Napp, Diego

Trentelman, H.L.

English

This paper deals with systems described by constant coefficient linear partial differential equations (n-D Systems) from a behavioral point of view. The performance functional is the integral of a quadratic differential form. In this case an appropriate tool to express quadratic functionals are 2-n variable polynomial matrices. We look for a characterization of the set of stationary trajectories and for the set of local minimal trajectories with respect to compact support variations, turning out that they are equal if the system is dissipative. Finally we implement these trajectories of the given plant as a (regular) interconnection of the plant and a controller

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   University of GroningenThe Behavioral LQ-problem for linear nD systemsNapp, Diego; Trentelman, H.L.Published in:EPRINTS-BOOK-TITLEIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Napp, D., & Trentelman, H. L. (2005). The Behavioral LQ-problem for linear nD systems. In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022The Behavioral LQ-problem for linear nD systemsDiego Napp and H.L. TrentelmanAbstract— This paper deals with systems described by constantcoefficient linear partial differential equations (n-D Systems)from a behavioral point of view. The performance functionalis the integral of a quadratic differential form. In this casean appropriate tool to express quadratic functionals are 2-nvariable polynomial matrices. We look for a characterization ofthe set of stationary trajectories and for the set of local minimaltrajectories with respect to compact support variations, turningout that they are equal if the system is dissipative. Finally weimplement these trajectories of the given plant as a (regular)interconnection of the plant and a controller.I. INTRODUCTIONGiven the plant and a quadratic differential form (in thefollowing abbreviated with QDF), the LQ-problem treatedin this paper, is to characterize the trajectories of the plantthat are stationary or optimal with respect to the integral ofthe QDF.The linear quadratic (LQ) control was initially developedfor input/state/output system with a performance functionalgiven by an integral of the input and the state with one inde-pendent variable (usually time). However, often the systemsdo not have a clear input/state/output structure, may containhigh order derivatives, or the cost may have derivatives inthe control variables. Moreover, many if not most of themodels of physical systems involve both time and spacevariables. The purpose of this paper is to approach the LQ-problem considering systems described by partial differentialequations without assuming any special representation (be-havioral approach) and considering the cost as an integral ofan arbitrary quadratic expression in the system variables andtheir derivatives.With this approach we can deal with systems that are not instate space form, and their dynamics depend on both timeand space. The main result of the first part of this paper canbe summarized as follows. We find an explicit representationof the behavior of the stationary trajectories and we provethat the set of local minimum trajectories are actually thesame set if the system is dissipative or empty otherwise.In the second part the so called synthesis problem is ad-dressed i.e. find an n-D system called the controller thatconstrains the plant behavior through a distinguished set ofvariables, namely, the control variables, in order to imple-ment the optimal trajectories. A representation of such acontroller is found.Research Institute for Mathematics and Computer Science, P.O. Box800, 9700 AV Groningen, The Netherlands diego@math.rug.nl,H.L.Trentelman@math.rug.nlII. MULTIDIMENSIONAL SYSTEMSIn behavioral system theory, the behavior is a subset ofthe space WT consisting of all trajectories from T, theindexing set, to W, the signal space. In this paper weconsider systems with T = Rn (from which the terminology“nD-system” derives) and W = Rw. We call B a lineardifferential nD behavior if it is the solution set of a systemof linear, constant-coefficient partial differential equations;more precisely, if B is the subset of C∞(Rn, Rw) consistingof all solutions toR(ddx)w = 0 (1)where R is a polynomial matrix