A new model transformation method and it

A new model transformation method and its application to extending a class of stability criteria of neutral type systems Quan Quan,Dedong Yang, Hai Hu, Kai-Yuan Cai National Key Laboratory of Science and Technology on Holistic Control, Beijing University of Aeronautics and Astronautics, Beijing 100191, P R China; phone: +86-010-82338464; Quan Quan’s e-mail:qq_buaa@asee.buaa.edu.cn Abstract This paper proposes a generalized equivalent model transformation method, which can recover methods proposed by Fridman et al and Bellen et al., for the stability analysis of a class of neutral type systems By using the proposed model transformation method, a class of existing stability criteria derived by the Lyapunov functional approach can be extended to less conservative ones in terms of nonlinear matrix inequalities Furthermore, procedures to solve these nonlinear matrix inequalities are also proposed Illustrative examples are presented to demonstrate the eÔectiveness of the proposed model transformation method Key words: Model transformation; Neutral type system; Nonlinear matrix inequality; Optimization Introduction Recently, many model transformation methods have been proposed for the stability analysis of neutral type systems [1],[2],[3] However, some of the transformed systems are not equivalent to the original systems In order to overcome the problem, Fridman et al proposed an equivalent augmented model as a "descriptor form" representation of the system [2],[3] Another Preprint submitted to Nonlinear Analysis Series B: Real World ApplicationsFebruary 7, 2010 similar equivalent model transformation method was proposed by Bellen et al [4] Both models proposed by Fridman et al and Bellen et al are in fact special cases of the 2-D model [5],[6],[7] Based on this fact, we introduce a slack matrix into the equivalent augmented model proposed by Fridman et al to form a generalized 2-D model Through choosing speci…c slack matrices, the proposed generalized model can recover equivalent models proposed by Fridman et al and Bellen et al The new model transformation method can be used to extend many existing stability criteria However, in order to demonstrate the eÔectiveness more explicitly, we only focus on extending a class of existing stability criteria By using this new model transformation method and the Lyapunov functional approach, the class of existing stability criteria can be extended to less conservative criteria in terms of nonlinear matrix inequalities In view of this, procedures to solve these nonlinear matrix inequalities are also proposed The eÔectiveness of the proposed model transformation method is demonstrated through illustrative examples The main contributions of this paper are: 1) a new model transformation method is proposed; 2) based on this new transformation method, this paper extends a class of existing stability criteria rather than just designing new Lyapunov functionals, and it is proven that the extended criteria can reduce the conservatism of the original criteria; 3) procedures to solve a class of nonlinear matrix inequalities are proposed The notation used in this paper is as follows Rn is the Euclidean space of dimension n X T is the transpose of matrix X: j j denotes the absolute value of a scalar and k k denotes the Euclidean norm or the matrix norm induced by the Euclidean norm max (X) denotes the maximal eigenvalue of matrix X X > (X < 0) represents that matrix X is a positive de…nite (negative de…nite) matrix In is an identity matrix of a speci…ed dimension n 0n m Rn m denotes a zero matrix A New Model Transformation