Systems & Control Letters

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1 Systems & ontrol Letters ( ) ontents lists available at ScienceDirect Systems & ontrol Letters journal homepage: A converse to the eterministic separation principle Jochen Trumpf a,, Harry L. Trentelman b a esearch School of Engineering, Australian National University, anberra, Australia b Johann Bernoulli Institute for Mathematics an omputer Science, University of Groningen, The Netherlans article info abstract Article history: Available online xxxx Keywors: Separation principle Output feeback Linear systems Behavioral approach In the classical theory of finite-imensional linear time-invariant systems in state space form the term eterministic separation principle refers to the observation that a stabilizing output feeback controller can be constructe by first constructing an asymptotic state observer that is then couple to a stabilizing state feeback controller. In this paper we iscuss the following converse problem: an every stabilizing output feeback controller be realize as interconnection of an asymptotic state observer an a stabilizing state feeback controller? We will provie an affirmative answer to this question (moulo a number of technicalities) in a behavioral setting an with the help of rational representations. 216 Elsevier B.V. All rights reserve. 1. Introuction The classical eterministic separation principle says that, given the plant ẋ Ax + Bu, y x, an asymptotic full state observer ˆx (A G)ˆx + Bu + Gy (1) with A G Hurwitz, an a stabilizing static full state feeback controller u F ˆx (2) with A + BF Hurwitz, the close-loop ynamics is given by x A + BF BF x, (3) e A G e where e ˆx x is the observer error [1]. It follows from the form of the system matrix in (3) that lim t1 x(t), i.e. the observerbase output feeback controller (1) an (2) is stabilizing. In fact, it is even internally or totally stabilizing since x implies both y an ˆx (since also e ), an hence also u. orresponing author. aresses: Jochen.Trumpfanu.eu.au (J. Trumpf), h.l.trentelmanrug.nl (H.L. Trentelman) / 216 Elsevier B.V. All rights reserve. The above principle is calle a separation principle because it allows to complete the task of constructing an output feeback controller with esirable properties (namely stability) by separately constructing a state observer an a full state feeback controller with that property. Another classical but unrelate separation principle is that of optimal stochastic control, see e.g. [2]. An obvious converse question is whether there are any other constructions of (totally) stabilizing output feeback controllers, or whether any such controller permits an interpretation as a series connection of a full state observer followe by a (possibly ynamic) full state feeback controller. A partial answer to this question was given by Schumacher using the geometric notion of compensator couples at the beginning of the 198s [3], but a full answer remaine elusive to this ate. In this paper we aress the converse question in a behavioral framework using both polynomial an rational representations of linear ifferential systems. We show that, uner mil assumptions on the to be controlle system with variables (x, u, y), any controllable, regular, totally stabilizing controller through the variables (u, y) can be separate into an asymptotic i/o-observer for x from (u, y) with variables (ˆx, u, y) an a regular, totally stabilizing controller with variables (ˆx, u) in the sense that the controllable part of the observer/(ˆx, u)-controller interconnection coincies with the given (u, y)-controller. The paper is organize as follows. Section 2 introuces our notation an collects relevant results from the theory of behaviors incluing basics on rational representations. In Section 3 we review the require material on stabilization in a behavioral framework. In Section 4, we evelop a convenient system representation that

2 2 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) is aapte to the problem treate in this paper. Section 5 contains the main result, Section 6 iscusses the special case of state space systems an Section 7 conclues the paper. 