Controller synthesis for L2 behaviors using rational kernel representations

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1 Controller synthesis for L behaviors using rational kernel reresentations Citation for ublished version (APA): Mutsaers, M. E. C., & Weil, S. (008). Controller synthesis for L behaviors using rational kernel reresentations. In Proceedings of the 47th IEEE Conference on Decision Control (CDC 008) : Mexico, Cancún, 9-11 December 008 ( ). Piscataway, NJ: Institute of Electrical Electronics Engineers (IEEE). DOI: /CDC DOI: /CDC Document status date: Published: 01/01/008 Document Version: Publisher s PDF, also known as Version of Record (includes final age, issue volume numbers) Please check the document version of this ublication: A submitted manuscrit is the version of the article uon submission before eer-review. There can be imortant differences between the submitted version the official ublished version of record. Peole interested in the research are advised to contact the author for the final version of the ublication, or visit the DOI to the ublisher's website. The final author version the galley roof are versions of the ublication after eer review. The final ublished version features the final layout of the aer including the volume, issue age numbers. Link to ublication General rights Coyright moral rights for the ublications made accessible in the ublic ortal are retained by the authors /or other coyright owners it is a condition of accessing ublications that users recognise abide by the legal requirements associated with these rights. Users may download rint one coy of any ublication from the ublic ortal for the urose of rivate study or research. You may not further distribute the material or use it for any rofit-making activity or commercial gain You may freely distribute the URL identifying the ublication in the ublic ortal. If the ublication is distributed under the terms of Article 5fa of the Dutch Coyright Act, indicated by the Taverne license above, lease follow below link for the End User Agreement: Take down olicy If you believe that this document breaches coyright lease contact us at: oenaccess@tue.nl roviding details we will investigate your claim. Download date: 15. Mar. 019

2 Proceedings of the 47th IEEE Conference on Decision Control Cancun, Mexico, Dec. 9-11, 008 Controller synthesis for L behaviors using rational kernel reresentations Mark Mutsaers Sie Weil Abstract This aer considers the controller synthesis roblem for the class of linear time-invariant L behaviors. We introduce classes of LTI L systems whose behavior can be reresented as the kernel of a rational oerator. Given a lant a controlled system in this class, an algorithm is develoed that roduces a rational kernel reresentation of a controller that, when interconnected with the lant, realizes the controlled system. This result generalizes similar synthesis algorithms in the behavioral framework for infinitely smooth behaviors that allow reresentations as kernels of olynomial differential oerators. I. INTRODUCTION The analysis of system interconnections is at the heart of many roblems in modeling, simulation control. Indeed, when focusing on control, the controller synthesis question amounts to finding a dynamical system (a controller) that, after interconnection with a given lant, results in a controlled system that is suosed to erform a certain task in a more desirable manner than the lant. Usually the control synthesis roblem is formulated as a feedback otimization roblem in which the lant controller interact through a number of distinguished channels that have been divided in inut- outut variables. The behavioral theory of dynamical systems has been advocated as a concetual framework in which esecially interconnection structures of dynamical system can be studied in an inut-outut indeendent setting. There are many concetual, edagogic ractical reasons for doing so we refer to 9, 10 for a detailed account on this matter. One key roblem concerning the interconnection of dynamical systems involves the question when a given dynamical system Σ K can be imlemented (or realized) as the interconnection of a dynamical system Σ P, that is suosed to be given, a second dynamical system Σ C, that is suosed to be designed. With the interretation that Σ P Σ K denote the lant- (desired) controlled system, this question is therefore equivalent to a synthesis question for the controller Σ C. Within the behavioral framework this question received a very comlete elegant answer for the class of linear time-invariant systems that admit reresentations in terms of olynomial difference or olynomial differential oerators 5, 6. A rather comlete theory has been develoed for Ir. Mark Mutsaers is PhD student with the deartment of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherls M.E.C.Mutsaers@tue.nl Dr. Sie Weil is associate rofessor with the deartment of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherls S.Weil@tue.nl This work is suorted by the Dutch Technologiestichting STW under roject number EMR.7851 Σ P w Σ C (a) Full interconnection. Fig. 1. Σ K P Interconnection roblems. K C (b) P C = K. such reresentations that covers, among other things, H, LQ H otimal control. It is the urose of this aer to reconsider the controller synthesis question for secific classes of linear timeinvariant L systems that admit reresentations in terms of rational functions. In doing so, we deart from the setting roosed in 11 of considering infinitely smooth trajectories as solutions of rational differential equations. Instead, we view rational functions in H as multilicative oerators on L functions define L systems through the kernel of such oerator. In this way, rational functions naturally define dynamical systems in the frequency domain offer distinct algebraic advantages over olynomial kernel reresentations. The aer is organized as follows. Section II contains the formulation of the main roblem that is discussed in this aer. In Section III some notational remarks about saces oerators are introduced. Sections IV V contain the introduction of L behaviors, the interconnection roblem a novel controller synthesis algorithm. An examle using this synthesis algorithm is given in Section VI. In the last section of this aer, the results of this aer are discussed some recommendations for further research on L systems are given. II. PROBLEM FORMULATION Following the behavioral formalism, a dynamical system 1 is described by a trile: Σ = (T, W, B), (1) where T R or T C is the time- or frequency-axis, W is the variable signal sace, which tyically contains inuts oututs will be taken to be a finite dimensional vector sace throughout, B W T is the behavior, that is defined in more exlicit terms in Section IV. Using (1) it is ossible to describe lants, controllers desired controlled systems (as Σ P, Σ C Σ K resectively). Fig. 1 illustrates the interconnection Σ K of two systems Σ P = (T, W, P) Σ C = (T, W, C). It is defined as Σ K = (T, W, P K) motivated by the idea that the /08/$ IEEE 5134

3 behavior of the interconnection satisfies the laws of both Σ P Σ C. Fig. 1(a) gives an illustration of the roblem treated in this aer, namely given a lant Σ P a desired controlled system Σ K, construct, if it exists, a controller Σ C that after interconnection with the lant results in the desired controlled system. We address this roblem for very secific classes of L systems. More secifically, we address the roblems of existence, (non-) uniqueness of controllers, together with the roblem to arametrize all controllers that establish a desired controlled system after interconnection. As mentioned in the introduction, earlier research, using infinitely smooth trajectories, has been carried out for this roblem 6, 9, 10. This aer contributes to the controller synthesis question by considering various L systems, reresented through rational oerators. A. Hardy saces III. NOTATION Hardy saces are denoted by H H, where = 1,,...,, defined by: H :={f : C C q f H H :={f : C C q f H < }, < }, where C := Re{s} > 0 C := Re{s} < 0, with s = σjω. So, functions in H are analytic 1 in C { } functions in H are analytic in C { }. The H saces are the classical Hardy saces 4. The norms of functions in H H are defined as: ( ) 1 lim f(σ jω) dω, 0 < <, f H = σ 0 lim f(σ jω), =, su σ 0 ω R ( ) 1 lim f(σ jω) dω, 0 < <, f H = σ 0 lim f(σ jω), =. su σ 0 ω R It is remarked that the tangential limits σ 0 in the above exressions exist, which makes the Hardy