1 Introduction QUADRATIC CHARACTERIZATION AND USE OF OUTPUT STABILIZABLE SUBSPACES 1

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1 QUADRATIC CHARACTERIZATION AND USE OF OUTPUT STABILIZABLE SUBSPACES Eugênio B. Castelan 2, Jean-Claude Hennet and Elmer R. Llanos illarreal LCMI / DAS / UFSC Florianópolis (S.C) - Brazil {eugenio, llanos}@lcmi.ufsc.br ; Fax: LAAS du CNRS Toulouse Cedex 4 - France hennet@laas.fr ; Fax: Abstract : This note treats the problem of stabilization of linear systems by static output feedback using the concept of (C, A, B)-invariant subspaces. The work provides a new characterization of Output Stabilizable (C,A,B)- invariant subspaces through two coupled quadratic stabilization conditions. An equivalence is shown between the existence of a solution to this set of conditions and the possibility to stabilize the system by static output feedback. An algorithm is provided and numerical examples are reported to illustrate the approach. Keywords : Output feedback, geometric approach, Lyapunov equations, quadratic stabilizability, convex programming. Introduction Stabilization of linear systems by static output feedback is recognized as an important and still open problem in control theory. A review of existing approaches and techniques to treat different versions of this problem can be found, for instance, in 4. Other recent results not covered by this survey paper are found in, 7, 9 and 0. A commom feature shared by different methods (Lyapunov, Riccati, LMI or Eigenstructure Assignment) is that the output feedback stabilization problem is equivalent to obtaining the solution of a coupled set of matrix equations. In particular, through the use of coupled Sylvester equations 3, the output feedback control problem can be decomposed into two stages : (i) determination of a (C, A)-outer detectable subspace, and (ii) inner stabilization of this subspace. This paper proposes a quadratic counterpart of this result, under a similar decomposition into two stages. In particular, we show that solutions for the first stage can be obtained as solutions of a reducedorder Lyapunov equation. The method then defines a set of coupled Lyapunov-like equations that can be used for construction of an output stabilizable (C, A, B)-invariant subspace, as an intermediate mechanism in the process of designing a static output feedback controller. Furthermore, the quadratic characterization of both stages by Lyapunov equations provides a convenient framework for the numerical resolution of the problem through orthogonal transformations and convex programming techniques. The second section recalls, with a geometric interpretation, a known result that relates the existence of a stabilizing static output feedback to the solution of coupled Sylvester equations. In the third section, the output stabilizing conditions are reformulated in the form of coupled Lyapunov equations. And some connections are established between this characterization and the quadratic stabilizability framework. An algorithmic procedure based on the use of orthogonal decomposition and convex programming techniques is proposed and discussed in section 4. This approach is illustrated in section 5 through numerical examples and some concluding remarks are finally presented. This work has been partially supported by CNPq/Brazil. A first version of this paper was presented at 8th IEEE Mediterranean Conference on Control and Automation - MED 2000, Greece, July Author to whom correspondence should be addressed.

