Regulation transients in discrete-time linear parameter varying systems: a geometric approach to perfect elimination

Transcription

1 Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 FrB084 Regulation transients in discrete-time linear parameter varying systems: a geometric approach to perfect elimination G Marro and E Zattoni Abstract This work encompasses the problem of the exact elimination of regulation transients for linear parameter varying systems in a straight geometric framework Discrete-time, stabilizable systems are specifically addressed Conditions for problem solvability are proved and the synthesis of the control scheme is illustrated in detail I INTRODUCTION Robust asymptotic regulation, achieved through the internal model principle as was established in the seminal papers 1, 2, is very effective in those situations where the systems involved are subject to sufficiently small parameter variations However, the problem of handling wide and abrupt parameter changes occurring in the regulated systems has been the object of a fair number of contributions in the more recent literature: linear parameter varying systems, jump linear systems, switching systems, bumpless systems are definitions extensively used to denote different classes of systems somehow affected by sensible changes in their parameters or structures Many different strategies have been proposed to deal with these classes of systems However, as to linear parameter varying systems, the geometric approach has proved to be a particularly congenial methodology 3, 4, 5, 6, 7, 8 Hence, many aspects have been analyzed in depth in the abovementioned articles, but, to the best of the authors knowledge, a complete discussion of the problem of the exact elimination of regulation transients aimed to discrete-time, stabilizable systems is lacking A geometric interpretation of the autonomous regulator problem was first presented for continuous-time controllable systems in 9 The extension of those concepts to discretetime stabilizable systems is discussed in the introductory Section II It is worth noting that removing from the present work the hypothesis of controllability which, by converse, was a standing assumption in 9, is of primary importance, given the nature of the problem tackled In fact, systems that switch from one configuration to another possibly include uncontrollable (stable) parts, since, as will be pointed out in Section III, the different configurations are required to be compatible, ie to have the same dynamic order Some basic ideas concerning the exact elimination of regulation transients in continuous-time systems subject to large parameter jumps were outlined in 5 However, in the abovementioned paper, possible parameter changes were allowed in the to-be-controlled system only: ie, not in the exosystem Moreover, addressing discrete-time systems G Marro and E Zattoni are with the Department of Electronics, Computer Science, and Systems, University of Bologna, Bologna, Italy gmarro@deisuniboit, ezattoni@deisuniboit appears more consistent with respect to the possible industrial applications that motivate the present work: these are, eg, digital control of chemical processes, where the compounds vary at pre-specified times, numerical control of tool-machines, where the profiles to be tracked can be obtained through a set of switched exosystems, or even flight control Furthermore, the discrete-time case seems, in some sense, more appealing than the continuous-time case in connection with some aspects of the problem solution In fact, the discrete-time solution, which, as will be shown in Section IV, includes a dynamic feedforward unit and a supervisory controller, is directly implementable as a unique digital controller: this implies an evident simplification of the realization Notation: The symbols Z, Z, R, R are used for the sets of integer numbers, nonnegative integer numbers, real numbers, nonnegative real numbers, respectively