Method Given the following neutral type system: (1) = F (A0 ; A1 ; x; x; _ t) with the initial condition: x (t) = (t) ; 8t [ ; 0] where F (X0 ; X1 ; x; v; t) , v (t) Dv (t ) x (t) ; v (t) Rn ; X0 ; X1 Rn X0 x (t) n X1 x (t ) f (x; t) ; f (x; t) Rn ; R: In (1), X0 = A0 ; X1 = A1 ; v (t) = x_ (t) : x (t) is the state vector, is the time delay, f (x; t) is a vector function which does not contain derivative terms of the state vector, (t) is a continuously diÔerentiable smooth vector valued function representing the initial condition function If D = 0; then system (1) degenerates into a retarded type system Without loss of generality, the purpose of this paper is to propose a new model transformation method to derive stability criteria for neutral type system (1) De…ne y (t) , x_ (t) Sx (t) (2) where S Rn n is the slack matrix which needs to be designed to obtain less conservative stability criterion This will be discussed later Since the vector function f (x; t) does not contain derivative terms of the state vector, substituting x_ (t) = Sx (t) + y (t) into (1) yields: = [Sx (t) + y (t)] D [Sx (t ) + y (t )] A0 x (t) A1 x (t ) f (x; t) : Consequently, system (1) is transformed into the following equivalent form: < x_ (t) = Sx (t) + y (t) (3) : = F (A0 S; A1 + DS; x; y; t) where y (t) can be treated as the ‘fast variable’as mentioned in [2] Since the transformed system (3) is equivalent to the original system, we will focus on the stability analysis of the transformed system in the following sections When choosing S = or S = A0 ; system (3) can recover the equivalent models proposed in [2] or [4], respectively Furthermore, by designing an appropriate slack matrix S, the conservatism of the criteria derived by the model transformation methods proposed in [2] and [4] can be eÔectively reduced A Method to Extend a Class of Stability Criteria The proposed model transformation method can help to design new Lyapunov functionals and then obtain new stability criteria as proposed in [2] However, in order to demonstrate the eÔectiveness more explicitly, we only focus on extending a class of existing stability criteria in this section First, by applying the proposed model transformation method, a simple application on extending an existing stability criterion is given Following the idea of this application, a generalized method in terms of a theorem is derived to extend a class of existing stability criteria Finally, a stability criterion proposed in [8] is extended to a less conservative one by using the generalized method 3.1 A Simple Application For simplicity, consider neutral type system (1) with f (x; t) 0; i.e (4) = F1 (A0 ; A1 ; x; x; _ t) where F1 (X0 ; X1 ; x; v; t) , v (t) Dv (t ) X0 x (t) X1 x (t ): The following criterion proposed in [2] is used to determine the delayindependent stability of neutral type system (4) Fact 1([2]) If there exist < P1 = P1T ; P2 ; P3 Rn < Q2 = QT2 Rn n n and < Q1 = QT1 ; such that: (A0 ; A1 ; P1 ) < (5) then the solution x (t; ) of neutral type system (4) is asymptotically stable, where 6 6 (X0 ; X1 ; P1 ) , 6 P1 = h X0T P2 + P2T X + Q2 P1 P3T P2 + P3T X0 P1 P2T + X0T P3 P3 + Q1 X1T P2 X1T P3 D T P2 D T P3 P1 P2 P3 Q1 Q2 i : P2T X1 P2T D 7 P3T X1 P3T D 7 Q2 0n n 0n n Q1 Fact is a special case of Corollary proved in [2] The outline of the proof of Fact is described as follows The Lyapunov functional is chosen to be V1 (x; x; _ t) ; where V1 (x; v; t) , x (t) P1 x (t)+ T Z t T v(s) Q1 v (s) ds+ t Z t t x (s)T Q2 x (s) ds: (6) The time derivative of V1 (x; x; _ t) is V_ (x; x; _ t) = 2x (t)T P1 x_ (t) + x(t) _ T Q1 x_ (t) + x (t)T Q2 x (t) )T Q1 x_ (t x(t _ )T Q2 x (t x (t ) ): Introducing a zero term (x; x; _ t)T F1 (A0 ; A1 ; x; x; _ t) = (7) into equation above yields V_ (x; x; _ t) = 2x (t)T P1 x_ (t) + x(t) _ T Q1 x_ (t) +x (t)T Q2 x (t) + = T )T Q1 x_ (t x(t _ )T Q2 x (t x (t ) ) (x; x; _ t) F1 (A0 ; A1 ; x; x; _ t) (A0 ; A1 ; x; x; _ t) where 1 (x; v; t) , P2 x (t) + P3 v (t) (X0 ; X1 ; x; v; t) , (x; v; t) , h (x; v; t)T T x (t) (X0 ; X1 ; P1 ) T v (t) x (t T ) (x; v; t) v (t T ) iT : This leads to Fact Now, we will apply transformed system (3) to the proof of Fact and obtain an extended stability criterion Design a new Lyapunov functional V1 (x; y; t) ; where v in (6) is replaced by y The time derivative of V1 (x; y; t) is V_ (x; y; t) = 2x (t)T P1 y (t) + y(t)T Q1 y (t) + x (t)T Q2 x (t) y(t )T Q2 x (t x (t )T Q1 y (t ) ) + 2x (t)T P1 Sx (t) where x_ (t) = Sx (t) + y (t) is used Introducing a zero term (x; y; t)T F1 (A0 (8) S; A1 + DS; x; y; t) = into equation above yields V_ (x; y; t) = 2x (t)T P1 y (t) + y(t)T Q1 y (t) + x (t)T Q2 x (t) + = 1 (A0 )T Q2 x (t x (t (x; y; t)T F (A0 )T Q1 y (t y(t ) ) + 2x (t)T P1 Sx (t) S; A1 + DS; x; y; t) S; A1 + DS; x; y; t) + 2x (t)T P1 Sx (t) (9) This leads to Corollary which is an extended result of Fact Corollary If there exist < P1 = P1T ; P2 ; P3 Rn Q2 = QT2 Rn n and S Rn (A0 n n , < Q1 = QT1 ; such that: S; A1 + DS; P1 ) + ~ (S; P1 ) < 0; (10) then the solution x (t; ) of neutral type system (4) is asymptotically stable, where ~ (S; P1 ) = P1 S + S T P 0n 3n 03n 3n 5: Remark If the original criterion (5) in Fact has a feasible solution, denoted by P ; then there also exists a feasible solution, i.e S = and P1 = P1 ; to the extended criterion (10) in Corollary However, this does not hold vice versa, i.e there may not exist a feasible solution to the original criterion in Fact when the extended criterion (10) has a feasible solution Therefore, the extended criterion is less conservative than the original criterion 3.2 A Method to Extend a Class of Stability Criteria In the preceding section, we use y to play the role of x: _ Compared with x; _ y has a freedom to choose the slack matrix S: An important step of obtaining the extended criterion is to substitute the zero term (8) for (7) in the proof of Corollary 1, where transformed system (3) is utilized Based on the idea of the application above, we will extend a class of existing stability criteria in this section To begin with, we need some properties for the existing stability criteria which need to be extended For the stability of neutral type system (1), a normative proof process is described as follows: Property (i) A nonnegative functional V (x; x; _ t) is designed as V (x; x; _ t) = Vm (x; x; _ t) + Va (x; x; _ t) where Vm (x; v; t) , Z T t T (t) x (t) P0 x (t) + (s) x(s) P1 x (s) ds t Z t T + (s) v(s) P2 v (s) ds t Z 0Z t T Va (x; v; t) , (s) v (s) P3 v (s) dsd t+ and < Pk = PkT Rn n and k (t) R are nonnegative functions, k = 0; 1; 2; 3: (ii) The time derivative of V (x; x; _ t) in (i) is calculated as follows: V_ (x; x; _ t) = @t Vm (x; x; _ t) + @x Vm (x; x; _ t)T x_ (t) + @t Va (x; x; _ t) where @t V , @V @t Rn and @x V , @V @x Rn : Along a given trajectory of system (1), F (A0 ; A1 ; x; x; _ t) = holds By introducing zero terms (x; x; _ t)T F (A0 ; A1 ; x; x; _ t) = (x; x; _ t)T z (x; x; _ t) = and a nonnegative term nn (x; t) 0; V_ (x; x; _ t) can be further written as V_ (x; x; _ t) m (11) (A0 ; A1 ; x; x; _ t) + @t Va (x; x; _ t) m (X0 ; X1 ; x; v; t) , @t Vm (x; v; t) + @x Vm (x; v; t)T v (t) + (x; v; t)T F (X0 ; X1 ; x; v; t) + + where (x; v; t) Rn ; and (x; t) R; nn nn nn z (x; v; t)T z (x; v; t) (x; t) (x; v; t) Rn ; z (x; x; _ t) for 8x; x_ Rn for 8x Rn : (x; t) (iii) For 8x; x_ Rn ; inequality (12) holds @t Va (x; x; _ t) a (x; x; _ t) : (12) By using (12), (11) becomes V_ (x; x; _ t) where (X0 ; X1 ; x; v; t) , m (A0 ; A1 ; x; x; _ t) (X0 ; X1 ; x; v; t) + a (13) (x; v; t) : Remark For simplicity, we introduce the form of nonnegative functional V (x; x; _ t) as in Property (i) By applying Property 1, we aim to show how to use the new model transformation in a proof In fact, the form can be changed according to actual situation and then the following results will be changed correspondingly The proof of Fact is a normative proof process described in Property Note that, in the proof of Fact 1, the terms (x; x; _ t)T as Va (x; x; _ t) ; z (x; x; _ t) and nn (x; t) are not introduced Remark Some proof processes of existing stability criteria seem to be slightly diÔerent from the normative process described in Property But if these proof processes can be normalized as the process described in Property 1, then Property still holds for them Take system x_ (t) = h (x; t) for example, the time derivative of V (x; x; _ t) along a given trajectory of system x_ (t) = h (x; t) is calculated as follows: V_ (x; x; _ t) = @t V (x; x; _ t) + @x V (x; x; _ t)T h (x; t) : (14) The method of introducing zero terms or nonnegative terms are not formally used as (11) in Property In fact, along a given trajectory of system x_ (t) = h (x; t) ; V_ (x; x; _ t) can also be rewritten as V_ (x; x; _ t) = @t V (x; x; _ t)+@x V (x; x; _ t)T x_ (t)+ (x; x; _ t)T [x_ If (x; x; _ t) = h (x; t)] : (15) @x V (x; x; _ t) is chosen, then the above equation becomes (14) This implies that the derivation process (14) is only a special case of (15) Remark Leibniz–Newton formula [9],[10] is often used to construct the zero term z (x; x; _ t) For example, Rt x_ (s) ds 0: t z (x; x; _ t) = x (t) x (t ) Theorem Suppose Property holds for (1) The time derivative of a given Lyapunov functional V (x; x; _ y; t) = Vm (x; y; t) + Va (x; x; _ t) 10 The outline of the proof of Fact is described as follows The Lyapunov functional V2 (x; x; _ t) is chosen to be V2 (x; x; _ t) = V2;m (x; x; _ t) + V2;a (x; x; _ t) where V2;m (x; v; t) , e t T + x (t) P x (t) + Z Z V2;a (x; v; t) , Z t e2 s xT (s) Q1 x (s) ds t t e2 s v T (s) Q2 v (s) ds Z t e2 s v T (s) Q3 v (s) dsd t (23) (24) t+ (x; y; t)T Therefore, Property (i) is satis…ed The term z (x; x; _ t) is not introduced and Property (ii) is satis…ed with 2e2 t P x (t) (x; x; _ t) = nn (x; t) = "1 a2 x (t)T x (t) "1 g1T g1 + "2 b2 x (t )T x (t ) "2 g2T g2 where (21) is utilized The derivative of V2 is bounded as V_ (x; x; _ t) 2;m where the concrete form of Z t t (A0 ; A1 ; x; x; _ t) + @t V2;a (x; x; _ t) (25) is not given here for simplicity By using Q Q 3 w (t) ; x_ (s)T Q3 x_ (s) ds w (t)T Q3 Q3 2;m the following inequality holds [8] t @t V2;a (x; x; _ t) = e T x_ (t) Q3 x_ (t) Z t e2 (s t) T x_ (s) Q3 x_ (s) ds t 2;a (26) (x; x; _ t) 14 where 2;a (x; v; t) , e2 w (t) = h t < : v (t)T Q3 v (t) + e x (t)T x (t )T iT w (t)T : = Q3 w (t) ; Q3 Q3 Q3 (27) By using (26), inequality (25) becomes V_ (x; x; _ t) 2;m = (A0 ; A1 ; x; x; _ t) + 2;a (x; x; _ t) (A0 ; A1 ; x; x; _ t) where (X0 ; X1 ; x; v; t) , e2 t (x; v; t)T h (x; v; t) , x (t)T x (t 2 (X0 ; X1 ; P2 ) )T v (t (x; v; t) )T g1T g2T iT : Therefore, Property (iii) is satis…ed The proof of Fact is a normative proof process as described in Property The following Corollary is an extended result of Fact by Theorem Corollary Considering the uncertain nonlinear neutral system (20), for given scalars > and matrices P; Q1 ; Q2 ; Q3 Rn > 0, if there exist