Linear matrix inequality approach for st

1 An LMI Approach for Stability Analysis of Linear Neutral Systems in a Critical Case Quan Quan, Dedong Yang, Kai-Yuan Cai Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100191, P R China Email: Quan Quan: qq_buaa@asee.buaa.edu.cn, Dedong Yang: dedongyang@gmail.com Kai-Yuan Cai: kycai@buaa.edu.cn Abstract This paper mainly focuses on the stability of a class of linear neutral systems in a critical case, i.e., the spectral radius of the principal neutral term (matrix H) is equal to It is dif cult to determine the stability of such systems by using existing methods In this paper, a suf cient stability criterion for the critical case is given in terms of the existence of solutions to a linear matrix inequality (LMI) Moreover, it is also shown that the proposed stability criterion conforms with a fact that the considered linear neutral systems are unstable when H has a Jordan block corresponding to the eigenvalue of modulus An illustrative example is presented to determine the stability of a linear neutral system whose principal neutral term H has multiple eigenvalues of modulus without Jordan chain This is infeasible in existing studies Index terms Linear neutral system, Stability criterion, LMIs, Critical case Introduction For clarity, we rst introduce a class of linear neutral systems x_ (t) where H x_ (t (1) ) = F (xt ) > is a constant delay, F ( ) is a linear functional and xt , x (t + ) ; [ on spectral radius of matrix H; the neutral system (1) can be classi ed into three cases: ; 0] Based (H) < 1; (H) > and (H) = 1: The case (H) < 1; namely matrix H is Schur stable, is a necessary condition for exponential stability of the linear neutral system (1) [1], [2] To the best knowledge of the authors, the case (H) > means that there are characteristic roots of the linear neutral system (1) with positive real part, so the system is unstable The last case (H) = is the critical case which is concerned in this paper Neutral systems in the critical case need to be considered in practice because they are in fact related to a class of repetitive control systems [3], [4] However, it is much more complicated to determine the stability of such systems because their characteristic equations may have an in nite sequence of roots with negative real parts approaching zero In recent years, stability problem of neutral systems in the critical case is investigated by frequency-domain methods [5], [6] (the interested readers could consult [5] and [6], and references therein, for the development on such a problem) As we know, the frequencydomain stability criteria will become more and more dif cult to verify as the dimension of matrix H increases Moreover, when H has multiple eigenvalues of modulus without Jordan chain, the analysis of non-exponential asymptotic stability is still an "open problem" [5, pp 426-427] The dif culty remains when time-domain methods are used In most of existing literature, the candidate Lyapunov functionals usually include a nonnegative term like kD (xt )k2 ; where D ( ) is called D operator [1, pp 286-287] and is de ned as D (xt ) = x (t) Hx (t ) for (1) In the case (H) < 1; it can be proved that the zero solution of D (xt ) = is asymptotically stable when kD (xt )k2 approaches zero asymptotically However, we cannot obtain the property in the critical case, thus cannot further analyse stability by investigating the tendency of kD (xt )k : On the other hand, the other type of stability criteria usually rely on the condition (H) < to prove the boundedness