©2006 Institute for Scientific Computing and Information INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 2, Number 3, Pages 398-420 DESIGN OF LINEAR FUNCTIONAL OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL-TYPE TYRONE FERNANDO AND HIEU TRINH Abstract This paper presents the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. The type of the observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated. First, conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector are derived. Then, based on the newly derived existence conditions, a procedure is given for the determination of the observer parameters. The results derived in this paper include a range of linear systems and extend some existing results of linear functional observers to linear neutral delay systems. A numerical example is given to illustrate the design procedure. Key Words, Functional Observers, Time-delay Systems 1. Introduction Time-delay systems, frequently encountered in various engineering systems, have been a subject of extensively studies over the years (see a recent survey paper by Richard, 2003). In particular, there has been considerable attention focusing on the stabilization and state estimation of a class of time-delay systems commonly referred to as neutral systems [1, 6, 11, 14, 21]. Examples of neutral-type systems include lumped parameter networks interconnected by transmission lines, systems of a turbojet engine, infeed grinding, continuous induction heating of a thin moving body, and reactor in a chemical engineering system (see, [10, 21]). As defined in [15], neutral systems are time-delay systems that have the same highest derivation order for some components of the state vector, x(t ) , at both time t and past time(s) t p < t . The presence of a retarded argument in the state derivatives increases mathematical complexity and makes the investigation of such equations more complicated than equations with delays only in the states [15, 21]. One of the difficulties stems from the fact that neutral systems almost always have neutral root chain (infinite spectrum) in a vertical strip of the complex plane. Thus, most of the literature dealing with the problem of feedback stabilization of neutral systems either assumed that there is no unstable neutral root chain [6] or the unstable neutral root chain is assigned to the left-hand side of the complex plane by using derivative feedback [14]. The control and stabilization of neutral systems is often based on the assumption that the entire state vector is available for state feedback control. As discussed in [21], Received by the editors March 9, 2006 398 DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 399 the observer design problem for neutral-type delay systems has not yet been fully investigated in the literature and remains to be important. By using a Lyapunov functional approach, Wang et al [21] examined the exponential stability of a class of neutral systems and presented a sufficient stability condition, expressed in terms of an algebraic matrix equation. Observer design can then be constructed based on the existence of a solution to an algebraic matrix equation. One of the attractive features of the approach in [21] is that it does not need to consider the effect due to the unstable neutral root chain since the stability of the observer’s error is only related to the solution of an algebraic matrix equation. On the other hand, a drawback of their approach is that there is a restriction imposed on the matrix associated with the delayed state derivatives; namely it is required that the norm bound of this matrix is less than one. This assumption is also required in the observer-based control for neutral systems [11]. The observers proposed in [11] and [21] are full-order observers and they include a delayed state (estimated) derivatives. On the other hand, there are many applications where estimation of an entire state vector is not really required. Rather, only a linear combination of the states or a partial set of the states are required for feedback control and/or system monitoring. Linear functional observers (see for example, [2, 13, 19, 20]) estimate linear functions of the state vector of a system without estimating all the individual states. As a result, the order of a linear functional observer can be significantly lower than that of a full-order state observer. Such functional estimates are useful in feedback control system design because the control signal is often a linear combination of the states, and it is possible to utilize a linear functional observer to directly estimate the feedback control signal. So far, the design of functional observers for neutral-type delay systems has received little attention and not yet been fully investigated in the literature. This motivates us to consider the functional observer design problem for a class of linear neutral systems. The main features of this paper can be summarized as follows: (i) the functional observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated; (ii) the functional observer does not include a delayed state (estimated) derivatives and thus makes it more attractive from the implementation point of view; (iii) there is no restriction imposed on the matrix associated with the delayed state derivatives and the system is allowed to be singular; (iv) the results derived in this paper include a range of linear systems and therefore may be regarded as extension of some existing results of linear functional observers to linear neutral delay systems. The organization of this paper is as follows. Section 2 presents a problem statement and a reason for the choice of the observer. The main results are given in Section 3. 400 T. FERNANDO AND H. TRINH Section 4 illustrates a numerical example. Section 5 concludes the paper. Appendix A provides the proof of the main results. Appendix B provides the simulation results. 2. Problem Statement Consider the following class of linear delay systems of the neutral-type [21] Ex& (t ) = Ax(t ) + Ad x(t − τ ) + Fx& (t − τ ) + Bu (t ), t > 0 , x(t ) = φ (t ) , ∀t ∈ [− τ ,0] , (1a) (1b) y (t ) = Cx(t ) , where x (t ) ∈ ℜ n , (1c) y (t ) ∈ ℜ p and u (t ) ∈ ℜ m are respectively the state, measured φ (t ) is a continuous vector-valued initial function and τ > 0 is a known constant time delay. Matrices E , A, Ad , F , B and C are output and input vectors. known real constant and of appropriate dimensions. In this paper, in contrast to [11] and [21], the restriction imposed on matrix F (i.e. || F ||< 1 ) is removed and also matrix E can be singular as well. Without loss of generality, let rank ( E ) = r ( r ≤ n) and rank (C ) = p . Let us define the following functional state vector, z (t ) ∈ ℜ q , where z (t ) = Lx (t ) , (1d) and L ∈ ℜq×n is a given constant matrix. Without loss of generality, it is assumed that ⎡C ⎤ rank ( L) = q and rank ⎢ ⎥ = ( p + q ) ≤ n . ⎣L⎦ The aim of this paper is to design reduced-order observers capable of asymptotically estimating any given function of the state vector, z (t ) ∈ ℜ q . Let us consider the following observer structure of order q for the system (1) ω& (t ) = Nω (t ) + Jy(t ) + J d y (t − τ ) + Hu (t ) , t > 0 ω (t ) = ρ (t ), t ∈ [− τ ,0] , zˆ (t ) = ω (t ) + My (t ) , where ω (t ) ∈ ℜ q , (2a) (2b) (2c) ρ (t ) is a continuous vector-valued initial function and zˆ (t ) denotes the estimate of z (t ). Matrices N , J, Jd , H and M are to be DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 401 determined such that zˆ (t ) converges asymptotically to z (t ) (i.e. zˆ (t ) → z (t ) as t → ∞ ). Remark 1: The type of the observer structure proposed in (2) is commonly referred to as “observer without internal delay” (see, for example, [5, 15]) as it involves the output knowledge at the present and delayed instants. In this paper, in addition to removing the restriction on matrix F ( || F ||< 1 ) and allowing matrix E to be singular, the observer (2) has a low order and does not have a delayed state (estimated) derivatives (cf. [11, 21]). Thus, from the implementation point of view, the observer (2) is more attractive. It should also be noted here that there is always a trade-off between complexity of the observer and its existence conditions, i.e. the simpler the observer structure, the more restrictive the existence conditions become [5]. In this paper, our focus is on the design of the observer (2) for the class of neutral-delay systems (1). We are interested in deriving existence conditions for the observer (2) and a procedure for the determination of the observer’s parameters such that zˆ(t ) converges asymptotically to z (t ) . 