The symbols C, C, C are used for the unit circle, the open set inside the unit circle, the open set outside the unit circle in the complex plane C Sets, vector spaces, and subspaces are denoted by capital script letters The quotient space of a vector space X over a subspace V X is denoted by X/V Matrices and linear maps are denoted by capital slanted letters The restriction of a linear map A to an A-invariant subspace J is denoted by A J The inverse image of a subspace V through a linear map B is denoted by B 1 V The spectrum, the image, and the kernel of A are denoted by σ(a), im A, and ker A, respectively The symbols A 1, A, and A are used for the inverse, the Moore-Penrose inverse, and the transpose of A The symbols I and O are used for an identity matrix and a zero matrix of appropriate dimensions The symbol x, with x X, is the Euclidean norm of x II PRELIMINARY GEOMETRIC RESULTS This section presents a geometric interpretation, specifically aimed to discrete-time stabilizable systems, of the autonomous regulator problem, ie the problem of finding a feedback regulator that guarantees asymptotic stability of the regulation loop and asymptotic tracking of the reference signal generated by an exosystem 1 The proofs of the statements are given in Appendix Refer to Fig 1 Let (A 1,B 1,C 1 ), with A 1 R n1 n1, B 1 R n1 p, C 1 R q n1, be a state space realization of the to-be-controlled system Σ p Let (A 1,B 1 ) be stabilizable Let A 2 R n2 n2, with σ(a 2 ) C C, be the dynamic matrix of the exosystem Σ e Let the reference signal r be obtained from A 2 through the matrix C 2 R q n2 Let A 3 R n1 n2 model the effect of the disturbance d on Σ p /07/$ IEEE 5170

2 Fig 1 Σ u v Σ e d Σ p Σ r Block diagram for the autonomous regulator problem Let the regulated system Σ be defined as the connection of Σ p and Σ e, so that, with x=x 1 x 2, x 1 R n1 and x 2 R n2, the state equations of Σ are where A1 A A = 3 O A 2 r y e _ ˆΣ x k1 = Ax k Bu k, (1) e k = Ex k, (2) B1, B= O, E = E 1 E 2, (3) and E 1 = C 1, E 2 = C 2 Let (A,E) be observable Let (N,M,L,K), with N R m m, M R m q, L R p m, K R p q, be a state space realization of the regulator Σ r, so that, with z R m, the state equations are z k1 = Nz k Me k, (4) v k = Lz k Ke k (5) Let the autonomous extended system ˆΣ be defined as the connection Σ and Σ r, so that, with ˆx=x 1 x 2 z, the state equations are where  = ˆx k1 =  ˆx k, (6) e k = Ê ˆx k, (7) A 1 B 1 KE 1 A 3 B 1 KE 2 B 1 L O A 2 O ME 1 ME 2 N, (8) Ê = E 1 E 2 O (9) The following symbols are introduced B, E, Ê stand for im B, ker E, ker Ê, respectively V = max V(A, B, E) is the maximal (A, B)-controlled invariant subspace contained in E: ie, the last term of the sequence V 0 = E, V i = A 1 (V i 1 B) E, with i=1,2,,ρ, where ρ is the least integer such that V ρ1 = V ρ S = min S(A, E, B) is the minimal (A, E)-conditioned invariant subspace containing B: ie, the last term of the sequence S 0 = B, S i = A(S i 1 E)B, with i=1,2,,ρ, where ρ is the least integer such that S ρ1 = S ρ R V is the reachability subspace on V : ie, R V = V S Z(A,B,E) and Z(A 1,B 1,E 1 ) are the sets of the invariant zeros of (A,B,E) and (A 1,B 1,E 1 ) X and ˆX are the state spaces of Σ and ˆΣ In the geometric context thus defined, the autonomous regulator problem is stated in the following terms Problem 1: Refer to the autonomous extended system (6) (9) Find a regulator (4) (5) such that an Â-invariant subspace ˆL exists, satisfying (i) ˆL Ê and (ii) σ(â ˆX/ ˆL) C The following lemma consists in a necessary and sufficient condition for solvability of Problem 1 referring to the autonomous extended system In order to express that condition, the subspace ˆP ˆX, extended plant, is introduced: ˆP =imˆp =im I O O O (10) O I Since  ˆP = ˆPŜ holds with A1 B Ŝ = 1 KE 1 B 1 L ME 1 N ˆP is Â-invariant and σ(â ˆP)=σ(Ŝ): ie, the internal eigenvalues of ˆP match the poles of the closed-loop system Lemma 1: Problem 1 is solvable if and only if, for some regulator (4) (5), an Â-invariant subspace Ŵ exists, such that (i) Ŵ Ê, (ii) Ŵ ˆP = ˆX, (iii) σ(â ˆX/ Ŵ ) C Remark 1: By construction, the subspace Ŵ satisfying the conditions of Lemma 1 is the minimal Â-invariant subspace satisfying the requirements of Problem 1 Moreover, it is not restrictive assuming for a basis matrix Ŵ of Ŵ the peculiar structure: Ŵ =imŵ =im X 1 I (11) Z The next theorem states a necessary and sufficient condition which directly refers to the regulated system Hence, the subspace P X, plant, is defined as P =imp =im I O, (12) Since AP =PS holds with S = A 1, P is A-invariant and σ(a P )=σ(a 1 ) Theorem 1: Problem 1 is solvable if and only if an (A, B)- controlled invariant subspace V exists, such that (i) V E and (ii) V P= X Remark 2: A basis matrix V of V has the structure: V =imv =im X1 I (13) Let Π denote the projection from the extended-system statespace ˆX to the regulated-system state-space X Direct inspection of the basis matrices of ˆP, P, Ŵ, V shows that P =Π ˆP and V =ΠŴ The necessary and sufficient condition expressed by the previous theorem is not constructive, since it does not provide any algorithm to compute the resolver V By converse, the next theorem gives a constructive, although only sufficient, condition In fact, its proof shows that V can be derived from V by complementation Theorem 2: Problem 1 is solvable if (i) V P = X and (ii) Z(A 1,B 1,E 1 ) σ(a 2 )= 5171

3 The algorithm below specifies how to compute the regulator matrices on the assumptions of Theorem 2 Algorithm 1: Consider the regulated system (1) (3) 1 Check (i) stabilizability of (A 1,B 1 ) and (ii) observability of (A,E) If (i) or (ii) do not hold, then stop 2 Set P =im IO Compute V, R V, and V P 3 Check (i) V P = X and (ii) Z(A 1,B 1,C 1 ) σ(a 2 )= If (i) or (ii) do not hold, then stop 4 Compute F such that (ABF)V V and σ((abf) RV ) σ(a 2 )= 5 Perform the similarity transformation T =T 1 T 2 T 3, with im T 1 = V P and im T 1 T 2 =V Hence, A F = T 1 (ABF)T = 6 Solve the Sylvester equation A F 11 A F 12 A F 13 O A F 22 O O O A F 33 A F 11 X XA F 22 = A F 12 and set V =im(t 1 X T 2 ) 7 Compute F such that (ABF)V V and σ(a 1 B 1 F 1 ) C 8 Compute G such that σ(agc) C 9 Set N = ABF GE, M = G, L=F, K =O The regulator obtained by means of Algorithm 1 has dynamic order n 1 n 2 It guarantees that the 2n 1 n 2 poles of the closed-loop system are placed in C (ie, it guarantees closed-loop asymptotic stability) Moreover, it satisfies the internal model principle since its eigenstructure includes that of the exosystem (ie, it ensures asymptotic tracking of the reference and rejection of the disturbance) To conclude this section, a brief mention to the, so-called, robust asymptotic regulation is worth As is known 1, robust asymptotic regulation is achievable by means of a regulator whose eigenstructure includes a replica of the exosystem eigenstructure for each regulated output Let nj nj J R denote the Jordan block of the exosystem and e 1 the first vector of the main basis of R nj The matrices A 2 and E 2 of the regulated system (1) (3) are modified into J O O } 1 O J O O } 2 A 2 = O O J } q and e e E 2 = 0 0 e q respectively, and the matrix A 3 is modified consequently The structure of the new regulated triple (A,B,E) implies that the geometric interpretation of the discrete-time autonomous regulator problem previously discussed is extensible to the more general context of robust regulation with no additional requirements, III PERFECT ELIMINATION OF REGULATION TRANSIENTS: PROBLEM STATEMENT In this section, a multivariable regulation scheme based on the internal model principle, like that presented in Section II, is considered However, it is assumed that the regulated system is subject to wide, instantaneous parameter variations, so that sets of regulated systems and corresponding regulators must be taken into account It is also assumed that the quantitative description of the abrupt parameter variations and the time of their