE RICCATI GROUP--I S S S S S S S S S S S S S S Finite Time Problems--I . The Orthogonal Group . Finite Time Problems--I Infinite Time Zero Terminal *~~~ S S S S Case ACTION OF THE RICCATI GROUP- II S S S S 5 5 S S S S S 5 5 The Algebraic Riccati Equation Approach S S 5 SYSTEM THEORETIC APPLICATIONS Spectral Theory of Linear Other Applications . dratic Optimal ntrol. REFERENCES BIOGRAPHICAL S S S S S S S S * SKETCH Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree CANONICAL FORMS of Doctor FOR LINEAR of Philosophy '- QUADRATIC OPTIMAL CONTROL PROBLEMS By PRAMOD PRABHAKAR KHARGONEKAR August, 1981 Chairman: Cochairman: Dr. R. Kalman Bullock Major Department Electrical Engineering This work acting with q on spaces uadratic concerned of various with types cost criteria. the Riccati of linear Canonical group optimal forms of transformations control complete problems systems of invariants are obtained. are studio ed for These canonical this group action and forms the Riccati several results transformation group are shown In particular, problems, to have spectral sever theory Riccati differential al important of linear equation, theoretic -quadratic positive optimal applications. control real transfer functions are inve solution "infinite stigated via the Riccati problem of time free transformation group. existence terminal state solution problems complete to the so called is given terms non- negativity a symmetric matrix. This solution is also related CHAPTER INTRODUCTION Linear -quadratic optimal control problem one of the most fundamental problems modern system theor Optimal control 1 and filtering problems to the in system control concepts theory eory was reach today. done ability e pione KALMAN obser ering work [1960]. vabilit on linear Some other -the concepts -quadratic papers which opt imal made substantial include There are contributions POPOV seven 1:19624) ral t WILLEMS books the theory [1971] of linear -quadratic J ONCKHEERE on the subject which optimal SILVERMAN we mention control [1978]. BROCKETT 1970] ANDERSON and MOORE [1971], and WILLEMS [1981]. Despite voluminous research on this problem, some aspec the th eory have not been fully Invest tigated. This di ssertation explores one such direc tion. Very which Action corre mathematical spends GL(n) some (the objec notion set of admit a group equivalence nx n nonsingular among transformations these objects. matri on the space n X n matrices underlying notion y conjugation equivalence i a very g this case ood example arises from this. change of bas The main mathematics problem such a situation to find quantities which remain invariant under transformation. In the example above, invariant The basic factors idea that eigenvalues are these invariant some the invariant quantities reveal quantities. the fundamental features of the mathematical objects under consideration. Mathematical example, tem th GL(n) eory provides acts on the set examples reachable this systems type in a of problems. natural way. This Again a action detailed corresponds investigation a change of this of basis problem from an in the state invariant space. theoretic point POPOV some of view 1:197 assoc iated. has revealed KAL;MAN 1:1974), problems. basic TANNENBAUM section 4 structure 1980] we bri of reachable an account efly scuss syst ems. of this some and the results The main problem +ln a O C a -I- station arises from analogous considera- ,1 irl 13 U .1. ~3~G research proposal. This group of transformations soon became known as the Riccati four group. components: Roughly state speaking, space the Riccati transformations, group input is built space transforma- tions, Riccati is to find linear state variable differential invariants feedback, equation. canonical change main forms problem of of variables dissertation for linear-quadratic optimal control problems of Chapters IV and under the action V show that of the Riccati group. we are substantially succe Our re ssful suits in resolving this problem. The results are very satisfactory for case equal Kronecker indices. We also treat case of strictly unequal Kronecker indices easy to explicitly. formulate. The results However, in the in any general concre case situation do not seem to techniques developed in this dissertation can be used obtain invariants canonical forms. We will now briefly describe the contents of the following chapters. Chapter transformation II is devoted group. a precise We analyze each formulation "component" of the Riccati Riccati group separate group. KALMAN The bas combine ideas four this chapter components are esse into a ntially single transformation contained purpose Chapter III is to review some well known material on the action of the state space group feedback group on the space reachable pairs. We define a new canonical form for the action of the feedback group called "dual Kronecker canonical form" . This canonical form There is based are two main on the duality contributions of the in Chapter associated ITT. Young We show that s diagrams. the set of reachable systems with iven set of unorder Krone cker indices an affine variety; a geometric quotient for the action of the general linear group on this space exists the geometric quotient turns out to a Euclidean invariants space, due to POPOV the quotient r19723. This re sult given should by the well known be viewed as DAVISON [1976] reachable pair in the Kronecker the stabilizer canonical in the feedback group form. Chapter IV contains the first results on the invariants canonical forms satis for the factory action for the of the Riccati case equal group. Kronecker The r results are very We also treat case strictly un equal Kronecker indices expli city. Some the results this chapter turn out t useful in Chapter system theoretic applications. In Chapter canonical we devel forms. Here a real another we assume symmetric approach that assoc solution. invariants iated algebraic Again, the results very satisfact case equal Kronecker indices. difficulties in the neral case are analogous the difficulties Chapter Chapter contains some theoret applications the results of Chapters IV and The main application spec tral theory linear-quadratic JONCKHEERE optimal and SILVERMANT control [197 problems. for the di This theory score -time was developed case. ey analyze the so called "infinite time free terminal state linear -quadratic optimal control problem certain a direct problem" from a boundedness self-adjoint calculation Hilbert of infima Hilbert of the spec space operators reduced rator. trum of point view. nonnegativity However, it turns out that We consider JONCKKEERE a We show that the continuous- SILVERMAN the existence 1978] time analog as formula solution the problem originally to the optimization [1971]. problem remains invariant under the action of the Riccati group. However, the Hilbert space operator defined JONCKHEERE and SILVERMAN [1978] does not remain invariant under the Riccati group. We first compute a Hilb space operator which is the continuous-time anal the operator defined JONCKHEERE and SILVERMAN [1978]. Riccati group leads to a major simplific ation in the derivation operator. then show formulated indices. Riccati equation has obtaining are space this operator is not tractable. considered WILLEMS of boundedness of infima can be reduced to checking; nonnegativity symmetric matrix. We rederive a generalized version of this result using certain developed simple results KALMAN, from the theory and NARENDRA [196 "minimal This give s energy a very control" simple system theoretic interpretation of the spec tral theoretic results. We also examine fast algorithms Popov's for the equency Riccati function, differe ntial itive equa tion real matrices due to KAILATH [1975] Under and CASTI [ certain mild 1974] in the light conditions, we complete Riccati transformation y parametrize scalar group. itive real functions. Further work on the fast algorithms from cati group scalar intere point positive view real seems promising. functions General the matrix nation case our results also considerable We close assumptions. this introduction Throughout with a few words this dissertation, on notation we will work with d standing the field real numb ers. denotes m-dime nsional real vector space consisting m-length column vectors with real entries. denotes set of mx n matrices over the field real numbers , b] denotes the Hilbert space A polynomial square called integrable a Hurwitz functions polynomial on the interval iff the roots are in the open left half plane. undefined terminolo from invariant is referred theory and algebraic to FOGARTY [1965], geometry MIUMFORD is standard [1965], for which TANNENBAUM the reader [1980]. rrXn R CHAPTER RICCATI TRANSFORMATION GROUP In this chapter, we shall introduce the Riccati transformation group, quadratic KALMAN which arises optimi [1975] naturally as zation in a problems research a group This propose of transformations group was originally Roughly of linear- formulated speaking, Riccati transformation group consists state space transformations, input space of variables transformations in the Riccati linear state equation. variable shall feedback, scuss each change these transformations group action. separately then combine ese into a single The Linear Quadratic Optimization Problem. The "dynamic for a continuous- optimization time, finite problem with dimensional, integral linear quadratic stem criteria is defined as follows minimize the functional (T)Qx(T) + 2x'(T)Su(T) u' (T)Ru() + x'(0)Pox(0), over all continuous valued input functions Ut r) with respect to the solutions n-th order continuous -time linear (1.2) it(c = Fx( ) + Gu(T), = c. Here are respect tively n x n real matrices assumed t definite. symmetric We also assume matrices that is always assumed is reachable. itive It should noted that 1n slTTi A+.nt the system as well as tn lTiT the quadratic rpE3 F,1 tsz cost It In PP4\ 1.11 ~n~nsnl nrnss criteria are niir T" r e1l ts time- wh i ph n X n X are (1. 