3. Main Results In order to deal with the term Fx& (t − τ ) in (1) and also for the convenience of design, let us transform the system (1) into the following descriptor form ~ ~ ~& E~ x (t ) = A~ x (t ) + Ad ~ x (t − τ ) + Bu(t ), t > 0 , (3a) ~ ~ x (t ) = φ (t ), t ∈ [− τ ,0] , (3b) ~ y(t ) = C ~ x (t ) , (3c) ~ z (t ) = L ~ x (t ) , (3d) ⎡φ (t )⎤ ⎡ x(t )⎤ ~ where ~ x (t ) = ⎢ ∈ ℜ 2 n , φ (t ) = ⎢ & ⎥, ⎥ ⎣φ (t )⎦ ⎣ x& (t )⎦ ~ ~ ~ E = [ E 0] , A = [ A 0] , Ad = [ Ad F] , ~ ~ C = [C 0] and L = [ L 0] . Remark 2: The idea of transforming time-delay systems into “descriptor form” was first introduced by Fridman [7], Fridman & Shaked [8, 9]. This approach enriches the studies of time-delay systems through singular systems techniques [15]. However, despite the rich results available for singular systems, to date, little attention has been 402 T. FERNANDO AND H. TRINH paid to address the problem of designing linear functional observers (2) for the time-delay descriptor systems (3). This paper thus offers some results in this area. Now, let X ∈ ℜq×n and define error vectors ε (t ) ∈ ℜ q and e(t ) ∈ ℜ q as ε (t ) = ω (t ) − XE ~x (t ) , ~ and e(t ) = zˆ (t ) − z (t ) . (4a) (4b) The following theorem provides a sufficient condition ensuring that zˆ (t ) converges asymptotically to z (t ) . Theorem 1: There exists an observer of the form (2) for system (3) so that zˆ (t ) → z (t ) as t → ∞ provided that the following matrix equations hold. ~ ~ ~ NXE + JC − XA = 0, N is Hurwitz, (5) ~ ~ J d C − XAd = 0 , (6) ~ ~ ~ XE + MC − L = 0 , H = XB . Proof: (7) (8) From (4a), (2) and (3), the following error dynamics equation is obtained ε&(t ) = ω& (t ) − XE ~x& (t ) ~ ~ ~ ~ ~ ~ x (t ) + ( J d C − XAd ) ~ x (t − τ ) + ( H − XB)u (t ) , t > 0 (9a) = Nε (t ) + ( NXE + JC − XA) ~ ε (t ) = ρ (t ) − XEφ (t ), t ∈ [− τ ,0] . ~~ (9b) From (4b), (2c) and (4a), the error vector e(t ) can be expressed as ~ ~ ~ e(t ) = ε (t ) + ( XE + MC − L ) ~ x (t ) . (10) From (9) and (10) it is clear that e(t ) → 0 as t → ∞ if equations (5)-(8) of Theorem 1 are satisfied. This completes the proof of Theorem 1. Thus, in order to design a linear functional observer (2), we will need to solve the DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE matrix equations (5)-(8) for the unknown matrices N , J , J d , X, 403 M and H . First, the following theorem provides necessary and sufficient conditions for the solvability of the matrix equations (5)-(7) of Theorem 1 (matrix H is obtained from (8) once matrix X is solved). Theorem 2: The matrix equations (5)-(7) of Theorem 1 are completely solvable if and only if the following two conditions hold Condition 1: 0 ⎡ CA ⎢ 0 C ⎢ ⎢( E − I n ) A − Ad rank ⎢ 0 C ⎢ ⎢ LA 0 ⎢ 0 L ⎣⎢ 0 0 −F 0 0 0 Condition 2: 0 ⎡( sL − LA) ⎢ CA 0 ⎢ rank ⎢ C 0 ⎢ ⎢( E − I n ) A − Ad ⎢⎣ C 0 Proof: 0 0 0 −F 0 C⎤ 0 ⎡ CA 0 ⎥⎥ ⎢ 0 C ⎢ E⎥ ⎥ = rank ⎢( E − I n ) A − Ad 0⎥ ⎢ 0 C ⎢ ⎥ L ⎢ ⎥ 0 L ⎣ 0 ⎦⎥ 0 − L⎤ ⎡ CA ⎥ ⎢ C ⎥ C 0 ⎢ 0 ⎥ = rank ⎢( E − I n ) A − Ad ⎥ ⎢ E⎥ C 0 ⎢ ⎥ ⎢ L 0 ⎦ 0 ⎣ 0 0 −F 0 0 C⎤ 0 ⎥⎥ E⎥ ; ⎥ 0⎥ 0 ⎥⎦ C⎤ 0 0 ⎥⎥ − F E⎥ , ⎥ 0 0⎥ 0 0 ⎥⎦ ∀s ∈ C, Re( s ) ≥ 0 . (11) 0 (12) The proof of Theorem 2 can be constructed based around the ideas from the two papers by Darouach [2, 3]. A complete proof is shown in the Appendix A. Upon the satisfaction of the Conditions 1&2 of Theorem 2, a procedure for the determination of matrices N , J , J d , X , H and M that satisfy equations (5)-(8) of Theorem 1 can be derived (the procedure will be derived from the proof of Theorem 2). Before we present an observer design procedure, let us first focus on Conditions 1&2 of Theorem 2 and deduce some simpler conditions for some special cases. ⎡E ⎤ rank ⎢ ⎥ = n . ⎣C ⎦ Clearly for this case, matrix E can be singular. CASE 1: Note that the assumption, ⎡E ⎤ rank ⎢ ⎥ = n , is also a well-known assumption used in the design of Luenberger-type ⎣C ⎦ 404 T. FERNANDO AND H. TRINH observers for descriptor systems (see, for example, [4, 16, 17]). ⎡E ⎤ Since rank ⎢ ⎥ = n , then there always exists a nonsingular matrix Q ∈ ℜ ( n+ p )×( n+ p ) ⎣C ⎦ such that ⎡E ⎤ ⎡ E ⎤ Q⎢ ⎥ = ⎢ 1 ⎥ , ⎣C ⎦ ⎣GE1 ⎦ (13) where E1 ∈ ℜ n×n , rank ( E1 ) = n and G ∈ ℜ p×n . Let us also define the following ⎡− Ad ⎤ ⎡ Qd 1 ⎤ ⎡( E − I n ) A⎤ ⎡ Q1 ⎤ Q⎢ ⎥ = ⎢Q ⎥ , Q ⎢ 0 ⎥ = ⎢Q ⎥ , and CA ⎣ ⎣ ⎦ ⎣ d2⎦ ⎦ ⎣ 2⎦ ⎡− F ⎤ ⎡ Q f 1 ⎤ Q⎢ ⎥, ⎥=⎢ ⎣ 0 ⎦ ⎣Q f 2 ⎦ (14) where Q1 , Qd 1 and Q f 1 are matrices of dimension (n × n) , and Q2 , Qd 2 and Q f 2 are matrices of dimension ( p × n) . Accordingly, the Conditions 1&2 of Theorem 2 are reduced to the following. ⎡E ⎤ Corollary 1: When rank ⎢ ⎥ = n , Theorem 2 is simplified into ⎣C ⎦ Condition 1: ⎡ Q2 − GQ1 ⎢ LA − L( E ) −1 Q 1 1 ⎢ rank ⎢ 0 ⎢ C ⎢ ⎢⎣ L Qd 2 − GQd 1 − L( E1 ) Qd 1 C −1 0 0 Q f 2 − GQ f 1 ⎤ ⎡Q2 − GQ1 Qd 2 − GQd 1 Q f 2 − GQ f 1 ⎤ − L( E1 ) −1 Q f 1 ⎥⎥ ⎢ ⎥ 0 0 C ⎥ ⎥ = rank ⎢ 0 ⎢ C ⎥ 0 0 ⎥ 0 ⎢ ⎥ ⎥ 0 0 L ⎣ ⎦ ⎥⎦ 0 (15) and Condition 2: ⎡ sL − LA + L( E1 ) −1 Q1 L( E1 ) −1 Qd 1 L( E1 ) −1 Q f 1 ⎤ ⎡Q2 − GQ1 Qd 2 − GQd 1 Q f 2 − GQ f 1 ⎤ ⎢ ⎥ ⎥ ⎢ C 0 0 0 0 C ⎥ = rank ⎢ ⎥, rank ⎢ ⎢ ⎥ ⎢ C 0 0 Q2 − GQ1 Qd 2 − GQd 1 Q f 2 − GQ f 1 ⎥ ⎢ ⎥ ⎥ ⎢ L 0 0 0 C 0 ⎦ ⎣ ⎣⎢ ⎦⎥ ∀s ∈ C, Re( s ) ≥ 0 . (16) DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 405 Proof: The RHS of (11) can now be expressed as follows 0 ⎡ CA ⎢ 0 C ⎢ rank ⎢( E − I n ) A − Ad ⎢ 0 C ⎢ ⎢⎣ 0 L ⎛ ⎡ Q1 ⎜⎢ ⎜ ⎢Q2 = rank ⎜⎜ ⎢ 0 ⎢ ⎜⎢C ⎜⎜ ⎢ ⎝⎣ L Qd 1 Qd 2 C 0 0 Qf 1 Qf 2 0 0 0 0 0 −F 0 0 ⎛ C⎤ ⎡( E − I n ) A − Ad ⎜ ⎥ ⎢ CA 0⎥ 0 ⎜ ⎢ Q 0 ⎤ ⎡ ⎜ ⎢ E ⎥ = rank ⎜ ⎢ C 0 0 I ( 2 p +q ) ⎥⎦ ⎢ ⎥ ⎣ ⎜ 0⎥ C 0 ⎢ ⎜⎜ ⎢⎣ 0 ⎥⎦ L 0 ⎝ E1 ⎤ In ⎡ GE1 ⎥⎥ ⎢ 0 0 ⎥⎢ 0 ⎥⎢ 0 ⎥⎢ −1 ⎢− ( E 1) Q1 0 ⎥⎦ ⎣ 0 In 0 − ( E 1) −1 Qd 1 0 0 In − ( E 1) −1 Q f 1 E1 ⎤ 0 0 0 ⎡ ⎢Q − GQ Q − GQ Q − GQ GE1 ⎥⎥ d2 d1 f2 f1 1 ⎢ 2 = rank ⎢ C 0 0 0 ⎥ ⎥ ⎢ 0 0 0 ⎥ ⎢ C ⎢⎣ L 0 0 0 ⎥⎦ ⎡Q2 − GQ1 Qd 2 − GQd 1 Q f 2 − GQ f 1 ⎤ ⎢ ⎥ 0 0 C ⎥. = n + rank ⎢ ⎢ C ⎥ 0 0 ⎢ ⎥ 0 0 L ⎣ ⎦ −F 0 0 0 0 E⎤ ⎞ ⎟ C ⎥⎥ ⎟ 0 ⎥ ⎟⎟ ⎥ 0⎥⎟ ⎟ 0 ⎥⎦ ⎟⎠ ⎞ 0 ⎤⎟ ⎟ 0 ⎥⎥ ⎟ 0 ⎥⎟ ⎥⎟ I n ⎥⎦ ⎟ ⎟ ⎠ (17) Similarly, the LHS of (11) can be expressed as 0 ⎡ CA ⎢ C 0 ⎢ ⎢( E − I n ) A − Ad rank ⎢ C 0 ⎢ ⎢ LA 0 ⎢ L 0 ⎢⎣ 0 0 −F 0 0 0 C⎤ 0 ⎥⎥ E⎥ ⎥ = n+ 0⎥ L⎥ ⎥ 0 ⎥⎦ ⎡ Q2 − GQ1 ⎢ LA − L( E ) −1 Q 1 1 ⎢ ⎢ 0 ⎢ C ⎢ ⎢⎣ L Qd 2 − GQd 1 − L( E1 ) −1 Qd 1 C 0 0 Q f 2 − GQ f 1 ⎤ − L( E1 ) −1 Q f 1 ⎥⎥ ⎥. 0 ⎥ 0 ⎥ ⎥⎦ 0 (18) From (17) and (18), Condition 1 of Theorem 2 is thus reduced to (15). It is also easy to show that the LHS of (12) can be expressed as 406 T. 0 ⎡( sL − LA) ⎢ CA 0 ⎢ rank ⎢ C 0 ⎢ ⎢( E − I n ) A − Ad ⎢⎣ C 0 0 0 0 −F 0 FERNANDO AND H. TRINH − L⎤ ⎡ sL − LA + L( E1 ) −1 Q1 L( E1 ) −1 Qd 1 L( E1 ) −1 Q f 1 ⎤ C ⎥⎥ ⎥ ⎢ C 0 0 ⎥, 0 ⎥ = n + rank ⎢ ⎢ Q2 − GQ1 Qd 2 − GQd 1 Q f 2 − GQ f 1 ⎥ ⎥ E⎥ ⎥ ⎢ 0 C 0 ⎥⎦ ⎢⎣ ⎥ 0 ⎦ ∀s ∈ C, Re( s ) ≥ 0 . (19) From (19) and (17), Condition 2 of Theorem 2 is thus reduced to (16). This completes the proof of Corollary 1. CASE 2: Matrix E = I n . For the case where matrix E = I n , then Conditions 1&2 of Theorem 2 are reduced to the following. Corollary 2: When E = I n , Theorem 2 is simplified into Condition 1: Condition 2: ⎡CA CAd ⎢0 C ⎢ ⎢ rank LA LAd ⎢ 0 ⎢C ⎢⎣ L 0 CF ⎤ ⎡CA CAd 0 ⎥⎥ ⎢0 C ⎥ LF = rank ⎢ ⎢ C 0 ⎥ 0 ⎥ ⎢ 0 ⎣L 0 ⎥⎦ ⎡( sL − LA) − LAd ⎢ CA CAd rank ⎢ ⎢ C 0 ⎢ 0 C ⎣ CF ⎤ 0 ⎥⎥ ; 0 ⎥ ⎥ 0 ⎦ (20) − LF ⎤ ⎡CA CAd ⎥ ⎢0 CF ⎥ C = rank ⎢ ⎢C 0 ⎥ 0 ⎢ ⎥ 0 ⎦ 0 ⎣L CF ⎤ 0 ⎥⎥ , 0 ⎥ ⎥ 0 ⎦ ∀s ∈ C , Re( s ) ≥ 0 . (21) Proof: Let us first show the proof for the first condition of Corollary 2 (i.e. (20)). Substituting E = I n into (11), the RHS of (11) can be expressed as follows 0 ⎡CA ⎢C 0 ⎢ rank ⎢ 0 − Ad ⎢ C ⎢0 ⎢⎣ L 0 0 0 −F 0 0 ⎛ ⎡ 0 − Ad C⎤ ⎜⎢ 0 0 ⎥⎥ ⎜ ⎢CA ⎜ C I n ⎥ = rank ⎜ ⎢ 0 ⎥ ⎢ ⎜⎢ C 0 0⎥ ⎜⎜ ⎢ ⎥ 0 0⎦ ⎝⎣ L −F 0 0 0 0 In ⎤ ⎡In C ⎥⎥ ⎢ 0 0 ⎥⎢ ⎥⎢ 0 0 ⎥⎢ 0 0 ⎥⎦ ⎣ 0 In 0 Ad 0 0 In F ⎞ 0 ⎤⎟ ⎟ 0 ⎥⎥ ⎟ 0 ⎥⎟ ⎥⎟ I n ⎦ ⎟⎟ ⎠ DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 0 ⎡0 ⎢CA CA d ⎢ = rank ⎢ 0 C ⎢ 0 ⎢C ⎢⎣ L 0 0 CF 0 0 0 In ⎤ ⎡CA CAd C ⎥⎥ ⎢0 C 0 ⎥ = n + rank ⎢ ⎢C 0 ⎥ 0⎥ ⎢ 0 ⎣L 0 ⎥⎦ CF ⎤ 0 ⎥⎥ . 0 ⎥ ⎥ 0 ⎦ 407 (22) Similarly, the LHS of (11) can be written as 0 ⎡CA ⎢C 0 ⎢ ⎢ 0 − Ad rank ⎢ C ⎢0 ⎢ LA 0 ⎢ 0 ⎣⎢ L 0 0 −F 0 0 0 C⎤ ⎡CA CAd 0 ⎥⎥ ⎢0 C ⎢ In ⎥ ⎥ = n + rank ⎢ LA LAd 0⎥ ⎢ 0 ⎢C ⎥ L ⎢ ⎥ 0 ⎣L 0 ⎦⎥ CF ⎤ 0 ⎥⎥ LF ⎥ . ⎥ 0 ⎥ 0 ⎥⎦ (23) From (22) and (23), Condition 1 of Theorem 2 is thus reduced to (20). The rest of the proof of Corollary 2 can be easily proven by following similar lines and therefore is omitted. This completes the proof of Corollary 2. ⎡C ⎤ Matrix E = I n and rank ⎢ ⎥ = n . ⎣L⎦ ⎡C ⎤ When rank ⎢ ⎥ = n , an (n − p) -order observer (2) for x(t ) can be obtained, ⎣L⎦ CASE 3: ⎡C ⎤ ⎡ zˆ (t ) ⎤ where xˆ (t ) = ⎢ ⎥ ⎢ ⎥. ⎣ L ⎦ ⎣ y (t )⎦ −1 For this case, Conditions 1&2 of Corollary 2 are reduced to the following. ⎡C ⎤ Corollary 3: When rank ⎢ ⎥ = n , Corollary 2 is simplified into ⎣L⎦ ⎡A F⎤ ⎡CA CF ⎤ Condition 1: rank ⎢ d ; = rank ⎢ d ⎥ 0 ⎥⎦ ⎣C 0⎦ ⎣ C Condition 2: ⎡ sI n − A − Ad rank ⎢⎢ C 0 ⎢⎣ 0 C − F⎤ ⎡CA 0 ⎥⎥ = n + rank ⎢ d ⎣ C 0 ⎥⎦ (24) CF ⎤ , 0 ⎥⎦ ∀s ∈ C , Re( s ) ≥ 0 . (25) 408 T. FERNANDO AND H. TRINH ⎡C ⎤ Proof: When rank ⎢ ⎥ = n , the RHS of (20) can be written as ⎣L⎦ ⎡CA CAd ⎢0 C rank ⎢ ⎢C 0 ⎢ 0 L ⎣ CF ⎤ 0 ⎥⎥ ⎡CA = n + rank ⎢ d 0 ⎥ ⎣ C ⎥ 0 ⎦ CF ⎤ . 