positive de…nite symmetric n , positive scalars "1 , "2 and matrix S Rn n such that (A0 S; A1 + DS; P2 ) + ~ (S; P2 ) < 15 (28) then system (20) is robustly exponentially stable with 2 T ~ S Q3 (A1 + DS) S T Q3 D 11 6 0n n 0n n ~ (S; P2 ) = 6 0n n 6 ~ 11 = P S + S T P + S T Q3 S + decay rate S T Q3 0n n 0n n 0n n S)T Q3 S + (A0 ;where T S Q3 7 0n n 7 0n n 7 0n n 0n n S T Q3 (A0 S) : Proof The candidate Lyapunov functional is chosen to be V2 (x; x; _ y; t) = V2;m (x; y; t) + V2;a (x; x; _ t) where V2;m (x; y; t) and V2;a (x; x; _ t) are de…ned in (23) and (24) respectively By Theorem 1, V_ (x; x; _ y; t) satis…es V_ (x; x; _ y; t) (A0 The left problem is to obtain 2;a = e2 t S; A1 + DS; x; y; t) + 2 (S; x; y; t) : (29) (S; x; y; t) : According to (27), we obtain (x; Sx + y; t) 2;a (x; y; t) h i T T T T x (t) S Q3 Sx (t) + 2x (t) S Q3 y (t) : After substituting y (t) = Dy (t ) + (A0 S) x (t) + (A1 + DS) x (t g1 + g2 into the above equation, we can obtain (S; x; y; t) = 2e2 t x (t)T P Sx (t) + [ = e2 t 2;a (x; y; t)T ~ (S; P2 ) 16 (x; Sx + y; t) (x; y; t) : 2;a (x; y; t)] )+ Consequently, (29) becomes V_ (x; x; _ y; t) T t e (x; y; t) h (A0 i ~ S; A1 + DS; P2 ) + (S; P2 ) (x; y; t) : The subsequently proof is the same as in [8], so it is omitted here Procedures to Solve Matrix Inequalities The extended stability criteria (10) and (28) are given in terms of existence of solutions to matrix inequalities such as (A0 S; A1 + DS; P) + ~ (S; P) < where ~ (0; P) = 0: When choose S = 0; the extended criteria (10) and (28) can recover the original criteria (5) in [2] and (22) in [8], respectively Since A0 ; A1 and D are constant matrices, we rede…ne M (S; P) , (A0 S; A1 + DS; P) + ~ (S; P) In order to obtain less conservative, the slack matrix S can be determined by solving the following optimization problem < : P;S (30) s:t: M (S; P) < I; P DP ; S R n n where DP denotes the feasible domain of parameter P: Assume opt1 is the minimal value of (30) with restriction S = and is the minimal value of (30) Obviously, opt2 opt1 : opt1 < and opt2 opt2 < imply that original criteria and extended stability criteria are satis…ed, 17 respectively If the original criteria have feasible solutions, i.e there exists a P DP such that extended criteria, i.e opt1 < 0, then there also exist feasible solutions to the opt2 < However, this does not hold vice versa, i.e there may not exist feasible solutions to the original criteria when the extended criteria have feasible solutions Therefore, the extended criteria are less conservative than the original criteria The LMI Control Toolbox in MATLAB 6.5 cannot be used to solve the optimization problem (30) because M (S; P) is a nonlinear matrix function with respect to S and P Therefore, calculation procedures need to be designed in order to obtain the optimal solution to the problem (30) For example: where 1; M1 (S; P1 ) = (A0 S; A1 + DS; P1 ) + ~ (S; P1 ) (31) M2 (S; P2 ) = (A0 S; A1 + DS; P2 ) + ~ (S; P2 ) (32) ~ and 2; ~ are de…ned in (10) and (28), respectively If matrix S is …xed, then M (S; P) is a linear matrix function with respect to P On the other hand, when P is …xed, M (S; P) may be a linear matrix function with respect to S as (31) or a nonlinear matrix function with respect to S as (32) The next step is to design calculation procedures to obtain suboptimal value ; i.e opt2 opt1 ; and corresponding suboptimal solution P and S with the aid of the LMI Control Toolbox 4.1 Procedure If M (S; P) is a linear matrix function with respect to S as (31), similar to coordinate descent methods [11, pp 53-55], the procedure to solve the 18 optimization problem as (31) is