2. Behaviors of linear ifferential systems In this paper we will make heavy use of the mathematical machinery of the behavioral approach to linear ifferential systems. A linear ifferential system is efine as a triple (, w, B) whose behavior B 1 (, w ) is the solution space of a finite set of higher orer constant coefficient linear ifferential equations Polynomial an rational kernel representations Behaviors of linear ifferential systems can be represente in terms of a real polynomial matrix (s) with w columns as ( )w, so that B w 2 1 (, w ) w. (4) The representation (4) is calle a polynomial kernel representation of B, an we often write B ker ( ). If 1 an 2 are two full row rank polynomial matrices, then they represent the same behavior B, i.e. B ker 1 ( ) ker 2( ), if an only if there exists a unimoular polynomial matrix U such that 2 U 1. For an extensive treatment of polynomial representations of behaviors we refer to [4]. Behaviors also amit representations in terms of real rational matrices. A etaile exposition on rational representations can be foun in [5]. Here we will give a brief review. ecall that any given real rational matrix amits a left coprime factorization into polynomial matrices. A factorization of a real rational matrix as P 1 Q with P, Q real polynomial matrices is calle a left coprime factorization if P Q is left prime (meaning that it has a polynomial right inverse) an et(p) 6. Following [5], if P 1 Q is such a left coprime factorization then we efine w to be a solution of ( )w if it is a solution of the ifferential equation Q ( )w. In other wors, we efine ker : ker Q, (5) which is well-efine since any two left coprime factorizations of iffer by a unimoular polynomial factor. For a given rational matrix, we call a representation of B as ( )w arational kernel representation of B an write B ker ( ). For aitional material on rational representations we refer to [6,7]. In this paper we will often assume that the rational matrices (s) use in kernel representations have full row rank over the fiel of real rational functions. This is equivalent to saying that the kernel representation is minimal, see [5,6]. As note before, two minimal polynomial kernel representations iffer by a unimoular polynomial factor. A similar statement oes not hol for rational representations. We will come back to this in the next subsection ontrollability an controllable part In the behavioral approach an important role is playe by the property of controllability. The efinition of controllability of a behavior B is well known, an can be foun in [4]. ontrollability of a behavior can be teste in terms of its full row rank rational kernel representations as follows: If B ker ( ) where (s) is a rational matrix, then B is controllable if an only if has no zeros. A behavior B is calle autonomous if it is a finite imensional subspace of 1 (, w ). In terms of its rational kernel representations B ker ( ) this property requires that has full column rank. Any behavior B amits a irect sum ecomposition as B B cont B aut, where B cont, calle the controllable part of B, is the largest controllable subbehavior of B, an B aut, calle an autonomous part, is an autonomous subbehavior of B. The controllable part is uniquely etermine by B. In terms of its rational kernel representations ( )w, the controllable part of B can be foun by factorizing Q with Q nonsingular rational an a left prime polynomial matrix. For any such factorization we have B cont ker ( ), see [5]. It was shown in [6] that if 1 an 2 are full row rank rational matrices, then there exists a nonsingular rational matrix Q such that 2 Q 1 if an only if 1 an 2 represent behaviors with the same controllable part, i.e. (ker 1 ( )) cont (ker 2 ( )) cont Elimination of variables We will now review the basics of elimination of variables. Suppose we have a behavior B in which the manifest variable is partitione into two parts as w (v, c). Let v ( )v + c( )c be a polynomial kernel representation of B. The space of trajectories that satisfies this equation is calle the full behavior. The space of tractories c that are compatible with the equation of the full behavior is calle the behavior with v eliminate an is given by B c : c there exists v such that v v + c c. (6) The elimination problem is to obtain a kernel representation of (6). Such a kernel representation can be obtaine as follows: first fin a unimoular polynomial matrix U such that v,1 U v where v,1 has full row rank. Next, apply the same unimoular matrix to c to obtain c,1 U c. c,2 Since ker v c ( ) ker U v c ( ), a new, more structure, polynomial kernel representation of B is then given by v,1 c,1 v. c A kernel representation of the behavior (6) is now given by c,2 ( )c (see [4]). The above construction to obtain the eliminate behavior (6) is only vali for polynomial kernel representations an uses unimoular premultiplication. Its counterpart for the case that we eal with rational kernel representations an, instea of unimoular premultiplication, we use premultiplication with a nonsingular rational matrix is more subtle an is ealt with in the following lemma. Lemma 2.1. Let v ( )v + c( )c be a rational kernel representation of B. Let Q be a nonsingular rational matrix such that v,1 Q v