occurrence are completely known a priori The control target is the exact elimination of the tracking error caused by switches, on the assumption that the regulation loop is in a zero-error steady-state condition when the first switch occurs Refer to Fig 2 Let L={1,2,,n l } denote a finite index set Let {Σ(l),l L}= {(A(l),B(l),E(l)),l L} be a finite set of mutually compatible regulated systems: ie, systems of the type of (1) (3), with equal values of n 1 and n 2, respectively Let (A 1 (l),b 1 (l)) be stabilizable and (A(l),E(l)) be observable for all l L Let {Σ r (l),l L}= {(N(l),M(l),L(l),K(l)),l L} be the set of the corresponding regulators: ie, regulators of the type of (4) (5) Let {ˆΣ(l),l L}= {(Â(l),Ê(l)),l L} be the set of the corresponding autonomous extended systems: ie, systems defined according to (6) (9) Let {Σ c (l),l L}= {(N 1 (l),l 1 (l)),l L} be the set of the corresponding feedforward dynamic compensators: ie, compensators modeled by z 1k1 = N 1 z 1k, (14) v 1k = L 1 z 1k, (15) with the same dimension of the state z 1 R v Let the regulated-system switching law be defined as ϕ : Z L, k l (16) Hence, the ordered set of the switching times is K = {k i Z : ϕ(k i 1) ϕ(k i ),k i <k i1, i Z } The overall system outlined so far is subject to the following conditions: C1 initial zero-error steady-state condition: e k =E(ϕ(k))x k =0, k Z,k<k 0,k 0 K; C2 continuity of the state of the to-be-controlled system: x 1ki =A 1 (ϕ(k i ))x ki 1 B 1 (ϕ(k i ))u ki 1, k i K In this context, the problem of perfect elimination of regulation transients is stated in the following terms Problem 2: Let the set of the regulated systems {Σ(l),l L} and the regulated-system switching law (16) be given Find a set of regulators {Σ r (l),l L}, a set of feedforward dynamic compensators {Σ c (l), l L}, and a switching policy ψ : K R n2 R m R v, k i (x 2ki,z ki,z 1ki ) (17) 5172

4 of the states of the exosystem and of the regulators, such that the closed-loop be asymptotically stable and e k =0, k Z IV PERFECT ELIMINATION OF REGULATION TRANSIENTS: PROBLEM SOLUTION The conditions for solvability of Problem 2 shown in this section are based on the properties of the robust controlled invariant subspaces relative to the set of the regulated systems 3, 4 Those properties, adapted to the discrete-time case, are briefly recalled below Definition 1: A subspace V R R n such that A(l)V R V R B(l), l L, is called a robust (A(l), B(l))-controlled invariant subspace Property 1: Let V R R n For any initial state x 0 V R and any l L, a control input sequence u k, k Z, exists, such that the corresponding state trajectory x k, k Z, completely belongs to V R if and only if V R is a robust (A(l), B(l))-controlled invariant subspace Property 2: Let V R R n For any l L there exists a matrix F(l) such that (A(l)B(l)F(l))V R V R if and only if V R is a robust (A(l), B(l))-controlled invariant subspace Property 3: The set of all robust (A(l), B(l))-controlled invariant subspaces contained in a subspace E R n is an upper semilattice with respect to, The supremum, V R = max V R (A(l), B(l), E) is called the maximal robust (A(l), B(l))-controlled invariant subspace contained in E Algorithm 2: The subspace VR = max V R(A(l), B(l), E) is the last term of the sequence { Vi,l = max V(A(l), B(l), V i,l 1 ), l=1,,n l, V i1,0 = V i,nl, i=1,2,, with the initial condition V 1,0 = E and the terminal condition V i1,nl = V i,nl The definitions E = E(l), Ê = l L l LÊ(l) are assumed in order to state the solvability conditions for Problem 2 Theorem 3: Problem 2 is solvable only if V R P = X, (18) where P is defined as in (12) Proof: Let Problem 2 be solvable, so that the regulation loop is asymptotically stable and the error is identically zero Hence, the state of the autonomous extended system (Â(l),Ê(l)) evolves on a Â(l)-invariant subspace Ŵ(l) such that (i) Ŵ(l) Ê, (ii) σ(â(l) ˆX/W(l) ) C, and (iii) Ŵ(l) ˆP = ˆX, by virtue of Lemma 1 with the additional constraint