designed as follows: Procedure Step 1: Set S0 = 0; k = 1; " (" > 0); DP1 : Step 2: k = k + Solve the following optimization problem < P ( ) : s:t: M (S ; P) I; P D k P1 and obtain the optimal solution Pk and the optimal value If k < 0; then output k = k; = k; P k: = Pk ; S = Sk and step out Otherwise go to Step Step 3: Solve the following optimization problem < & S ( ) : s:t: M (S; P ) &I k and obtain the optimal solution Sk and the optimal value & k : If & k < 0; then output k = k; = k; P = Pk ; S = Sk ; and step out Otherwise go to Step Step 4: If j( k & k )/ ( k + & k )j < "; then k = k and step out Otherwise, set Sk+1 = Sk and go to Step In Step of Procedure 1, S is …xed to …nd a new solution P that minimizes the objective function of ( ) In Step 3, P is …xed to repeat the process for S; and so on Since M (S; P) is a linear matrix function with respect to S when the other variable is …xed (vice versa), optimization problems ( ) and ( ) in Procedure can be solved by using function "gevp" in the LMI Control Toolbox Obviously, holds, hence opt2 = &k k is the suboptimal solution 19 k &0 opt1 Example Consider a two-dimensional system (4) with D=4 0:4 0 0:4 ; A0 = 1 ; A1 = 0:1 0:1 5: 0:1 0:1 For the above example, when choosing S = in criterion (10), i.e using criterion (5) in [2], there does not exist a feasible solution Note that the extended criterion (10) still does not have a feasible solution when choosing S = A0 : Therefore we follow Procedure to …nd an appropriate S such that criterion (10) has a feasible solution Let " = 0:001, < 0:1I2 P1 100I2 0:1I2 Q1 100I2 DP1 = P1 : 0:1I2 Q2 100I2 P1 ; P2 ; P3 ; Q1 ; Q2 R2 and following Procedure for criterion (10), we obtain k = 1, and S =4 P3 = 0:0727 0:1653 0:1218 6:74 3:16 0:0912 1:49 5:70 P2 = Q1 = 5:08 4:31 3:78 0:23 1:04 4:48 0:23 2:69 = = P1 = Q2 = ; 1:3802 6:23 2:38 2:38 6:79 0:99 0:44 0:44 1:45 Therefore, system (4) in Example is asymptotically stable Procedure cannot be used for criterion (32), because, in this case, M2 (S; P) is a nonlinear matrix function with respect to S when P is …xed In view of this, Procedure is proposed 4.2 Procedure M (S + S; P) can be written as M (S + S; P) = M (S; P) + L ( S; S; P) + o k Sk2 20 5 where M (S; P) does not include with respect to S; L ( S; S; P) is a linear matrix function k Sk!0 0: For example, if M (S + o k Sk2 S when S and P are …xed and lim S; P) = where P; S; S Rn P (S + S)T P S) + (S + (S + k Sk = S+ T S) (S + S T S) P (S + n ; then T PS + S P S M (S; P) = T T S S PS P S + ST P S L ( S; S; P) = T T T S S PS S P S 0n n 0n n o k Sk2 = T 0n n S P S For (32), we have M2 (S + S; P2 ) = M2 (S; P2 ) + L2 ( S; S; P2 ) + o k Sk2 where 6 6 L2 ( S; S; P2 ) = 6 6 L11 L12 L22 T S Q2 D T S Q2 T S Q2 L23 L24 L25 0n 0n n 0n n 0n n 0n n 0n n 21 n 7 7 7 7 S) L11 = L12 = (A0 S)T Q2 S h S T Q2 (A1 + DS) + P D + (A0 S)T Q2 D + S T Q2 (A0 S) AT0 Q3 D L22 = S T DT Q2 + Q3 (A1 + DS) + (A1 + DS)T Q2 + L23 = S T DT Q2 + Q3 D L24 = L25 = S T DT Q2 + We try to obtain 2 i S Q3 D S Q3 S with the aid of LMI Control Toolbox at each iteration in order to update S: De…ne a region k Sk2 around the current iterate of S within which the linear matrix function M (S; P) + L ( S; S; P) can be trusted to be an adequate representation of M (S + S; P) : Similar to the idea of trust-region methods [11, pp 65-68], the procedure to solve the optimization problem (32) is designed as follows: 22 Procedure Step 1: Set S0 = 0; k = 1; "(" > 0); ; DP2 ; Step 2: k = k + Solve the following optimization problem < P ( ) : s:t: M (S ; P) I; P DP2 k v; s; d; i and obtain the optimal solution Pk and the optimal value If k < 0; then output