3 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) 3 where v,1 is a full row rank polynomial matrix. Partition c,1 Q c c,2 an assume that c,1 is also a polynomial matrix. Then the controllable parts of ker c,2 ( ) an B c are equal. Proof. Define the rational matrix by v,1 : c,1. (7) Since Q ( v c ), we have that B cont (ker ( )) cont. Let c,2 P 1 2 Q 2 be a left coprime factorization. Then ker c,2 ( ) ker Q 2 ( ), an also 1 I v,1 c,1 P 2 Q 2 is a left coprime factorization, so we have v,1 ker ker c,1. Q 2 Since v,1 has full row rank, we have (ker ( )) c ker Q 2 ( ) ker c,2 ( ). This yiels (B c ) cont (B cont ) c ((ker ) cont ) c ((ker ) c ) cont ker c,2 cont, where we have omitte the ifferentiation symbol. Here we have use the fact that the operation of taking the controllable part of a behavior, an eliminating a variable from a behavior commute, see e.g. Lemma in [8]. We now stuy the relate question whether the triangular structure in (7) amits a similar triangular structure in a representation of the controllable part. More specifically, assume that the behavior B is represente by the rational kernel representation associate with (7), oes there exist a representation of its controllable part with compatible triangular structure. This inee turns out to be the case as is shown in the next lemma. Lemma 2.2. onsier the behavior B represente by the full row rank rational representation 1 v,1 c,1 B v A. (8) c Assume that v,1 an c,1 are polynomial, an v,1 has full row rank. Then there exist polynomial matrices v,1, c,1, c,2, an rational matrices Q 11, Q 12 an Q 22, with Q 11 an Q 22 square nonsingular, such that v,1 c,1 Q 11 Q 12 v,1 c,1 Q 22 an such that 1 v,1 c,1 B v A c is a representation of the controllable part B cont of B. Proof. First note that since v,1 an c,1 are polynomial matrices, an v,1 has full row rank, c,2 ( )c is a kernel representation of B c. Factorize v,1 c,1 Q11 Q 12 v,1 c,1 (9) Q 21 Q 22 v,2 c,2 where the first factor is nonsingular rational an the secon left prime polynomial. The secon factor yiels a kernel representation of B cont. We now eliminate v from B cont : by premultiplying the secon factor in (9) with a suitable unimoular polynomial matrix we obtain v,1 c,1 U11 U 12 v,1 c,1, (1) U 21 U 22 v,2 c,2 with v,1 full row rank. Then c,2 ( )v is a representation of (B cont ) c. By combining (9) an (1) we obtain v,1 c,1 Q 11 Q 12 v,1 c,1 (11) Q 21 Q 22 for suitable rational matrices Q ij. From (11) we obtain Q 21 v,1. Since v,1 has full row rank this yiels Q 21. Also we rea that c,2 Q 22 c,2. Since (B c ) cont (B cont ) c, the controllable part of ker c,2 ( ) is equal to ker c,2( ). Since both c,2 an c,2 have full row rank, this implies that Q 22 must be square. Then also Q 11 must be square, an both must be nonsingular Inputs an outputs Often, the manifest variable w of a behavior B is partitione as w col(u, y), an accoringly the rational kernel representation takes the form u ( )u + y( )y. This partitioning is calle an input output partition of the behavior if u is input an y is output, see [5]. In terms of the full row rank rational matrices this is equivalent to the property that y is a nonsingular rational matrix. In general a behavior amits many input output partitions. However, the sizes of u an y (enote by m(b) an p(b), respectively) are uniquely etermine by the behavior. It is well known that if ( )w is a kernel representation of B (either polynomial or rational), then the number of outputs p(b) of B is equal to rank. 3. Stabilization in a behavioral framework 3.1. Stabilization by full interconnection Given a behavior B (calle the plant) with polynomial kernel representation ( )w an a behavior (calle a controller) with polynomial kernel representation ( )w, the interconnection of B an is simply their intersection B \. learly, the interconnection has the kernel representation 1 B A w. The interconnection is calle a regular interconnection if p(b\) p(b) + p(). If both an have full row rank, then the interconnection is regular if an only if the matrix (12)

4 4 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) has full row rank. The problem of stabilization by full interconnection is to fin, for B, a controller such that their interconnection is regular, an lim t1 w(t) for all w 2 B \. The controller is then calle a stabilizing controller for B. If both an have full row rank, then is a stabilizing controller for B if an only if the polynomial matrix (12) is Hurwitz, see [9,1]. Next we will prove that for each stabilizing controller for a given plant, also its controllable part is a stabilizing controller. Lemma 3.1. Let be a stabilizing controller for B. Then cont is a stabilizing controller for B. Proof. Let ( )w be a minimal polynomial kernel representation of B an let ( )w be a minimal polynomial kernel representation of. Then the polynomial matrix (12) is