in n indeterminates ξi,i = 1, . . . , n, and ddx = (∂∂x1, . . . , ∂∂xn ). We call (1) akernel representation of B and write B=ker(R). We denotethe set consisting of all linear differential nD-systems withw external variables by Lwn. However, there are many otherways to represent an n-D system. One is using some auxiliaryvariables, called latent variables, that appear in order toexpress basic physical laws. Let us mention a few: internalvoltages and currents in electrical circuits in order to ex-press the external port behavior; momentum in Hamiltonianmechanics in order to describe the evolution of the position;prices in economics in order to explain the production andexchange of economic goods, etc... Hence, B is the subsetof C∞(Rn, Rw) consisting of all functions w for which thereexits  ∈ C∞(Rn, Rw) such thatR(ddx)w = M(ddx). (2)Here R and M are polynomial matrices in n indeterminatesξi, i = 1, . . . , n, and again ddx = (∂∂x1, . . . , ∂∂xn ). We call(2) a latent variable representation of B and the variable is called the latent variable.III. QUADRATIC DIFFERENTIAL FORMSA quadratic differential form (QDF) is a quadratic formin the components of a function w ∈ C∞(Rn, Rw) andits derivatives. An appropriate tool to express quadraticfunctionals are 2n-variable polynomial matrices.In order to simplify the notation, we denote the vectorx := (x1, . . . , xn), the multi-indices k := (k1, . . . , kn) andl := (l1, . . . , ln), and use the notation ζ := (ζ1, . . . , ζn), ξ :=(ξ1, . . . , ξn) and η := (η1, . . . , ηn). Let Rw1×w2 [ζ, η] denotethe set of real polynomial w1 × w2 matrices in the 2nindeterminates ζ and η; that is, an element of Rw1×w2 [ζ, η]is of the formΦ(ζ, η) =∑k,lΦk,lζkηlProceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005TuB05.10-7803-9568-9/05/$20.00 ©2005 IEEE 2889where Φk,l ∈ Rw1×w2 ; the sum ranges over the nonnegativemulti-indices k and l, and is assumed to be finite. Such2n-variable polynomial matrix induces a bilinear differentialform LΦLΦ : C∞(Rn, Rw1) × C∞(Rn, Rw2) −→ C∞(Rn, R)LΦ(v, w) :=∑k,l(dkvdxk)T Φk,ldlwdxlwhere the k-th derivative operator dkdxkis defined as dkdxk:=∂k1∂xk11. . . ∂kn∂xknn( similarly for dldxl). Note that ζ correspondsto differentiation of terms to the left and η refers to the termsto the right.The 2n-variable polynomial matrix Φ(ζ, η) is called sym-metric if w1 = w2 =: w and Φ(ζ, η) = Φ(η, ζ)T . Notethat the former condition is equivalent to LΦ(w1, w2) =LΦ(w2, w1) for all w1, w2. We denote the symmetric ele-ments of Rw×w[ζ, η] as Rw×wS [ζ, η]. If Φ is symmetric then itinduces also a quadratic functionalQΦ : C∞(Rn, Rw) −→ C∞(Rn, R)QΦ(w) := LΦ(w, w)We will call QΦ the quadratic differential form associatedwith Φ.Definition 1: Given Φ ∈ Rw×wS [ζ, η]. The plant B is said tobe QΦ-dissipative if∫RnQΦ(w)dx ≥ 0 ∀w ∈ B with compact support (3)Dissipativity states that the system absorbs energy (in spaceand time) during any history in B that starts and ends withthe system at rest.Definition 2: Let Φ′ ∈ R×S [ζ, η]. QΦ′ is said to be averagenon-negative if∫RnQΦ′ ()dx ≥ 0for all  ∈ C∞(Rn, R) of compact support.(4)IV. STATIONARY AND LOCAL MINIMADefinition 3: Let Φ ∈ Rw×wS [ζ, η]. The trajectoryw ∈ C∞(Rn, Rw) is called stationary withrespect to∫RnQΦ(w)dx if for all ∆ ∈C∞(Rn, Rw) of compact support, we have∫RnLΦ(∆, w)dx = 0. (5)Oviously, w ∈ C∞(Rn, Rw) is stationary if and only if forall ∆ ∈ C∞(Rn, Rw) of compact support we have∫RnQΦ(w + ∆) − QΦ(w)dx =∫RnQΦ(∆)dx. (6)Theorem 4: Let Φ ∈ Rw×wS [ζ, η]. The set of stationarytrajectories with respect to∫RnQΦ(w)dx consists of allw ∈ C∞(Rn, Rw) that satisfyΦ(− ddx,ddx)w = 0 (7)Proof: If w is stationary then∫RnLΦ(∆, w)dx =0 for all ∆ ∈ C∞(Rn, Rw) with compact support. Thenintegrating by parts we obtain∫RnLΦ(∆, w)dx =∫Rn∆T Φ(− ddx,ddx)wdx, (8)Since this holds for all ∆ ∈ C∞(Rn, Rw) with compactsupport, w is stationary if and only if (7) holds.Now we look at the local minimum trajectories and theirrelation with stationary ones.Definition 5: Let Φ ∈ Rw×wS [ζ, η]. The trajectory w is calleda local minimum for∫RnQΦ(w)dx with respect to compactsupport variations if∫RnQΦ(w + ∆) − QΦ(w)dx ≥ 0, (9)for all ∆ ∈ C∞(Rn, Rw) of compact support.Theorem 6: Let Φ ∈ Rw×wS [ζ, η]. If QΦ is average non-negative then the set of local minimum trajectories is equalto the set of