of kx_ (t)k [7, pp 336-337], [8, pp 157-158] Unfortunately, it is dif cult to obtain the boundedness of kx_ (t)k in the critical case as well (see the beginning of Section 4.1) Therefore, the existing stability criteria cannot cover the critical case easily In fact, most of existing stability criteria have implicitly assumed (H) < [9],[10],[11],[12] In this paper, we mainly investigate the critical case of a class of linear neutral systems A suf cient delay-independent stability criterion for the critical case is given in terms of the existence of solutions to an LMI This makes the proposed criterion quite feasible with the aid of a computer Then, by the proposed criterion, an existing criterion is extended to determine the stability of a scalar linear neutral system in the critical case Finally, it is shown that the proposed criterion conforms with a fact that the considered linear neutral system is unstable when H has a Jordan block corresponding to the eigenvalue of modulus [5, pp 394,415] An illustrative example shows the effectiveness of the proposed criterion and gives an alternative to handle the "open problem" according to [5, pp 426-427] Notation The notation used in this paper is as follows Rn is Euclidean space of dimension n k k denotes the Euclidean norm or a matrix norm induced by the Euclidean norm C ([ continuous n-dimensional vector functions on [ ; 0] : (X) and ; 0] ; Rn ) denotes the space of (X) denote the spectral radius and the minimum eigenvalue of matrix X, respectively X T and X are used for the transpose and conjugate transpose of matrix X: tr (X) denotes the trace of matrix X: X > (X 0; X < 0; X 0) denotes matrix X is a positive de nite (positive semide nite, negative de nite, negative semide nite) matrix In is the identity matrix with dimension n: "0" denotes a scalar or a zero matrix (vector) of appropriate dimension "#" in matrices denotes the term which is not used in the development Sometimes, the dimension of a matrix will not be mentioned when no confusion arises 4 Problem Formulation and Preliminary Results For simplicity, we consider a special case of (1) as follows x_ (t) H x_ (t ) = A0 x (t) + A1 x (t (2) ) with the initial condition x (t) = where x (t) Rn , (t) ; 8t [ ; 0] > is a constant delay and H; A0 ; A1 Rn n are constant system matrices (t) is a continuously differentiable smooth vector valued function representing the initial condition function for the interval of [ ; 0] The purpose of this paper is to derive a stability criterion in terms of LMIs for the linear neutral system (2) with (H) 1, especially for the critical case In this paper, we not consider the case of mixed retarded-neutral type systems, i.e., when H 6= 0; det (H) = 0; and limit ourselves to one principal neutral term as in [5] Before proceeding further, we have the following preliminary results (the proofs are all shown in the Appendix): Lemma For any negative semide nite matrix Lemma For any T;H Rn Q = QT Rn then Q > 0; i.e n n n ; if 'kk = 0; then 'kj = and Q < 0( : ; if H is nonsingular and there exist matrices < P = P T Rn such that # (P + T Q) H 7 E=6 # H T QH Q n ; (3) (Q) > 0; where E = E T Lemma For any given < Q = QT Rn H T QH Rn ; n; where 'ij corresponds to the element in the ith row and jth column of 'jk = 0; j = 1; T = 0) ; then (H) < ( Lemma If there exist matrices where GGT = In ; then GT QG Q = 0: n ; if there exists a matrix H Rn n such that 1) : Q = QT Rn n and G Rn n such that GT QG Q Remark Lemma indicates that for any given Q > 0; if (H) = and the inequality H T QH Q