The canonical controllers and regular interconnection

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1 Systems & Control Letters ( The canonical controllers an regular interconnection A.A. Julius a,, J.C. Willems b, M.N. Belur c, H.L. Trentelman a Department of Applie Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschee, The Netherlans b K.U. Leuven, Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium c Department of Electrical Engineering, Inian Institute of Technology Bombay, Powai, Mumbai , Inia Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlans Receive 28 April 2004; receive in revise form 30 November 2004; accepte 1 December 2004 Abstract We stuy the control problem from the point of view of the behavioral systems theory. Two controller constructions, calle canonical controllers, are introuce. We prove that for linear time-invariant behaviors, the canonical controllers implement the esire behavior if an only if there exists a controller that implements it. We also investigate the regularity of the canonical controllers, an establish the fact that they are maximally irregular. This means a canonical controller is regular if an only if every other controller that implements the esire behavior is regular Elsevier B.V. All rights reserve. Keywors: Behaviors; Behavioral control; Regular interconnection; Regular controller; Canonical controller; Implementability 1. Introuction Control problems, seen from the behavioral systems theory point of view, amount to fining a controller, which when interconnecte with the plant in a specifie way yiels the esire behavior [14,1]. The problem may be formulate as follows. Consier a plant to be controlle which has two kins of : Corresponing author. Tel.: ; fax: aresses: a.a.julius@math.utwente.nl (A.A. Julius, jan.willems@esat.kuleuven.ac.be (J.C. Willems, belur@ee.iitb.ac.in (M.N. Belur, h.l.trentelman@math.rug.nl (H.L. Trentelman. an control. A controller is a evice that is attache to the control an restricts their behavior. This restriction is impose on the plant, such that it affects the behavior of the (see Fig. 1. The resulting behavior is calle the controlle system. In this paper we iscuss the properties of a special type of controllers, the so-calle canonical controllers [11,10,16]. We are particularly intereste in their regularity properties. The concepts of canonical an regular controllers will be formally introuce later in this paper. While the behavioral approach sees control as interconnection, the more common point of view in control theory is to view a controller as a feeback /$ - see front matter 2005 Elsevier B.V. All rights reserve. oi: /j.sysconle

2 2 A.A. Julius et al. / Systems & Control Letters ( W CONTROLLER full control CONTROLLED SYSTEM Fig. 1. Control in the behavioral approach. processor that accepts the plant sensor outputs as its inputs an prouces the actuator inputs as its outputs. In [14], this paraigm is calle intelligent control, as the controller acts as an intelligent agent capable of reasoning how to react to sensory observations. The main avantages of the behavioral over the classical feeback point of view are: (i Its practical generality. In many control systems, the controller (i.e. the evice ae to the plant system to obtain a esire behavior oes not act as a sensor/actuator evice. Dampers, heat fins, acoustic noise insulators are examples of such evices. (ii Its theoretical simplicity. Control in the behavioral setting has been introuce in [14], an subsequent evelopment inclues the work in [6,17,9]. The reaer is referre to [12] for further motivation an etails. Throughout this paper, we shall enote the control as c an the as w. These take their value at any given time from their respective signal spaces C an W. Let W an C enote the set of all trajectories of the w an c that are a priori possible, before we have even moelle the plant. In ynamical systems, W an C are typically the set of (smooth signals from the time axis to the signal spaces W an C. In iscrete event systems, the time axis is iscrete an the signal spaces are typically calle alphabets. We shall now iscuss our problem from a purely set theoretic point of view. The behavioral moel of the plant system that captures the relevant relation between w an c is calle the full plant behavior, which is enote by P full. Naturally, we assume that P full is C Fig. 2. The relation between P full, P, C, an K. containe in W C. The full plant behavior consists of all signal pairs (w, c compatible with the plant ynamics. If we project the full behavior on W, weobtain the so calle manifest behavior, which is enote as P. Thus, P := {w W c C such that (w, c P full }. A controller C is a subset of C, containing all signals c allowe by the controller. The controlle behavior is then efine as K := {w W c C such that (w, c P full an c C}. The relationship between the full plant behavior, the manifest behavior, the controller an the controlle behavior is capture in Fig. 2. In this framework, the control problem can be formulate as to fin a controller C that yiels a esire controlle behavior D. Hence the controller C shoul yiel the controlle behavior K = D. We call D the esire controlle behavior. If it is possible to fin a controller C that yiels K = D, then D is sai to be implementable 1 or implemente by C. Further, if a given esire D is implementable, we say that the control problem is solvable. 1 In the literature the term achievable is sometimes use instea of implementable. C