0 ⎥⎦ (26) Similarly, the LHS of (20) is ⎡CA CAd ⎢0 C ⎢ rank ⎢ LA LAd ⎢ 0 ⎢C ⎢⎣ L 0 ⎡A = n + rank ⎢ d ⎣C CF ⎤ 0 ⎥⎥ ⎡CAd ⎢ LF ⎥ = n + rank ⎢ LAd ⎥ ⎢⎣ C 0 ⎥ ⎥ 0 ⎦ F⎤ . 0 ⎥⎦ ⎛ ⎡ ⎡C ⎤ −1 CF ⎤ ⎜⎢ LF ⎥⎥ = n + rank ⎜ ⎢ ⎢⎣ L ⎥⎦ ⎜⎜ ⎢ 0 ⎥⎦ ⎝⎣ 0 ⎤ ⎡ ⎡C ⎤ 0 ⎥ ⎢ ⎢ ⎥ Ad ⎥ ⎢⎣ L ⎦ I p ⎥⎦ ⎣⎢ C ⎡C ⎤ ⎤ ⎞⎟ ⎢L⎥F ⎥⎟ ⎣ ⎦ ⎥ 0 ⎦⎥ ⎟⎟⎠ (27) From (27) and (26), Condition 1 of Corollary 2 is thus reduced to (24). Now, the LHS of (21) can be expressed as follows ⎡( sL − LA) − LAd ⎢ CA CAd rank ⎢ ⎢ 0 C ⎢ 0 C ⎣ ⎛ − LF ⎤ ⎜ ⎡G1 ⎥ CF ⎥ ⎜⎢ = rank ⎜ ⎢ 0 0 ⎥ ⎜⎢ 0 ⎥ ⎜⎣ 0 ⎦ ⎝ ⎡ sI n − A − Ad = rank ⎢⎢ C 0 ⎢⎣ 0 C ⎡L⎤ where [G1 G2 ] = ⎢ ⎥ . ⎣C ⎦ − G2 sG2 0 Ip 0 0 − F⎤ 0 ⎥⎥ , ∀s ∈ C, Re( s ) ≥ 0 , 0 ⎥⎦ ⎡( sL − LA) − LAd 0 ⎤⎢ CA CAd ⎥ 0 ⎥⎢ ⎢ 0 C I p ⎥⎦ ⎢ 0 C ⎣ − LF ⎤ ⎞ ⎟ CF ⎥⎥ ⎟ ⎟ 0 ⎥⎟ ⎥⎟ 0 ⎦⎠ (28) −1 From (28) and (26), Condition 2 of Corollary 2 is thus reduced to (25). This completes the proof of Corollary 3. DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE CASE 4: 409 Matrices E = I n and F = 0 . This case is the same as the problem of designing linear functional observers without internal delay [3] for the following time-delay systems ⎧ x& (t ) = Ax(t ) + Ad x(t − τ ) + Bu (t ), t > 0 ⎪ ⎪ x(t ) = φ (t ), ∀t ∈ [− τ ,0] ⎨ ⎪ y (t ) = Cx (t ) ⎪⎩ z (t ) = Lx (t ). (29) Accordingly, by letting F = 0 in Corollary 2, the following Conditions, which are the same as those derived by Darouach [3], are obtained. Corollary 4: When F = 0 , Corollary 2 is simplified into Condition 1: Condition 2: CASE 5: ⎡CA CAd ⎤ ⎡CA CAd ⎤ ⎢0 C ⎥⎥ ⎢0 ⎢ C ⎥⎥ rank ⎢ LA LAd ⎥ = rank ⎢ ; ⎢C 0 ⎥ ⎢ ⎥ 0 C ⎢ ⎥ ⎢ ⎥ 0 ⎦ ⎣L ⎢⎣ L 0 ⎥⎦ ⎡ sL − LA − LAd ⎤ ⎡CA CAd ⎤ ⎢ CA ⎥ ⎢0 C ⎥⎥ CA d ⎥ , ∀s ∈ C , Re( s ) ≥ 0 . rank ⎢ = rank ⎢ ⎢ C ⎢C 0 ⎥ 0 ⎥ ⎢ ⎢ ⎥ ⎥ 0 ⎦ C ⎦ ⎣ 0 ⎣L (30) (31) ⎡C ⎤ Matrices E = I n , F = 0 and rank ⎢ ⎥ = n . ⎣L⎦ This case corresponds to the problem of designing an (n − p) -order observer (without internal delay) for the time-delay system (29). Accordingly, by letting F = 0 in Corollary 3, Conditions 1&2 of Corollary 3 are reduced to the following. Corollary 5: When F = 0 , Corollary 3 is simplified into Condition 1: ⎡A ⎤ ⎡CA ⎤ rank ⎢ d ⎥ = rank ⎢ d ⎥ ; ⎣C ⎦ ⎣ C ⎦ (32) 410 T. Condition 2: FERNANDO AND H. TRINH ⎡ sI n − A − Ad ⎤ ⎡CA ⎤ 0 ⎥⎥ = n + rank ⎢ d ⎥ , ∀s ∈ C , Re( s ) ≥ 0 . rank ⎢⎢ C ⎣ C ⎦ ⎢⎣ 0 C ⎥⎦ (33) ⎡CA ⎤ ⎡C ⎤ Matrices E = I n , F = 0 , rank ⎢ ⎥ < n and rank ⎢ d ⎥ = n . ⎣ C ⎦ ⎣L⎦ This case corresponds to the problem of designing a q -order ( q < (n − p) ) linear CASE 6: functional observer (without internal delay) for the time-delay system (29) under the ⎡CA ⎤ assumption that rank ⎢ d ⎥ = n . ⎣ C ⎦ ⎡CA ⎤ Similar to the Case 1, since rank ⎢ d ⎥ = n , then there always exists a ⎣ C ⎦ 2 p×2 p nonsingular matrix P ∈ ℜ such that ⎡CA ⎤ ⎡ C ⎤ P⎢ d ⎥ = ⎢ 1 ⎥ , ⎣ C ⎦ ⎣ DC1 ⎦ (34) where C1 ∈ ℜ n×n , rank (C1 ) = n and D ∈ ℜ( 2 p−n )×n . Let us also define the following ⎡CA⎤ ⎡ P ⎤ P⎢ ⎥ = ⎢ 1 ⎥ , ⎣ 0 ⎦ ⎣ P2 ⎦ where P1 ∈ ℜ n×n and P2 ∈ ℜ( 2 p −n )×n . (35) Accordingly, Corollary 4 is equivalent to the following. ⎡CA ⎤ Corollary 6: When rank ⎢ d ⎥ = n , Corollary 4 is simplified into ⎣ C ⎦ Condition 1: P2 − DP1 ⎡ ⎤ ⎡ P2 − DP1 ⎤ ⎢ LA − LA (C ) −1 P ⎥ d 1 1⎥ ⎢ = rank ⎢⎢ C ⎥⎥ ; rank ⎢ ⎥ C ⎢⎣ L ⎥⎦ ⎢ ⎥ L ⎣ ⎦ (36) DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE Condition 2: Proof: ⎡ sL − ( LA − LAd (C1 ) −1 P1 )⎤ ⎡ P2 − DP1 ⎤ ⎥ ⎢ ⎢ ⎥ rank ⎢ P2 − DP1 ⎥ = rank ⎢ C ⎥ , ∀s ∈ C , Re( s ) ≥ 0 . (37) ⎥ ⎢ ⎢⎣ L ⎥⎦ C ⎦ ⎣ The RHS of (30) can be expressed as ⎛ ⎡CA CAd ⎤ ⎜ ⎢0 ⎥ P C ⎥ = rank ⎜⎜ ⎡⎢ rank ⎢ ⎢C 0 ⎥ ⎜⎣0 ⎢ ⎥ ⎜ 0 ⎦ ⎣L ⎝ ⎛ ⎡ P1 ⎜⎢ ⎜ P = rank ⎜ ⎢ 2 ⎢ ⎜ ⎢C ⎜ L ⎝⎣ 411 C1 ⎤ DC1 ⎥⎥ ⎡ I n ⎢ 0 ⎥ ⎣− (C1 ) −1 P1 ⎥ 0 ⎦ ⎡ ⎡CA⎤ ⎡CAd ⎤ ⎤ ⎞ ⎡ P1 ⎟ ⎢P 0 ⎤ ⎢⎢ ⎢⎣ 0 ⎥⎦ ⎢⎣ C ⎥⎦ ⎥⎥ ⎟ ⎢ 2 rank = ⎥⎟ ⎢C I ( p+q ) ⎥⎦ ⎢ ⎡C ⎤ 0 ⎥⎟ ⎢⎢ ⎥ ⎢ ⎟ ⎥⎦ ⎠ ⎢⎣ ⎣ L ⎦ ⎣L ⎞ ⎡ 0 ⎟ ⎢ P − DP 0 ⎤⎟ 1 ⎢ 2 = rank ⎢ C I n ⎥⎦ ⎟⎟ ⎢ ⎟ ⎣ L ⎠ C1 ⎤ DC1 ⎥⎥ 0 ⎥ ⎥ 0 ⎦ C1 ⎤ ⎡ P2 − DP1 ⎤ DC1 ⎥⎥ = n + rank ⎢⎢ C ⎥⎥ . (38) 0 ⎥ ⎢⎣ L ⎥⎦ ⎥ 0 ⎦ Similarly, it can be easily shown that the LHS of (30) can be expressed as ⎡CA CAd ⎤ P2 − DP1 ⎤ ⎡ ⎥ ⎢0 C ⎥ ⎢ −1 ⎥ ⎢ LA LA C P ( ) − d 1 1⎥ rank ⎢ LA LAd ⎥ = n + rank ⎢ . ⎥ ⎢ C ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢C L ⎦ ⎣ ⎥ ⎢⎣ L 0 ⎦ (39) From (39) and (38), Condition 1 of Corollary 4 is thus reduced to (36). Now, it can be easily shown that the LHS of (31) can be expressed as ⎡ sL − LA − LAd ⎤ ⎡ sL − ( LA − LAd (C1 ) −1 P1 )⎤ ⎢ CA ⎥ CA ⎥ ⎢ d ⎥ = n + rank ⎢ P2 − DP1 rank ⎢ ⎥ , ∀s ∈ C , Re( s ) ≥ 0 .