2 2 Preliminaries The considered linear time-invariant systems are described by ẋ(t) = Ax(t) + Bu(t) () y(t) = Cx(t) (2) where : x X R n, u U R m, y Y R p. It is also assumed that B is full column-rank, C is full row-rank and that (C, A, B) is stabilizable and detectable. The studied problem is to find a static output feedback control law u(t) = Ky(t) (3) such that σ(a + BKC) C, or equivalently, the closed-loop system is asymptotically stable. Let R v v be such that span( ) = and T R (n v) v be a left annihilator of, i.e : Ker T = Im. The following theorem relates the existence of a stabilizing static output feedback control law (3) to the solution of two coupled-sylvester equations. Theorem 2. : There exists a static output feedback matrix K : Y U such that σ(a + BKC) C if and only the following conditions hold true for some matrices ( R n v, H v R v v, W R m v ), (T R n v n, H T R n v n v, U R n v p n v) and for some positive scalar v n: where: rank( ) = v and rank(t ) = n v. A H = BW, with σ(h ) C (4) T A H T T = UC, with σ(h T ) C (5) T = 0 (6) Ker C Ker W (7) Ker B T Ker U (8) The above result has been presented and exploited under different forms mainly in the literature related to the eigenstructure assignment by output feedback (see 4). The equations (4),(5) and (6) are recognized as coupled-sylvester equations and describe some geometric properties of subspace = Im, that can be summarized as follows (see 3): under the stability constraint imposed on matrix H, (4) means that = Im( ) must be a (A, B)-inner stabilizable subspace, that is there exists F such that (A + BF ) is asymptotically stable. Dually, (5) means that Ker T has to be a (C, A)-outer detectable subspace, that is there exists L such that (A + LC) (R n /Ker T ) is asymptotically stable; thus, under the coupling condition (6), (4) and (5) mean that = Ker T is an Output Stabilizable (or simply OS) (C, A, B) invariant subspace, as defined in 3. Notice also that inclusion conditions (7) and (8) can be equivalently replaced by: KC = W (9) T BK = U (0) From the above comments we see that the statement of theorem 2. is essentially equivalent to the one of theorem 3.2 stated in 3. For algorithmic purposes, we shall assume in the sequel that is a p-dimensional subspace. This assumption has been also adopted in many works that, explicitly or implicitly, use the coupled-sylvester equations for eigenstructure assignment by output feedback 8. In particular, based on the conditions of theorem 2., the following algorithm, proposed in, leads to a stabilizing output feedback when Kimura s condition n < m + p is verified : Step : Solve (5) to find a matrix T R n n p such that ( ) T rank = n Ker T Ker C = {0} () C 2

3 Step 2: Solve (4), taking into account that must verify (6) and rank( ) must be equal to p. Step 3: By construction () guarantees that rank(c ) = p and K can be computed by: K = W (C ) (2) Steps and 2 can be solved by standard eigenstructure assignment techniques. In particular, solutions for equation (5) such that () is verified can be obtained from an auxiliary reduced order system which is defined from an appropriate decomposition of output matrix C (see the Appendix). Once T has been determined, step 2 can be solved by using the non-square zero equation : A λi I B T 0 vi w i = 0 0 (3) where v i and w i, i =,..., p, form the columns of and W, respectively. Remark 2. To guarantee that the eigenvalues λ i of step 2 are freely assignable, the system matrix A λi B P (λ) = must have full normal row rank: rank(p (λ)) = 2n p, λ. Otherwise, system T 0 (A, B, T ) has invariant zeros, in which case they must be used to obtain the (A, B)-invariance of Ker T 3. Notice however that, under Kimuras condition, m + p > n, P (λ) does not lose rank for almost all triples (A, B, T ) 2. Furthermore, for particular pathological cases, where the corresponding P (λ) has