saces well defined normed saces, cf. 4. B. Rational functions Units The refixes R U denote, resectively, rational functions units in the Hardy saces H H as in RH := {f H f is rational}, RH := {f H f is rational}, 1 A function is analytic if it is comlex differentiable. UH :={U RH U 1 RH }, UH :={U RH U 1 RH }. Note that units are necessarily square rational matrices. C. Lalace transformation The Lalace transform L : L (R, R q ) L (C, C q ) defines an isometry between the L Hilbert sace the inner roduct sace L : L := H H = {f : C Cq f < }, which inherits the following norm: f = f(jω) H f(jω)dω, the inner roduct on comlex valued functions: f,g = f(jω) H g(jω)dω. Any element w L can be uniquely decomosed as w = w w, where w := Π w, with Π : L H, w := Π w, with Π : L H. Here, Π Π denote the canonical rojections from L onto H H, resectively. D. Maings in RH RH Elements of RH RH (also known as stable- anti-stable functions of RH stable RH antistable ) define oerators in the following manner. Let Θ RH define Θ : L L by: (Θw)(s) := Θ(s)w(s), where w L, which is the usual multilication or Laurent oerator in the frequency domain 4. Similarly, let Ψ RH define Ψ : L L by: (Ψw)(s) := Ψ(s)w(s), where w L. When restricted to the domains H or H, these oerators define functions as: Lemma 3.1: Let Θ RH, with ossible domains L, H H. Then Θ : L L, Θ : H H, Θ : H L. Similarly, let Ψ RH, with ossible domains L, H H. Then Ψ : L L, Ψ : H L, Ψ : H H. The roof of this lemma more details about Hardy saces can be found in

4 ŵ(t) t First, behaviors associated with maings P from the sace of rational stable Hardy functions are discussed. For any P RH, the following three dynamical systems are defined: (ˆσ τ ŵ)(t) ˆσ τ ˆσ τ (ˆσ τ ŵ)(t) Σ P := (C, C q, P(P)), Σ P, := (C, C q, P (P)), Σ P, := (C, C q, P (P)), (a) t E. τ-shift oerators Fig.. τ-shift of ŵ = L 1 {w} with w H. We define the τ-shift oerator ˆσ τ on a signal ŵ : R R as: (ˆσ τ ŵ)(t) = ŵ(t τ), where ŵ is the inverse Lalace transform of w, which is an element of L, H or H (so, ŵ = L1 {w}). A τ-shift is called a left-shift if τ < 0 (which means that the signal shifts left with resect to the time axis) is named a right-shift if τ > 0 (so, the signal shifts right with resect to the time axis). For any τ R, we introduce the shift oerators σ τ : L L, σ τ : H H σ τ : H H by defining: (σ τ w)(s) = e sτ w(s), e sτ w(s), τ > 0 (σ τ w)(s) = e sτ τ (w(s) ŵ(t)e st dt), τ < e sτ (w(s) ŵ(t)e st dt), τ > 0 (σ τ w)(s) = τ e sτ w(s), τ < 0 resectively. Obviously, σ 0 is the identity ma. Note that σ τ : L L defines an isometry (for all τ R) that σ τ : H H σ τ : H H define isometries only if τ 0 τ 0, resectively. When interreted in the time domain, a left- right-shift for a signal w H are illustrated in Fig.. Definition 3.1: A subset P of L (or H or H ) is said to be left-shift invariant if σ τ P P for all τ < 0. The set P is said to be right-shift invariant if σ τ P P for all τ > 0. IV. RATIONAL REPRESENTATIONS OF BEHAVIORS In the revious section, stable- anti-stable rational oerators have been introduced on Hilbert saces. In this section we will associate behaviors as linear shift invariant subsets of L, H H defined through the null saces of these oerators. Throughout this section, the variables w are elements of L, H or H. t where P(P) := {w L Pw = 0} = ker P L, P (P) := {w H Pw = 0} = kerp H, (b) P (P) := {w H Pw H } = ker Π P H. Here, Π is the canonical rojection that is introduced before. For these sets we have the following roerties: Lemma 4.1: For P RH, the behaviors P(P), P (P) P (P) are linear right-shift invariant subsets of L, H H, resectively. A system Σ with either of these behaviors is called an L right-shift invariant system. Definition 4.1: The classes of all linear rightshift invariant systems in L, H H that admit reresentations as the kernel of a rational element P RH are denoted by M, M M. Similarly, for any ˆP RH, the following three dynamical systems are introduced as: Σ ˆP := (C, C q, P( ˆP)), Σ ˆP, := (C, C q, P ( ˆP)), (3a) Σ ˆP, := (C, C q, P ( ˆP)), where the behaviors are given by: P( ˆP) := {w L ˆPw = 0} = ker ˆP L, P ( ˆP) := {w H ˆPw H } = ker Π ˆP H, (3b) P ( ˆP) := {w H ˆPw = 0} = ker ˆP H, As introduced before, Π : L H is the canonical rojection. Lemma 4.: For ˆP RH the behaviors P( ˆP), P ( ˆP) P ( ˆP) are linear left-shift invariant subsets of L, H H, resectively. A system Σ with either of these behaviors is called an L left-shift invariant system. Definition 4.: The classes of all linear left-shift invariant systems in L, H H that admit reresentations as the kernel of a rational element ˆP RH are denoted by L, L L. Now dynamical systems (1) can be described using L behaviors, some roerties, using rational elements, are introduced: 5136