3 A.A. Julius et al. / Systems & Control Letters ( 3 The remainer of this paper is organize as follows. In Section 2 we present two constructions for the canonical controllers an their properties in the setting of general behaviors. In Section 3 we iscuss the concept of implementability, particularly for LTI behaviors. Section 4 is evote for iscussion on the concept of regularity. In Section 5, we stuy the behavior implemente by the canonical controller an use the result from Section 4 to establish its regularity. control CANONICAL CONTROLLER DESIRED CONTROLLED BEHAVIOR 2. The construction an properties of the canonical controllers In the previous section we explaine that we work in the generality in which the plant has two types of, the an the control, an further that a controller can put a restriction on just the control. This restriction is propagate through the plant to the. The iea of the canonical controller uses the internal moel principle in the following way [11,10]. We use a plant moel that has the same behavior as the plant. The propagation of information explaine in the previous paragraph is then reverse by interconnecting the plant moel to the esire behavior D using the 2. This is how we construct the canonical controller. Fig. 3 illustrates this construction. Notice that the wor is mirrore to highlight the fact that the interconnection is reverse. The behavior of the canonical controller obtaine using this construction is enote as C canonical. C canonical := {c C v W such that (v, c P full an v D}. (1 Fig. 4 provies a block iagram showing how C canonical is applie. In [11,10], it is proven that for a class of plants, which are homogeneous in the plant an control, the control problem is solvable (i.e. D is implementable if an only if the canonical controller C canonical implements D. We now efine the homogeneity property. 2 A construction similar to the canonical controller has been use in [7] Fig. 3. The construction of C canonical. Mirrore text reflects the iea that the interconnection is reverse. Definition 1. A full plant behavior P full is sai to have the homogeneity property if for any w 1,w 2 W an c 1,c 2 C, the following implication hols: (w 1,c 1, (w 1,c 2, (w 2,c 1 P full (w 2,c 2 P full. Homogeneity can also be unerstoo as follows. The behavior P full can be seen as a relation between W an C. A relation is calle inepenent if it can be written as a Cartesian prouct of its projections on the relate omains. The behavior P full has the homogeneity property if it can be written as a isjoint union of inepenent relations. In particular, if W an C are linear spaces an P full is a linear subspace, then P full has the homogeneity property. The following theorem captures an important property of the canonical controller C canonical. Theorem 2. If the full plant behavior P full has the homogeneity property, then the canonical controller C canonical implements the smallest implementable behavior containing P D. Proof. Denote the behavior implemente by C canonical as K. From (1, we can infer that K P D. To show that C canonical implements the smallest implementable behavior containing P D, consier any other controller C that implements K such that K P D. We shall prove that K K. Take any element w K. We are going to show that w K. If w P D, then w K, since K P D.

4 4 A.A. Julius et al. / Systems & Control Letters ( DESIRED CONTROLLED BEHAVIOR control CANONICAL CONTROLLER Fig. 4. The canonical controller C canonical in action. If w/ P D, there exists a c C an w P D such that both (w, c an (w,c are elements of P full. Now we are going to show that w/ K is a contraiction. Suppose that it is true, then {c C (w, c P full } C =. By the homogeneity property, we also have that {c C (w, c P full }={c C (w,c P full }. Thus, the following relation is also true: {c C (w,c P full } C =. This implies w / K, which is a contraiction. Remark3. Theorem 2 also tells us that for every control problem involving a plant with the homogeneity property, the smallest implementable behavior containing P D exists. Obviously, a necessary conition for implementability of D is D P. This fact, combine with Theorem 2 gives C canonical its special property. Corollary 4. Assume the full plant behavior has the homogeneity property. Then, the canonical controller C canonical implements D if an only if D is implementable (that is, if an only the control problem is solvable. As alreay note, if P full is a linear behavior, then it has the homogeneity property. Hence, although seemingly restrictive, the class of behaviors with the homogeneity property is, in fact, fairly large, an most importantly, it captures the class of linear time-invariant behaviors, the subject of Sections 3 5. We give LTI behaviors as examples of behaviors with the homogeneity property. For that of behaviors without the homogeneity property, refer to the plant behavior epicte in Fig. 2. There is, in fact, a secon canonical controller that is of interest, an has been introuce in [16]. The canonical controller, enote as C canonical, is efine as follows, C canonical := {c C v such that (v, c P full, an (v, c P full v D }. (2 In other wors, this canonical controller accepts a control-variable trajectory c if an only if every tobe-controlle- trajectory v that can be paire with c is accepte in the esire behavior. Clearly, whatever behavior is implemente by this controller, it must be containe in D. In fact, we have the following theorem. Theorem 5. The canonical controller C canonical implements the largest implementable behavior containe in D.