holds, then max H T QH Q = Lemma also implies that if (H) > 1, then H T QH Q 0 does not hold for all Q > 0: Main Results In this section, a delay-independent stability criterion (Theorem 1) in terms of an LMI is proposed for the linear neutral system (2) with (H) Then, an existing criterion is extended to determine the stability of a scalar linear neutral system in the critical case (Theorem 2) Finally, we prove that the proposed delay-independent stability criterion does not hold when matrix H has a Jordan block corresponding to the eigenvalue of modulus (Theorem 3) 4.1 A stability criterion The condition (H) < usually plays a role to show kx_ (t)k being bounded This is a very important step to show asymptotical stability of neutral type systems [1, pp 296-297],[7, pp 330-331, 336-337],[8, pp 157-158] If we have obtained that kx (t)k is bounded, then kx_ (t) H x_ (t )k (kA0 k + kA1 k) sup kx (t)k t2[0;1) by (2) Consequently, kx_ (t)k is bounded by applying (H) < This is not true in the critical case Taking this into account, we need to seek another condition to replace the boundedness of kx_ (t)k : To begin with, we need De nition ([13, p 123]) Suppose g (t) : [0; 1) ! R We say that g (t) is uniformly continuous on [0; 1) if for any " > there exists > such that jg (t + h) g (t)j < " for all t on [0; 1) with jhj < : Barbalat's Lemma ([13, p 123]) If the differentiable function f (t) has a nite limit as t ! 1; and if f_ is uniformly continuous, then f_ (t) ! as t ! 1: Uniform continuity is often awkward to assert from the de nition A very simple suf cient condition for a differentiable function to be uniformly continuous is that its derivative is bounded By this condition, many proofs are to show the boundedness of the derivative rather than its uniform continuity, although the latter in fact may play the same role as the former In the following proof, we will need to show the uniform continuity from the de nition Theorem The solution x (t; ) of (2) is asymptotically stable, if H is nonsingular and there exist matrices < W = W T Rn n ; < P = P T Rn n ;0 + LW LT where AT0 P + P A0 + S1T QS1 =6 H T (P + QS1 ) Q = QT Rn n such that (4) P + S1T Q H 7;L = H T QH Q T In R2n n ; S1 = A0 + H A1 : Proof The proof is composed of three propositions: Proposition is to show x (t; ) L1 [0; 1) ; Proposition is to show x (t; ) L2 [0; 1) ; Proposition is to show that kx (t; )k2 is uniformly continuous If the three propositions are satis ed, then the solution x (t; ) of (2) is asymptotically stable Z t kx (s; )k2 ds; then f_ (t) = kx (t; )k2 : Since The outline of the proof is as follows Let f (t) = kx (t; )k is continuous by Proposition 3, f (t) is a differentiable function Moreover, f (t) has a nite limit as t ! by Proposition and f_ (t) is uniformly continuous by Propositions It follows that lim x (t; ) = by Barbalat's Lemma The solution x (t; ) is stable by Proposition [7, p 352, Theorem t!1 2.3], therefore the solution x (t; ) of (2) is asymptotically stable [7, p 330] Next, we will prove the three propositions above one by one Proposition 1: x (t; ) L1 [0; 1) : If H is nonsingular, then the neutral system (2) can be rewritten as x_ (t) + H De ne z (t) , x_ (t) A1 x (t) = H x_ (t )+H A1 x (t ) + A0 + H A1 x (t) : S0 x (t) ; then the equation above becomes z (t) = Hz (t ) + S1 x (t) (5) where S0 = H A1 and S1 = A0 + H A1 : Choose a candidate Lyapunov–Krasovskill functional to be Z T V (t) = x (t) P x (t) + t (6) z (s) T Qz (s) ds t where < P = P T Rn n and Q = QT Rn n : Note that x_ (t) can be represented as x_ (t) = S0 x (t) + z (t) ; then the time derivative of V (t) is calculated as follows V_ (t) = x (t)T S0T P + P S0 x (t) + 2x (t)T P z (t) + z (t)T Qz (t) z (t )T Qz (t ): Substituting (5) into the above equation yields V_ (t) = Y (t)T (7) Y (t) T where Y (t) = T x (t) z (t : Since T ) V_ (t) Y (t)T LW LT Y (t) = Since W > 0; hence V_ (t) LW LT by (4), the equation (7) becomes 0: It gives V (t) x (t)T W x (t) : (8) V (0) : From (6), x (t) is bounded as sup kx (t)k (9) b1 t2[0;1) p where b1 = V (0)/ (P ): Therefore, x (t; ) L1 [0; 1) : Proposition 2: x (t; ) L2 [0; 1) : Integrating both sides of (8) from to t; we obtain V (t) Since V (t) and (W ) > 0; hence lim Therefore Z V (0) Z t!1 t Z t (W ) Z t t kx (s)k2 ds kx (s)k2 ds kx (s)k2 ds: V (0)/ V (0)/ min (W ) : Consequently, (W ) : kx (s)k2 ds has a limit as t ! by (10), i.e x (t) L2 [0; 1) : Proposition 3: kx (t; )k2 is uniformly continuous (10) Since (t) is continuously differentiable, the solution x (t; ) is continuously differentiable except maybe at the points t0 + k ; k = 0; 1; [1, p 25, Theorem 7.1] Then, by Newton-Leibniz Formula, we have x (t + h) x (t) = Z t+h x_ (s) ds: t where h > without loss of generality Utilizing (9) and x_ (t) = S0 x (t) + z (t) ; we have kx (t + h)k2 kx (t)k2 2b1 kx (t + h) x (t)k Z t+h 2b1 kx_ (s)k ds t Z t+h kS0 x (s) + z (s)k ds = 2b1 t Z t+h kz (s)k ds : 2b1 b1 kS0 k h + (11) t Using the Cauchy-Schwarz inequality ha; bi Z t Since + LW LT Z t+h 0, hence kz (s)k ds ds Z t+h kz (s)k ds t (12) : (Q) > by Lemma Then noticing (6), we obtain t2[0;1) t+h t sup where b2 = V (0)/ ha; hb; bi ; we obtain Z t t kz (s)k2 ds b2 (Q) : Thus (12) becomes Z t+h t kz (s)k ds N p p b2 h where N = b h/ c + 1; b h/ c represents the nearest integer of h/ Therefore, the inequality (11) becomes kx (t + h)k2 kx (t)k2 2b1 b1 kS0 k h + N This implies that kx (t; )k2 is uniformly continuous Remark If + LW LT < 0, then (H) < by Lemma Therefore, if + LW LT rather than p p b2 h : < 0: As a result, we obtain H T QH Q < 0: This implies (H) = 1; then the matrix inequality (4) must have the form + LW LT < 0: When the conditions of Theorem are satis ed, the solution x (t; ) of (2) is exponentially stable in the case with (H) < [2], whereas the solution x (t; ) is non-exponentially stable in the critical case [5, p 413] Remark In Theorem 1, the condition Q Remark If can be changed to Q > by Lemma (H) > 1, then Theorem does not hold by Lemma (or refer to Remark 1) 4.2 A scalar case Now, let us consider a scalar linear neutral system x_ (t) hx_ (t ) = a0 x (t) + a1 x (t (13) ) where h; a0 ; a1 R: Verriest and Niculescu gave the following result: Lemma ([14]) The scalar neutral system (13) is delay-independent asymptotically stable if (i) a0 < 0; (ii) jhj < 1; (iii) ja1 j < ja0 j : By Theorem 1, the extension of Lemma for the critical case is given as follows Theorem The scalar neutral system (13) is delay-independent asymptotically stable if (i) a0 < 0; (ii) jhj 1; (iii) ja1 j < ja0 j : Proof When jhj = 1; (4) can be written as 2a0 p + (a0 + h a1 ) q + w [p + (a0 + h a1 ) q] h 7 [p + (a0 + h a1 ) q] h where p; q; w R are all positive numbers Thus, if the following condition > > < 2a0 p + (a0 + h a1 )2 q < > > : (14) (15) p + (a0 + h a1 ) q = holds, then (14) holds with a suf ciently small positive number w This implies that the scalar linear neutral system (13) is asymptotically stable with h2 = 1: Solving (15) yields (i) a0 < (iii) ja1 j < ja0 j : 10 Combining the above results and Lemma 5, we can conclude this proof Remark When jhj = 1; the characteristic equation of