Ŵ(l) l LÊ(l) Fig 2 Σ(l) u v Σ e (l) Σ p (l) Σ r (l) ψ r y Digital Controller Block diagram for exact elimination of regulation transients In fact, Ŵ(l) is the locus of the extended-state trajectories corresponding to a zero-error steady-state condition for any initial state of the exosystem, in the presence of the additional constraint Then, V(l)=ΠŴ(l), ie the projection of Ŵ(l) into the regulated-system state-space X,isan(A(l), B(l))- controlled invariant subspace such that (i) V(l) VR and (ii) V(l) P= X, by virtue of Theorem 1 with the additional constraint V(l) l L E(l) Since Problem 2 is assumed to be solvable for any regulated system in the considered set, the previous propositions hold for all l L In particular, the latter propositions (i) (ii), true for all l L, imply (18) The following theorem consists in a set of sufficient conditions for solvability of Problem 2 According to a practice well-settled in the geometric context, the structural aspect and the stabilizability aspect are evaluated separately The structural condition is the same that was shown to be necessary in Theorem 3 The stabilizability condition deals with the location in the complex plane of the invariant zeros Z(A 1 (l),b 1 (l),e 1 (l)) of the regulated plants {(A 1 (l),b 1 (l),e 1 (l)), l L} The proof is constructive, since it provides the set of the regulators, the set of the feedforward dynamic compensators state and the state switching policy ψ Theorem 4: Problem 2 is solvable if (i) VR P = X and (ii) Z(A 1 (l),b 1 (l),e 1 (l)) C for all l L Proof: In condition (i), VR is the maximal robust (A(l), B(l))-controlled invariant subspace contained in E Condition (ii) implies that Z(A 1 (l),b 1 (l),e 1 (l)) σ(a 2 (l))= for all l L, since σ(a 2 (l)) C C for all l L Hence, conditions (i) (ii) imply that, for any l L,an(A(l), B(l))-controlled invariant subspace V(l) such that V(l) VR and V(l) P= X can be derived through a complementation algorithm like that considered in the proof of Theorem 2 Consequently, for any l L, the procedure outlined in the proof of Theorem 1 yields a regulator (N(l),M(l),L(l),K(l)) such that the Â(l)-invariant subspace Ŵ(l) ˆX of the autonomous extended system (Â(l),Ê(l)) projects into the (A(l), B(l))-controlled invariant subspace V(l) X of the regulated system (A(l),B(l),E(l)) Let ϕ:z L be such ϕ(k 0 )=l 1 and ϕ(k 0 1)=l 2, with k 0 K and l 1,l 2 L By virtue of condition C1, a state of _ e 5173

5 the exosystem corresponding to the desired value of the reference signal at the initial time k =1can be fixed Hence, due to the particular structure of the basis matrix considered for Ŵ(l 1 ), namely W(l 1 )= X 1 (l 1 ) I Z(l 1 ) the whole extended state at the initial time k =1 ˆx 1 is fixed Since, the extended system state trajectory evolves on Ŵ(l 1 ) for k = 1,,k 0, the whole extended state at the time k 0 is ˆx k0 = Â(l 1) k0 1ˆx 1 By similar arguments, the extended state corresponding to perfect regulation for the extended system ˆΣ(l 2 ) would x 1 k0 x 2 k0 z k 0 be ˆx k 0 = Â(l 2) k0 1ˆx 1 Let ˆx k0 = and ˆx k 0 = (x 1 k0 ) (x 2 k0 ) (z k 0 ) The state of the exosystem and that of the regulator can be arbitrarily imposed at the switching time k 0 through the state switching law ψ Instead, the state of the to-be-controlled system is subject to condition C2 Consequently, the differnce between the actual state and the ideal state must be steered appropriately Let δx 1k0 =x 1k0 x 1 k0 Let Π 1 denote the projection from the regulated system state-space X to the to-be-controlled system state-space X 1 As to the state variation, δx 1k0 V 1 =Π 1 (V R P), since both V(l 1 ) V R and V(l 2) V R The subspace V 1 is a robust (A(l), B(l))-robust controlled invariant subspace since it is the intersection of the maximal (A(l), B(l))- robust controlled invariant subspace contained in E and a robust (A(l), B(l))-controlled invariant subspace for any B(l) Moreover, V 1 E 1 = l L E 1 (l) Furthermore, V 1 is internally stabilizable