5 A.A. Julius et al. / Systems & Control Letters ( 5 DESIRED CONTROLLED BEHAVIOR control CANONICAL CONTROLLER Fig. 5. The canonical controller C canonical in action. Proof. Denote the behavior implemente by C canonical as K. It is quite obvious that K D. To show that C canonical implements the largest implementable behavior in D, consier any other controller C that implements K such that K D. We shall prove that K K. Take any element w K. There exists a c C such that K W p full (w, c P full, {v W (v, c P full } K D. (3 K From (2 an (3, we can infer that c C therefore w K. canonical an Remark6. Using Theorem 5, we can also infer that for every control problem (not necessarily the ones with homogeneity property, the largest implementable behavior containe in D exists. As a consequence of Theorem 5, the canonical controller C canonical possesses the following special property. Corollary 7. The canonical controller C canonical implements D if an only if D is implementable (that is, if the control problem is solvable. Fig. 5 illustrates the action of C canonical. Notice that the connectors are replace with symbols enoting implies. For comparison, the behaviors C canonical C canonical Fig. 6. Comparison between the C canonical an C canonical. implemente by the two canonical controllers are shown in Fig Implementability of linear behaviors In the remaining of the paper we shall restrict our attention to LTI ifferential behaviors. This class of behaviors has been iscusse quite extensively in the literature, see [13,6,12], for example. In the following we give a brief introuction to the subject to make this paper self-containe. C

6 6 A.A. Julius et al. / Systems & Control Letters ( We use the symbol L w to enote the class of LTI ifferential 3 systems with w. These are ynamical systems Σ = (R, R w, B, where the behavior B can be expresse as the solutions of a system of ifferential equations ( R w = 0. (4 Here the polynomial matrix R R w [ξ]. Furthermore, the behavior B is efine to be the set of all smooth solutions to the ifferential equations, { B := w C (R, R w ( } R w = 0. (5 The ifferential equation (4 is calle a kernel representation of B an sometimes we write B = ker R(/. A kernel representation is calle minimal if the rank of the polynomial matrix is equal to the number of its rows. Often, the behavior is efine through auxiliary. In this case, we use the term manifest for the of interest, an latent for the auxiliary ones. If B L w+l is a system involving the manifest w an the latent l then it can be proven (see [13,6] that the manifest behavior B w efine by B w := {w C (R, R w l C (R, R l such that (w, l B} is also an element of L w. This result is referre to as the elimination theorem. Remark8. The choice of the unerlying function space C (R, R w is mae for mathematical convenience. An alternative that is quite commonly use is to regar the behavior as the collection of weak solutions of the ifferential equation (4, which are elements of L loc 1 (R, Rw, the space of locally integrable functions. We refer to [6] for further exposition on this issue. We return to the control problem iscusse in Section 1. For linear time-invariant ifferential systems, the control problem can be formulate as follows. The plant behavior P full L w+c is expresse in terms of 3 The analysis also hols if ifference equations were use instea of ifferential equations; however, we restrict our attention to ifferential equations in orer to ease the exposition. the w an the control c. The controller behavior C is an element of L c. The controlle behavior K efine by K := {w C (R, R w c C such that (w, c P full }, is an element of L w (as a consequence of the elimination theorem. For linear ifferential systems, the implementability question becomes: Question. Given P full L w+c, which behaviors K L w can be implemente by using a suitable controller C L c? The answer to this question is summarize in the following theorem. Theorem 9. Given P full L w+c, the behavior K L w is implementable if an only if N K P, (6 where N L w is the hien behavior efine by N := {w C (R, R w (w, 0 P full }, an P is the manifest plant behavior efine by P := {w C (R, R w c C (R, R c such that (w, c P full }. This result was first publishe in [15] an subsequently use in [2,5,8 11]. It is important to notice that the controller that implements K is usually not unique. Generally speaking, the controllers that implement the same behavior may have very ifferent properties. 4. Regular interconnections In the previous section, we have been iscussing interconnection of linear ifferential behaviors without consiering any further restrictions. We now introuce a notion of compatibility in the control problem. In orer to motivate it, consier the following example.