system (13) has an in nite sequence of roots with negative real parts approaching zero As a result, it is dif cult to determine the stability of system (13) with jhj = by using frequency-domain methods 4.3 A special case For simplicity, let = f j j j = 1; a Jordan block corresponding to 2 (H)g : The linear neutral system (2) is unstable when H has [5, pp 394,415] In this section, we will show that Theorem does not hold in the case The Jordan blocks corresponding to or have two forms as follows [15, pp 82-83] 07 61 7 7 Dr = Rr r 17 7 2 I2 C( ) 6 C ( ) Dr = 6 I2 C( ) Lemma If DrT Qr Dr Qr and 7 7 7 R2r 7 2r cos ( ) ;C ( ) , sin ( ) (16) sin ( ) 7: cos ( ) (17) Qr = QTr ; then Qr is singular Proof See in Appendix Theorem If matrix H has a Jordan block corresponding to 1, then Theorem does not hold Proof The key point of this proof is to show that Theorem holds with Q matrix H has a Jordan block corresponding to 1: rather than Q > when But this is a contradiction by Lemma 11 Suppose, to the contrary, that Theorem holds when matrix H has a Jordan block corresponding to Then H T QH Q is satis ed If matrix H has a Jordan block corresponding to 1, then H can be transformed into the real Jordan canonical form [15, p 83] (18) SJ HSJ = HJ Jo 7 and Dr has the form as in (16) or (17) Pre-multiplying and post-multiplying where HJ = Dr H T QH Q by SJT and SJ respectively, we obtain HJT QJ HJ JoT QJ;11 Jo QJ = DrT QTJ;12 Jo QJ;11 JoT QJ;12 Dr QTJ;12 DrT QJ;22 Dr QJ;12 7 QJ;22 (19) QJ;11 QJ;12 7 Since DrT QJ;22 Dr QJ;22 by (19), QJ;22 is singular where QJ = SJT QSJ and QJ = QTJ;12 QJ;22 by Lemma 5, consequently, QJ is singular This implies that Q is singular, i.e., Theorem holds with Q rather than Q > Remark By Theorem 3, Theorem conforms with the fact that the system (2) is unstable when H has a Jordan block corresponding to [5, pp 394,415] An illustrative example Consider the linear neutral system (2) in the critical case with 2 7 ; A0 = H=6 0:4 0:1 7: ; A1 = 5 0:4 0:1 In this example, all eigenvalues of H are 1: By the stability criterion (4), we obtain the following solution 3 3:5025 P =6 2:1255 The eigenvalues of 2:1255 2:3021 0:4519 7;Q = ; W = 0:1I2 : 5 5:9123 0:4519 7:1216 + LW LT are ( 12:0220; 3:7504; 0; 0) Therefore, the system considered in this example is asymptotically stable independent of delay 12 Remark H in the system considered in the example has multiple eigenvalues of modulus without Jordan chain The stability analysis of such system is still an "open problem" according to [5, pp 426-427] However, the stability of the system can be determined by the stability criterion (4) Conclusions Asymptotic stability of neutral type systems, especially in the critical case, is studied and a stability criterion in terms of LMIs is proposed It is also shown that the proposed stability criterion conforms with the fact that the considered linear neutral system is unstable when H has a Jordan block corresponding to the eigenvalue of modulus Furthermore, the proposed criterion can help to determine the stability of the case where H has multiple eigenvalues of modulus without Jordan chain This gives an alternative to handle the "open problem" according to [5, pp 426-427] R EFERENCES [1] Hale J.: `Theory of Functional Differential Equations' (Springer-Verlag, New York, 1977) [2] Mondie S., Kharitonov V.L.: `Exponential estimates for retarded time-delay systems: An LMI approach', IEEE Trans Automat Control, 2005, 50, (2), pp 268–273 [3] Hara S., Yamamoto Y., Omata T., Nakano M.: `Repetitive control system: a new type servo system for periodic exogenous signals', IEEE Trans Autom Control, 1988, 37, (7), pp 659–668 [4] Quan Q., Yang D., Cai K.