since its internal unassignable eigenvalues are a subset of the invariant zeros of the to-be-controlled system, which are stable by assumption Consequently, the state variation δx 1k0 can be driven asymtotically to the origin along a trajectory in V 1, hence invisible at the output, by means of a suitable control action Such control action can be obtained through a replica of the internal dynamics of V 1 in the feedforward dynamic unit In particular, let F 1 (l 2 ) be such that (A 1 (l 2 )B 1 (l 2 )F 1 (l 2 )) V 1 and σ((a 1 (l 2 )B 1 (l 2 )F 1 (l 2 )) V1 ) C Perform the similarity transformation T =T 1 T 2, where T 1 is such that im T 1 = V 1 Then, in the new basis, the matrices A F 1 (l 2 )=T 1 (A 1 (l 2 )B 1 (l 2 )F 1 (l 2 ))T 1 and F 1(l 2 )=F 1 (l 2 )T 1 have the structures: A A F1,11 F 1 (l 2 )= (l 2) A F (l 1,12 2) O A F (l, 1,22 2) F 1(l 2 )= F 1,1(l 2 ) F 1,2(l 2 ) Therefore, the matrices of the feedforward dynamic unit are defined as N 1 (l 2 )=A F 1,11 (l 2), L 1 (l 2 )=F 1,1(l 2 ), and the state is set to z 1k0 = T 1 δx k0 The criterion at the basis of the state switching policy ψ is thus completely defined The structural condition (i) of Theorem 4 is rather extensive, particularly when the digital systems involved are, indeed, sampled-data systems, where, normally, the condition V B = X holds Instead, the stabilizability condition (ii) is, in a sense, more restrictive, since it circumscribes the field of applicability to sets of minimum-phase switched systems By converse, it is worth noting that these conditions, which can be easily checked by means of the standard routines of the geometric approach (see eg 10), are strong in the sense that whenever they are satisfied, the control scheme suggested in the proof of the theorem guarantees that the tracking error is identically zero for any possible switching law ϕ known a priori V CONCLUSIONS In this paper, conditions for solvability of the problem of the exact elimination of regulation transients in discrete-time, stabilizable, linear parameter varying systems have been proved A constructive procedure for the synthesis of the complete control scheme has been illustrated in detail The conditions are sharp and easy to check, in practice, by means of reliable numerical algorithms Future investigations will focus on the application of these conditions to the Hamiltonian systems associated to the switched systems through an H 2 criterion This should guarantee the minimization of the l 2 norm of the tracking error, which represents a valid resort whenever the conditions discussed in this work are not directly satisfied by the switched systems APPENDIX This appendix collects the proofs of the lemmas and theorems set forth in Section II Proof of Lemma 1: If Assume that, for some regulator (4) (5), an Â-invariant subspace Ŵ exists, satisfying conditions (i) (iii) Then, Problem 1 is solvable since conditions (i) (ii) of the problem statement are satisfied with ˆL=Ŵ Only if Assume that Problem 1 is solvable Then, for some regulator (4) (5), an Â-invariant subspace ˆL exists, satisfying conditions (i) (ii) of the problem statement Let Ŵ be the subspace of the non-strictly stable modes of  Then, Ŵ is Â-invariant subspace, σ(â Ŵ ) C C, and σ(â ˆX/ Ŵ ) C Hence, condition (iii) directly follows by construction Moreover, by definition of ˆP, σ(â)=σ(a 2) σ(â ˆP), where σ(a 2 ) C C by assumption and σ(â ˆP) C since Problem 1 is solvable by means of some regulator (4) (5) Hence, condition (ii) follows from σ(â Ŵ )=σ(a 2) and σ(â ˆX/ Ŵ )=σ(â ˆP) Finally, (i) is proved by contradiction In fact, assuming Ŵ ˆL implies σ(â Ŵ ) C C, due to condition (ii) of the problem statement Hence, Ŵ ˆL and Ŵ Ê due to condition (i) of the problem statement Proof of Theorem 1: If Let V be an (A, B)-controlled invariant subspace satisfying conditions (i) (ii) Let V denote a basis matrix of V 5174