7 A.A. Julius et al. / Systems & Control Letters ( 7 Let P an C be efine as the following: { P := w C (R, R w } 2 w 2 w = 0, { C := w C (R, R w w } + w = 0. We can easily verify that P is an autonomous behavior [6], that is, any trajectory in P is completely characterize by its past. Moreover, this behavior has unstable exponential trajectories. Thus, P is an unstable autonomous behavior. Now, consier the interconnection P C, which consists of the trajectories in the intersection of P an C, i.e. P C. (Here, enotes the interconnection of two systems, while enotes the intersection of the behaviors of the two systems. For the above example, notice that the unstable trajectories in P (the trajectories that are not boune as t o not belong to the interconnection P C. Moreover, since all trajectories in P C are stable (i.e., lim t w(t = 0 for all elements w in P C weinfer that the interconnection of P an C yiels a stable behavior. Therefore, if we o not a any further restrictions on the amissible controllers, it is perfectly possible that an autonomous unstable behavior is stabilize. Such controllers may be impossible to implement. More on this can be foun in [14,3,10]. In orer to cope with this, we introuce the concept of compatibility. With this concept, interconnections like the one in the previous paragraph are iscounte. The control problem then becomes to fin a compatible controller instea of just to fin any controller. A notion of compatibility for behavior interconnections in a general sense, not limite to just LTI systems, has been stuie in [3]. For LTI systems, this general notion is relate to the concept of regular interconnection, that has been introuce before in [14]. Consier a behavior B L w. Let R(/w=0 be a minimal kernel representation of B. Being minimal, R has at least as many columns as it has rows. Let g be the number of rows. The number of columns is obviously w. Therefore, g w. This means that we are always able to select g columns from R to form a square polynomial matrix with nonzero eterminant. Notice that the selection is generally not unique. If we group together the components of w corresponing to the g selecte rows an call them y, an o similarly to the remaining w g components an call them u, we en up with partitioning w into output y an input u. The reason u is calle input is because it is free, in the following sense. For any choice of C input trajectory u, we can always fin an output trajectory y such that (u, y B. Notice that the number of inputs an outputs are properties of the behavior, an are inepenent of the minimal kernel representation use to represent the behavior. Therefore, we can efine two maps m an p such that m : L w {0, 1,...,w} an p : L w {0, 1,...,w}, which give the number of inputs an the number of outputs of a given behavior, respectively. Obviously, for any behavior B L w,we have that m(b + p(b = w. Definition 10. The interconnection of two behaviors B 1 an B 2 is sai to be regular if p(b 1 + p(b 2 = p(b 1 B 2. In a sense, regularity implies that the set of equations governing the ynamics of both behaviors are inepenent of each other. We shall now apply regularity to the control problem. Recalling the efinitions of the full plant behavior P full an the controller behavior C, we efine the full controlle behavior K full as K full := {(w, c P full c C}. The interconnection between the plant an the controller is regular if p(p full + p(c = p(k full. If this is the case, then the controller C is calle a regular controller. It can be shown that a controller is regular if an only if it can be realize as a (possibly non-proper transfer function from the output to the input of P full. Therefore, a controller is regular if it can be viewe as an intelligent controller that processes sensor outputs into actuator inputs. See [14] for more etails. Given the formulation of regular interconnection, we recast the question of implementability of K in the previous section into the question of regular implementability of K.