not full normal row rank, other solutions T for step may be found such that = Ker T can be internally stabilized. This can be done, for instance, by slightly changing the eigenvalues chosen to be placed through step. 3 A New Quadratic Stabilizability Characterization A quadratic characterization of theorem 2. can be obtained by replacing the (A, B)-inner stabilizability and the (C, A)-outer detectability requirements by the two equivalent quadratic Lyapunov stability conditions 6 : σ(h ) C Π = Π > 0 such that ΠH + H Π = Q, Q = Q > 0 (4) σ(h T ) C Γ = Γ > 0 such that H T Γ + ΓH T = Q T, Q T = Q T > 0 (5) Theorem 3. : There exists an output feedback K : Y U such that σ(a + BKC) C, if and only if the following conditions hold true for some full rank matrices R n v and T R n v n, with v > 0, such that T = 0 : (i) Q = Q > 0, Q R p p, there exist P = P 0, P R n n and Y R m n such that AP + P A + BY + Y B = Q (6) P > 0 ; T P T = 0 (7) Y = W Π for some W Π R m p (8) (ii) Q T = Q T > 0, Q T R n p n p, there exist S = S 0, S R n n and Z R n p such that (iii) A S + SA + C Z + ZC = T Q T T (9) T ST > 0 ; S = 0 (20) Z = T U Γ for some U Γ R n p p (2) Ker CP Ker Y (22) Ker B S Ker Z (23) 3

4 Proof : Necessity : Consider that X is an OS(C, A, B)-invariant subspace and, hence, that the coupled Sylvester equations (4), (5) and (6) are verified. Let us first show the necessity of part (i). For any Q = Q > 0, the quadratic stability condition given by (4) holds true, and we get : ( ΠH + H Π ) = ΠH + H Π = Q 0 (24) From (4) we also have A + BW = H, which can be used in (24) to obtain Π A + ΠW B + A Π + BW Π = Q (25) Thus, by setting { P = P = Π and Y = W Π, and by considering that rank( ) = p and that Π > 0 = Π > 0 T Π, (25) can be equivalently replaced by (6), (7) and (8). T = 0 Using similar arguments, we can show the necessity of part (ii). Thus, from (5) and (5), for any Q T = Q T > 0 we obtain : T (H T Γ + ΓH T )T = A T ΓT + C U ΓT + T ΓT A + T ΓUC = T Q T T (26) By setting S = T ΓT and Z = { T ΓU, (26) can replaced by (9) and (2). Furthermore, since rank(t ) = T T n p and Γ > 0, we also have ΓT T > 0 T ΓT = 0. The necessity of part (iii), (22) and (23), follows from conditions (7) and (8), respectively, by taking into account the above definitions of P, Y, S and Z. Sufficiency : Consider that parts (i), (ii) and (iii) are verified. By definition, and T have, respectively, full column and row ranks. Thus, from (7) and (20), we get : P = P = P > 0 = P = ( ) P ( ) (27) T ST = S = S > 0 = S = T (T T ) S(T T ) T (28) We shall now show that the verification of part (i) implies that = Im is a (A, B)-inner stabilizable invariant subspace. Thus, similar arguments can be used to show that = Ker T, which is implied by the coupling condition, is also a (C, A)-outer detectable subspace. From (27) and (28) we have: P = T Π From (22), there exists K such that Thus, by substituting (30) into (6), we get T 0 for Π = Π = ( ) P ( ) > 0 (29) KCP = Y (30) P (A + BKC) + (A + BKC)P = Q (3) Let us now take into account (29), the similarity equation (A + BKC) T = T Ā Ā 2 Ā 2 Ā 22 (32) and the fact that Q = T Q T 0 (33) 4

5 From the quadratic equation (3), we obtain : ( T ΠĀ ΠĀ 2 Ā Π Ā 2 Π 0 Hence, (34) implies that: ) T = T Q T 0 (34) = Im is (A + BKC)-invariant, since Ā2Π = 0 iff Ā2 = 0 ; and is inner-stabilizable, since ΠĀ + ĀΠ = Q < 0 implies the stability of (A + BKC). To complete this section, we present some remarks complementary to theorem 3.. Remark 3. : From the proof of theorem 3., it can be seen that any pair of matrices (P, S) solution to parts (i) and (ii) of theorem 3. verifies: Ker S = Im = Im P, or equivalently: SP = 0 (35) Remark 3.2 Let us consider the following coupled quadratic equations that have been used to analyze the output stabilization problem: AP + PA + BW + W B < 0, P > 0 (36) SA + A S + YC + C Y < 0, S > 0 (37) SP = I n (38) The coupling condition (35) captures the major conceptual difference between the quadratic output stabilizing conditions obtained in theorem 3. and the coupled quadratic conditions expressed above. Indeed, the obtained stabilizing conditions are directly related to the geometric properties: Ker S = Im = Im P, while the coupled conditions (36), (37) and (38) are mainly related to the invariance and contractivity of an ellipsoidal set defined from a Lyapunov function v(x) = x Sx. Thus, in our opinion, theorem 3. bridges a gap between the geometric approach and the quadratic stabilizibility to treat the static output feedback stabilization problem. This is particularly clear in the algorithm of the next section. Remark 3.3 The previous theoretical results allow to readily associate a quadratic Lyapunov function to the closed-loop system. Consider a solution (P, S,, T ) to the conditions in theorem 3. and let K be the corresponding stabilizing output feedback matrix. Define matrices T R p n and R n (n p) as: T {}}{{}}{ T = ( ) +D l T ; = T (T T ) + D l (39) where D l R p (n p) is such that: D l H T H D l = (A + BKC)T. Then, v(x) = x Sx, where S = T T Π 0 T = 0 Γ T P 0 0 S (40) is strictly decreasing along the trajectories of the closed-loop system ẋ(t) = (A + BKC)x(t), since v(x) = x ( (A + BKC) S + S(A + BKC) ) x < 0 x 0. Furthermore, S can also be written as: S = T P 0 0 Γ T = P 0 0 S. 5

6 4 An algorithm for output feedback stabilization Theorem 3. gives a necessary and sufficient condition for the existence of a stabilizing output feedback. This result involves coupled matrix equations that are non-linear and non-convex in the considered decision variables T,, P and S. However the quadratic characterization of theorem 3. may be used to construct OS(C, A, B)-invariant subspaces that lead to stabilizing output feedback matrices K. Based on these results, and in the light of the Syrmos and Lewis algorithm outlined in section 2, the following basic procedure is proposed to compute a stabilizing static output feedback when m+p > n : Step : Find a decomposition C M M 2 M M M 2 M 2 = I n ; = C 0 and set A2 A 22 = M M 2 AM 2, where Solve the reduced-order generalized Lyapunov equation (4) to find S 2 and S 22 = S 22 > 0 : Set T = Step 2 : A 22S 22 + S 22 A 22 + A 2S 2 + S 2 A 2 = Q T, for Q T > 0 ; (4) S 2 S 22 M M 2. If the triple (A, B, T ) has non-minimum phase (infinite or unstable) invariant zeros then repeat step for a new decomposition of C (see remark 4.4); Otherwise, go to step 2. Compute as an orthogonal basis of Ker T, i.e: = I p ; : Solve equations (6), (7) and (8) to find P 0, Y and W Π ; T = 0 (42) Step 3 : Compute the stabilizing output feedback matrix as the unique solution of: KCP = Y KC P = W Π, since = I p. 4. Some remarks about the proposed basic procedure The steps of the above basic algorithm can be justified as follows: Remark 4. The key step of the algorithm is the construction of an adequate matrix T in step. This construction is based on the use of a reduced-order system, represented by the pair (A 2, A 22 ), obtained from the representation of the open-loop system in the basis formed by the columns of matrix M = M M 2. It is important to remark that the reduced-order Lyapunov equation (4) is always solvable since, by assumption, the pair (A 2, A 22 ) is detectable. Hence, any feasible solution for (4) produces a candidate matrix T such that Ker T Ker C = {0} and such that Ker T = is (C,A)-outer detectable (for details, see lemma 7. in the Appendix). Notice that the resolution of (4) is only necessary for finding a stabilizing output feedback. This has been pointed out by corollary in 9, but it can be also interpreted under the terms of remark 2., because the (A, B)-inner stabilizability of Ker T = is guaranteed if and only if the candidate T R n p n verifies ( ) A λi B normal rank = 2n + p, λ C T 0 6