Hurwitz. amits a factorization Q, with Q nonsingular polynomial an where ( )w is a minimal representation of the controllable part cont of. Then from the fact that I Q it is easily seen that the matrix must be Hurwitz as well. The above can be extene to stable rational kernel representations for the controller. We o however ahere in this paper to polynomial kernel representations of the plant B. It is easily seen that if the plant B has full row rank polynomial kernel representation ( )w an the controller has full row rank rational kernel representation ( )w then is a stabilizing controller if an only if the rational matrix (12) is nonsingular an has all its zeros in. A rational matrix is calle a stable rational matrix if all its poles lie in. A square nonsingular stable rational matrix is calle miniphase if its inverse is again stable, equivalently if all its zeros lie in. If ( )w is a rational kernel representation of the controller with a full row rank stable rational matrix, then it is a stabilizing controller if an only if the rational matrix (12) is miniphase, see [5]. For a given plant B, a stabilizing controller exists if an only if it is stabilizable. A efinition of stabilizability of behaviors can be foun in [4]. If B is given by the full row rank rational kernel representation ( )w, then B is stabilizable if an only if ( ) has full row rank for all 2 +, see [5]. In the following lemma we will give a parametrization of all stabilizing controllers for a given plant B. Lemma 3.2. Let B be stabilizable, an let ( )w be a minimal polynomial kernel representation. Let be a stable rational matrix such that (13) is miniphase. Then a controller ker ( ) with full row rank rational is a stabilizing controller for B if an only if there exists a rational matrix F an a square nonsingular rational matrix G with all its zeros in such that F + G. Proof. Let F + G with F rational an G square nonsingular an all its zeros in. We have I, F G which is clearly square nonsingular an has all its zeros in. Hence ker ( ) is a stabilizing controller. onversely, let ker ( ) be a stabilizing controller, an full row rank. Then is square nonsingular an has all its zeros in. Define 1 Q11 Q 12 :. Q 22 Q 21 This prouct is clearly square nonsingular an has all its zeros in. Also, it yiels Q 11 + Q 12, which implies that (I Q 11 Q 12 ). This clearly implies Q 11 I an Q 12. Now efine F : Q 21, an G : Q 22. Then G is square nonsingular an has all its zeros in an, finally, F + G Stabilization by partial interconnection In general, the plant B has two types of variables, the variable w to be controlle, an the variable c, calle the interconnection variable, through which the plant can be interconnecte with a controller. More specific, if the plant B has the polynomial kernel representation w ( )w + c( )c, an if a controller has the polynomial kernel representation ( )c, then the interconnection of B an through c is efine as the behavior B ^c : {(w, c) (w, c) 2 B an c 2 }. As before, the interconnection is calle regular if p(b) + p() p(b ^c ). In this paper we will consier the problem of total stabilization for the plant B. A controller is calle a totally stabilizing controller for B if the interconnection is regular an if lim t1 (w(t), c(t)) for all (w, c) 2 B ^c. If both plant an controller are represente by a minimal polynomial kernel representation, then is a totally stabilizing controller for B if an only if w c (14) is Hurwitz. Again, the above can be extene to stable rational kernel representations for the controller. We o however again ahere to polynomial kernel representations of the plant B. If ( )w is a rational kernel representation of the controller with a full row rank stable rational matrix, then it is a totally stabilizing controller if an only if the rational matrix (14) is miniphase. If is a full row rank rational matrix (not necessarily stable) then it is a totally stabilizing controller if an only if (14) is a square nonsingular rational matrix with all its zeros in. We will now eal with the question uner what conitions there exists a totally stabilizing controller for a given plant B. These conitions involve stabilizability an etectability. Given the plant behavior B with variable (w, c), we say that w is etectable from c if (w, ) 2 B implies lim t1 w(t). If B is represente by the polynomial kernel representation w ( )w + c( )c, then w is etectable from c if an only if w ( ) has full column rank for all 2 +, see [4]. The following lemma gives necessary an sufficient conitions on the plant B for the existence of a totally stabilizing controller. Lemma 3.3. Given a plant B with variable (w, c), there exists a totally stabilizing controller if an only if B is stabilizable an w is etectable from c in B.