1) J(u( I. f . (1.3) c, N) 1 k=-N (k)Qx(k) + 2x'(k)Su(k) + u'(k)Ru(k)] + x'(0)Pox(0) over all valued input sequences u(k), with respect to the n-th order discret e-time linear system. (1.4) = Fx(k) + Gu(k), x(-N) = c. , x(k) Here that is not assumed are as before. positive The only difference definite. linear quadrati optimi nation problems defin ed above have been studied extensively over the last two decades. The literature surrounding these problems grown enormously. reader referred seminal WILLEMS paper [1971] of KALMAN [1960]. . JONCKHEERE and also KALMAN [1964] SILVERMAN POPOV refer [1964], ences cited there. It is well known that the solution the above continuous-time problem optimal base control on the (matrix) input Riccati differential equation. iven u~&r) x ( r), = L(r) = Fx (r) - + Gu(r), X,( t where LtC'r) -l(G'P(r) is the solution of the Riccati differential equation (1.6) -JP( r) = F'P(-r) + P(t)F - (P(P')G Here we implicitly assume that the solution to (1.6) exists. + S'), -l(P(7)G C, J(u( ) U~ ( dx+( 7) dx* d2 Chapter VI for a discussion of the general case. A similar feedback solution base on the Riccati differe nce equation can be given for the discret e-time problem. thus see that the optimi zation problem as well as its solution can be fully Since features described the initial of the optimi in terms state zation the "problem data" plays blem, no spec 1 role we do not consider in various it in the "problem data" be identified Thus with every linear quadratic optimization a six tuple Conve problem rsely, six tuple of matrices of appropriate sizes, reachable, are symmetric positive definite then we can define a linear quadratic optimization problem ociat ed with In view these ementary serva tions, we define (finite time) optim nation data space (1.7) := ((F, >( _ _mxm X R X R reachable, symmetric, positive definite). Similar considerations also apply discrete-time linear quadratic optimization probl ems. In particular we have discrete-time optimization data space RInn := (-F, R~ nxn x R reachable, symmetric). An important problem closely related to the linear-quadratic opt imiza- tion problem is so-called infinite time zero terminal state linear- quadratic optimal control problem defined a follows Minimize (1.8) =io[x'(T)Qx(T) + 2x'(T)Su(T) u'(T)Ru(-) ]d'r ass can >(~ ><~ )<~ t &~ : (F nXn R F , J(u(.) over possible functions m R -valued, Necessary a continuous, d sufficient square integrable conditions input for the existence solution the above this infinite problem are well known. time zero terminal In particular, state problem exists a solution if and only if the algebraic Riccati equation (1.9) +PF - (PG + + S) admits a rea 1 symmetric solution. case (1.9) a real symmetric solution, it has a unique real symmetric solution such that eigenvalues - GR-(G'P aGR (C PP are in the (clo s e&) left half plane; Re(A(F GRGP + -GR (G'P+ WILLEMS 1971] an exce llent reference for a detailed analysis of the algebraic Riccati equation. It is clear from above discussion that the infinite time zero terminal matrices data space state problems are completely In view for the infinit time zero specific this we terminal ed b define state quintuple the optimi zation problems DS := ((F, X R _~ _~ Nt tXm reachable, symmetric, positive definite. Analogous considerations apply to the discrete-time case. Riccati Transformation Group. purpose the above dynamic this section optimization introduce problems some which transformations o naturally from the problem statement and the associate feedback solution based on the Pic reati di fferen. al. erniaBti nfl -l~pc st) nXn R :- T the n the linear quadratic optimization problem (1.1) transforms minimize -1i c, -= t [z'(T)T'QTz(T) o + 2z'(T)T'Su(T) u'( )Ru(T)]dz z' (O)T'P Tz(O), subject to the linear dcz(T) dcT system --TFTz(T) Gu(Tr), Thus, the problem data transform , T'QT, Riccati diff and only i equation a erential T'P(T-)T ssoc iated T'PoT). equation Furthermore associated is the solution with VT, 1, the solution with to the Riccati T'QT, differ T' PoT). ential It is easy eck that optimal control inputs for the original problem transformed problem are same. Conse quently, state space basis chan do not alter basic features linear quadratic optimization problems. Summarizing, GL(n) acts on the data space (2.1) x GL(n) 1TT, T-G, T , T'QT, T'S, T' PoT). Finally, reachable we note pairs that has b the action een considered GL(n) from on the seven space ral points view in the literature. TANNNBAIJM [1980] and the See, for example reference cited POPOV there. 1972 ], KALMMAI summarize [1974], some ~~~~c' ~ ~ ~ ~ ~ i-i- ---I-rrf ~ x(t) J(U() D : H(T n L~. - I, \ rm on_.~ ---- f--_ *>n -*IK- -1^~ Change of basis in the input space. This transformation similar to the state space transformation scusse d above. mx m nonsingular matrix. We introduce new input variables v(.) v( r) Proceeding as this leads to the transformation (2.2) D X GL(m) '- V'RV, Linear state variable feedback. This perhaps the most nontrivial transformation of the original problem. Suppose we choose N; t := u(t) where n matrix and u(.) continuous -valued function The linear system then transforms = (F + GL)x(-r) + Gu(T), = c. We also have X'Q) + 2x' (T)Su(T) + u'(T)Ru(T) = X'Qra) + SL + L' S' + L'RL]x(r) + 2x'(r)( + L'R)x(T) + v'(C)RV(t). Thus the orig := (F inal + GL, problem data + L'RL tran + SL + L'S + L'R, sform to It is routine verification that a solution the Riccati differential equation asso ciated with if and only P(T) is the solution (whenever the Riccati diffe they exist). rental The optimal equation associated feedback matrices with L(d) L(dL) -1 V- -x r dx( D : Po) '. R Ut ~t) Summarizing, linear state variable feedback induces transformation (2.4) nRlxn x R - + GL, + SL + L'S + L'RL, + L'R, It has een recog nized. prev iously in liter nature that edba transformation does not alter most the basic features the dynamic optimi [1975] zation blem. JONCKHEERE and See, for example, SILVERMAN [197 SILVERMAN etc. [197 Feedback MOLINARI tran sformation stabilize system JONCKHEERE and SILVERMAN we have problem does differential shown above not alter equation the form under does n linear eedbac .ot alter solution and the quadratic the Ri optimal optimi ccati zation feedback matrices related a simple trans lati as in (2.5). of variables the Riccati differential equation. This tra ns linear differ formation quadratic ntial arises optimi from zation fact blem that given solution erms to the the Riccati equation (1.6) -dP(T) dir = F'P(T) + P(t )F P(T) + S)R 1(P(r)G + S) Suppose we perform the change of variables P(rT) := P(r) where obtain a N: U the differential CO ns tant symmetric matrix. equation A 5) ) = F'P =FP + P(r)F + F'M + a ('P(T)G + MG) use are (Q H (F pt ~) As is clear from this trans formed equation, the Riccati differential equation assoc iated with the data + F'M + MF, + MG, t rari formed b MG). problem Furthermore cost assoc iated with + F'M + + 2x'(T)( + MG)u(r) 7)Ru( m) 3dm + x'(0)( - M)x(O). Recalling that x(r) is governed = Fx(-r) + Gu(t), we have + 2x'(T)Su.(T) + u' (T)Ru(t) ( 'r)MX~rC))3d + x'(o)P - x' (O)I x(O) - c'Mc. Thus if and only every fixe initial an infima state exists an infima Further exists an input function u(s) fixed minimizes initial if and state only Finally, P(T) minimizes a solution each and only optimal also A P(r) feedback preserves a solution in matrices most the b are the asic (1.6), same. atur Thus, the associated this tran the problem. sformation conclude, Sym(n) be the additive group nx n symmetric matrices then have (2.6) the transformation X Sym(n) dT + I( =~f D: t S, 7:)(Q J(u( ) J,(u( ) This transformation occurs seven ral different contexts in system theory literature. In the study stability and optimality this transformation arises POPOV [197 In the study stochastic reali CANDY this [1975] optimal zation [197 transformation consider control. problem, this transformation has the closely related occurs a similar in ANDERSON ransformat ion been problem [1967] for the cons idered positive BERNHARD inverse real matrices, COHEN problem We will now combine the four different types transformation scuss ed above . This will cons titute the action of the Riccati group on the data space Consider the set (2.7) GL(n), GL(m), ym(n) ). We define a binary operation between any two elements given o (T2, 2' L2' := (TTT2, V'V2, 1 2 VL 1 2 + LT 2, 1 2s + (T) - T'). It is easy check that with the binary operation a group. We just inverse define note that , O,0) (T-. V serves as the identity and , V-ILT, - T'MT). Po)' ^ A' where now mXn R + F'M + + T'(S + MG)L + L' (S + MG)'T + L'RL, := V'RV, + MG) + L'R)V, + M)T. In an Riccati entire group analogous on the manner space we can define corresponding the action of the to the infinit e-time zero terminal stat problems where are as above. It is easy check that the function satisfies the following rules: (2.9) and , 72 cp(d., (2.10) P (cl, =c. In fact the binary operation was defined precisely that (2.9) hold. Invariant s, Group Actions, Canonical Forms. an abstract that acts denoted multiple the right) catively iff there a set. a map such that have w1 ) =i(~z Wj, ii(z, 2), = z. (Here algebraic is the identity transformation The triple space is called the (right) w): w in W} i call ed an action called =- i'( *-= T' H(i, Y2 MF)T -t DS: cl(z, Ccl(zl Now let other said to be invariant (under a function. the action Then iff for An invariant is called a complex te invariant iff for implies ri~i( orbit In this of finding set theor etic a set some The set is a complete setup, that there invariant a trivial parametrizes orbit said solution space. parametrize surjective. the problem e can choose to be the set general, very orbits useful. It is mor e However desirabi such a soluti to have with some math ematical structure. group called an algebra group an algebraic ety, ->W: W i-) the -1 two are maps C: W x W morphisms. Now suppose r-W W2 algebr and aic group an algebra vari such that Z W a morphism. We then say that acts morphically. In such a situati variety. invariant excellent theoretic on, we may require This problem theory pioneer erence aspec the parametri one of central MEJMFOBID an expository account of linear system to be an algebraic problems TANNENBAUM f geometric the modern 1980] geometric is an invariant theory. action is a right action D follows directly from (2.10). machine group seem owev ,there metric main have are certain invariant difficulty algebraic lies ometric difficulties theory in the struc to the fact ture. in directly action that applying Riccati does not principal r eason for this symmetric seems matrix be the is well condition known > 0. The positivity equivalent certain polynomial inequalities the entries Thus, it does not seem to be space vari -* f( f(i ,W)) -IW: CL ( However, whether is linearly reductive seems to be a difficult question as well. The next three chapters are concerned with answering questions invariants canonical forms for the action We do some aspects of modern invariant theory, but only after certain preliminary reductions of the original problem to somewhat more mannageable situations. use CHAPTER III. STATE SPACE AND FEEDBACK GROUPS purpose this chapter is to review some relevant results about the action reachable of the systems. state space The results eedback groups described here will on the space be the main preliminary the action material of the Riccati obtaining group. invariants This chapter canonical is divided forms into sections: first we di scuss invariants canonical forms for the action of the state feedback group. several years. Most space group, and next of these results We present a new result we di have on the scuss n known action e action for the of the past state space group which be viewed as a g geometric counterpart some previously known results on invariants due to POPOV [197 Let us define space of reachable systems := ((F, Rnxn reachable); denotes of all reachable teams with states inputs basis - As changes GL(n) prec acts ending chapter, group state space the following way: x GL(n) i- (T-1FT, T-1G) . It is easy to verify that a right action GL(n) feedback group defined := ((T, GL(n), GL(m), Rrtn)n3 a group under the multiplication given L1) o(T2, S(1T2' according r. , acts 21 -.1 -,1 -t C: nXm R * ,, right action identified with reason that subgroups leads us groups of the Riccati study actions GL(n) group; these can be in fact, groups this on the naturally is the main space reachable systems. Invariants for the State Space Group Action. The action several points of view. GL(n) complete has been studied. set of invariants extensively canonical from forms, and a geome tric quotient have been discovered. We shall describe some of these results the interested HAZEWINKEL reader KALMAN very briefly. referred [197 detailed POPOV HAZEWINKEL [1972], 1977], proofs KALMAN TANNENBAUM discussion [1972, 1974], 1980], BYRNES GAUGER C J977] and others. Define the reachabilit y matrix map (4.1) p: Z R t (G, F1G =. p(F, Number the columns p(F, lexicographically (from left to right) - 1)1, - l)m. nice selection defin a subset v of these indices size such that belongs then belongs < i. For a nice selection v define :nXn PV(F, where p (F, denotes nX n submatrix p(F, consisting the columns results of KALMAN indexed [1974] by the nice selection is that reachable of the if and