-Y., Jiang J.: `Repetitive control by output error for a class of uncertain time-delay systems', IET Proc.-Control Theory Appl., 2009, has been accepted [5] Rabaha R., Sklyarb G.M., Rezounenkoc A.V.: `Stability analysis of neutral type systems in Hilbert space', J Differ Equ., 2005, 214, (2), pp 391–428 [6] Rabaha R., Sklyarb G.M.: `On a class of strongly stabilizable systems of neutral type', Appl Math Lett., 2005, 18, (4), pp 463–469 [7] Kolmanovskii V., Myshkis A.: `Introduction to the Theory and Applications of Functional Differential Equations' (Kluwer, Boston, 1999) [8] Kolmanovskii V., Nosov V.R.: `Stability of Functional Differential Equations' (Academic Press, London, 1986) [9] Fridman E.: `New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems', Syst Control Lett., 2001, 43, (4), pp 309–319 13 [10] Han Q.-L.: `On delay-dependent stability for neutral delay-differential systems', Int J Appl Math Comp Sci., 2001, 11, (4), pp 965–976 [11] Niculescu S.-I.: `On delay-dependent stability under model transformations of some neutral linear systems', Int J Control, 2001, 74, (6), pp 609–617 [12] Leite V.J.S., Peres P.L.D., Castelan E.B., Tarbouriech S.: `On the robust stability of neutral systems with time-varying delays' Proc 16th IFAC World Congress, Prague, Czech Republic, 2005 [13] Slotine J.-J E., Li W.: `Applied Nonlinear Control' (Prentice-Hall, New Jersey, 1991) [14] Verriest E., Niculescu S.-I.: `Delay-independent stability of linear neutral systems: A Riccati equation approach', in Dugard L., and Verriest E (Ed.): `Stability and Control of Time-Delay Systems' (Springer-Verlag, London, 1998), pp 92–100 [15] Laub A.J.: `Matrix Analysis for Scientists & Engineers' (SIAM, Philadelphia, 2005) Appendix A Proof of Lemma Without loss of generality, take 6 =6 11 12 13 T 12 'kk 23 T 13 T 23 33 for example, where 11 ; 7 7 ; 'kk = 0; 13 ; 33 T 12 = '1k '(k ; 1)k 23 = 'k(k+1) are matrices with appropriate dimensions Assume 'kn 12 6= to the contrary that there exists a unitary matrix U with an appropriate dimension such that UT where and 6= 0: So we can choose T Choosing v = T (U ) T 12 U where T 12 U (20) = 6= to satisfy T > 0: ; we have vT v = 12 T UT 11 U +2 T UT T 12 U 12 is a scalar and v 6= By using the equation (20), the equation above becomes vT v = T UT 11 U +2 T : (21) 14 Since T > 0; hence we can choose 0: Therefore, 12 > T UT 11 U T to make v T v > 0: This contradicts with the fact = 0: Using the similar method, we can also prove 23 = 0: B Proof of Lemma Suppose, to the contrary, that (Q) = 0: Then there exist two cases: Q = and Q 6= 0: If Q = 0; then (3) becomes # PH 7 E=6 # 0: Consequently, P H = by Lemma Since < P = P T ; we obtain H = This contradicts with nonsingularity of H: Therefore, the remainder of proof only needs to consider Q 6= 0: For Q 6= 0; there exists a unitary matrix U Rn where yields =6 n where (22) U QU T = 7 with > 0: Pre-multiplying and post-multiplying (3) by E such that T U (P + T Q) HU T # =4 # U H T QHU T U QU T and T respectively U 7 : Furthermore, in light of (22) and the fact U T U = In , we have =6 U E T U P + T U T U Q U T U HU T # =6 T T T T T # U H U U QU U HU U QU ~ # P~ + T~ H =4 ~T H ~ # H ~ = U HU T ; P~ = U P U T and T~ = U T U T : where H (23) 15 ~ 11 H ~ 12 H ~ ~ ~T H ~ ; the term H By rewriting H as H = ~ 21 H ~ 22 H ~T by (23), hence H 12 ~ 12 = 0: In this case, P~ + T~ hence H ~ H12 (24) ~T On the other hand, H 12 ~ can be written