5 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) 5 Proof. Let w ( )w + c( )c be a minimal polynomial kernel representation of B. By unimoular premultiplication we can obtain an alternative kernel representation in the triangular form 1 w,1 c,1 B w A, (15) c with w,1 full row rank. Using the assumption that w is etectable from c we fin that, in fact, w,1 ( ) has full column rank for all 2 +. As a consequence, w,1 is square an Hurwitz. By stabilizability of B, c,2 ( ) has full row rank for all 2 +. Thus, the behavior represente by c,2 ( )c itself is stabilizable, so there exists a full row rank polynomial matrix such that c,2 is Hurwitz. learly then w,1 c,1 is Hurwitz, so the controller ker ( ) totally stabilizes B. onversely, assume that there exists such that w c (16) is Hurwitz. Then obviously ( w ( ) c ( )) has full row rank for all 2 +, so B is stabilizable. Now, assume that (w, ) 2 B. Then, clearly, also (w, ) 2 B ^c. Since the controller is totally stabilizing we must have w(t) as t 1, so in B, w is etectable from c. 4. A convenient system representation As announce in Section 1, in this paper we will consier linear ifferential systems P with system variable (x, u, y), where x is interprete as the variable to be controlle, an (u, y) as the interconnection variable through which P can be interconnecte with a controller. We will make the following assumptions on P. (A1) P is stabilizable, (A2) x is etectable from (u, y), (A3) u is input with (x, y) output, (A4) im(y) apple im(x). We will now briefly iscuss the above assumptions. First note that by Lemma 3.3, assumptions (A1) an (A2) are necessary an sufficient for the existence of a totally stabilizing controller for P. In Section 2.3 it was explaine that starting from a minimal polynomial kernel representation x x + u u + y y of P, we can always obtain a new, more structure, minimal polynomial kernel representation for P of the form x x,1 u,1 y,1 u, (17) y in which x,1 has full row rank. We will stuy how our assumptions are reflecte in the polynomial matrices in the representation (17). First note that (A2) hols if an only if x,1 has full column rank for all 2 +. Since x,1 has also full row rank we therefore have that assumption (A2) is equivalent with x,1 being Hurwitz. Next, assumption (A3) is equivalent to the conition that the submatrix x,1 y,1 y,2 is square an nonsingular. In particular this implies that y,2 is square an nonsingular. Finally, as a consequence of assumption (A4) we have that the matrix y,1 is tall, meaning that its number of columns oes not excee its number of rows. By applying Lemma 2.2, there exist polynomial matrices x,1, u,1, y,1, u,2, y,2 an rational matrices Q 11, Q 12 an Q 22, with Q 11 an Q 22 square nonsingular, such that x,1 u,1 y,1 Q 11 Q 12 x,1 u,1 y,1 Q 22 an such that v x,1 u,1 y,1 u (18) y is a representation of the controllable part P cont of P. Obviously, uner the assumptions mae above x,1 an y,2 are both nonsingular an y,1 is tall. emark 4.1. eferring to the iscussion at the beginning of Section 3.2, every regular, totally stabilizing controller for P through (u, y) has a polynomial (an hence stable rational) representation ker u y such that x,1 u,1 y,1 u y is Hurwitz (an hence miniphase). Since 1 1 x,1 u,1 y,1 Q 11 Q 12 Q 22 A x,1 u,1 y,1 A u y I u y this implies that u,2 u,2 u y has full row rank. We will use this fact in the proof of Proposition A converse to the separation principle The following theorem is the main result of this paper. It provies a converse to the classical eterministic separation principle. Theorem 5.1. onsier a system with variables (x, u, y) such that Assumptions (A1) (A4) from Section 4 hol. Then for every controllable, regular, totally stabilizing controller through (u, y) there exists an asymptotic i/o-observer O for x from (u, y) with variables (ˆx, u, y) an a regular, totally stabilizing controller K for the plant system with variables (ˆx, u) such that O ^(ˆx,u) K (u,y),cont. (19) A few remarks are in orer before we prove this theorem. Fig. 1 illustrates the separation of the controller into the two blocks O an K. Assumptions (A1) an (A2) are clearly necessary for totally stabilizing controllers through (u, y) to exist (Lemma 3.3). Assumptions (A3) an (A4) will turn out to be sufficient conitions