stationary trajectories. If QΦ is not averagenon-negative, then the set of local minimum trajectories isempty.Proof: It is easy to see that∫RnQΦ(w + ∆) − QΦ(w)dx= 2∫RnLΦ(w, ∆)dx +∫RnQΦ(∆)dx(10)Suppose QΦ is average non-negative. If∫RnLΦ(w, ∆)dxis not zero for all ∆ of compact support, then one canalways choose a ∆ in such a way that∫RnLΦ(w, ∆)dx +∫RnQΦ(∆)dx is negative since LΦ is linear in ∆ and QΦ isbilinear in ∆. Once we have∫RnLΦ(w, ∆)dx = 0 for all ∆,then we clearly have that w is local minimum. Now supposeQΦ is not average non-negative. Using the same argumentsas before we have that∫RnLΦ(w, ∆)dx must be zero aswell. Then the set of local minimum is empty.V. INTERCONNECTIONWe now discuss the issue of control as interconnection. Sincea plant behavior B ∈ Lwn consists of all trajectories satisfyinga set of differential equations, one would like to restrictthis space of trajectories to a desired subsystem, K ⊂ B.This restriction can be effected by increasing the numberof equations that the variables of the plant have to satisfy.These additional laws themselves define a new system, calledthe controller (denoted by C). The interconnection of thetwo systems (the plant and the controller) results in thecontrolled behavior K. After interconnection, the variableshave to satisfy the laws of both B and C. In this section wewill look at two types of interconnections, full and partial.The full interconnection of B and C is defined as the systemwith behavior B∩C. Note that B∩C is again an element ofLwn. A given behavior K ∈ Lwn is called implementable withrespect to B by full interconnection if there exists a C ∈ Lwnsuch that K = B ∩ C. The full interconnection of B and Cis called regular, ifp(B ∩ C) = p(B) + p(C).2890Where p is equal to the rank of the polynomial matrix inany kernel representation of B. Let Rw = 0 and Cw = 0be kernel representations of B and C respectively. We knowthat the full interconnection of B and C is regular if and onlyif B + C = C∞(Rn, Rw), (see [6]) . Regular interconnectionexpresses the idea of ”restricting what is not restricted”.In a regular interconnection, the controller imposes newrestrictions on the plant; it does not reimpose restrictitionsthat are already present. In this sense, the controller in aregular interconnection has no redundancy. We call optimalcontroller, the controller that implements local minimumtrayectories. We would want the controlled behavior to havenon-zero trayectories (i.e. non-trivial), and so we look fornontrivial controllers.Theorem 7: Given QΦ average non-negative and B = ker R,there exit nontrivial optimal controllers if[R∂Φ]is no ZRP (zero right prime), that is, there does not exits apolynomial matrix S such that im (R( ddx ))= ker (S). If so,C = Ker ∂Φ is one controller. If it is ZRP only the trivialcontroller is optimal.Where ∂Φ(ξ) = Φ(ξ, ξ).Proof: See [5].VI. CONTROLLABILITY AND OBSERVABILITYOne of the properties of behaviors which is very convenient,in particular for LQ problems, is controllability.Definition 8: A system B ∈ Lwn is said to be controllable iffor all w1 , w2 ∈ B and all sets U1 ,U2 ⊂ Rn with disjointclosure, there exist a w ∈ B such that w |U1 = w1 |U1 andw |U2 = w2 |U2 .There are a number of characterizations of controllabilitybut the one useful for our purposes is the equivalence ofcontrollability with the existence of an image representation.Consider the following special latent variable representation:w = M(ddx)with M ∈ Rw×[ξ¯] . Obviously, by the elimination theorem,see [3], its manifest behavior B is a linear differential n-D system again, i.e. B ∈ Lwn. Such special latent variablerepresentations often appear in physics, where the latent vari-ables in a such representation are called potentials. Clearly,B = im(M( ddx )). For this reason this representation is calledan image representation of its manifest behavior.Theorem 9: (See [3]), B ∈ Lwn admits an image representa-tion if and only if it is controllable.Now, we will consider a useful property of n-D systems,observability. For this property one needs to split the vari-ables of the system in two sets; the first set of variables areinterpreted as the observed variables an the other is the setof ’to be deduced’ variables.Definition 10: Let B ∈ Lwn with manifest variable w, w =(w1, w2) be a partition of