(40) ⎢ C 0 ⎥ ⎥ ⎢ C ⎢ ⎥ ⎦ ⎣ C ⎦ ⎣ 0 From (40) and (38), Condition 2 of Corollary 4 is thus reduced to (37). This completes the proof of Corollary 6. 412 T. CASE 7: FERNANDO AND H. TRINH Matrices E = I n , Ad = 0 and F = 0 . This case is the same as the well-known problem of designing linear functional observers for the following linear time-invariant systems (see, for example, Darouach, 2001; Tsui, 1986) ⎧ x& (t ) = Ax(t ) + Bu (t ) ⎪ ⎨ y (t ) = Cx(t ) ⎪ z (t ) = Lx(t ). ⎩ (41) Accordingly, by letting Ad = 0 in Corollary 4, the following Conditions, which are the same as those derived by Darouach [2], are obtained. Corollary 7: When Ad = 0 , Corollary 4 is simplified into Condition 1: ⎡CA⎤ ⎡CA⎤ ⎢ LA⎥ ⎢ ⎥ rank = rank ⎢⎢ C ⎥⎥ ; ⎢C ⎥ ⎢⎣ L ⎥⎦ ⎢ ⎥ ⎣L⎦ Condition 2: ⎡CA⎤ ⎡( sL − LA)⎤ ⎥ ⎢ rank ⎢ CA ⎥ = rank ⎢⎢ C ⎥⎥ , ∀s ∈ C , Re( s ) ≥ 0 . ⎢⎣ L ⎥⎦ ⎢⎣ C ⎥⎦ (42) (43) An Observer Design Procedure Upon the satisfaction of the Conditions 1&2 of Theorem 2, a procedure for the determination of matrices N , J , J d , X , H and M that satisfy equations (5)-(8) of Theorem 1 can be summarized in the following design procedure. Design Procedure Step 1: Obtain matrices N1 and N 2 from (A15). Step 2: Use (A14) to derive the matrix gain Z and a stable matrix N . Step 3: Use (A13) to obtain [ M T X J d ] . Step 4: From (A4), obtain matrix J , where J = (T + NM ) . Step 5: Finally, obtain matrix H from (8). DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 413 4. A Numerical Example In this section, an example is presented to demonstrate the results of this paper. Let us consider the linear differential-delay system (1) with ⎡2 1⎤ ⎡ − 2 − 3⎤ E = I2 , F = ⎢ , τ = 2 , C = [1 0] , A = ⎢ ⎥ ⎥, ⎣0 0 ⎦ ⎣ 0. 5 − 4 ⎦ ⎡1⎤ and B = ⎢ ⎥ . ⎣1⎦ ⎡1 − 0.5⎤ Ad = ⎢ 0 ⎥⎦ ⎣0 For this example, the assumption that || F ||< 1 which was used in [11, 21] does not hold and therefore their design procedure can not be used. Let us now use the results of this paper to design a first-order observer (2) to estimate the second state, x2 (t ) , of the system, i.e. z (t ) = Lx (t ) = [0 1]x(t ) . ⎡C ⎤ ⎡1 For this example, we have E = I 2 , rank ⎢ ⎥ = rank ⎢ ⎣L⎦ ⎣0 case falls within the special Case 3. It is easy to check that Corollary 3 hold. Accordingly, a first-order observer for x2 (t ) 0⎤ =2. 1 ⎥⎦ Clearly, this the Conditions 1&2 of can be easily designed. By following the observer design procedure presented in Section 3, the following is obtained Step 1: Matrices N1 and N 2 are obtained as N1 = −4 and N 2 = 0 . Step 2: For this example, the pair ( N 2 , N1 ) is detectable (but not observable) and we have a stable matrix N = −4 . Thus, we can set the matrix gain Z as Z = 0 . Step 3: From (A13), [ M T X J d ] = [0 0.5 0 1 0] . This gives M = 0 , T = 0.5 , X = [0 1] and J d = 0 . Steps 4&5: J = 0.5 and H = 1 . It is easy to confirm the satisfaction of equations (5)-(8) of Theorem 1 by substituting the above derived matrices M , N , J , J d , X and H into (5)-(8). The observer design is completed and a first-order observer is obtained for x2 (t ) , where w& (t ) = −4w(t ) + 0.5 y(t ) + u (t ), t > 0 , ω (t ) = ρ (t ), t ∈ [− τ ,0] , xˆ 2 (t ) = w(t ) . (Note that the above observer is also known as a memoryless state observer [12, 18] since matrix J d = 0 ). 414 T. FERNANDO AND H. TRINH The following simulation study was carried out with the control input signal u (t ) is as shown in figure (1). Figure (2) shows the simulated responses of x2 (t ) and xˆ2 (t ) . Figure (3) shows the error state ~ x2 (t ) = x2 (t ) − xˆ2 (t ) . The initial conditions for the system and observer were taken to be x(t ) = 0 and ω (t ) = 1 , t ∈ [−2,0]. Figures (2)-(3) clearly show that the state estimation error converges to zero. Appendix B shows the simulation results. 