6 6 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) Fig. 1. The separate controller of Theorem 5.1. for solvability of the matrix equation that is equivalent to (19), cf. Proposition 5.2. Ieally, we woul like to rop the assumption of controllability for the controller an the corresponing restriction of the theorem statement to the controllable part of the separate controller. This assumption an restriction are, however, necessitate by our use of rational representations. The general theory will likely require more avance algebraic methos an our inability to generalize the current proof using stanar polynomial or rational methos might help to explain why such a result was never obtaine in the classical state space literature (cf. [3]). Note, however, that if a controller is totally stabilizing then its controllable part is also totally stabilizing (Lemma 3.1). In orer to prove Theorem 5.1, we first erive a matrix equation formulation of conition (19). Proposition 5.2. onsier a system P with variables (x, u, y) an such that assumptions (A1) an (A2) from Section 4 hol. epresent P as in (17). onsier a controllable, regular, totally stabilizing controller ker u y through (u, y), where the polynomial representation of has been chosen as in emark 4.1. Then there exists an asymptotic i/o-observer O for x from (u, y) with variables (ˆx, u, y) an a regular, totally stabilizing controller K for P with variables (ˆx, u) such that (19) hols if an only if there exists a rational matrix Y an a stable rational matrix T such that y,1 Y YT y. (2) y,2 Proof. The proof of Proposition 5.2 procees in four major steps. In a first step we parametrize all observers O(S 1, S 2 ) for x from (u, y) using the authors recent internal moel principle for observers [11] an the algebraic generalization thereof [12]. Here, S 1 an S 2 are matrix parameters. In a secon step, we use Lemma 3.2 to provie a parametrization K(Y 1, Y 2, X) of all fully interconnecte regular stabilizing controllers for P where Y 1, Y 2 an X are matrix parameters. In a thir step we use Lemma 2.1 to show that the separate controller resulting from the interconnection of O(S 1, S 2 ) an K(Y 1, Y 2, X) has the same controllable part as if an only if we choose Y 2 an X appropriately. In a fourth an last step we characterize when the parameter Y 1 can be chosen such that K(Y 1, Y 2, X) is a controller with variables (ˆx, u) as require. Step 1. Since the system P is stabilizable, its anti-stabilizable part is equal to its controllable part, cf. Theorem 2.3 in [11]. But then O ker ˆˆx ˆu ˆy is a full row rank polynomial representation of an asymptotic i/o-observer for x from (u, y) if an only if ˆˆx is Hurwitz an P cont O, cf. Theorem 5.6 in [11] an orollary 16 in [12]; equivalently, ˆˆx ˆu ˆy S 1 S 2 x,1 u,1 y,1 S 1 u,1 + S 2 u,2 S 1 y,1 + S 2 y,2 with S 1 polynomial an Hurwitz, an S 2 polynomial. We write O(S 1, S 2 ) to inicate the epenence of O on the matrix parameters S 1 an S 2. Step 2. By Lemma 3.2, K ker K x K u K y is a fully interconnecte, regular, stabilizing controller for P if an only if Q K x K u K y Z 1 Z 11 Q 12 2 Q 22 x,1 u,1 y,1 + X u y, where Z 1 an Z 2 are rational an X is rational an non-singular square with only stable zeros. Define Y 1 Y 2 : Z 1 Z 2 Q 11 Q 12 Q 22 x,1 I an observe that the reparametrization Z 1 Z 2 $ Y 1 Y 2 is bijective. Hence K K(Y 1, Y 2, X) ker K x K u K y is a fully interconnecte, regular, stabilizing controller for P if an only if K x K u K y Y 1 Y 1 1 x,1 u,1 + Y 2 u,2 + X u Y 1 1 x,1 y,1 + Y 2 y,2 + X y (21) with Y 1 an Y 2 rational an X rational an non-singular square with only stable zeros. Step 3. ompute I Y I S 1 u,1 + S 2 u,2 S 1 y,1 + S 2 y,2 