as H ~ H12 by # # ~ =6 H ~ 22 # P~22 H P~ + T~ # # 7: =6 ~ T 1H ~ 12 # H 12 ~T H ~ H ~T H ~ Since H becomes > 0; (25) P~11 P~12 7 : Substituting (24) and (25) into (23) yields where P~ = T P~12 P~22 E T # # # # 6 ~ 22 # # # P~22 H =6 6 # # # # # # # 7 7 7 7 0: ~ 22 = by Lemma If P~22 is nonsingular, then H ~ 22 = which implies This inequality implies P~22 H ~ 11 H ~ This contradicts with the nonsingularity of H: If P~22 is singular, then it contradicts H =4 ~ 21 H with P > 0: Therefore Q > 0: C Proof of Lemma Suppose, to the contrary, that for any given Q > there exists a matrix H such that H T QH Q < Use H to denote an eigenvalue of H where j eigenvector vH 6= such that HvH = H vH : Since H T QH Hj = (H) and (H) ; then there exists an Q < 0; we have vH H T QH Q vH < 0: Consequently, (H)2 vH QvH < 0: (26) 16 If (H) = 1; then the inequality (26) becomes < which is a contradiction; On the other hand, if (H) > 1; then (26) becomes vH QvH < which contradicts with Q > 0: Therefore, if there exists a matrix H such that H T QH Q < 0; then (H) < 1: Similarly, we can also prove that for any given Q > 0; if there exists a matrix H such that H T QH 0; then Q (H) D Proof of Lemma Since tr GT QG Q = tr GT QG tr (Q) = tr QGGT tr (Q) = tr QGGT Q and GGT = In ; we obtain that tr GT QG (27) Q = 0: The equation (27) implies that the sum of the elements on the main diagonal of GT QG Moreover, since GT QG Q 0; every main diagonal element of GT QG Q is smaller than or equal to zero Therefore, we can conclude that every diagonal element of GT QG according to Lemma 1, we have GT QG Dk Dr in (16) or (17) has a recursive form as Dk = Therefore, DrT Qr Dr Qk = Qr Q is zero Furthermore, Q = E Proof of Lemma form as DkT Qk Dk Q is zero DkT Qk Dk Qk # implies that D2T Q2 D2 # 7 ; k = 2; # ; r: So DkT Qk Dk Qk has the # Qk # 7, where Qk = ; k = 2; 5 # # # Q2 ; r 0: If Q2 is singular, then we can conclude Qr is singular The remainder of the proof is to show that Q2 is singular 17 1 7 ; then D2T Q2 D2 (i) If D2 = Q2 D2T Q2 D2 can be represented by q11 7 Q2 = q11 q11 + 2q12 (28) q11 q12 7 ; q11 ; q12 ; q22 R: If the inequality (28) holds, then q11 = by Lemma This where Q2 = q12 q22 implies that Q22is singular I2 C( ) ; then (ii) If D2 = C( ) D2T Q2 D2 3 d1 d2 7 Q2 = dT2 # q11 q12 7 ; q11 ; q12 ; q22 R2 ; d1 = C ( )T q11 C ( ) where Q2 = T q22 q12 C ( )T q12 C ( ) q12 : The above inequality implies d1 q11 and d2 = C ( )T q11 + 0: Since C ( ) C ( )T = I2 and q11 0; hence d1 = by Lemma Consequently, d2 = by Lemma 1, i.e C ( )T q11 + C ( )T q12 C ( ) q12 = 0: Pre-multiplying C ( ) on both sides of the above equation and using C ( ) C ( )T = I2 , we have q11 = q12 C ( ) + C ( ) q12 : Then tr (q11 ) = tr [q12 C ( )] + tr [C ( ) q12 ] = tr [q12 C ( )] + tr [q12 C ( )] = 0: (29) 18 Since q11 0; every diagonal element of q11 is larger than or equal to zero Consequently, similar to the proof of Lemma 4, we get q11 = by (29) This implies that Q2 is singular  ... Verriest E., Niculescu S.-I.: `Delay-independent stability of linear neutral systems: A Riccati equation approach' , in Dugard L., and Verriest E (Ed.): `Stability and Control of Time-Delay Systems'... terms of an LMI is proposed for the linear neutral system (2) with (H) Then, an existing criterion is extended to determine the stability of a scalar linear neutral system in the critical case... exponential stability of the linear neutral system (1) [1], [2] To the best knowledge of the authors, the case (H) > means that there are characteristic roots of the linear neutral system (1) with