7 Remark 4.2 Step 2 is used to solve the set of conditions (6), (7) and (8) for = Ker T. It is important to recall that these conditions are generically solvable in the sense that the system matrix P (λ) has full normal row rank for almost all triples (A, B, T ), when m + p > n. Thus, a feasibility test has been set at the end step so that the computations involved in step 2 are carried out only when a stabilizing solution is feasible (see remark 2.). Numerical experiments, reported in the next section, show the effectiveness of the proposed approach. Remark 4.3 Standard convex programming techniques can be used to find feasible solutions for the involved Lyapunov-like conditions 2. Although in the present study, asymptotic stability is the only closed-loop requirement, the degrees-of-freedom existing in the quadratic approach given here could be exploited to achieve other requirements as, for instance, regional pole placement. Remark 4.4 The decomposition used in step is not unique. In particular, it can be obtained computationally through orthogonal transformations. Then, the set of all decompositions (44) can be described from a chosen orthogonal matrix M as CMD = C M M 2 D 0 D 2 D 2 = C 0 for all non-singular matrices D R p p and D 2 R n p n p. Furthermore, for any particular choice of matrix D, the new (detectable) reduced order pair can be easily computed: A n 2 D A n = 0 A2 22 D 2 D2 D A 2 = 22 D A 2D 2 (D2 A 22 D 2 A 2 )D 2 It is important to remark that it is not a simple task neither to determine an a priori good decomposition M nor to define an optimization criterium guaranteeing that the numerical resolution of (4) will produce an adequate matrix T (see also 9). Thus, for computational purposes, the above parameterization can be used when a particular choice of decomposition M does not produce an adequate matrix T. Hence, matrix D can be used to define a new reduced-order pair and to produce a perturbed solution to equation (4) so that step 2 becomes feasible. Although this trial and error scheme has no guarantee of numerical convergence, its use produces good results in practise, as shown in the next section. Remark 4.5 The given algorithmic procedure is not directly adapted for the more difficult cases where m+p n, although a try-and-error search may be carried-out to find a good T, through the definition of decompositon M. In the case m + p = n, P (λ) is square and almost all triple (A, B, T ) has p finite invariant zeros 2. In this case, the basic procedure can produce stabilizing solutions only if T found in step generate p minimum-phase (asymptotically stable) invariant zeros, which have to be used to solve (3) 3. Finally, it is worth recalling that stabilizing solutions can be also computed from the dual system (A, C, B ) and that even for the less restrictive case m + p n, the procedure can be systematically used to compute ν-order stabilizing dynamic output compensators, by recovering the Kimura s condition, with ν > n (m + p). 5 Examples The examples reported in this section were solved with the aid of the sofware Scilab (developed at INRIA, France). In all the tests, the matrix decomposition in step has been firstly obtained from the QR decomposition of matrix C. Then standard convex programming techniques have been applied to find feasible solutions for the coupled quadratic conditions without additional requirements. (43) 7