only p (F, is nonsingular some nice selection (4.2) REMARK. each nice selection define open affine subsets X Rnxm (F, tlXn R It is easy to see that GL(n) acts in such a that : i. is stable under the action GL(n). Further, : v a nice selection) constitutes an affine open cover TATINENBAUM tJ-98O2 shows that GL(n) acts and hence on each with closed MUMFORD orbits. TANITENBAUM Using then metric proves that invariant a geometric theory quotient the action Finally shown , using that the action out to be a GL(n) the techniques "patch up" v GL(n) smooth, irreducible, exists of modern such a exists. quas for each algebraic way that This nice selection geometry, geometric geometric iprojective quoti variety it is quotient ent turns of dimension However, we will not work directly with this ometric quoti ent. Following section, POPOV called 1972], we will the Kronecker now describe nice selection, how a can be special nice naturally associated with every reachable a reachable pair. Consider the ordered set of vectors ' ~(g1, Fgl', " ., Fn-lg where is the column the matrix In other words, cons ists of columns p(F, list ed from the left to the right A vector said to an antecedent another vector im + the list < pm + can now occurs define some well known the left invariants FPg F g in qof the pa of the pair DEFINITIONS. be in an integer, Then j-th unordered Kronecker index is the smallest eger such that is linearly dependent upon antecedents. unordered Kronecker S..,t Integers indices obtained K2, ..., K of the pair ( by reordering are called t The integers in a * a a - n-l F II .. condition =1, for the sake of notational and We will computational often make simplicity. this assumption Let us define E: rank := ((F, = m}. be ih with unordered Kronecker indices There Kronecker a ni i ndic e selection s, defined naturally as follows: ass ociated with a (double) the unorder index longs called chas the definitions the Kronecker it is easily section seen of the pair that (t'4) det PYK(F, 0. We will now describe some of the main r results POPOV the invariants of the action GL(n) reachable pair. For the sake of notational simplic rank belo that there exists Then a unique Corollary of (indexed) real of POPOV numbers tJ1972J states = 1, = 1, - 1, = 0, =1, min(K., + 1, - 1); = 0, = 1, min (K., 1 such (4,6) that Ki F -g. Sa. Fk k ijk where for each the summation with respect runs over index set F gi on above this list index defined above. its ante depends set by cedents. Nijk] Note only upon where determines that the ind the unordered = (K1, the linear ex set of Kronecker dependence S, tJjk indices; s in the we denote K). M cardinality denote the lexicographicallyy) ~~~~.~h L.I --.4- , -) m; ~aijk: t '"') cr(F, I,, \ _L L orbit under the action GL(n) if and only the unordered Kronecker indices = t(F2' , G) , G2) are the same Further, = (K , is an m-tuple positive integers satisfying -tuple such that a(F, real number then are unorder there exists Kronecker a pair indices = Cy. Roughly a complete, will speaking, unordered independent now describe Kronecker invariants algebro- indices for the geometric cons action structure titute GL(n) set of reachable pairs S(K1, havin ..., Km) * *4m same uno a fixed s rder Kronecker integers indices satisfyin = n. 4(K) subset defined 4(K) the unordered Kronec mndi Km. m now have (4.8) THEOREM. = K ,1 S.., K ) as above. Then 4(K) has the structure an affine vari The function a: 2(K) is a regular surj active map. pair N R K) constitute a geo metric quot i 4(K) under action GL(n). PROOF. first some notation. x Rnxm each int Anti(F, G eger denote the submatrix consi Rnxn sting the columns inde the nice Ant.(F, Define selec tion consists satisfying + mj of antecedent + mK. which In other words, are the (polynomial) ... + K .. + K are ces f K~ n, : (F a(F1' Ha(F, Here denotes the (fixed) number of columns Ant.(F, 1 denote the set of integers We will now give equations such which give that as a subset x Rnxm We start noting that (4.4) det PK (F, is a necessary condition Ant.(F, condition that F .g s For any Kig gi should linearly linearly matrix F to be in dependent dependent m.(B) J i 2(K). upon upon denote e additional antecedents the columns j-th lar minor - i, ... Then we must have (4.9) m.(Anti(F, FKig.) = 0, for all suffic sible ient It is order easily seen that *... that are be the unordered necessary Kronecker tnti Following the standard proce dure of algebraic geometry, we now introduce projective a new variable variety) Then the (real) 2( K) solutions isomorphic a quasi- the equations (4.10) det PVk the Equations (4.9). This proves our assertion that 4(K) has the structure of an affine variety We will now show that a: Z(K) a surje results ctive of POPOV regular [1972] map. Surjectivity as described in follows Theorem directly (4.7). from prove that I. nran r1 r mmii nlm ces ~Xn R ~h nw fh s~ f j r ^ is regular for each index in the index Now, ~(PvF, a regular map on 2(K). It follows that 22(K) ( (i, j) -th entry (pvK(FI G))-1 is also ijk (4.12) a regular (implicitly) FKigi i for each given ijFk denote ition of G)) F gk n PvK(F, Multiplying (4.12) q-th row (pVKQE', we ge = a. ijk As p..' s a regular and map. FKigi Hence are regular a sure 2(K), active it follows regular that map. Finally, , G ) S(K) belong to the same orbit under the action GL(n) if and only G2). It follows directly from the definition metric quotient, (see MUMFORD [1965],) that RNK) a ge ometric quotient. The above reformulation result should results viewed as of POPOV 1972)]. geometric invariant We see that theoretic a disjoint union of affine varieties corresponding various choices of unordered Kronecker indic quotient with on each of these affine respect the action varieties GL(n) ex: a ge ists. metric In our work, 2( K), Kig R: G))-l de t (Pij : (cp,,, 'Sqn)F a, 1Jk (F1' Cr(F1' ~~jF2~ w v m geometric quotient for the action GL(n) Z(K) can be deduced Am a much Section with osed FOGARTY it then quotient quotient easier way. Corollary orbits. [1965].) follows exists can be from It is known 1.6] Recall Having MUMFORD The point shown that from TANNENBAUM GL(n) that GL(n) proved that pages here course acts [1980, Part and hence a linearly r Z(K) -350] is eductive is an affine that a geo that Z(K), group. variety, metric geometric N) BK). (4.114) REMARK. Results the above theorem should compared with certain results of HAZEWINKEL [1980]. It is shown there that construct general, tions the well "discontinuous known "control However, canonical one restricts forms"I are attention these constru continuous. actions It is 2(K), well known then that these cons tructions these constructions are indeed are closely related regular to the invariant maps 4(K). s OX. Thus We have the continuity shown that a, s are of the constructions canonical forms 2(K) is natural. Action of the Feedback Group. now turn our atte nation to the action the feedback group This problem has een complete ely r resolved due to the deep and striking results of BRUNOVSKY [1970], ROSENBROCK [1970], KALMAN [1972 ], POPOV [1972], WOIIIAM 1974], others. The main purpose this section describe start with the stabilizer a theorem that in the feedback group describes a complete in a pair invariants under the action of the feedback group. THEOREM. The ordered Kronecker indices constitute a complete set of invariant of the action of the feedback group various proofs of this result, the reader is referred to the papers mentioned above be in with ordered Kronecker indices := block diag ... Fm] := block diag ... gm ; m^_ f^ where are K. X1 1 matrices of the form F The pair In fact, called a triple the Kronecker that tran canoni 1 iform form sfhrins can be cons truct using the invaria ijk In other words, there a regular S(K) such that transforms to the Kronecker canonical form. our purposes, however, it is more useful choose a slightly different canonical form under the feedback group. In order to describe this canonical r := K * form, we shall first set a given some of ordered notation. Kronecker indices indices Define the Young' matrix mx n matrix assoc iated with this set of give Y (i, K i, O, otherwise. Then the dual Kronecker indices are defined 21+ m j.'Ji = 1i, map -t rf X Ki, be the unique integers such that "r(1) r(l)+l * - * = r(q-1) r q-1)+l r(q) - m. To each eger we can uniquely associate another eger PB(i) defi (i)) Finally, define r(2) r(q-l)' We note that case the Kronecker indices are all equal; = n/m, then - m, =1, r(1) =11. As we will see i the following, case of equal Kronecker indices is relatively uncomplicated and leads very nt results. be in Then with it is order easy Kronecker see that indices feedback (5.2) equivalent Here "0" in Aj it. entry matrix consisting zeroes, - IT is the 1 x +l 1+1 matrix the form I -. - 2 .. f = K =... ... > K > m - , G ) K2 "2 m, r(l) (1) consisting zeroes. case the Kronecker indices are equal, we have SF2 We call see that , G ) * G , G) the dual Kronecker can be obtained canonical from form. ff15s easy to a permutation state variables Here again, there exists a regular map, ("built" out of d: Z(K) Sd(F, such that d(F, dissertation, some transforms , G ) choice of Krone the stabilizer in the will cker eedback denote the indices group . We dual now Kronecker turn a pair Throughout canonical this form our attention , G ). Stabilizer in the canonical feedback group form) has been the pair obtained the Kronecker WANG and DAVISON [197 BROCKETT [1977 ], however F1I[MRMA'NN turns and WILLEMS out that [1979]. Ricca group the stabilizer problem, the dual Kronecker canonical form) is more conve nient. be ih the feedback group. We partition SF iT'r a, rJk where matrix. T.h The stabilizer x2.,' X cr subgroup ,G) SG is character following THEOREM. , G) be i the dual Kronecker canonical form as described above. Then a triple , L ) belongs to the stabilizer , G) and only , L ) satisfies: - 0, dat V.i S0, - 1, S.., = 0, >1, where B(i) where where is a Xc"k matrix satisfying - 1 + r(k - 1)), r>i where B( i) finally, = 0, = 2, r(i-l) (I. 1 S.., The proof of this result omitted since it is a very tedious cumb ersome a very but straightforward simple corollary of the direct above verifi cation. theorem which We will describes now state > j, B( i) ~(j), (5.4) COROLLARY. * ,G in the dual Kronecker canonical form with Kronecker ii) belongs indices to the ... = K stabilizer = n/m. ,G*) Then a triple if and only = O, det V S0. The above corollary is much easier to verify directly. In the following Corollary chapters, (5.4) will we shall make extensive be especially useful use of these since in this results. case results are easy to formulate 'n an explicit form. K2 = V, CHAPTER ACTION OF THE RICCATI GROUP--I In this chapter we shall develop an approach which leads to a complete Riccati invariants group on certain canonical forms subsets for the action of the "optimization data of the space" The re suits this chapter are most satisfactory for the case of equal Kronecker indi ces. In other cases the results are similar; however seems to be difficult formulate these results in closed form due notational and which also runs computational into somewhat complexity. analogous An alternative approach difficulties is developed the next in Chapter chapter. VI where Some of the results we discuss stem of this chapter will theoretic be used applications interpretations of the action the Riccati transformation group. This chapter is divided into four parts. First we scuss finite-time invariant problems, theory next we q-tuples give an expos squar e itory account matrices under the orthogonal group, we then the infinite reconsider h on zon-zero finite final time state problems, problems. and finally we consider Finite Time Problems-- I. We recall that the Riccati group acts on the data space ((F, where = T-FT + T-1GL, = T-IGV, + F'M+ + T'(S + MG)L + i2 (S + MG) T + L'RL, -- V'RV, = (T'(S + MG) + L'R)V, -'( h H(F, MF) T We start with the following (6.1) PROPOSITION. be ih and let ordered Kronecker indices Then there exists a quadruple in the Riccati group such that some Further, any two six-tuples belong to the same orbit if and 0) only if the ordered (and hence Kronecker =F2 indices there = G exists are equal, a triple in the stabili zer in the feedback group , G such that (6.2) 'QIT 1- 'S'T 1 = (T ) 1s 'RV 1 (6.4) = (V C) 'RIV [Here *G ,G ) is the dual Kronecker canonical form explained Chapter PROOF. By the techniques of Chapter there exists a triple such that = T -T + T-GL, = T V.