6 Let F, partitioned into F 1 F 2, be such that (ABF)V V and σ(a 1 B 1 F 1 ) C (A 1,B 1 ) is stabilizable by assumption Let G be such that σ(age) C (A,E) is observable by assumption Then, the regulator (4) (5) with N =ABF GE, M = G, L=F, K =O solves Problem 1 In fact, the matrices (8) (9) of (6) (7) become A BF  =, Ê = E O, GE A BF GE and the conditions (i) (iii) of Lemma 1 are satisfied with Ŵ =im V V, as can be easily checked by performing the similarity transformation ˆT = ˆT 1 I O = I I Only if Let Problem 1 be solvable Then, for some regulator (4) (5), an Â-invariant subspace Ŵ exists satisfying conditions (i) (iii) of Lemma 1 As observed in Remark 1, a basis matrix of Ŵ has the structure Ŵ = X1 IZ Let V =imv = X1 I Then, Â-invariance of Ŵ implies (A, B)-controlled invariance of V In fact, Ŵ is Â- invariant subspace if and only if a matrix, say Ŝ, exists such that ÂŴ = ŴŜ With the partition of  and Ŵ considered in (8) and (11), the first two row blocks of the previous identity can also be written, in compact form, as (ABKE)V =V Ŝ, which, in turn, is equivalent to (A, B)- controlled invariance of V Moreover, Ŵ Ê implies condition (i) In fact, 0=ÊŴ =E 1 E 2 O X1 IZ = EV Finally, (ii) comes from the direct inspection of the basis matrices V and P Proof of Theorem 2: Assume that conditions (i) (ii) hold The A-invariance of P and the peculiar structure of B imply that P is (ABF)-invariant for any F In particular, let F be such that (ABF)V V and σ((abf) RV ) σ(a 2 )= Consequently, also V P is (ABF)-invariant Consider the similarity transformation T =T 1 T 2 T 3, with im T 1 = V P and im T 1 T 2 =V Then, in the new basis, A F =ABF has the structure A A F = T 1 F 11 A F 12 A F 13 (ABF)T = O A F 22 O O O A F 33 Note that σ(a F 22 )=σ(a 2 ), since σ(a 2 )=σ(a F V ) σ(a F V /V P) ie, the poles of the exosystem are the subset of the internal eigenvalues of V external to V P Moreover, σ(a F 11 )=σ(a F RV ) Z(A 1,B 1,E 1 ), since R V V P and Z(A 1,B 1,E 1 )=Z(A,B,E) σ(a 2 ) ie, the invariant zeros of (A 1,B 1,E 1 ) are the subset of the invariant zeros of (A,B,E) external to the exosystem By virtue of condition (ii) and arbitrary placement of σ(a F RV ), A F 11 and A F 22 have no common eigenvalues Hence, the Sylvester equation A F 11 X XA F 22 = A F 12 is solvable and its solution is unique Let V =imv =im(t 1 X T 2 ) Then, by construction, V is an (A, B)-controlled invariant subspace satisfying conditions (i) (ii) of Theorem 1 Hence, it solves Problem 1 REFERENCES 1 B A Francis, The linear multivariable regulator problem, SIAM Journal on Control and Optimization, vol 15, no 3, pp , May B A Francis and W M Wonham, The internal model principle of control theory, Automatica, vol 12, pp , G Basile and G Marro, On the robust controlled invariant, Systems & Control Letters, vol 9, no 3, pp , September G Conte, A M Perdon, and G Marro, Computing the maximum robust controlled invariant subspace, Systems & Control Letters, vol 17, no 2, pp , August G Marro and A Piazzi, Regulation without transients under large parameter jumps, in Proceedings of the 12th Triennial World Congress of the International Federation of Automatic Control, vol 4, Sydney, Australia, July 1993, pp G Balas, J Bokor, and Z Szabó, Invariant subspaces for LPV systems and their applications, IEEE Transactions on Automatic Control, vol 48, no 11, pp , November J Bokor and G Balas, Detection filter design for LPV systems a geometric approach, Automatica, vol 40, pp , , Linear parameter varying systems: a geometric theory and applications, in Selected Plenaries, Milestones and Surveys, P Horacek, M Simandl, and P Zitek, Eds Prague: 16th IFAC World Congress, July 4-8, 2005, pp G Marro, Multivariable regulation in geometric terms: Old and new results, in Colloquium on Automatic Control, ser Lecture Notes in Control and Information Sciences, C Bonivento, G Marro, and R Zanasi, Eds Berlin / Heidelberg: Springer, 1996, vol 215, pp G Basile and G Marro, Controlled and Conditioned Invariants in Linear System Theory Englewood Cliffs, New Jersey: Prentice Hall,