8 5. Randomly generated examples First, random triples (A, B, C) have been generated, verifying the Kimura s condition m + p > n, with n = 5, m = p = 3, and such that A is unstable. The basic procedure computed stabilizing solutions in 99.6% of cases, without the need of iterations over step. In the examples not solved by the basic algorithm, only one iteration was necessary to obtain a stabilizing solution. This iteration was carried Ip 0 out using remark 4.4, with D =, for a random matrix D 2. Thus, under the Kimura s D 2 I n p condition, the algorithm was always successful and had a better performance than the performance reported in 0 for the cone complementarity linearization algorithm. Table summarizes the computational experiments for randomly generated examples. In particular, it shows the effectiveness of the proposed algorithm to compute both static and dynamic stabilizing solutions when Kimura s condition is either verified or recovered. Furthermore, the data in the last row show that the approach may also be used to search for stabilizing solutions even when Kimura s condition is not verified. Systems order Compensator s order Number of examples Succes Basic procedure iteration 2 iterations n = 5 m = p = 3 ν = n = 5 m = 2 ; p = 3 ν = n = 5 m = 2 ; p = 2 ν = n = 4 m = p = 2 ν = Table : Summary of randomly generated examples 5.2 A numerical example The objective of the next example is to describe the numerical results obtained at each step. Consider the following data 8 : A = ; B = The two triples (C, A, B) considered bellow are controllable and observable and verify m + p = 5 > n. Also, to obtain closed-loop eigenvalues with real parts less than α = 2, computations are done using (A + αi) instead of A. st Case : Consider first the following output matrix: C = Using an orthogonal matrix M that performs step, a matrix T = found in step 2 : T = S 2 S 22 M M 2 is readily =

9 It implies, in step 3: =. A feasible solution for step 4 is then found: P = Y = ; W Π = The corresponding stabilizing output feedback, K =, gives: σ(a + BKC) = { ; ; ± j }, where the eigenvalue corresponds to step 2, through (47). 2 nd Case : Consider now the output matrix used in 8 : C = Using the orthogonal matrix M = , convex programming techniques do not produce a feasible solution in step 4, due to the particular matrix T selected in step : T = In this case, the corresponding triple (A, B, T ) has unstable invariant zeros: { ; 0 }. However, a stabilizing solutions exists (see 8). Using remark 4.4, we define a new D decomposition in step, from: =. We now obtain: D 2 D D T = S 2 S 0 M 22 D 2 D 2 M 2 = This new T is such that the corresponding triple (A, B, T ) has no invariant zeros. Hence, a feasible solution for step 4 can now be obtained. Using = 0 0, we get : W Π = , Y = P = , and The corresponding stabilizing output feedback, K =, gives: σ(a BKC) = { ; ; ± j }. 9

10 6 Concluding Remarks The concept of (C, A, B)-invariance has been revisited to obtain a quadratic characterization of the socalled Output Stabilizable (C, A, B)-invariant subspaces, which play a fundamental role in output feedback stabilization of linear systems. The paper has focused both on theoretical and algorithmic aspects. In particular, a necessary and sufficient condition for the existence of a stabilizing output feedback has been obtained in terms of two coupled Lyapunov-like equations. An algorithm has also been proposed to compute a stabilizing output feedback matrix under Kimura s condition m + p > n. Some extensions of the proposed approach are currently under investigation. In particular, an interesting approach consists of replacing strict pole placement by regional pole placement, using, for example, the conditions given in 4 and 5. Among the remaining open questions for future studies, one may consider the use of the existing degrees of freedom to include additional control requirements and the possibility to treat in a systematic way the more difficult case mp n. 7 Appendix Lemma 7. : Under the assumption that (C, A) is detectable, there always exist S = S 0 and Z verifying (9), (20) and (2) for some T R n p n, with rank (T ) = n p, such that Ker T = is (C, A)-outer detectable and Ker T Ker C = {0}. x Proof : Consider the following change of basis : x = M M 2, M x R n p, M 2 R n (n p), 2 where M = M M 2 is a