-1 Choose M = P Then cP((F, Po) A r S, = (F a A. 0, ... > S)'R L cp((F + (L Suppose there exists a quadruple such that (P((*; Then -1 * = TT y i r1 -1 * + T GIL, 1' -1 * = T GIV. 1 Hence, be the dual = G* feedb by feedback i same as those Kronecker * nvariance, the Kronecker Gj). canonical form, Hence ack group (T, G ,G ) (Fi, it follows belongs indices Gl) that f (F S* (F2, = F2 the stabili 2' G2 G2) =: F zer must in the Further, 0 = T'(0 Since where + M)T nonsingular , L ) = T'MT. Hence ongs the stabili , G ) Now the Equations 6.4) JI ow immediately from the definition Converse Kronecker Kronecker Further, feedback hold, then indi (F1, since ces, canoni if there group it fo form have ( F, it follows exists such flows dire ctly that from that in the stabili Equations definite .2), the are * zer same ordered in the dual = G2 in the (6.3), (6.10 that *k , 0)) Hence, to the same (F2' belong orbit. This proposition shows that the problem of the action Riccati S~~~~~~~~~~ jj*. *.II.f V~~AA- -- =X* _ L, =: P ---- ----- 1 t 0), L i,, 1 n ir Thus, it is Kronecker Kronecker natural indices. indices. to study this The simplest problem for various among these cases choices is the case of equal Let us define a subset of the optimization problem data space := ((F, : the ordered Kronecker indices are equal). It is obvious that is stable under the action of the Riccati group. be in such that Kronecker indices =1% S= K = n/m. Propos ition there exists such that cp((F, -(F Further, feedback we need to group consider , G ) the action on matrices of the stabilizer of the nx n matrix and any nx m matrix be the mxm matrices such that With this notation, we now have (6.5) PROPOSITION. such that a -- -- are / t &, K2 Po) &, 1 II Further, two such six tuples with equal Kronecker indices belong to same orbit if and only if there exist orthogonal matrix such that (6.6) V'Q. i QijV 2 = Qij , t~, s2 ~ k' = 1, PROOF. Proposition (6.1), there exists a quadruple such that cp((F, mXin nonsingular matrix such that V'RV =1. Such a V exists since a pos itive definite symmetric matrix. Define * CV, Then S..,i it is easy see that ,o) -(F Suppose there exists such that cp((F , QI = (F , 2 , 0). Then Propos ition belongs stabilizer the feedba group G , G) Hence L = O. Further, implies that ortho onal. Corollar y (5.4), =4'i ag (V, ..., V) Equations (6.6) follow directly from Equations and (6.5). Conversely (6.6) holds, then is any it follows mx m directly from orthogonal matrix such the definition that that cp((F I, = (F , Q , 0). where T := diag This completes the proof. (F L, w m r cp((F 0) (6.1) 0)) 0) , R_ reduced to the action of the orthogonal group on matrices given well, group digre under (6.6). the problem on a bunch ssion and It will seen later is essentially that of matrices. describe action In view the invariant that in some the action of this theory other cases of the orthogonal we make q-tuples a brief matrices orthogonal group. The Orthogonal Group purpose of this section to describe certain somewhat recent results on the invariants the action the ortho onal group on a space SIBIRSKII matrices [1967], Most which these were later result redis are originally covered due to an extended form PROCESS relevant involving We will our problems. other state only those For detail classical groups, proofs see SIBIRISKII results which and relat [1967] are ed results PROCESS [1976]. be the space q-tuples m X matrices. group O(m) mX m (real) ortho nal matrices acts as g iven X: W x 0(m) A ), q i- (V'A V, V'A2V, The problem is to find invariants the action P1W) denote the ring see that real R[W] valued polynomial functions is isomorphic to the ring It is easy of polynomials indeterminate with real coeffi clients. be the subring R[W] consisting of those functions which remain invariant under action 0(m) 14W) belongs and only A )) 9. = f((V'A V, 1 V'A V, 2 for all 0(m). Fv rv rv ii] r:3C,1.---aa-4 n-. -a-a. * (A1' f((n,, 1.,,,,,,,,.L? n function tr f: A')). q now have the main THEOREM. The ring SIBIRSKII is ge 1967, nerated Lemma as an PROCESS -algebra Theorem the elements tr f where a monomial in noncommutative indeterminate q-tuples ..., A) with and deg (B , - 1. Further, belong same orbit if and only tr f ((A1, = tr f ((BI, for all monomials in the above mentioned generating set. be all the distinct monomials in the noncommutative variables 1' 2' egree - 1. iven q- tuple = (A1, ..., A ) Tr A := (tr Then, light above theorem, Tr A a complete invariant for the action O(m) is known group. that Thus, affine vari immediately Theorem ety. that gives the orthog the action Theorem ring o: explicit onal group a linearl of FOGARTY f invariant formulae a line y re early re ductive [196 duct ive group on an it follows is finitely for a set generated. nerators. now consider O(m) the problem of a private canonical forms communication, for the action Process indicated I' I SI -I. tr f2(A) , tr fN(A)). R: t B,, ) i B ~~ tr(f(A1' t fN f,(n) (7.1) Ir I .m i I r. I I subsets the Euclidean topology) For any integer define - (A = (Al, the (real) eigenvalues + A.) 1 are distinct). Now, be in an orthogonal matrix such that V'(A. + A!) = diag *.., Further V'A V 1i = + A.) = diag + V'(A Al) + a skew-symmetric matrix. We call a matrix diagonal matrix quasi-skew- (with decrease symmetric entries) is the and a sum skew-symmetric matrix. matrix Thus, any such can be that quas transformed i-skew- by an orthogonal symmetric. be such that are quas i-skew-s ymmetric. belong to same orbit under O(m), then there exists orthogonal matrix such that particular, V'A.V = B., = 1, V'A.V 2l = B. Since parts are quasi- and 1 are skew-symmetric, equal. Since it follows the diagonal that the diagonal entries distinct, it then follows from elementary linear algebra that = diag - 1. m ...> n , are ~ t 1 Finally, we note that each a dense open sub set of For, belongs if and only if the characteristic polynomial + A.) 1 has no repeated root Using resultants it is easy show that the complement an algebraic set of strictly lower pension. Finite Time Problems--II now resume our discussion of Section was shown Proposition (6.5) that for the case equal Kronecker indices , every orbit contains a six-tuple of the form denote ((n/m) + (n/m)) -.=: q-tuple consi sting m x m matrices Qij' ordered in some immediately from Proposition fixed Theorem way. (7.1) It that now follows Tr(Q, forms a complete invariant. More preci sely we have (8.1) exist THEOREM. such be ih Let ( that under denot acti Tr (Q, Then is in the the Riccati same there orbit group Then , S, -* TR[(F, a well S(F1, - 1' -. defined function. Further, 1'P 1> 01 belong same orbit if and two six-tuples - (F2, only TR[dl] 2' $2' = TR[d PROOF. The theorem follows immediately from Propositions (7.1). We not that using techniques outlined at the end of the previous section, we can obtain canonical forms for the action of the Riccati group. T7. --!1-I-r -- 1 1I -1 _1 TR: D1 P o) &t N R - - 0) (F, -_ - 1- r II .. -, L the ordered Kronecker := ((F, indices K > m - In this case we shall outline a proce dure that leads a canonical form (and hence group a complete In orde: set of invaria r to motivate for the this procedure, action of the Riccati we start with example. = 1. The dual Kronecker canonical form , L) belo to the stabilizer of the feedback group , G ) and only vii V11 0 t V11 1 #0, v21, arbitrary. of the lower sition Propo form triangular we can restrict A ,Q, matrix our attention given such to six-tuples there exists that ) 'RV Then, = I. transforms are ... > A , Q, 8.3) Then under action transforms 1. I where (Ii) 'ST + Let us consider the transformation that undergoes: (s. denotes ij-th entry 31' s -4 ~J~2 + ts -4 s2J~., -s Choose can be Then transformed This shows that any such element that Further (with there =0) , Q2 , I, bel same orbit if and 2, only exist Tn other words. )I with a canonical 45 I, + (L (T + (T + (L + t, -S2 +1 , 1 i Cons equently, modulo the action of such signature matric the entries in the above form constitute a complete of invariants. In brought general, into we start the form noting that any , Q, R, S, 0) element a suitable can element the Riccati group. Here SG) denotes the dual Kronecker canonical form as sociated with Kronecker indices K > Kg 1 2 *** K > m - Spec ializing v* *. V L ) Theorem belo (5.5) this stabili case zer it is in th easy feedback see that group ,G) if and only Lr], 0O, = 1, T. ., itive satisfy definite, their the formulae e exists an in Theorem upper (5.3 triangular Since matrix such that =1. Then, with in the stabilizer e appropriate *(T , , G ), T , such that transforms A , Q, , 0) Furthermore, some elements , Q2 if there , 0) exists belong , L) to the same orbit in the stabilizer under if and only in the feedback group ~~iif nh th ) 'RV *^ Ct' erlFh (6.5) hold. is lower triangular, this implies that = dia ..., - 1. Let us consider the action such elements the feedback group as given Partition - [S' 1' si]. r" Choose action It is then easy to verify that Under -Os. 1 + Til) i ... + T'.S ri + L!' i' =1, AL free. be the set of indices Then we can recurs ively such that solve for the L ij-th entry such that the (ij)-th entry zero start with It is clear that can be chosen so that the entries correspond- indices are zero. can easily verified that this choice fixes r(r-l)' T(r-1) T21. Next we consider + r(r-l) Since is fixed we can find trivialize certain entries which corres pond indi ces Clearly, this procedure Hence that can be recursively we may assume entries complete to obtain that corresponding the desired in such a indices form are zero. us denote the set of such matrices It is straightforward, though tedious, verify that p I' r-1 + 1 r-2) &, r IQi /V r A = diag = diag - 0, we have (8.1*) =Q2 In other words, with a canonical form modulo the action signature matrices as described in (8.4). now define an equivalence relation pair s matrices two pairs n x n symmetric of matrices , sl) S1. S matrices with 2 in in are said be sign equivalent if there exist such that Equations (8.4) hold. Clearly, sign equivalence an equivalence relation. denote the equivalence class of the pair denote the collection of these equivalence sses. It is obvious that each consists a finite collection of pairs matrices. can now state the main (8.5) Then THEOREM. there exist matri ces with same orbit SC (F, element such that under acti of the Riccati group. Further, the assignment A [Q qs: D2 a well defined qs(dl) function = qs( from onto Finally, if and only for any belong same orbit under the Riccati Group. The above theorem gives a complete set of invariants in terms - 11 fl .1 -I- 2-- C ), m f j, =3-1 &r Eai) ' (E1' -t Qs: " " -- " rr 11 17 * L- Thus we see that Theorems (8.1) (8.5) solve the problem of invariant s distinct of these two the action o canonical Kronecker cases. forms for the indices. Analy the orthogonal group and cases neral neral of equal case case the "affine Kronecker indices a "combination" requires action" combining However, seems very difficult to formulate any general results along this direction. entirely differ In the next ent approach which chapter, also leads we shall consider invariants canonical forms, runs into analo gous diffi culties. Infinite Time-Zero Terminal State Case-- discus the action in Chapter the Riccati this problem arises group when on quintuples we consider In this section, we restrict our attention the subset data space which corres ponds to optimization problems with equa Kronecker luciA ces. have not been able to extend techniques this section neral case obtain invariant ca nonica forms. the next chapter, we shall reexamine this blem om an entire different point view which leads invari ants and canonical forms. Let us define be the subset .