8 8 A.A. Julius et al. / Systems & Control Letters ( Question. Given P full L w+c, which behaviors K L w can be implemente by using a suitable regular controller C L c? It turns out that regular implementability involves controllability of the plant. For more on the concept of controllability from the behavioral systems theory point of view, we refer the reaer to [6]. In fact, it has been proven in [4,14] that every implementable behavior K of a controllable plant P is regularly implementable. A necessary an sufficient conition for regular implementability, even when the plant may not be controllable, is given in [2]. Theorem 11. Given a full plant behavior P full L w+c. Denote the hien behavior an the manifest plant behavior as N an P respectively. Let P controllable be the controllable part of P. The behavior K L w is regularly implementable if an only if (1 K is implementable, i.e. N K P an (2 K + P controllable = P. Regular implementability of a behavior K implies the existence of at least one regular controller that implements it. In general, given a regularly implementable behavior, there exist irregular controllers that implement it. The question that we aress in the rest of this section is: Uner what conitions on the plant P full an the controlle behavior K, can we conclue that every controller that implements K is regular? It turns out that the answer to this question oes not epen on K, but just on the plant. Define the control variable plant behavior P c L c as follows: P c := {c w such that (w, c P full }. We have the following result. Theorem 12. Let P full L w+c be the full plant behavior. Given any controlle behavior K L w, every controller C L c that implements K is a regular controller if an only if P c = C (R, R c. Proof. Let ( R w + M ( c = 0 be a minimal kernel representation of P full. Note that P c = C (R, R c is equivalent to R having full row rank. (If Take any controller C L c. Let C(/c= 0 be its minimal kernel representation. Since R has full row rank, it follows that ( ( R M ( 0 C ( ( w = 0 c is a minimal kernel representation of K full. Therefore, p(k full = rank R + rank C = p(p full + p(c. Hence the controller is regular. (Only if Suppose that P c = C (R, R c. Let P(/c = 0 be a minimal kernel representation of P c. Note that P = 0. It follows that if we choose a controller that has the same minimal kernel representation as P c, then the resulting interconnection is not regular. 5. Control with the canonical controller Let us revisit the formulation of the control problem for linear time-invariant systems. We are given a full plant behavior P full. Let ( ( R w + M c = 0 (7 be a minimal kernel representation of P full. We are also given a esire controlle behavior D, whose minimal kernel representation is D(/w = 0. The behavior of the first canonical controller C canonical Lc is efine as C canonical := {c C (R, R c v W such that (v, c P full an v D}. Obviously, a kernel representation for this controller can be obtaine by eliminating v from the following kernel representation: ( ( R M ( ( D ( v = 0. (8 0 c

9 A.A. Julius et al. / Systems & Control Letters ( 9 The behavior of the secon canonical controller is efine as C canonical := {c C (R, R c v { } (v, c Pfull, an such that. (v, c P full v D (9 Recall the fact that LTI behaviors satisfy the homogeneity property. For linear time-invariant ifferential systems, these two canonical controllers are essentially equivalent, as shown in the following theorem. Theorem 13. The following three statements are equivalent: (i C canonical = C canonical. (ii The secon canonical controller C canonical is not empty. (iii The hien behavior N is containe in the esire controlle behavior D, i.e. N D. Proof. We prove (i (ii (iii (i. (i (ii: Notice that the zero trajectory is always containe in C canonical, therefore C canonical is never empty. (ii (iii: We first prove that for any c 1 an c 2 in C canonical, their linear combinations are also in. Notice that from efinition (9, it is not C canonical clear that this is the case. Suppose that c 1 an c 2 are in C canonical. There exist w 1 an w 2, both in D, such that (w 1,c 1 an (w 2,c 2 are both in P full. Now take any linear combination α 1 c 1 + α 2 c 2. For any w such that (w, α 1 c 1 +α 2 c 2 P full, the following reasoning hols: (w, α 1 c 1 + α 2 c 2 P full linearity of P full (w α 1 w 1 α 2 w 2 + w 1,c 1 P full, property of C canonical (w α 1 w 1 α 2 w 2 + w 1 D, linearity of D w D. Hence, if C canonical is nonempty, it is obvious that the zero trajectory is inclue in C canonical. Let K canonical be the controlle behavior implemente by C canonical. Since 0 C canonical, we have that N K canonical. From Theorem 5 we also know that K canonical D. Hence N D. (iii (i: Takeanyc C canonical. There exists a w D such that (w, c P full. Take any other w such that (w,c P full, then we also have (w w, 0 P full. Therefore (w w N D. By the linearity of D, we conclue that w D an therefore c C canonical. We have shown that C canonical