non-singular matrix such that : C M M 2 = C 0, with C R p p and rank(c ) = p (44) M The inverse of matrix M is denoted : M = M 2 with M R n p, M2 R n n p. In this basis, the open-loop system takes the form x A A = 2 x B + u (45) x 2 A 2 A 22 x 2 B 2 y = C 0 x x 2 where the matrices involved have the appropriate dimensions. As shown in 6, detectability of the pair (C, A) implies detectability of the pair (A 2, A 22 ). Thus, for any given matrix Q T = Q T > 0, let S 22 R n p n p, with S 22 = S 22 > 0 and S 2 R n p p be solutions to the generalized reduced-order Lyapunov equation (4). Let L 2 = R n p p be such that and consider also a Choleski decomposition of S 22 = S 22 > 0 given by: (46) S 22 L 2 = S 2 (47) S 22 = T 2T 2 (48) Thus, from (4), (47) and (48), we get: T 2 T 2 (A 22 + L 2 A 2 ) + (A 22 + A 2 L 2 ) T 2 T 2 = Q T, where, by construction, σ (A 22 + L 2 A 2 ) C. Since T 2 is, by definition, non-singular, we can define the asymptotically stable matrix H T R n p n p from the similarity relation T 2 (A 22 + L 2 A 2 ) = H T T 2 (49) 0

11 Furthermore, since C is invertible, a matrix U R n p p can always be computed from: UC = (T 2 L 2 A + T 2 A 2 H T T ) (50) Thus, by defining T = T 2 L 2, (49) and (50) can be replaced by Hence, T = A T T A 2 2 A 2 A 22 T T 2 M M 2 is such that ; H T T T 2 = U C 0 (5) Ker T is (C, A)-outer detectable, since H T = (A + LC) X / is such that σ(h T ) C for L = T (T T ) U ; Ker T( Ker C = {0}, ) since the invertibility of T 2 implies that T T T rank = 2 M C C 0 M 2 = n. Furthermore, let Γ = Γ > 0 be the unique solution of H T Γ + ΓH T = Q T. Thus, the following matrices Z and S = S T 0 verify (9), (20) and (2) for T defined above : Z = M M 2 T 2 UΓ S and S = M M S 2 M S S 2 S 2 S 22 M 2 2, with T = S 2 S 22 T 2 Γ T T 2. References A.T. Alexandridis and P.N. Parakevopoulos, A new approach to Eigenstructure Assignment by Output Feedback. IEEE Trans. Automatic Control; ol. 4, No 7, pp , S.P. Boyd, L. El Ghaoui, E. Feron and. Balakrishhnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, E.B. Castelan and J. C. Hennet, Eigenstructure Assignment for State Constrained Linear Continuous Time Systems. Automatica; ol. 28, No 3, pp , E.B. Castelan E. R. Llanos illareal and J. C. Hennet, Output Feedback Design by Couple Lyapunovlike Equations. Accepted to the Ifac Barcelone, E.B. Castelan E. R. Llanos illareal. A. S e Silva and S. Tarbouriech, Regional Pole Placement by Output Feedback for a Class of Descriptor Systems. Accepted to the Ifac Barcelone, C.T. Chen, Linear system theory and design : New York, Holt, Rinehart and Winston, INC, C.A.R. Crusius and A. Trofino, Sufficient LMI Conditions for Output Feedback. IEEE Trans. Automatic Control; ol. 44, No 5, pp , L.R. Fletcher, J. Kautsky, G.K.G Kolka and N.K Nichols, Some necessary and sufficient conditions for eigenstructure assignment. Int. J. Control; ol. 42, No 6, pp , J.C. Geromel, C.C. de Souza and R.E. Skelton, Static Output Feedback Controllers: Stability and Convexity. IEEE Trans. Automatic Control; ol. 43, No, pp , L. El Ghaoui, F. Oustry and M. AitRami, A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems. IEEE Trans. Automatic Control; ol. 42, No 8, pp. 7-76, 997.

12 .L. Syrmos and F.L. Lewis, Output Feedback Eigenstructure Assignment Using Two Sylvester Equations. IEEE Trans. Automatic Control; ol. 38, No 3, pp , Syrmos and F.L. Lewis, Transmission Zero Assignment Using Semistate Descriptions. IEEE Trans. Automatic Control; ol. 38, No 7, pp. 5-20, L. Syrmos and F.L. Lewis, A Bilinear Formulation for the Output Feedback Problem in Linear Systems. IEEE Trans. Automatic Control; ol. 39, No 2, pp , L. Syrmos, C.T. Abdallah, P. Dorato and K. Grigoriadis, Static Output Feedback - A Survey. Automatica; ol. 33, No 3, pp , W.M. Wonham, Linear Multivariable Control, a Geometric Approach. Springer erlag, New York, Heidelberg, Berlin,