*- ((F, : the ordered Kronecker indices are equal). We start choosing a Hurwitz X(s) = sr polynomial - als - t2s example oose -(S + 1)r Given a reachable pair with Kronecker indices - n/m there st a triple Pc i i group such that T-1T -+ TGL + T GL = T-GV T GV -G where is given -IX &, r m r-1 m CaI' 1 i =: F + GL. It is easy verify that the chara cteri stic polynomial and hence the eigenvalues are in the left half plane now have a preliminary (9.E1 LEMMA. A triple belongs to the stabilizer of the feedback group if and = a nonsingular mXm only matrix, = diag L = 0. PROOF. 971 + T-1GL = + TGL = T-'GV then T-1T T+ TG *LT + T-1G L= F +GL. It follows that T~F + T-1 + T G +L - VL ] , L) +LT a Corollary SG ) G) (5.4), if and belongs -VL ) (T, only to the stabili belongs +LT a to the -VL) zer if and only stabilizer belongs to the stabilizer = an mx m nonsingular matrix, F However, belongs = diag * ri then the stabilizer LT = cc if and VL . a; Thus, only = an nonsingular matrix, = diag The above lemma shows that abili zer the feedback group dual Krone cker canonic al form , G ) same as the stabilizer of the feedback group Returning to the consideration of the Riccati group action DS1, we first have the following (9.2) PROPOSITION. transformed to a quintuple quintuple (] of the form can be Further, . belong belong to the same orbit under the Riccati group if and only if there exists orthogonal matrix such that V') sV diag PROOF. Choose a triple such that -ITT + T-1GL = -'cv *C =0 Sbe such that ' Q, Sl ) Since matrix 1 is positive such definite, that there = 1. exists mym nonsingular With = diag we have _ S2 t &t V~ some Since the eigenvalues are in the left half there plane, exists it is well known in the theory n x n symmetric matrix of Liapunov such equation that that i )'M + = Qn. - - Then 4lr((F = (F ,Q2, , 0, - Mi). Combining all the transformations above, there exists the Riccati group such that - (F where - MG. This proves the first part of the propo sition. Now, , I, belong same orbit under if and. only if there exists such that (9.3) PT 1 T + T-1G L=F (9.4) -1 * T GV (9.5) T'((F )'M + S MF )T + T'(S1 + MG)L + L'(S1 + MG)'T + L'L= (9.6) + MG) + L') (9.7) = I. Equations (9.5) (9.4) are equivalent = diag (V, * ,M)) JI( (F, = 0. Again, since it follows the eigenvalues from the theory are in the apunov equation open that left half M= O. plane, Finally, equivalent north nal. Thus, Equations (9.3) through (9.7) T'S are equivalent = S2 some ortho onal matrix = diag ..., V) .1 ence, belong same orbit under if and only if there exists an ortho onal matrix such that diag V')sV see that the problem of Ric cati group ac tion on i esse ntially reduces to the action the ortho gonal group on a space matrices. can now apply res ults of S section t a complete set of invariants canonical forms for the action Given matrix we partition sl]. r denote r- tuple mx m matrices can now state the following (9.8) THEOREM. be in Then there exists nx m matrix such that , R, (F , O, same orbit under action of the Riccati group. TRS[(F, denote Then TRS: = S2 -- [S, I, 'M+ t "') H TR PROOF. The theorem follows immediately from Proposition (9.2) and (7.1). We note that using the techniques outlined at the end Section we can obtain certain dense canoni open forms under the ortho onal group subs The idea of using Liapunov equation generalizes to finite time problems as well. We state the result A =(F, (9.9) THEOREM. be in and let an arbitrary = (F, Hurwitz polynomial egree orbit Then under there such exists that charact eristic polynomial PROOF. mXm nonsingular matrix such that A V'RV =1. By well known results on state-feedback, there exists m xn matrix such that -GL) = X(s). ROSENBROCK [1970] for a proof of this.) It now follows from the definition that A - (F, It follows from the theory of Liapunov equation that there exists n symmetric matrix such that cp(d, -(F, some For the forms case of equal invariants Kronecker following the t indict technique s we can obtain developed canonical in this chapter. Tn nava-tno1 4 a i-~ n4 rt~ ~' A r~ e n4 lnA 4 r OOnnn' flll-tol =: d fn nnnonal kn~.tMtIIYI I-< Gin; I Fnnmn Cn nk n: nlrn nnC fnn~ C CHAPTER ACTION OF THE RICCATI GROUP--II In this This invariants approach chapter , we shall canonical is based. on the develop an alternative forms asso for the action ciat ed algebraic approach of the Riccati Riccati obtain- group. equation. Under symmetric closed fo cases in the feedba assumption solution, rm results equal and neral ck group case plays that this algebraic we derive Riccati invariants for invariants strictly unequal arise a very due to equation has canonical canonical Kronecker the fact critical role that - jus indi forms ces. forms. only f a real We get or the The difficulties the stabilizer t as in Chapter in the This stabilizer it difficult subgroup, in general, formulate appears neral results very complicated making on invariants. Some results which are di of this chapter scusse have d in the next direct chapter syst theoretic appli cations The Algebraic Riccati Equation Approach start y defining certain subsets of the optimization data spaces quintuple recall that algebraic (10.1) Riccati + PF + Q equation (ARE) - (PG = 0. -l~pG+ define the ARE with has a real (10. 1) symmetric associated solution). In an analogous way, we have * ((F the ARE (10.1) has a real symmetric solution). -t Ir II 0 a* - t S) n 11 rm 1 ir I I (10.2) belongs Riccati PROPOSITION. if and group only := (F, if there exists be ih -- (T, Then in the such that cp(d, -= (F, , 0, Po ' o some such that the eigenvalues are in the closed left half plane. PROOF. symmetric exists solution. a unique belongs Furth solution ermore such then it i that known the ARE that (10.1) in this a real case there with - 1(MG + -RI(MG+ + GL) has eigenvalues in the closed left half plane. (See WILLEMS [1971, Theorem Then F' M+ ME' + MG)L + MG)' + L'RL, + F'M + MF - (MG + $146. + and also + MG) + L'R = 0. Therefore, with T = I, =1, we have cp(d,(T, M)) =(F, where + GL = - GR-1MG + -O R (MU+- - M. Finally, was chosen such that the roots are in the closed &, L, + L'( r Lt R, G, & + some A r, R, Then definition we have (10.3) = T'(Q + F'M + MF)T+ + MG)L + L'(S +MG) T + L'RL = 0, (10.4) Since = (T'(S + is nonsingular + L'R)V , (10.4) = 0. implies (10.5) = -T'( + MG)R Substituting = T'(Q into (10.3), + F'M + we have + MG)R + MG)'T - T'(S + MG)R + MG) T+T + MG)R + F'M + - (S + MG)R + MG) ')T. nonsingular, it follows that a solution to the ARE (10.1). Hence, belongs The above group Riccati propos interpretation equation ition may regarded as of the condition be solvable. We shall that the Riccati assoc consider transformation iated algebraic consequences o this result problems on spec tral as developed rece eory ntly of linear-quadratic JONCKHEERE optimal S ILVERMAN control [197 Thus for each in the orbit such there that exists the ei a sixtuple values are in the closed left of the Riccati half plane group on Therefore we may re for the analysis strict our attention the action to elements of the form such that the roots are in the closed left half plane. The conditions under which any two such s ixtuDles belong to same orbit under are Given in the following =T'( R, Po) h ~ (F, (P (d, (lo. 5) MF)T T' +MG) = (F2, and 1 ---- same nonsingular be ih are orbit matri in the closed under x T left if and and an hand. only mx m be such plane. if there nonsingular that Then the eigenvalues belong exists matrix such that T-'FT (10.7) -1 T G V V'R V = G2' = R2 PROOF (10.7) holds. Suppose there It then follows exist from nonsingular matrices the definition such that that cp((F1, ol~' = (F2, 02) Conversely, suppose there exists = (T, such that cP(d.1, Then we have V'R V Furthermore, T' (FM + MF1) T + T'MG L 1 + L' GMT 1 + L'R L = 1 + L'R )V 1 = 0. is nonsingular, it follows that - 2'MG B Consequently T'(F{M + - MGIR 'GtIYT = is nonsingular, satisfies F'M + 1 - MG R 1 -l GM = F~3' d2' (T'MG1 As the e nvalue s are the left half plane, it follows that eige value s - GR 1 SG;M are also in the closed left half plane. Thus, a real symmetric solution to the ARE (10.1) associate with such the do sed left in the closed left half half that plane. plane, the ei Since values the eig G1 -1 G'M 1 "1 envalues n x n matrix consi are are also sting zeroes also one such solution. However , by uniqueness such a solution WILLEMS [1971, Theorem we must have = 0. Consequently, Thus, T- F1T 1 - F2, -1 T G V 1 V'R V 1 = R2. thus see that every element can be brought into the form a suitable such a form is unique up state element space the Riccati transformations group. (GL(n)) Further, the input space transformati ons (GL(m)). We will now show that these actions are intimately related with stabilizer of the feedback group. We state the result in the following (10.8) PROPOSITION. be a reachable pair. Then there nonsingular matrices such that T-1FT - G L, some n matrix Furthermore, m x n matrices there exist nonsingular matrices such that (10.9) *-1F V (F *1) T - G L1)T - G L2, ) -1 GV 2 ) G V if and only if there exists such that , L ) belongs the stabili zer in the feedback group *G ,G) *-l1 * ) (L1T + L*). see = G *E G2' t R1' T ol -1GV T-FT Defining + -1GL + T GL5 := V-1L , j* T1TT = F* T-1GV we have - G L, T-1V T GV Equations hold (T ) ) if and. *(V G (V only if - LT ) *)- 1 *v* ') G V Now, (10.10) holds stabilizer holds in the if and only if if and o feedback there nly group exists SV* L V L ) LT )* belongs Equivalently, to the (10.9) in the stabilizer the feedback group such that * - LLT V L2 = L * -1 * ) (TTT *+ ). + L). us now return to the consideration of the action of the Riccati group Using the above propos ition, we now have following (10.11) exist n are in THEOREM. Let , R, natrices the closed left A = (F, such o - half G, Q, that plane A A 4- the eige and there be in Then value s exists exists 7 such there - GL that S(FL, , 0, PO). 