5 Theorem 4.1: Let P,K RH let P (±) = P (±) (P) K (±) = K (±) (K) be as defined in (). Then the following statements are equivalent: i K P, ii K P, iii K P, iv F RH such that P = FK. Moreover, K = P K = P K = P U UH such that P = UK. The roof of this theorem can be found in the Aendix. Also anti-stable maings can be used in the reresentations, which yields the following theorem: Theorem 4.: Let ˆP, ˆK RH let P (±) = P( ˆP) K (±) = K( ˆK) as in (3). Then the following statements are equivalent: i K P, ii K P, iii K P, iv ˆF RH such that ˆP = ˆF ˆK. Moreover, K = P K = P K = P Û UH such that ˆP = Û ˆK. The roof of this theorem is similar to the one of Theorem 4.1 therefore is not included in this aer. V. CONTROLLER SYNTHESIS A. Full Interconnection roblem For each of the above classes of L systems, the synthesis roblem defined in Section II can now be formally stated as follows: Problem 5.1: Given two linear left-shift invariant systems Σ P Σ K in the class L (or L or L ). i Verify whether there exists Σ C L (L or L ) such that P C = K. Any such system is said to imlement K for P by full interconnection through w (Fig. 1(a)). ii If such controller exists, find a reresentation C 0 RH of Σ C = (T, W, C) in the sense that C = kerc 0 (or C = ker Π C 0 or C = kerc 0 ). iii Characterize the set C ar of all C RH for which Σ C = (T, W,ker C) imlements K for P. A similar roblem formulation alies for the model classes M, M M. Our synthesis algorithm is insired by the olynomial analog that has been treated in 5, 6 leads to exlicit rational reresentations of behaviors C that imlement K for P. Theorem 5.1: Given the systems Σ P = (T, W, P) Σ K = (T, W, K) in the class L (±) (or M (±) ). i There exists a controller Σ C = (T, W, C) L (±) (or M (±) ) that imlements K for P by full interconnection if only if K P. ii Whenever one of the equivalent conditions of item i holds, the set C ar of all ossible kernel reresentations of controllers that imlement K for P by full interconnection is given in Ste 5 of Algorithm 1 below. The roof of Theorem 5.1 is insired by the olynomial analog in 5 6 is given in the next subsection. B. Algorithm The following algorithm results in the exlicit construction of all controllers Σ C that solve Problem 5.1 for the class L of L systems. A similar algorithm alies for the solution of Problem 5.1 for the model classes L, L M (±). Algorithm 1: Given P,K RH that define the systems Σ P Σ K as in (3). Aim: Find all C RH that define the system Σ C = (T, W, C) L with C = kerc, such that C imlements K for P in the sense that P C = K by full interconnection. Ste 1: Verify whether K P. Equivalently, verify whether there exists a maing F RH such that P = FK. If not, the algorithm ends no controller exists that imlements K for P. Ste : Determine a unit U UH which brings F into column reduced form: F = FU = F 1, 0, where F 1 RH is square of full rank. Ste 3: Extend the matrix F with W = 0, I such that Λ = F W = F1 0 0 I belongs to UH. Factorize W = WU with W = WU 1. Ste 4: Set Σ C = (T, W, C) where C = kerc 0 C 0 = WK RH. The controller Σ C then belongs to L imlements K for P. Ste 5: Set C ar = {Q 1 P Q WK Q 1,Q RH,Q full rank}. Then C ar is a arametrization of all controllers Σ C = (T, W, C) that imlement K for P by ranging over all kernel reresentations C = ker C with C C ar. Proof: Proof of Theorem 5.1: i ( ): This is trivial. ( ): If K P, then there exists a F as in Theorem 4.1 or 4.. In the controlled behavior K, the restrictions of the lant as well as the restrictions alied by the controller have to be satisfied:, K = ker P C = ker K = ker(λk), where Λ UH, where Λ = col(f,w) with W unknown. The extended matrix Λ in ste 3 is a multilication of a unit U with Λ, so Λ has to be a unit. Therefore: K = ker P C = ker ( F W K), so, C = WK which results in C = ker C = ker{wk}. ii One can aly a multilication with a unit Q UH : K = ker ( I 0 Q 1 Q }{{ C P } ) = ker ( ) P Q 1PQ WK, Q 5137