Y 1 Y 1 1 x,1 u,1 + Y 2 u,2 + X u Y 1 1 x,1 y,1 + Y 2 y,2 + X y S 1 u,1 + S 2 u,2 S 1 y,1 + S 2 y,2 (Y 2 Y S 2) u,2 + X u (Y 2 Y S 2) y,2 + X y By Lemma 2.1, (O(S 1, S 2 ) \ K(Y 1, Y 2, X)) (u,y),cont ker (Y 2 Y S 2) u,2 + X u (Y 2 Y S 2) y,2 + X y an since u,2 y,2 u y has full row rank, (O(S 1, S 2 ) \ K(Y 1, Y 2, X)) (u,y),cont if an only if Y 2 Y S 2 an X is rational an non-singular square with only stable zeros. Step 4. It only remains to characterize when K(Y 1, Y 1 1 x,1 S 1 1 S 2, X) has variables (ˆx, u) only, i.e. when K y Y 1 1 x,1 y,1 + Y 2 y,2 + X y Y 1 1 x,1 y,1 + Y S 2 y,2 + X y. Setting Y X 1 Y 1 1 x,1 an T S 1 1 S 2 shows that this is equivalent to Eq. (2). We can now complete the proof of Theorem 5.1 by showing that, uner the conitions of the theorem, Eq. (2) has a solution. Proof of Theorem 5.1. Assumption (A3) implies that y,2 is invertible. By Assumption (A4), the matrix y,1 1 y,2 is tall an hence there exists a left-invertible rational matrix B such that T : B y,1 1 y,2

7 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) 7 is stable rational: let p 1 1 q U 1 y,1 1 y,2 V p r B q r A be in Smith McMillan form then B can be chosen as p 1 1 q B : U p r V 1, B q r A I making T polynomial an hence stable rational. Let B L be a leftinverse of B, i.e. B L B I. Let Y : y 1 Y YT as require. y,1 y,2 y,2 BL then y 1 y,2 BL y,1 y 1 y,2 BL (B y,1 1 y,2 ) y,2 y In the above theorem, the only constraints place on the controller K are that it is itself regular an totally stabilizing, an that it connects to the plant only through (x, u); K is written here with variables (ˆx, u) since it is intene to be combine with the observer O. In particular, it is not assume a priori that the controller K acts by feeback or that it is even static in the classical sense. The latter woul amount to the aitional conitions Kˆx I an K u F where F is a constant matrix, yieling the controller representation u F ˆx. It can be shown that Eq. (2) oes not always have a solution uner these aitional conitions. 6. Example: the state space case eturning to the state space case that we alreay briefly iscusse in Section 1, assume that our plant P with system variable (x, u, y) is represente by ẋ Ax + Bu, y x, where x(t) 2 n, u(t) 2 m an y(t) 2 p. For simplicity, assume that the pair (A, B) is controllable an that the pair (, A) is etectable. Let be a regular totally stabilizing controller through (u, y) an assume that is controllable. Let have polynomial kernel representation u u + y y. In this section we will ientify an asymptotic i/o-observer O for x through (u, y) an a regular totally stabilizing controller K through (x, u) such that the controllable part of their interconnection equals. learly, a minimal polynomial kernel representation of P is given by x I A B u (22) I y an this reveals that, uner the above assumptions on the matrix pairs (A, B) an (, A), the behavior P is controllable (so also stabilizable) an that x is etectable from (u, y) in P. Obviously, in P, u is input an (x, y) is output, so the assumptions (A1), (A2) an (A3) of Section 4 hol. Assume that in aition we have p apple n so that also (A4) hols. Since P is controllable it is equal to its controllable part P cont. We will now first bring the kernel representation (22) in the require upper iagonal form. Let (si A) 1 L 1 2 L 1 be a polynomial left coprime factorization, an let N 1 an N 2 be polynomial matrices such that N1 N 2 L 1 L 2 is unimoular. Pre-multiplying (22) by this unimoular matrix, we see that an upper iagonal polynomial kernel representation of P P cont is given by N 1 ( x I A) + N 2 N 1 B N 2 u. L 1 B L 2 y Denote : N 1 (si A)+N 2 an note that this polynomial matrix is Hurwitz. In orer to ientify a suitable observer/controller pair we first solve the nonlinear equation (2), that in this case takes the form N2 Y YT y. L 2 Since assumptions (A1) (A4) hol, this equation inee has a solution pair (Y, T) with Y rational an T stable rational (cf. the Proof of Theorem 