C canonical. The converse is obvious from the efinitions of the canonical controllers. This theorem implies that if the control problem is solvable, then the two canonical controllers are equal. Motivate by this theorem, we shall consier in the subsequent iscussion only the first canonical controller C canonical. The question what controlle behavior is actually implemente by the canonical controller is answere by the following theorem. Theorem 14. Consier P full L w+c an D L w. The controlle behavior K implemente by the canonical controller C canonical Lc is K = N + D P with N the hien behavior an P the manifest plant behavior. Proof. Let the kernel representation of P full be given by (7, an let D be represente by D(/w = 0. We then know that a kernel representation of C canonical can be obtaine by eliminating w from (8. Therefore, K is the manifest behavior (with w as the manifest variable of the behavior represente by R ( M ( 0 0 M ( R ( [ ] w 0 0 D ( c = 0. v Notice that K is then also the manifest behavior (with w as the manifest variable of the behavior represente by R ( M ( R ( 0 0 D ( [ ] w v c = 0. (10 v Now efine w := w v, we can see from (10 that the ynamics of w is ecouple from that of c an v. Furthermore, the behavior of w is exactly N (see Section 3. The secon an thir rows of (10 inicate that the behavior of v, which is obtaine by eliminating c, isd P. From here, using the fact

10 10 A.A. Julius et al. / Systems & Control Letters ( that w = w + v, we obtain K = N + D P. This result is not unexpecte. In fact, we can see it as an application of Theorem 2 to the special case of LTI behaviors. Similarly, we can apply Corollary 4 to LTI systems to obtain the following corollary. Corollary 15. The canonical controller C canonical L c implements D L w if an only if D is implementable, i.e. N D P. So far we have seen that when D is implementable, the canonical controller C canonical implements it. However, as we have seen in Section 4, implementability alone may not be goo enough. In the following we shall aress the issue of regularity of the canonical controller C canonical. Theorem 16. Given a full plant behavior P full L w+c an a esire controlle behavior D L w. Assume that D is implementable. The canonical controller C canonical implements D regularly if an only if P c = C (R, R c. Proof. (If Follows irectly from Theorem 12. (Only if Without loss of generality, we can assume that P full has a minimal kernel representation of the following form. [ R1 ( M 1 ( 0 M 2 ( ][ ] w = 0, c with both R 1 an M 2 having full row rank. The kernel representations of N an P c are then given by R 1 (/w=0 an M 2 (/c=0, respectively. Since N D, we are able to fin a suitable full row rank matrix F(/ such that F(/R 1 (/w= 0isa minimal kernel representation of D. Therefore a kernel representation of the canonical controller C canonical can be obtaine by eliminating v from ( R ( 1 M 1 ( 0 M 2 F ( ( R1 0 [ v c ] = 0. Since R 1 has full row rank, we easily obtain the following kernel representation of C canonical (possibly non-minimal. [ ( M ] 2 F ( ( c = 0. M1 We see that C canonical always repeats some laws of P full, namely the rows in M 2. Thus C canonical is regular only if M 2 is the zero matrix, which implies P c = C (R, R c. This result, combine with Theorem 12, tells us that the canonical controller is maximally irregular, in the sense that if there exists any irregular controller that implements the esire behavior D, then the canonical controller is irregular too. 6. Concluing remarks The iea of the canonical controller in the behavioral framework is attractive because of its simplicity of construction an also since it formalizes the internal moel principle without unue recourse to the equations with which the plant is escribe. This approach of builing systems without using the equations explicitly unerlines the representation free nature of behavioral theory. Some specific issues are summarize here to highlight the main results of the paper. We efine two canonical controllers. For the case of linear time-invariant behaviors (or more generally, for plants with homogeneity property, when the esire behavior D is implementable, the canonical controllers are the same (see Theorem 13. However, they can iffer when D is not implementable. In this situation each of the two canonical controllers are extreme in a certain sense. The first canonical controller implements the smallest implementable behavior that contains P D. The secon canonical controller C canonical implements the largest implementable behavior containe in D. These statements were formulate an prove in Theorems 2 an 5, respectively. For the case of linear time-invariant behaviors, the implementability of D implies non-emptiness of C canonical (Theorem 13. When C canonical is nonempty, it is a linear subspace of C (R, R, an hence contains the zero trajectory. Thus, we have a necessary conition for implementability of D, namely if the zero trajectory belongs to C canonical.