0 Further, ,' P * = (F ) in -GL, 11' such 0, that = (F2 eigenvalues C- ~I' 3 - G2 j 2L - GL1 * (10.10) = F =* cA + (T .. r " L L - F2 1= G2 = G ), there exists in the stabilizer the fe edback group , G ) such that (10.12) (10.13) )'RV R V ) p (10.14) PROOF. = (F, that position (10.2), belongs there exist P under under to the orbit such Further , by Propo sition(10.8) there exist nonsingular matrices such that T-TT - G L, T GVv Hence, -= (FL' , 0, where = V'RV = T'P T. O This proves the first part of the theorem. under Propos if and ition(l0.6) only if there exist belong nonsingular to the same matrices orbit such that -1 * T (F1 - G L)T 1 1 (10.15) -1 * T G V 1 T' PolT V'R V 1 R2 Hence (F - Z the order 2L:2 Kronecker are indices same. Hence X GL 1 121 = F2 = F Further, Proposition "I n(10.8) Equation (10.15) holds if and only *c -X- + *), -(LT T 01 = G = 2 L=: F (V (T = G2 (P(d, *) -1( * = (v ) (L1T S'RV o2 T )'P The above theorem shows that problem of Riccati group action reduced to the action of the stabilizer of the feedback group on matrices scuss ed in Chapter stabilizer depends only on the Kronecker indices. This leads us to study the problem for various equal Kronecker indi choices ces of Kronecker the simplest indices. to analyze Again, case as in Sections Following case the techniques strictly unequal developed Kronecker in Section indices. we can also In fact treat the results here closely parallel results of Sections Before analyzing special cases, we have following COROLLARY. (lo.i6 there exist h .. -Ci, a matrices such A that be in Then - F* := F the eigenvalues -GL in the closed left half plane and := (FL, belongs to the orbit PROOF. Note that ordered Kronecker indi lower belongs triangular to the choose Q = S L- such that matrix stabilizer there , G ). - G L , the eigenvalues are exists By Th such eorem that (10.11), we may for an appropriate in the left half plane. choose a lower triangular matrix such that V ARV* V 'RV =1. Then , 0, - (FL' + L ), can are Po) , cp((~ L= (V *) LT Noting that -1^ T TT = FL, the result follows. We start with case of equal Kronecker indices. := ((F, the Kronecker indices are equal). Given any m xn matrix n xn matrix be partitioned L = [L where := n/m. Now, Theorem (10.11) for = (F1 DlR' there exist matrices such that = (F - G L2, , O, belongs matrix to the orbit such that V' R2V under be an mXm nonsingular = I. := diag Then it is easy see that p(d2, = (F , 0, R;1 Le], mRRd.2 := Tr(P where denote are following the the collection notation developed of matrices in Chapter I Sections now state main (10.17) THEOREM. TRR: ~~N Then a well defined function. Further, D1R' same TRR[ d] orbit = TR [d2] if and only belong under PROOF. the argument just before the theorem there exists an element under = (F under of the form - G L, = (F )' Po2) Further, G 0, - G L , I, if and only if there 'It 0, I, long exists every the the orbit same orbit stabilizer the feedback group such that (10.18) and (10.12) and (10.14) hold. Since the Kronecker indices *G , G) equal, it follows from Corollary (10.18) that mx m orthogonal matrix, = 0, = diag thus, only see that belong to the same orbit under = (V )'L1T w can *) 'V are v") G Le, where mxm orthogonal matrix and = diag V*). p ,I now follows from eorem (7.1) that a well defined function inR[a) ~ThR if and only belong to same orbit under This Esse theorem ntially, very similar the problem to Theorem same: (8.1) analyze of the previous action chapter. of the ortho onal group on a space of matrices. case strictly unequal Kronecker indices can be treat ed using the techniques developed in Section Intuitively speaking, plays the role plays the role Here we state the results without proofs since the proofs follow the pattern Section We start defining := ((F, the Kronecker indices ... K now have (10.19) THEOREM. -(Ps AQ A , Q, R -0 element D2R. Then there matri ces with such that eigenvalues -GL are in the closed left half plane = (F -GL , 0, belo the orbit under Riccati group. Further assignment A P d we [P a well Ap(dl) orbit defined Sp(d2) under the function. if and Riccati only Finally, belong to the D2R~ same group. Here we are following the notation of Section eore m (10.19) should compared with Theorem (8.5) Roughly speaking, in the approach are Qs : Theorems (8.5) (10.19) give a complete set of invariants in terms finite collection of matrices. We conclude this section with case of infinite time-zero terminal state problems. We start with (10.20) there THEOREM. exists mx n the eigenvalue s = (F1, matrix + G L) be in matrix and an are in the closed left DSR. Then such that half plane d= (F d = ( DS s R - + G L, belongs to the F1 uch -that t that t ;he eigenvalues = (F S2 - GL ) 1 same orbit - G2L2' and Further, )p 2, - G L ) 2 2 0) are in the closed left half plane, belong same orbit under only are if th same, ordered (hence, exists Kronecker indices , G in the stabilizer = G2 of the there feedback group SG ) such that )'R1V 1 = R2. -" 2/ = (V *1 -IT ) (LIT *+ + L). PROOF. Recall as it does observation, proof that except follows the Riccati that from group the last the proof term acts is omitted. With same this of Theorem (10.11) Again, unequal one can analyze Kronecker indices. the special The results cases are very equal and strictly similar to those obtained in Chapter Section We omit the details. =: F CHAPTER VI. SYSTEM THEORETIC APPLICATIONS In this chapter, we will examine some of the system theoretic applications interpretations of the Riccati transformation group the techniques developed in the last two chapters. The main application optimization of our results problems, dev is to the spectral eloped recently by theory of linear-quadratic JONCKHEERE and SILVERMAN [1978]. We will show that their results can be considerably extended simplified we will infima show using by using infinite the Riccati the Riccati time transformation group free terminal that group. the bounded state case In particular, ness is equivalent the positivity Hilbert space nx n operators which mmetric are matrix. central We will objects show that of study JONCKHEERE SILVERMAN [197 are not invariant under Riccati transformation group. We will examine the behavior of Popov's frequency function under the Riccati transformation group. We will also show that sitive real transfer functions the Riccati transformation group are closely related. 11. Spectral Theory of Linear Quadratic Optimal Control spec tral theory linear quadratic optimal control problems ently been developed JONCKHEERE and SILVERMAN [1978] relate the problems of boundedness of the infima to the spectra of certain Hilbert space operators. Here we will examine these problems the light of Riccati continuous-time versions transformation of the discret group. e-time will analyze problems inve stigated JONCKHEERE and SILVERMAN 1978] Our techniques and results apply equally well to the discrete-time problems. We start that we will precisely consider. defining the dynamic Following WILLEMS [1971 optimization ], we define problems the infinite time asso free ciated terminal with state = (F, linear quadratic optimization problem as follows. Infimize rec J (u(.)) -f o10 x' (t) + 2x'(t)Su(t) + u'(t)Ru(t) ]dt + x'(0) Pox(0), subject to the linear dx(t)/dt = Fx( system t) + Gu(t), lim x(t) t->--o over = 0, u(.) oo, O] := (f: o, 0] : f(t) I square integrable o, 0]}. Here that the infimum the boundedness taken the infima as the infimum of of infinite time limits It turns free terminal state problems of critical importance in the exist ence solutions the Riccati differe ntial equations. We state the rele vant results the following THEOREM. = (F, be in Then following statement s are equivalent. For the associate infinite ed with time free terminal state problem Jr(u(.)) where e the infimum is taken over The Riccati differential equation associated with (11.5) - dP(t)/dt = F'P(t) + P(t)F - (P(t)G + S)R -l(p(t)G + P(0) Lm(. in( m 2- S, (11. ,R" (ii i) For all the finite time linear quadratic optimization problem associate with has a solution. For all near quadratic the ( optimi zero initial nation state) blem assoc finite iated time with a solution. Sketch follows ANDERSON between esse of {i 1.) ntially and MOORE from 1971, (iii) "'>r UOiF' V ertn the class Section follows cal r results (iii) KALMAN details. rom the reach 1960 Equivalence ability implication -4(1) follows from a rivial modifi cation result of WILLEMS 1971, eore m 1] now prove finite time linear - (iv) . quadratic optimi Suppose nation that for blem some associate with does not have a solution. Then there exists a sequence Cu1 11j=~, ~i such that lim J(u. * ar 1n - 0 Define m O, fo (uit), : u^(t) Then belongs cc, o] for all Now with x(t) we have Hence, inf J (u(-)) - .CO This contradicts The result, thus, follows. .~~~-.~I .IA f I -t Jf(u(')) o0 F' 1 I Lmt _ 2 7(u( ._ J_ ^ mr r n n * I - of this section the general show that it is case both order natural and. necessary to t a satisfactory consider theory. In fact, it turns out that contains most critical information for the boundedness of infima. The next theorem shows that the existence of solutions to the various dynamic Riccati optimization problems remains invariant under the action group. (11.4) TIIEOBEM. = (F, be in = (F, be in in the Riccati group < 0. Then a solution to the infinite time free terminal state respectc tively, finite time, infinite time zero terminal state) linear-quadratic optimization problem as social with spec tively, exists if and only a solution to the same problem a associated with in a, (respectively, P(d, , 7)) exists. terminal PROOF. state It follows linear from quadratic Theorem (11.2) that optimization the infinite problem a associated time with free ,P0) has a solution if and only if the associated Ri ccati differ ntial equation (11.3) has a solution for all <0. However, P(t) with T' (P(t) with is the solution then - M)T cp(cl, differential it is to the Riccati easy check (by a solution Converse equation differential direct Riccati P(t) associate equation associate substitution) differential is the solution ed with the n that equation associated the Riccati ) -l(t)T- is the solution It now follows the Riccati from differential Theorem (11.2) that equation as the infinite time sociated free with terminal state problem as problem associate sociated ed with with cp(d, has a solution has a solution. if and only if the same The proofs for the finite time problems the infinite time zero terminal state problems are similar. I-nv 1- A.1..- -I... S <1 -1-.- ---A A--a --C O' a R, O a(d, CP d, I- -- I 1.1. -.1 ~.. _ - orbit This s imple observation allows us to extend and simplify the results of JONCKHEERE SILVERMAN 1978] as we will demonstrate in the following results. any point Then, Theorem (9.9) such of that := F Section - (F, +GL it follows that 'I, is asymptotically there belong stable. exist s to Hence matrices the orbit where in the light Theorems (11.2) and (ll.4), we , I, need. where only asympt consider otically points of stable. the form (Note that we have dropped from as the actual canonical form of does start not play with optimization an important the infinite problem time associate role in the following free terminal with state a six tuple development.) linear in the quadr form atic asymptotically stable. Then the associate cost criterion (11.5) = [012x -m (t)Su(t) + u' (t)Ru(t)]dt subject to the linear x(t) = Fx(t) system + Gu(t), lim x(t) t-4-- = 0. It is well known that the solution to the above equation is given x(t) F(t-to) (t == e 'ux(t t + fte o F(t-)Gu) i. Gut ) dr. is asymptotically stable, taking limit -4 0i we get (11.6) x(t) Substituting (11.6) into (11.5) we get = =(F, A , Q, S, Po) + x'(0)P x(0), J,(u()) =ItF(t-I) Gu(r)dr. where (11.7) Em(- c, 0], (11.8) (11.9) : L(-2 -emL 2 =1, (11.10) (pu) (t) (1. 11) (P2u)(t) := f w(t -CO - T)u(T)dT, (11.12) (p u) (t) -F'tp eo Po~e Gu(T) dT, w(t) SI * 0.1 0 FtG -F' tS > 0 <0 =- 0. 0, 0], let us define := fu'(t)v(t)dt. Then (11.14) in this Jf(u) notation, =