5. Conclusion This paper has presented the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. Sufficient conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector have been derived. A procedure for the determination of the observer parameters has been given. The results derived in this paper extend some existing results of linear functional observers to include linear neutral delay systems. References [1] Clarkson, I.D., & Goodall, D.P. (2000). On the stabilizability of imperfectly known nonlinear delay systems of the neutral type. IEEE Transactions on Automatic Control, 45, 2326–2331. [2] Darouach, M. (2000). Existence and design of functional observers for linear systems. IEEE Transactions on Automatic Control, 45, 940-943. [3] Darouach, M. (2001). Linear functional observers for systems with delays in state variables. IEEE Transactions on Automatic Control, 46, 491-496. [4] Darouach, M., & Boutayeb, M. (1995). Design of observers for descriptor systems. IEEE Transactions on Automatic Control, 40, 1323-1327. [5] Fairman, F.W., & Kumar, A. (1986). Delay-less observers for systems with delay. IEEE Transactions on Automatic Control, 31, 258–259. [6] Fiagbedzi, Y.A. (1994). Feedback stabilization of neutral systems via the transformation technique. International Journal of Control, 59, 1579–1589. [7] Fridman, E. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. System & Control Letters, 43, 309–319. [8] Fridman, E., & Shaked, U. (2002a). A descriptor system approach to H∞ control of linear time-delay systems. IEEE Transactions on Automatic Control, 47, 253–270. [9] Fridman, E., & Shaked, U. (2002b). An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 47, 1931-1937. [10] Kolmanovskii, V.B., & Nosov, V.R. (1986). Stability of Functional Differential Equations. New York: Academic. DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 415 [11] Kou, J.M., Lien, C.H., Fan, K.K., & Hsieh, J.G. (2004). Delay-independent observer-based control for a class of neutral systems. Journal of Dynamic Systems, Measurement, and Control, 126(4), 896-898. [12] Leyva-Ramos, J., & Pearson, A.E. (1995). An asymptotic modal observer for linear autonomous time lag systems. IEEE Transactions on Automatic Control, 40, 1291-1294. [13] Luenberger, D.G. (1971). An introduction to observers. IEEE Transactions on Automatic Control, 16, 596-602. [14] O’Connor, D., & Tarn, T.J. (1983). On stabilization by state feedback for neutral differential difference equations. IEEE Transactions on Automatic Control, 28, 615-618. [15] Richard, J.P. (2003). Time-delay systems: An overview of some recent advances and open problems. Automatica, 39, 1667 – 1694. [16] Shafai, B., & Carroll, R.L. (1987). Design of a minimal-order observer for singular systems. International Journal of Control, 45, pp. 1075-1081. [17] Shields, D.N. (1992). Observers for descriptor systems. International Journal of Control, 55, 249-256. [18] Trinh, H., & Aldeen, M. (1997). A memoryless state observer for discrete time-delay systems. IEEE Transactions on Automatic Control, 42, 1572-1576. [19] Trinh, H. (1999). Linear functional state observer for time-delay systems. International Journal of Control, 72, 1642-1658. [20] Tsui, C.C. (1986). On the order reduction of linear functional observers. IEEE Transactions on Automatic Control, 31, 447-449. [21] Wang, Z., Lam, J., & Burnham, K.J. (2002). Stabilty analysis and observer design for neutral delay systems. IEEE Transactions on Automatic Control, 47, 478-483. Appendix A: Proof of Theorem 2 First, by letting ⎧ E = E~ − I u , I u = [ I n ⎪⎪ ~ ⎤ ⎡ A ⎨ ⎥, ⎪ Au = ⎢ ⎪⎩ ⎢⎣0 n×2n ⎥⎦ 0 n×n ] (A1) ~ then matrix A can be expressed as ~ ~ A = ( E − E ) Au . Substituting (A2) and (7) into (5), the following equation is obtained (A2) 416 T. ~ ~ NL = L Au − [ M where FERNANDO AND H. TRINH T ⎡C~Au ⎤ ⎢ ~ ⎥ X ]⎢ C ⎥ , ⎢EA ⎥ ⎣ u⎦ (A3) T = J − NM . (A4) Post-multiply both sides of (A3) by the following full-row rank matrix ~ S = [ L+ ~~ ( I 2 n − L+ L )] = [ S1 S2 ] , (A5) ~ ~ ( L+ denotes the generalized matrix inverse of L ) yields the following two equations ~ N = L Au S1 − [ M and [M T T ⎡C~Au ⎤ ⎢ ~ ⎥ X ] ⎢ C ⎥ S1 , ⎢EA ⎥ ⎣ u⎦ (A6) ⎡C~Au ⎤ ⎢ ~ ⎥ ~ X ] ⎢ C ⎥ S 2 = L Au S 2 . ⎢EA ⎥ ⎣ u⎦ (A7) Equations (A7), (6) and (7) can be written in an augmented matrix equation as follows [M where T X Jd ]Ω = Ψ , (A8) ~ C⎤ ⎥ 0⎥ ( 3 p + n )×6 n ~⎥ ∈ ℜ E ⎥ 0 ⎦⎥ ⎡C~Au S 2 ⎢ ~ CS2 Ω=⎢ ⎢EA S ⎢ u 2 ⎣⎢ 0 0 0 ~ − Ad ~ C ~ Ψ = [ L Au S 2 ~ 0 L ] ∈ ℜ q×6 n . (A9) and From the above equations, the knowledge of [ M (A10) T X J d ] is necessary and DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE 417 sufficient for the determination of matrices N , J and H . From (A8), a solution for [M T X J d ] exists if and only if the following condition holds ⎡Ω ⎤ rank ⎢ ⎥ = rank (Ω) . ⎣Ψ ⎦ (A11) Now, it is easy to show that the following condition ⎡C~Au ⎢ ~ ⎢ C ⎢EA rank ⎢ u ⎢ 0 ⎢ L~A ⎢ ~u ⎢⎣ L 0 0 ~ − Ad ~ C 0 0 ~ C⎤ ⎡C~Au ⎥ 0⎥ ⎢ ~ ~⎥ ⎢ C E ⎥ = rank ⎢ E Au ⎢ 0⎥ ⎢ 0 ~⎥ L⎥ ⎢ L~ ⎣ 0 ⎥⎦ 0 0 ~ − Ad ~ C 0 ~ C⎤ ⎥ 0⎥ ~ E⎥ ⎥ 0⎥ 0 ⎥⎦ (A12) is equivalent to the condition (A11). (Note: To show that (A12) is equivalent to (A11), ⎡ S1 ⎢ post-multiply both sides of (A12) by a full row-rank matrix ⎢ 0 ⎢⎣ 0 ~ ~ ~ Then by substituting (A1), E = [ E 0] , A = [ A 0] , Ad = [ Ad S2 0 0 I 2n 0 0 0⎤ 0 ⎥⎥ ). I 2 n ⎥⎦ ~ F ] , C = [C 0] and ~ L = [ L 0] into (A12), Condition 1 of Theorem 2 is obtained. Therefore upon the satisfaction of (11), a general solution to (A8) is [M T X J d ] = ΨΩ + + Z ( I ( 3 p+n ) − ΩΩ+ ) , (A13) where Z ∈ ℜq×( 3 p+n ) is an arbitrary matrix. Substituting (A13) into (A3) yields N = N1 − ZN 2 , where (A14) 418 T. FERNANDO AND H. TRINH ⎡C~Au S1 ⎤ ⎢~ ⎥ C S1 ⎥ ~ N1 = L Au S1 − ΨΩ + Γ , N 2 = ( I ( 3 p+n ) − ΩΩ+ )Γ and Γ = ⎢ . ⎢EA S ⎥ u 1 ⎥ ⎢ ⎢⎣ 0 ⎥⎦ (A15) In (A14), matrix N is Hurwitz if and only if the pair ( N 2 , N1 ) is detectable, i.e. ⎡ sI − N1 ⎤ rank ⎢ q ⎥ = q , ∀s ∈ C , Re( s ) ≥ 0 . ⎣ N2 ⎦ (A16) In the following, we will show that that the following condition ~ ⎡C~Au − L⎤ ⎢ ~ ~⎥ C ⎥ ⎢ C ⎥ 0 = rank ⎢ E Au ⎢ ~⎥ E ⎥ ⎢ 0 ⎢ L~ ⎥ 0 ⎦ ⎣ ⎡( sL~ − L~Au ) 0 ⎢ ~ 0 C A u ⎢ ~ rank ⎢ 0 C ⎢ ~ − Ad ⎢ E Au ~ ⎢ 0 C ⎣ 0 0 ~ − Ad ~ C 0 ~ C⎤ ⎥ 0⎥ ~⎥ E , ∀s ∈ C , Re( s ) ≥ 0 , ⎥ 0⎥ 0 ⎥⎦ (A17) is equivalent to the condition (A16). First, post-multiply the RHS of (A17) by a full ⎡ S1 ⎢ row-rank matrix ⎢ 0 ⎢⎣ 0 ⎡C~Au ⎢ ~ ⎢ C rank ⎢ E Au ⎢ ⎢ 0 ⎢ L~ ⎣ S2 0 0 I 2n 0 0 0 0 ~ − Ad ~ C 0 0⎤ 0 ⎥⎥ to give I 2 n ⎥⎦ ~ C⎤ ⎥ 0⎥ ~ E ⎥ = q + rank (Ω) . ⎥ 0⎥ 0 ⎥⎦ The LHS of (A17) can be expressed as follows (A18) DESIGNING LINEAR OBSERVERS FOR TIME-DELAY SYSTEMS OF THE NEUTRAL TYPE ⎡( sL~ − L~Au ) 0 ⎢ ~ 0 ⎢ C Au ~ rank ⎢ C 0 ⎢ ~ − Ad ⎢ E Au ~ ⎢ C 0 ⎣ ⎡ sI − L~Au S1 = rank ⎢ q Γ ⎣⎢ ⎡ sI q − N1 = rank ⎢⎢ N 2 + ⎣⎢ ΩΩ Γ ~ ⎛ ⎡( sL~ − L~Au ) − L⎤ 0 ⎜⎢ ~ ⎥ ~ ⎜ C ⎥ 0 ⎢ C Au ⎜ ~ C 0 ⎥ = rank ⎜ ⎢ 0 ⎢ ~ ~ ⎥ ⎜ E ⎥ − Ad ⎢ E Au ⎜⎢ ~ ⎜ C 0 ⎥⎦ 0 ⎝⎣ ⎛ ⎡I q ⎜⎢ − Ψ⎤ ⎥ = rank ⎜ ⎢ 0 Ω ⎦⎥ ⎜⎜ ⎢ ⎝⎣ 0 ⎛ ⎡ sI q − N1 0⎤ ⎜ ⎥ 0 ⎥ = rank ⎜ ⎢⎢ N 2 ⎜⎢ + Ω⎦⎥ ⎝ ⎣ ΩΩ Γ ~ − L⎤ ~ ⎥ C ⎥ ⎡ S1 ⎢ 0 ⎥⎢ 0 ⎥ ~ E ⎥ ⎢⎣ 0 0 ⎥⎦ ⎤ ~ ⎥ ⎡ sI − L Au S1 ( I ( 3 p + n ) − ΩΩ )⎥ ⎢ q Γ ⎥⎣ ΩΩ + ⎦ ΨΩ + + S2 0 0 I 2n 0 0 419 ⎞ ⎟ 0 ⎤⎟ ⎟ 0 ⎥⎥ ⎟ I 2 n ⎥⎦ ⎟ ⎟ ⎟ ⎠ ⎞ − Ψ⎤ ⎟ ⎥⎟ Ω ⎦ ⎟⎟ ⎠ ⎞ 0⎤ 0 ⎤⎟ ⎡ Iq ⎥ 0 ⎥⎢ ⎥⎟ − Ω + Γ I 6n ⎦ ⎟ ⎣ Ω ⎦⎥ ⎠ ⎡ sI − N1 ⎤ = rank ⎢ r ⎥ + rank[Ω] , ∀s ∈ C, Re( s ) ≥ 0 . ⎣ N2 ⎦ (A19) It is clear from (A18) and (A19) that (A17) is equivalent to (A16). Finally, and ~ ~ ~ again, by substituting (A1), E = [ E 0] , A = [ A 0] , Ad = [ Ad ~ F ] , C = [C 0] ~ and L = [ L 0] into (A17), Condition 2 of Theorem 2 is obtained. This completes the proof of Theorem 2. Appendix B: Simulation Results Figure (1): Input signal u( t ) 420 T. FERNANDO AND H. TRINH Figure (2): Responses of x 2 ( t ) (solid line) and x$ 2 ( t ) (dashed line). Figure (3): Response of the error state ~ x2 ( t ) Tyrone Fernando, Department of Electrical Electronic and Computer Engineering, University of Western Australia. Email: tyrone@ee.uwa.edu.au. Dr Fernando is a senior lecturer at the University of Western Australia. He received his BE (Hons) and PhD degrees from University of Melbourne in 1990 and 1996, respectively. His research interests are in the areas of Control Systems and Biomedical Engineering. Hieu Trinh School of Engineering and Information Technology, Deakin University, Australia. Dr Trinh received the B.E. (Hons.), M.Eng.Sc., and PhD degrees from the University of Melbourne, Melbourne, Australia, in 1990, 1992 and 1996, respectively. He is currently a Senior Lecturer at Deakin University. His research interests are in the areas of control, robotics and power systems.