w∈ Lwn .Then w2 it is said tobe observable from w1 in B if given any two trajectories(w1′, w2′), (w1′′, w2′′) ∈ B we have that w1′ = w1′′implies w2′= w2′′.So, observability only becomes an intrinsict property of thebehavior after a partition of the manifest variable w is given.Although we can divide the set of variables in many ways,one natural way to do it, is when one is looking at a latentvariable representation of the behavior is to ask whether thelatent variables are observable from the manifest variables.If this is the case we call the latent variable representationobservable. For controllable 1-D systems it can be shownthat there always exists an observable image representation.This is not true for n-D systems. ¿From now on, we willassume that the plant is controllable and we will assumeit has an observable image representation B := ImM( ddx ).If w = M( ddx ) is an observable image representation ofB, then w ∈ B has compact support if and only if thecorresponding  ∈ C∞(Rn, R) has compact support.Given a 2n-variable polynomial matrix Φ(ζ, η), suppose w =M( ddx ) is an observable image representation of B, we canthen consider the induced 2n variable polynomial matrix Φ′defined byΦ′(ζ, η) := MT (ζ)Φ(ζ, η)M(η) (11)Remark 1: Note that B is QΦ-dissipative if and only if QΦ′is average non-negative.We replace QΦ(w) by QΦ′ () in the performance functional(or equivalently replace Φ(ζ, η) by MT (ζ)Φ(ζ, η)M(η)),and obtain an LQ problem in which the dynamic variablew is replaced by the unconstrained variable .Remark 2: Given a 2n-variable polynomial matrix Φ(ζ, η),suppose w = M( ddx ) is an observable image representationof B. Then it can be shown that there exist a left inverseM†(ξ) ∈ Rw×•[ξ] of M . Therefore from w = M( ddx ),M†(ddx)w=. (12)For more details see [1].Theorem 11: Let Φ ∈ Rw×wS [ζ, η] and a behavior B = ker Rand suppose B has also w = M( ddx ) as an observableimage representation. Assume that the polynomial matrixΦ′(ζ, η), satisfy det(Φ′(−ξ, ξ)) = 0. The set of stationarytrajectories of K with respect to∫RnQΦ(w)dx, is regularlyimplementable with respect to B by full interconnection,and a controller C = Ker C that regularly implements K isrepresented by:MT (ddx)Φ(− ddx,ddx)w = 0 (13)Proof: The proof that C is the controller which im-plement the stationary trayectories follows by sustitution. Tosee that the interconnection is regular we need to check thatB + C = C∞(Rn, Rw). Now, a kernel representation of B+ C is obtain as folows: Consider[RC]and let[N L]be such that ker[N L]= im[RC].Then obviously NR = −LC and according to ([6],lemma 2.14), B + C = ker(NR). In our case we2891have C(ξ) = MT (ξ)Φ(−ξ, ξ) so we get N(ξ)R(ξ) +L(ξ)MT (ξ)Φ(−ξ, ξ) = 0. Hence for every  ∈ C∞(Rn, R)we get L( ddx )MT ( ddx )Φ(− ddx , ddx )M( ddx ) = 0. Recall thatΦ′(−ξ, ξ) = MT (−ξ)Φ(−ξ, ξ)M(ξ) which is assumed tobe nonsingular. Hence L(ξ) = 0. Thus NR = 0 andB + C = C∞(Rn, Rw). This proofs that the interconnectionis regular.Remark 3: It can be shown that M†( ddx )Φ′( ddx ,ddx ) = wrepresents also a controller that regularly implements K.REFERENCES[1] Cotroneo, T., ”Algorithms in Behavioral systems theory”, Thesis,2001.[2] Belur,M.N. and Trentelman,H.L., “Stabilization, pole placement, andregular implementability”, IEEE Trans. Automatic Control, 2002.[3] Pillai, H.K. and Willems, J.C., ”Lossless and dissipative distributedsystems”, SIAM J. Control Opt., vol 40, no. 5, pp 1406–1430, 2002.[4] Rapisarda, P. and Trentelman, H.L., ”Linear Hamiltonian behaviorsand bilinear differential forms”, submitted to SIAM J. Control Opt.,2002.[5] Napp, D. and Trentelman, H.L., In preparation.[6] Rocha, P. and Wood, J., ”Trajectory control and Interconnection of 1Dand nD Systems”, SIAM J. Control Opt., 2001.[7] van de Schaft, A.J. , ”Achievable behavior of general systems”,Systems and Control Letters, Vol 49,pp 141-149, 2003.[8] Trentelman, H.L. and Willems, J.C., ”On quadratic differential forms”,SIAM J. Control Opt., vol. 36, no. 5, pp. 1703-1749, 1998.[9] Willems, J.C. and Polderman, J.W., ”Introduction to MathematicalSystem Theory”, Springer, New York 1998.2892

The Behavioral LQ-problem for linear nD systems

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The Behavioral LQ-problem for linear nD systems

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