k = k; k; P = k: = Pk ; S = Sk ; and step out Otherwise go to Step Step 3: Solve the following optimization problem > & > > < S > > s:t: M (Sk ; Pk ) + L ( S; Sk ; Pk ) > : and obtain the optimal solution &I; k In S ST In ( ) Sk and the optimal value & k : Go to Step Step 4: (i) If j( Set & k )/ ( k =( k + & k )j < "; then k = k and step out & k ) ; where k )/ ( k k (ii) Otherwise if set Sk+1 = Sk + (iii) Otherwise if set Sk+1 = Sk + v = M (Sk + [very successful, < Sk and s k k+1 = i k; where [successful, < Sk and k+1 = k: s v < 1] ; then 1: Go to Step i v Sk ; Pk ) : < 1] ; then Go to Step (iv) Otherwise [unsuccessful] ; then set Sk+1 = Sk and where < d k+1 = d k; < 1: Go to Step In Procedure 2, optimization problems ( ) and ( ) are both problems of eigenvalue minimization under LMI constraints which can be solved by using 23 function "gevp" in the LMI Control Toolbox Remark In optimization problem ( ) of Procedure 2, the following inequalities are equivalent k In S ST In , k Sk2 k by using the Schur Complement [12] If the size of trust region k is too small, the algorithm may miss an opportunity to take a substantial step that will move it much closer to the minimizer of the objective function of ( ) If the size of trust region M (S + k is too large, M (S; P) + L ( S; S; P) may be far from S; P) in the region, so the size of trust region k k may be larger than & k or should be changed according to k Therefore, (in Step (i)), i.e the performance of the algorithm in previous iterations Theorem By using Procedure 2, i.e = opt2 k opt1 ; is the suboptimal solution Proof (By induction) Since = opt1 ; then opt1 algorithm steps out at the k s iteration, then we assume opt1 holds Without loss of generality, we assume holds If the k k 0 opt1 holds at the k s iteration and the algorithm does not step out In this case, we need to prove that k = max k+1 k [M (Sk ; Pk )], otherwise if tradicts with the fact that k &k = max k > max still holds [M (Sk ; Pk )] ; then it con- is the minimizer of the optimization problem ( ) in Procedure Otherwise if with the constrain M (Sk ; Pk ) opt1 k < k I max [M (Sk ; Pk )] ; then it contradicts Similarly, [M (Sk ; Pk ) + L ( Sk ; Sk ; P )] : 24 Since M (Sk ; Pk ) + L (0; Sk ; Pk ) = M (Sk ; Pk ) ; we have &k = = If j( & k )/ ( k k max [M (Sk ; Pk ) + L ( Sk ; Sk ; P )] max [M (Sk ; Pk ) + L (0; Sk ; P )] max [M (Sk ; P )] = + & k )j < " (Step (i)), then step out (which is not con- sidered in this proof) Otherwise if ( & k ) > 0; then k )/ ( k k by j( k & k )/ ( k k+1 (33) k + & k )j = max = max k > < k s > (Step (ii) and (iii)), i.e (since & k k by (33) and & k 6= k " > 0) In this case, we have M Sk+1 ; Pk+1 [M (Sk + max Sk ; Pk )] = k [M (Sk+1 ; Pk )] < k Otherwise (Step (iv)), we have k+1 Therefore, Since = max = max k+1 opt2 M Sk+1 ; Pk+1 [M (Sk ; Pk )] = k k+1 ; max k = [M (Sk+1 ; Pk )] k opt1 : otherwise it contradicts with the fact that minimal value of (30) Therefore, opt2 = k opt2 is the opt1 ; i.e is the suboptimal solution Remark This implies that if the original criteria as (22) have feasible solutions, i.e opt1 < 0; then the extended criteria as (28) also have feasible solutions which can be obtained through Procedure However, this does not hold vice versa This will show in Example 25 Example Consider (20) with neutral system 2an uncertain3nonlinear 2 0:3 5, a = 0:05; b = ; A1 = 5, A0 = D=4 1 0:9 0:3 0:1; = 1: The problem is to determine whether system (20) is exponentially stable with decay rate = 0:1: There does not exist a feasible solution for criterion (22) proposed in [8], i.e criterion (28) with S = Let " = 0:001; d = 0:5; i = 0:1; v = 