11 A.A. Julius et al. / Systems & Control Letters ( 11 We then aresse the issue of regularity of a controller with respect to a given plant. It turns out that the conition of guarantee regularity of every controller that implements a given esire behavior, is a property of just the plant, an is inepenent of the given esire behavior. The issues of regularity of every controller an the canonical controllers are relate by the results in the final section. Here we showe that, given a plant, irregularity of any controller implies irregularity of the canonical controller, an hence we terme the canonical controller as maximally irregular. Acknowlegements This research is supporte by the Dutch NWO Project No , Compositional Analysis an Specification of Hybri Systems, by the Belgian Feeral Government uner the DWTC program Interuniversity Attraction Poles, Phase V, , Dynamical Systems an Control: Computation, Ientification an Moelling, by the KUL Concerte Research Action (GOA MEFISTO 666, an by several grants an projects from IWT-Flaners an the Flemish Fun for Scientific Research. References [1] M.N. Belur, Control in a behavioral context, Ph.D. Thesis, University of Groningen, June [2] M.N. Belur, H.L. Trentelman, Stabilization, pole placement an regular implementability, IEEE Trans. Automat. Control 47 ( [3] A.A. Julius, A.J. van er Schaft, Compatibility of behavior interconnections, in: Proceeings of the European Control Conference, Cambrige, IEE, Lonon, September [4] J.W. Polerman, Sequential continuous time aaptive control: a behavioral approach, in: Proceeings of the 39th IEEE Conference on Decision an Control, Syney, IEEE, New York, 2000, pp [5] J.W. Polerman, I. Mareels, A behavioral approach to aaptive control, in: J.W. Polerman, H.L. Trentelman (Es., Mathematics of System an Control, Festschrift on the occasion of the 60th birthay of J.C. Willems, Groningen, 1999, pp [6] J.W. Polerman, J.C. Willems, Introuction to Mathematical Systems Theory: A Behavioral Approach, Springer, New York, [7] P. Rocha, Regular implementability of nd behaviors, in: Proceeings of Mathematical Theory of Networks an Systems, University of Notre Dame, [8] H.L. Trentelman, A truly behavioral approch to the control problem, in: J.W. Polerman, H.L. Trentelman (Es., Mathematics of System an Control, Festschrift on the occasion of the 60th birthay of J.C. Willems, Groningen, 1999, pp [9] H.L. Trentelman, J.C. Willems, Synthesis of issipative systems using quaratic ifferential forms: part II, IEEE Trans. Automat. Control 47 ( [10] A.J. van er Schaft, Achievable behavior of general systems, Systems Control Lett. 49 ( [11] A.J. van er Schaft, A.A. Julius, Achievable behavior by composition, in: Proceeings of the 41st IEEE Conference on Decision an Control, Las Vegas, IEEE, New York, 2002, pp [12] J.C. Willems, jwillems. [13] J.C. Willems, Paraigms an puzzles in the theory of ynamical systems, IEEE Trans. Automat. Control 36 ( [14] J.C. Willems, On interconnections, control an feeback, IEEE Trans. Automat. Control 42 ( [15] J.C. Willems, Behaviors, latent, an interconnections, Systems Control Inform. (Japan 43 ( [16] J.C. Willems, M.N. Belur, A.A. Julius, H.L. Trentelman, The canonical controller an its regularity, in: Proceeings of the IEEE Conference on Decision an Control, Hawaii, December 2003, pp [17] J.C. Willems, H.L. Trentelman, Synthesis of issipative systems using quaratic ifferential forms, Part I, IEEE Trans. Automat. Control 47 (