5.1). Next, factorize T S 1 2 S 1, with S 1 polynomial an Hurwitz, an S 2 polynomial. This yiels an asymptotic i/o-observer O for x from (u, y) with polynomial kernel representation ˆx S 1 (S 1 N 1 + S 2 L 1 )B S 1 N 2 + S 2 L 2 u. y It is inee easily verifie that if (x, u, y) 2 P an (ˆx, u, y) 2 O, then the error e : ˆx x satisfies S 1 e, so e(t) as t 1. Next we ientify a regular totally stabilizing controller K through (x, u). In general, all totally stabilizing controllers for P are parametrize by (21), with Y 1, Y 2 rational an X nonsingular rational with only stable zeros. Here we take X : I m, Y 1 : Y, Y 2 : YT. This yiels the regular totally stabilizing controller K with rational kernel representation x Y Y(N 1 + TL 1 )B + u. u Now interconnect the observer O S 1 (S 1 N 1 + S 2 L 1 )B S 1 N 2 + S 2 L 2 ˆx u y with K ˆx Y Y(N 1 + TL 1 )B + u u

8 8 J. Trumpf, H.L. Trentelman / Systems & ontrol Letters ( ) through the variables (ˆx, u). Then the controllable part of the (u, y) behavior of this interconnection is equal to the given controller as esire. 7. onclusions We have shown that uner the mil assumptions (A1) (A4) from Section 4 the converse of the eterministic separation principle hols true, namely that for a linear ifferential system with variables (x, u, y) any controllable, regular, totally stabilizing controller through the variables (u, y) can be separate into an asymptotic i/o-observer for x from (u, y) with variables (ˆx, u, y) an a regular, totally stabilizing controller with variables (ˆx, u) in the sense that the controllable part of the observer/(ˆx, u)-controller interconnection coincies with the given (u, y)-controller. Together with the fact that the controllable part of a totally stabilizing controller is also totally stabilizing an combine with the internal moel principle for observers an the structure of the interconnection, this result implies that any such controller must contain an internal moel of the controllable part of the to be controlle system. This observation shoul have interesting consequences for robustly stabilizing output feeback controllers. A characterization of those (u, y)-controllers that can be separate into an asymptotic observer an a feeback (ˆx, u)- controller is the topic of future work as is a generalization to not necessarily controllable (u, y)-controllers an the case im(y) > im(x). Acknowlegment This work was supporte in part by the Australian esearch ouncil through the A Discovery Project DP Geometric observer theory for mechanical control systems. eferences [1] H.L. Trentelman, P. Antsaklis, Observer-base control, in: Encyclopeia of Systems an ontrol, Springer, 214, p. 6. (no pagination). [2] T. Georgiou, A. Linquist, The separation principle in stochastic control, eux, IEEE Trans. Automat. ontrol 58 (1) (213) [3] J.M. Schumacher, Dynamic feeback in finite- an infinite-imensional linear systems (Ph.D. thesis), Vrije Universiteit te Amsteram, [4] J.. Willems, J.W. Polerman, Introuction to Mathematical Systems Theory: A Behavioral Approach, Springer, [5] J.. Willems, Y. Yamamoto, Behaviors efine by rational functions, Linear Algebra Appl. 425 (2) (27) [6] S.V. Gottimukkala, S. Fiaz, H.L. Trentelman, Equivalence of rational representations of behaviors, Systems ontrol Lett. 6 (2) (211) [7] S.V. Gottimukkala, H.L. Trentelman, S. Fiaz, ealization an elimination in rational representations of behaviors, Systems ontrol Lett. 62 (8) (213) [8] M.N. Belur, ontrol in a behavioral context (Ph.D. thesis), ijksuniversiteit Groningen, 23. [9] J.. Willems, On interconnections, control, an feeback, IEEE Trans. Automat. ontrol 42 (3) (1997) [1] M.N. Belur, H.L. Trentelman, Stabilization, pole placement, an regular implementability, IEEE Trans. Automat. ontrol 47 (5) (22) [11] J. Trumpf, H.L. Trentelman, J.. Willems, Internal moel principles for observers, IEEE Trans. Automat. ontrol 59 (214) [12] I. Blumthaler, J. Trumpf, A new parametrization of linear observers, IEEE Trans. Automat. ontrol 59 (214)