6 where Q 1,Q RH Q is full rank. Then all ossible rational functions C can be arametrized by C ar as in ste 5. VI. EXAMPLE: QUADRATIC COST In this examle, the lant behavior P of an unstable lant Σ P is described by the state-sace realization: { ẋ(t) = Ax(t) Bu(t), x(0) = x 0, (4) y(t) = x(t), where x(t) R x, u(t) R u, A R x x B R x u. The desired controlled behavior K consists of all airs (u,y) L (R, R u x ) that minimize the cost function: J(x 0,x(t),u(t)) = 1 0 x T (t)qx(t) u T (t)ru(t)dt subject to the system equations (4) of the lant model Σ P. Here, 0 Q R x x, 0 < R R u u. As discussed in 3, 8, this controlled behavior can be written as a dynamical system Σ K with the state-sace realization: ẋ(t) = (A BR 1 B T S) x(t), x(0) = x 0, u(t) = R 1 B T S x(t), (5) y(t) = x(t), where S is a solution of an Algebraic Ricatti Equation. The numerical values used for those matrices are the following: A = , B = 0 4 1, Q = , R = 0 0, S = , α,β R, α β. The dynamical systems are secified by trajectories in the time domain, but we are interested in L behaviors using rational kernel reresentations as system reresentations. Because the controlled system is autonomous, the left-shift invariance roerty is required, which restricts us to use antistable maings for P,K also C. This results in: P(s) = I (sia) 1 B RH, K(s) = (si(abr 1 B T S))(sIαI) 1 0 RH R 1 B T S(sIβI) 1 (siβi), 1 where w(s) = y(s) u(s) T, α,β > 0 α β. Due to the requirement, the anti-stable oles α β are introduced. Of course, no ole-zero cancellation should occur when α β are chosen. Using those reresentations, the full interconnection algorithm can be alied to the roblem: Ste 1: The first ste in the full interconnection algorithm is to verify whether K P, which should be the case. Equivalently, we need to verify whether there exists a F(s) RH such that P(s) = F(s)K(s): F(s) = Γ(s) Λ(s) RH, where Γ(s) = I(IsA) 1 B(IsβI)R 1 B T S(sIβI) 1 Λ(s) = (IsA) 1 B(IsβI). (IsαI)(sI(ABR 1 B T S)) Ste : The next ste is to column reduce F(s). This can be done using algorithms as in, which results in: 0 sα (s4) F(s) = sβ s4 0 RH, which can be column 0 sα s4 0 0 reduced to F(s) = F(s) U(s) = F 0 (s) 0, where sα (s4) F(s) = sβ s4 0 0 RH U(s) = sα s sβ sα sα sβ UH. Ste 3-4: Then, as discussed in Ste 4 of Algorithm 1, a ossible controller behavior C = ker C 0 is exressed as: s6 1 sα sα C 0 (s) = W(s)K(s) = 0 0 RH. 1 1 sβ 8 sβ 0 1 sβ As mentioned before, the behavioral framework does not require a searation of the variable w into inuts oututs. This can be seen in the result above, because the controller restricts the oututs of the lant in the first row when a searation in the variable sace is made. So, mathematically the interconnection with this controller results in the desired controlled behavior, but this reresentation is not directly imlementable for a real system, because oututs of a lant can t always be used as inuts. For a ractical reason, we consider the arametrization of all controllers that imlement Σ K in the next ste. Ste 5: Another controller can be found using the matrices Q 1 (s) Q (s) as defined in Algorithm 1. When these matrices are chosen to be: 1 s4 Q 1 = sb 1 1 sb 0 0 RH 1 sa Q = the resulting controller is equal to: sb 0 1 C(s) = Q 1 (s)p(s) Q (s)c 1 (s) 4 sb = 15 1 sb sb 0 RH, 1 1 sb 8 1 sb 0 sb RH, In fact, this controller allows a feedback imlementation as it is equivalent to the general LQR, namely: u(t) = x(t). Note: The values in the Ricatti solution S the values in the estimated rational exressions are rounded to integers for simlification. VII. CONCLUSIONS AND RECOMMENDATIONS We considered the roblem of controller synthesis for secific classes of L functions. Oerators in the classes RH of stable rational functions RH of anti-stable rational functions define linear right-shift invariant L behaviors linear left-shift invariant L behaviors by considering their kernel saces. Given two L systems Σ P Σ K we solve the question to synthesize a third L system Σ C that realizes 5138