the Hilbert space operator defined above. We note that + P2 whereas a Wiener-Hopf Sis type a "Hankel like" operator (with operator. the "kernel" Thus, we have R6(t) a decompo + w(t) sition the sum of a Wiener-Hopf operator and a "Hankel like" operator. This decomposition corresponds to the decomposition obtained -Lp(- := u(t), m 2 '- P1 : = G' ~, ol, (11. 13) theoretic significance of these decompositions remains unclear. case, Hopf it is clear operator) from corresponds the definitions that to the integral the Wiener- quadratic criteria whereas Hankel like matrix natural case, tool to "account" operator) the final corresponds to final state weighing matrix for the so called "Hankel state weighing is the perturbation" (see JONCIGIEERE: and SILVERMAN [197 now have the following simple (11.15) LEMMA. a bounded linear self adjoint operator. PROOF. self adjoint. Clearly, Since is a linear asymptotic ally operator stable, a bounded it follows from definition that there exist such that I w(t) U It then follows from GOHBERG and KRE flN 1960, page that bounded operator. asymptotic ally stable we can define .- IIf~e * -O It is easy to check that Hence is also 5 a bounded operator. Since =' it follows from YOSIDA [1964, page 197] that self adjoint. Finally, p u> = 5~ v'(t)G' -F't t o Po '(T)G'e -F' pofO P^f . our , 0], o -FtGG' -F' tdtJ( t) Ifl L (- -P 'G Gut t) d~dt -Ft Gu(t)dtd.e Hence is self adjoint. Thus, is the sum of bounded linear self adjoint operators Hence, a bounded linear self adjoint operator. now analyze the boundedness the infima for infinite time free terminal state linear quadratic optimization problems in terms the Hilber t space operator In particular, we have (11.16) THEOREM. = (F, be in such that is asymptotically stable. Then the following statement are equivalent. The Riccati differential equation (11.5) associated with has a solution for < 0. The operator associated with is positive semi- definite. PROOF. Note that Jf(u) f , pu> where is a bounded linear self adjoint operator. Hence, standard re suits on th spectral eory self adjoint operators, if and only is positive semidefinite The conclusion follows from Theorem (11.2). This sets along the the sta lines ge for a detailed of JONCKHEERE and investigation S ILVERMAN on the spectrum [1978, 1980]. will not pursue this avenue here as that our main purpose. fact, a direct analysis JONCKHEERE and defining another SILVERMAN Hilbert of the spectrum of [1978, space 1980] overcome operator whose seems these spectrum be difficult. difficulties differs from that is not Pf only necessary zero. compute itself. However, in the light the spectrum of It may be easier to of Theorem ass oc elated (11.4), with check the positivity somS, some other point on the orbit a matter of fact, turns out to be the case. be in It follows directly from the results of WTTIJLEM. r1IQ7 . Theorems I. C *t I that it is necessary can S, h (F, h P o) to admit a real symmetric solution in order that the Riccati differential equation(11.5) associated with have a solution for < 0. However, solution, if the algebraic then Riccati equation (11.17) has we are in the setup of Chapter a real symmetric In particular, Corollary (10.16) stable, , O, (but implies that ) belongs not be there exist the orbit asymptotically matrices such stable). that such F : However, that + GL order us e stable. which give this the operator We will we need first the resulting operator section. theoretic restrict turns assume our attention out to interpretations that n to asymptoti of these asymptotically those cally points stable. consideration We will later = (F, be in such that asymptotically stable. It follows directly from our previous results that the operator in this case (11.18) =PL where are as in(l. and (11.12). We will now show that itive semidefinite (resp ectively, definite), if and only a certain definite). n symmetric e main result matrix based itive on the semidefinite following respectively, simple (11.19) LEMMW. be any two Hilbert spaces a: H1 two bounded linear operators. be the identity complex number 1, elongs operators. nectrum of Then, only belon to the spec trum of +c43. PROOF - 1) a complex -1 Now, Now, number suppose in the spectrum of does not belong t the spectrum of +c43a. Then -1)1 an invertible operator. It follows that is also -A invertible. is easy F i H2 -~ HL H1 + ga -, H1 " H2 -t Bcx. - ag GcrB But this implies that - 1) I2 is invertible. This contradicts fact is in spec trum + g2 Hence, must belong ctrum of czf3. converse follows a symmetrical argument. We now return the operator to the analysis given in (11.12) of the operator can be written Notice that (11.20) = P4P5 ' where (11.21) : u -Po0 e Ic-0 -Ft Gu(t)dt, : Rn (11.22) i( 2(- 03 0 x p x, x)(t) := G'e -F't We now apply the result of Lemma (11.19) to check the positivity (11.25) THEOREM. = (F, , 0, be in such that is asymptotically stable. Define the matrix -FtGGf (11.21*) Then, the operator PfK associate with positive semidefinit e (resp actively, definite if and only if the eil values + P 0 are nonnegative (resp actively, itive). PROOF. Note that is the identity operator where are given by (11.21), (11.2 asymptotically now apply stable, the result are of Lemma (11.19) bounded linear . It follows operators. that (real) can number belongs the spectrum of if and only if it belongs spectrum of -F't dt. n := 10 03 Ba PqP However, -FtGG eGG'e (p5%) (x) -.F't = PoiX. Consequently, the spectrum consi of the eige values of the matrix + P modulo Hence sitive semidefinite (respectively, are nonnegative definite respect if and tively, only if the eigenvalues positive). Thus, assoc neces we see that ated. sary and with suffi in order a six tuple ent to check = (F, check the the positivity the operator it is positivity the eigenvalues + P T. 0 Even though are symmetric matrices may not a symmetric matrix. Therefore, a prior it is not obvious that + P7 real itive eige values definite. . Note Further, that reachabilit we may write +Po 0 + Po). Thus, a pos + P7 1 0 itive can be written as definite matrix. a product can now use a symmetric the classic matrix cal results regular pencils GANTMACHER [1959, quadratic Chapter forms. The reader Section referred In particular, we have (11.25) PROPOSITION. n Xn real symmetric matrices itive nonnegati symmetric (resp matrix definite. actively + Po) Then, itive' eige if and positive values only + P T) if the semidefinite n real efinit PROOF. We can write + Po7) +Pw) + Po7T) +pw)* o) (n (TT it follows that from GANTMACHER the eigenvalues [1959, + Pw 7) Chapter Section are nonnegative Theorem 10] (respectively, positive) if and only positive semidefinite (definite). Note that +P~) a real symmetric matrix. now summarize some of the main results section the following (11.26) Riccati for all a real THEOREM. differential only symmetric equation(11.5) a if the solution. assoc Now, ssoc lated suppose -o) lated. algebraic that be in with Riccati the algebraic Then as a solution equation has Riccati equation exist orbit then associated such the Riccati for all with that is stable. differ if and positive ential only a rea symmetric = (F, case equation if the semidefinite solution. ) belongs asymptoti associated with d n real symmetric cally Then there to the stable, has a solution matrix where 7T *= -F'tt r dt. Some Ao) remarks are in order here The trans mainly r formation requires A , Q, from a real symmetric solution a pos proce there itive dure algebraic definite find. are several Riccati matrix. solutions efficient equation course, to the there algebraic algorithms finding a s no known Riccati actually square finite equation. compute root However, the stabilizing solution t completely the algebraic clarify and Riccati equation. extend the invest case, tion started our results JONCKHEERE S ILVERMAN [1978]. We note that our condition that be asymptotically stable is sati sfied those points generically D for which the data the algebraic space Riccati corresponding equation has a real symmetric solution. Aq - 'I l~ a % -f .I.-- --- !-. -a r a -i4 (~, hG 'FtGG' ?,C ,,,, hLnt: nn f\ - ,,, L,,1 L1,,,,, L,' _. control KALMAN, NARENDRA~ [1963]. We now proceed to describe these considerations. = (F, be in consider the zero initial state) finite time linear quadratic optimization problem a associated with = 0, (for < 0). Minimize u'(t)u(t)dt subject to the linear system dx(t)/dt = Fx(t) + Gu(t), = 0. Clearly, energy"' state x the first whereas th term in the e second t erm Now, cost criterion corres then ponds the solut corresponds a weighing ion to this to "control on the final problem, obviously, using the then results on minimal energy control above problem as developed can be KALMAN solved , HO, NAREITDRA tJ196 Let us define, for each 7(t) -F-rGG -F' d drt. Since reachable it follows that w(t) itive definite for all Consider an arbitrary the set U(t, of input but fixed functions vector Ll[t, such that tran time with sfers Then the state at time system from Corollary the state Section zero to the state KALMAN, and NARENDRA implies that u' ()u(T)dT can now state the following (11.27) THEOREM. be a reachable pair. Then + x'(0)P x(0), = x wT(t) J(u( :(0) w lim r(t) - t- -co exists. is asymptotically stable -1 = 7v then PROOF. be any define vector arbitrary but fixed Lm[t ->Rm A u(t) evident that for each A and u U(tl, L(t) xo) = u(t) U(t elongs It Xo) follows that mrn U(tin X U(t2,xo) ^2 Now, from the result of KALMAN, and NARENDRA [1965, Corollary Section quoted above, we have x'o(t 0 < xot(t ) above inequality holds for all Since 7(t) positive definite for all it follows that X'o(t) 0 -1 X 0 Thus, X'Tn(t)- 0 a monotonically increasing function bounded below Therefore lim x'(Trt) t-a--o0 exists each Consequently, - I_ \ -l now U(tl,xo U(t,, u (( 't)U( T) dr u'( Z)U( T)dT. is asymptotically stable, then by the definition lim w(t) t.-*-oo = 7> 0. elementary results of analysis, we may conclude that lim r( t) - t->-co =77 now ready to state our main (11.28) solution "a THEOREM. to the Riccati differential equation be in exists Then, for all only if the algebraic Riccati equation assoc lated with solution. Suppose a solution. belongs that algebraic Then there o the orbit Riccati equation a exist such Furth ermore, ssoc that the Ri iated = (F, with ccati differential if and only equation nxn ass ociated real with mmetric a solution matrix for all itive semi- definite PROOF. (11. . The exist The first part ence the theorem such that follows from Theorem , I, longs the orbit follows from Corollar .= (F, G, y (10.16) Chapter It now follows from Theorem (11.4) that the Riccati differential equation associated with has a solution for all if and only if the Riccati differential equation associate ed with has a solution for Now, asso ciated with Theor em (11.2), has a solution Riccati differe for all ntial only equation if for all <0 , the ( zero initial state) finite time linear quadratic optimi zation problem a ssoc iated with a solution. The cost criteria this case = fou 1('r)U()d'r x'(0)Pox(0). are (F t G, t, J(u() inf ueL*[t,0] J(u( jnfb xc:Rn x' (7(t) Hence, this finite time optimization problem has a solution if and only (T(t) -1 itive semidefinite. Theorem (11.27) implies that <)-1 or -- 1` Thus , again y Theorem(11.27), (r( t) positive semidefinite for all if and. only itive semidefinite. Conse quently, the Riccati diff erential equation assoc lated with a solution particular, if and only asymptotic ally itive stable semidefinite then Note that Theorems (11.27) and (11.28) their imply the results Theor em(ll. 26). Roughly speaking, we conclude that Riccati differential equation Riccati decomposes equation (o into parts equivalently, one part Wie ner-Hopf consi operator algebraic the other part consists minimum energy control with final state weighing. The conditions well it is understood. quite easy solvability of e, for example, check positive algebraic WILIJEMS~ Riccati [1971]. equation are On the semidefiniteness other hand, nx n symmetric matrix. This can done for example, y computing all principal minors. (11.29) before REMARK. that has a solution the Riccati = (F, differ for all be in ential only equation (11. ) if the algebraic It was noted associated with Riccati equation associated natural op with erator has a real theoretic symmetric interpretation. solution. This result Here we will describe relevant the lines results without of ,TJONCKHEREP proofs. The proofs andf RTT.VEPRMAN r o7 ( I I be constructed along '.iiin.a the standard techniques )x. o- where is the is asymptotically sum of stable. Then, iven associated by (11.10) - (11.12). a Wiener-Hopf can be class. that Therefore that Hopf shown order rator. that Using is a complete 5 prov techniques is contain that be positive operators, (for whi itive from stability operator perturbation ctrum it is semidefinite semidefinite. see GOHBERG the theory KBETIN 1960] the trace theory necessary of Wiener- ROSENBLUM the spectrum + p2) given values "frequ ency function" (taME - F)-1G - F') -Is, for all Hence, is positive semidefinite if and only for all well known equation as equival sociated nonnega ent to with tivit the solvability thus see that algebraic order that itive Riccati semidefinite equation have it is necess that assoc a solution. We close this section examining the behavior of Popov' frequency domain function + S'(sI -F)-1G + G'(- - F')-S - F) associated with = (F, under the action of the Riccati group. be ih := (d, operator can then spectrum of the asymptotic continuous in the for all Riccati iated algebraic + G'(- - F') Q( := R + Pe) + Pe) i P2) ~( W) + G' ~(W) is (LU) T, b (11.50) =v( -' sl- F')-1L') ( s)(I - L(sI - F)-1G)V. case, the algebraic Riccati equation associated with has a real symmetric solution, then there exist matrices such that -(F, =1. Equation (11.30) which shows that is well known this corresponds to a spec be closely related tral factor to algebraic zation Riccati equation. 