0:7; i = 0:3; = 1:5, D P2 > > 0:1I2 > < = P2 0:1 > > > : > P1 ; Q1 ; Q2 ; Q3 100I2 > > = "1 ; "2 100; "1 ; "2 R > > > P ; Q ; Q ; Q R2 ; 1 and following Procedure for criterion (28), we obtain k = 2; = 0:0568 and S =4 Q2 = 0:14 0:01 0:015 0:22 15:74 1:22 1:22 1:46 P =4 46:85 0:19 0:19 Q1 = 3:62 4:89 1:20 Q3 = 1:20 1:79 80:26 1:67 1:67 4:61 "1 = 99:99; "2 = 66:90 Therefore, system (20) is exponentially stable with decay rate = 0:1: Remark By selecting appropriate slack matrices through Procedure and Procedure 2, feasible solutions of the extended stability criteria can be obtained for Example and Example 2, whereas the original criteria (5) and (22) are not applicable here The improvement of the proposed model transformation method over its predecessors is demonstrated 26 Conclusions By introducing a slack matrix into the equivalent augmented model proposed by Fridman et al., a new model transformation method is proposed in this paper for the stability analysis of a class of neutral type systems By using the new model transformation method and following the Lyapunov functional approach, a class of existing stability criteria are extended to less conservative ones in terms of nonlinear matrix inequalities Procedures to solve the nonlinear matrix inequalities that represent the extended criteria are also proposed Two illustrative examples indicate that the two extended stability criteria have feasible solutions when the corresponding original criteria are not applicable Acknowledgement This work was supported by the Innovation Foundation of BUAA for PhD Graduates References [1] S.-I Niculescu, Delay EÔects on Stability: A Robust Control Approach, Springer, Berlin (2001) [2] E Fridman, New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems, Systems Control Lett 43 (2001), pp 309-319 [3] E Fridman, U Shaked, Delay-dependent stability and H1 control: constant and time-varying delays, Internat J Control 76 (2003), pp 48-60 27 [4] A Bellen, N Guglielmi, A.E Ruehli, Methods for linear systems of circuits delay diÔerential equations of neutral type, IEEE Trans Circuits Syst 46 (1999), pp 212-216 [5] J.P Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003), pp 1667-1694 [6] P Agathoklis, S Foda, Stability and the matrix Lyapunov equation for delay diÔerential systems, Internat J Control 49 (1989), pp 417-432 [7] E Rogers, K Galkowski, D.H Owens, Control Systems Theory and Applications for Linear Repetitive Processes, Springer, Berlin (2007) [8] Y Chen, A Xue, R Lu, S Zhou, On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations, Nonlinear Analysis 68 (2008), pp 2464-2470 [9] Y He, M Wu, J.-H She, G.-P Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems Control Lett 51 (2004), pp 57-65 [10] M Wu, Y He, J.-H She, New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE Trans Automat Control 49 (2004), 2266-2271 [11] J Nocedal, S.J Wright, Numerical Optimization, Springer, New York (1999) [12] S Boyd, L.E Ghaoui, E Feron, V Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia (1994) 28 ... transformation method is proposed; 2) based on this new transformation method, this paper extends a class of existing stability criteria rather than just designing new Lyapunov functionals, and it. .. the model transformation methods proposed in [2] and [4] can be eÔectively reduced A Method to Extend a Class of Stability Criteria The proposed model transformation method can help to design new. .. introducing a slack matrix into the equivalent augmented model proposed by Fridman et al., a new model transformation method is proposed in this paper for the stability analysis of a class of neutral