7 Σ K in the sense that the full interconnection of Σ P Σ C satisfies K = P C. Necessary sufficient condition for the existence of an L system Σ C is the inclusion K P. An exlicit controller synthesis algorithm for the class of all controllers that imlement an L controlled system for an L lant has been derived. An examle is given to demonstrate the algorithm for the construction of a rational reresentation of C. This aer only introduced the case when the lant behavior P controller behavior C are fully interconnected, which is not always the case. Therefore, some further research has to be done for the artial interconnection case using those classes of rational functions. Studies already started for infinite smooth continuous behaviors in 6. In this case, disturbances like noise can be taken into account, which may yield in robust control roblems. Proof: (iv {i,ii,iii}): APPENDIX PROOF OF THEOREM 4.1 iv i: K, P L, so w L : If P = FK take a w K. Then, v = Kw = 0, so also Pw = FKw = Fv = 0. This imlies that P(s)w(s) = 0, so w P, K P. iv ii: K, P H, so w H : This roof is identical to the case when K, P L. iv iii: K, P H, so w H : Again, if P = FK w K, one can say that v = Kw H hence Pw = FKw = Fv. Now, F RH, so Fv H. So, Pw H which imlies using () that w P, which is equal to K P. (iv {i,ii,iii}): iv i: K, P L, so w L : Using the definition of K, one can write: K ={w L Kw,v L = 0 v L } ={w L w,k v L = 0 v L } = (im K ), where K : L L is the dual- or adjoint oerator in RH defined by K (s) = K T (s 1 ). Something similar can be alied to the lant behavior. So, K P imlies that P K using the revious definition of K, this results in (im P ) (im K ), where the bar denotes the closure in L. For rational oerators the latter imlies that: (im P ) (im K ), because in that case the images are closed. Then we can say that for some e i, P e i im K, so there exists a v i such that: P e i = K v i. This can be extended to a set of v i s, such that: P = K X with X = (v 1,...,v ) RH RH. Then, we can rewrite this to P = X K, where F is equal to the dual oerator X RH. iv ii: K, P H, so w H : This roof is similar to the one in the revious item, excet that now the H inner roduct is used. However, H inherited this inner roduct from L. iv iii: K, P H, so w H : Now, K can be written as: K ={w H Π Kw,v H = 0 v H } ={w H w,k Π v = 0 v H H } = ( im K Π, ) where K Π are adjoint oerators. This can also be done for the lant behavior P. As in item (iv i), P K, so: (im P Π ) (im K Π ). Then there exists a X RH such that P Π = K Π X. So, one can say that Π P = X Π K, where F = X. Equality condition: Using the revious items, one can say that P = K if only if P = U 1 K K = U P with both U 1 U in RH. Moreover, if U 1 U satisfy these conditions, then P = U 1 U K K = U U 1 P. If P K are full rank, we find that U 1 = U 1, which comletes the roof. REFERENCES 1 Madhu Nagraj Belur, Control in a behavioral context, PhD Thesis Rijksuniversiteit Groningen, 003. Th.G.J. Beelen, G.J. van den Hurk C. Praagman, A new method for comuting a column reduced olynomial matrix, Systems & Control Letters, number 10, ages 17-4, D.W.G. Bijnen, Model reduction on constrained otimally controlled systems, Graduation aer, Eindhoven University of Technology, Bruce A. Francis, A course in H Control Theory, Lecture Notes in Control Information Sciences, Singer-Verlag, Jan Willem Polderman Jan C. Willems, Introduction to Mathematical Systems Theory: A Behavorial Aroach, Sringer, H.L. Trentelman R. Zavala You C. Praagman, Polynomial Embedding Algorithms for Controllers in a Behavorial Framework, IEEE Transactions on Automatic Control, vol. 5, ages 1-6, Sie Weil Anton Stoorvogel, Rational reresentations of behaviors: Interconnectability stabilizability, MCSS, vol. 10, ages , Sie Weil, Jochem Wildenberg, Leyla Özkan Jobert Ludlage, A Lagrangian Method for Model Reduction of Controlled Systems, Proceedings of the 17 th IFAC World Congress, Seoul, Jan C. Willems, On Interconnections, Control Feedback, IEEE Transactions on Automatic Control, vol. 4, ages , Jan C. Willems, The Behavioral Aroach to Oen Interconnected Systems, IEEE Control Systems Magazine, vol. 7, ages 46-99, Jan C. Willems Y. Yamamoto, Behaviors defined by rational functions, Linear Algebra Its Alications, Volume 45, ages 6-41, 007. e i = 0,..., 1,..., 0, with the 1 on the i th osition. 5139