12. Other Applications This section relationship mainly between intended Riccati tran o show that formation there a close group and itive real matrices. show that so called "positive real lemma" rederiv obtain a Finally, Riccati ed from complete we make differential the Riccati transformation parametrization some remarks on the equation scalar fast CASTI gr oup point itive algorithms tJ-97h23 view. real We also functions. for solvin KAILA2JI [19731 z(s) Z(s) said to mx m sitive matrix real rational functions. Then real elements z(s) are analytic in the region Re(s) of the complex plane, the matrix itive semidefinite for all complex numbers such that where the complex conjugate transpose Z(s). The property of positive re alness turns out to be critical importance network sive network theoretic aspects synthesis. positive NEWCOMB real functions.) 1966] Using for the certain preliminary synthesis steps, (see ANDERSON VONGPANILTERD + z( "i1(9) t 0) "acs, z(oo) := lim Z(s) S~-- O transfer function. analytic exists. Further In other words, z(cm) + z(co) a proper is nonsingular. Since assumptions rational, are easily using seen elementary r e equivalent ealization theory existence the above of matrices of appropriate sizes such that + H(sI where ca non nonsingular asymptomati call stable. In the light of these iderations we may restate the problem as f follows. nx nx m mx m real matrices respect vely canonical, asympt otically stable nons ingular. Find conditions such that (12.1) := J + J' + H(sl - + G + c'(s I-F') >0, for all the complex plane such that It is well known that half in thi plane case itive semidefiniteness the cl osed right repla (12.2) >0, for all (12.1) hold Then defi" t ha t a necessary ujte However condition notice that the P . Hence, opoy frequency a resu It diue function assoc to WTLLEM ; ated. 1971, with eorem we see that is pos itive real only itive definite the algebraic c. ati equation associated with a real symnie trick solution. now we are in the up of Chapter In particular, proo Proposition(10.2) implies that algebraic a real Riccati symmetric equation solution associate if and with if there = (F, exists 1, , H') , H') =I R, iw) for half some plane. such Rewriting that (12. the eigenvalues explicitly, are in the left we get CJ2A4) (12.5) (12.6) V'RV F'M + + J')V MF = L'RL, iG = -L'R, (12.7) + GL, Equations (12.4) - (12.7) are easily seen to be equivalent to the called "P05 itive real and ANDERSON lemma" 1967]. equations This shows due to YAKUBOVIC that the positive [1962], real KALMAN lemma is closely related the Riccati transformation group. now consider spec case of scalar positive real functions. In this case be normal 1/2. Thus, (12.8) = 1/2 + h(sI - F)-1 Further, control without canonical loss of generality be taken to be in form: (12.9) Then the characteristic polynomial (12.10) ~s) := det (sI -F) =3 - a' now consider the following: problem. . ea I, + M a,, ,,,, a U~ U V'(J * I such matrice Choose that n vector are in the (close left given such half plane (12.8) that is positive values be the unique real. + gt) solution to the Liapunov equation (12.11) F'M + t vi. stence uniqueness or M F'T M = f e e follows from asymptotic stability FTt.) fact, define (12.12) It is easy to verify that = 1/2 + h/(sI a positive real function. can now state the following (12.15) THEOREM. defined fixed by (12.9), be asymptotically real numbers, stable. Define the set * - *( = 1/2 + h(sI is positive real). be the set the eigenvalues are in the (cbos left half plane). Then -4: a-H*1> a bijection. -t gR Ixn R PROOF. From the arguments iven above, it is clear that well defined. Suppose belongs Then there exist matrices such that F'M + IvF = 2%, the eigenvalues are in the closed left half plane. Hence belongs This proves that surjective. suppose that there exist such that h (29 hA -- h( = h. The n there exist symmetric matrices such that - 'Mi =- i, F'M. + M.F 1 = V2 iei, = i, Hence, are solutions to the algebraic Riccati equation F'P + + h') = 0, such that the eigenvalues both are in the closed right half plane. However , by WILLEMS 1971, eorem such a solution is unique, and, ence, . This proves that ective. This note that theorem completely answers the main step in obtaining a the question rais formula ed above. involves solution of the (linear) Liapunov equation (12.11). Using Kronecker e2 he' closed Combining this form expression formula with (12.12), we can as a subset obtain a is the set such that the eigenvalues + gi are in the closed left half plane. Using the classical stability theory and Hurwitz determinants, can be described by a system of polynomial inequalities. (See, such also that KAL~MAN (h2, [1979].) We note that there is not observable exist one such choice. Define is observable). := (2 Then, - g'M, belongs F) some if and observable. suitable belongs only Thus, polynomial be describe ed as a subset inequalities. We close this section with a remark on the fast algorithms solving [1974]. differential Riccati differential equation = (F, equation (11.3) associated due to KAILATH be in with [197 Consider The main CASTI the Riccati observation KAILATH [1973 ] CASTI [1974] is that r := rank = rank + P F + Q o - (PoG + 0 then the Riccati nonlinear differe differ ntial ential equations. equation may be replaced In particular, by a set of much smaller It turns Riccati further than out that tran then this formation reduction this rank group. Riccati eads to a considerable invariant This shows equation along that under saving of computation. action one cannot the lines of the achieve of KAILATH t 197 3) CASTI [1974] using Riccati transformation group. However, completely satisfactory explanation of this reduction still remains elusive. REFEENCES B. D. O. ANDERSON 1967] "A system theory criterion for positive real matrices" SIAM J. Control, 171-182. B. D . O. ANDERSON B. MOORE [1971] Linear Optimal Control, Prentice-Hall, Englewood Cliffs, B. D 0. ANDERSON S. VONGPANITLERD [1975] Network Analysis Synthesis , Prentice-Hall, Englewood Cliffs, P. BERNHARD and G. COHEN [197 "Etude d'une problem reduction function command de la t quentielle optimal avec int une ce problem" ervenant dans application a Revue RAIRO, July R. W. pages BROCKETT CJ1970) Finite Dimensional Linear Systems, Wiley, York. 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Transactions solutions to linear on Automatic matrix Control, . MUMFORD V. M. POPOV 1964] "Hyperst several Elektrotek. ability and optimality control functions". Re et Energ., of automatic V. Roum. 9-6~ Sci. systems with Tech., 1972] "Invariant description of linear time- invariant controllable systems" SIAM J. Control, 2-264. [19735] Hyperstability of Control Systems, Springer, Heidelberg. C. PROCESS "The [197 invariant Mathematics , 19: eory 306- n x n matrices" Advances 381. ROSENBLUM "A concrete operators", spectral American theory for self Iv~zth ema tic s adjoint Toeplitz 87: 709-718. H. H. ROSENBROCK 1970] State Space and Multivariable Theory, Nelson, London. SIBIRSKII [1967] "On unitary Dokl., 8: 5 and -40. orthogonal invariants" Soviet Math. M. SILVERMAN "Discr asymptotic Riccati equations properties, : alternative system Control and Dynamical ems, rneory Vol. 12 algorithms, interpretations" (edited by Leondes Academic Press York. A. TANNENBAUM 1980] Invariance and Sys Aspects, to appear Mathematics. 81 oa 4- ory: Algebraic in the Springer Lecture Geometric Notes ges S. H. WANG and E. J. DAVISON I WILLEMS tJ197Z2- "Least Riccati AC-16: squares equation 621-634. stationary optimal co n", IEEE Transactions ntrol on Automa the algebraic tic Control, P1-981) Linear Optimal Control and Filtering, forthcoming. W. M. WONHAM [1974] Linear Lecture Multivariable Notes Control: in Economics A Geometric Approach, Mathematical Systems , Springer Berlin. YAKUBOVIC [1962] "The control solution theory" of certain matrix Dokl. Akad. inequalities Nauk, SSSR, 1)45: in automatic 1304-1307. K. YOSIDA 1965] Functional Analysis, Springer, Berlin. BIOGRAPHICAL SKETCH Pramod Prabhakar KHARGONEKAR was born on August ,19 in Indore, INDIA, Bachej&: to Prabhakar KH r of Technology ARGONEKAR degree and Leela K electrical HARGONEKAR. engineering He obtained from the Indian Institute of Technology, Bombay in 1977. He obtained his Master Science degree in mathematics in 1980 from the University of Florida. certify that conforms adequate, I have to acceptable in scope read standards quality, this study and scholarly that prese as a dissertation in my opinion nation and fully the degree Doctor of Philosophy. Dr. R. Graduate Electri E. Kalman, Rese arch Chairman Professor cal Engineering certify that conforms adequate, to acceptable scope have read standards quality, study an scholarly as a di sser that pres station in my entation for the opinion and fully gree Doctor of Philosophy. Dr. T. E Bullock, Co-Chairman Prof essor of Electrical Engineering certify that have read this study that in my opinion conforms adequate, to acceptable in scope standards quality, of scholarly pre as a dissertation sentation fully for the degree Doctor of Philosophy. Dr. R; L. Long Assistant Professor of Math tics certify that conforms adequate, to acceptable in scope and have read standards quality, this study an of scholarly .that prese as a dissertation my opinion nation and fully for the degree Doctor of Philosophy. ' 4 4- Arl me. v~ R& ~La/~L certify that conforms adequate, to acceptable scope have read standards quality, this study that of scholarly pre as a dissertation se in my nation opinion for the degree fully of Doctor of Philosophy. C/ tfL2 V. Shaffer Prof essor of Electrical Engineering This dissertation of Engineering and was submitted was accepted to the Graduate as partial Faculty fulfillment of the College of the requirements for the degree of Doctor of Philosophy August, 1981. Dean, Dean, College Graduate of Engineering School I ,~kk/ e~c-~ UNIVERSITY OF FLORIDA 1262 08394 207 7 |

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