Finite-dimensional observers for delay systems

1986 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003 Technical Notes and Correspondence_______________________________ Finite-Dimensional Observers for Delay Systems Yawvi A. Fiagbedzi and A. Cherid Abstract—A finite-dimensional observer theory is formulated for a class of point delayed systems. The observer is characterized by the fact that it directly generates a finite-dimensional feedback controller for this class of systems. The theory is based on a preliminary approximation theory which allows the delay system to be exponentially approximated by a system of ordinary differential equations. Index Terms—Delay system, differential-difference equation, finitedimensional approximation, finite-dimensional observer. I. INTRODUCTION The aim is to construct a finite-dimensional observer for a delay system described by Sd : x_ (t) =A0 x(t) + A1 x(t 0 r) + B0 u(t) y (t) =C0 x(t) (1.1) (1.2) where t > 0 denotes time, overdot represents derivative with respect to t and r 2 (0; 1) is the system delay. Also, x(t) 2 n , y (t) 2 p , u(t) 2 m ; A0 ; A1 2 n2n , B0 2 n2m and C0 2 p2n . Define the segment function, xt , by xt () = x(t + ) where 0r    0. Using the segment function notation with t = 0, the system initial condition is given by x0 () = 1 (); 0r   < 0 n 0 ) and  2 1 0 x0 (0) = x(0) = 0 (1.3) where  2 L2 ([0r; 0]; . For brevity, the initial function will be denoted by  = ( ;  ) 2 L2 ([0r; 0]; n ) 2 n . Also, the initial value problem (IVP) defined by Sd and its initial function, , will be denoted by (Sd ; ). Following [1] and [12], there exists a unique solution to (Sd ; ). It is the (unique) function, x(1; n ); which is defined on [0r; 1), absolutely continuous for t  0, satisfies (1.1) almost everywhere (a.e.) on (0; 1) and satisfies (1.3). It also depends continuously on the initial data. For t  0, the state of (Sd ; ) is given by (xt (1; ); x(t; )) 2 L2 ([0r; 0]; n ) 2 n . Work on infinite-dimensional observers can be found, for example, in [2], [14], [15], and the references therein. For ease of implementation, however, finite-dimensional observers are to be preferred. Exploiting the finiteness of the system open loop poles in any given right half of the complex plane [8, Lemma 4.1, p. 18], a finite-dimensional observer is constructed in [11]. It is well known (see, for example, [6] and the references therein) that general feedback controllers for Sd employ the infinite-dimensional state (xt (1; ); x(t; )): Thus, a finite-dimensional observer per se is only half the solution of the complexity problem associated with the feedback control of Sd . This work aims at constructing a finite-dimensional observer based on a preliminary approximation of the delay system. Furthermore, the resulting observer has the advantage of generating, up to a constant gain, the feedback controller investigated in [4]–[6]. 1 n Manuscript received December 7, 2002; revised May 12, 2003 and June 30, 2003. Recommended by Associate Editor M. E. Valcher. The authors are with the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail:[email protected]). Digital Object Identifier 10.1109/TAC.2003.819282 The paper is organized as follows. Section II summarizes the background material and the standing assumptions. The finite-dimensional approximation of (Sd ; ) is contained in Section III. It begins with the introduction of an auxiliary, finite-dimensional initial function,  ; which gives rise to the auxiliary system, (Sd ;  ). Employing a controllability type assumption, it will be shown that the state of (Sd ;  ) admits a finite-dimensional representation and also approximates the state of (Sd ; ) exponentially. The approximation of (Sd ; ) by a system of ordinary differential equations (ode) is achieved through expressing (Sd ;  ) as a system of ode. This approximation theory was motivated by [7], [10] and forms the basis for the finite-dimensional observer theory in Section IV. A numerical example is given to illustrate the theory followed by concluding remarks in Section V. II. BACKGROUND MATERIAL, ASSUMPTIONS ()=0 Consider (1.1) with u t and the given initial condition (1.3). Following [12], let A denote the infinitesimal generator of the strongly continuous semigroup of the solution operators. Then the spectrum of  A fs 2 s g A is a set of eigenvalues given0by where s sI 0 A0 0 e rs A1 is the characteristic matrix of Sd . Let  0 < 1 be specified. Define the closed right half of the complex plane bounded by s 00+ as + fs 2 s  00 g. For each 0  ,  A \ 0 0 is a finite set (Lemma 4.1 [8, p. 18]). In view of this, and following [4], one can construct a real Jordan matrix J 2 M 2M such that +  J  A \ 0 . Associated with J 2 M 2M is a modal M 2n matrix Q 2 such that the pair J; Q 2 M 2M 2 M 2n satisfies the generalized left characteristic matrix equation (glcme) 1( ) = = ( )= : det1( ) = 0 0 Re = 0 ( ) : Re ()= ( ) ( JQ ) = QA + e0rJ QA : 0 ( ) G() = e0 r 1 ( )2 0r    0: For a given pair J; Q satisfying the glcme, define G  ( + )J QA1 ; (2.1) 2 M n by (2.2) Then, the transformation [4], [5] reduces 0 () (; ) (2.3) (Sd ; ) to the IVP z_ (t; ) = Jz (t; ) + QB u(t); t > 0 z (0; ) = Q + G() ()d: (2.4) ( ; ) = Qx(t; ) + z t  0r G  xt   d 0 0 0 1 0r (2.5) The order of J is governed by the choice of 0 and similarly for Q. The construction and the properties of J and Q are detailed in [4] and [5] with examples. The following assumptions will stand throughout the rest of this note. H1) For 0 sufficiently large, rank Q n. H2) Let W 2 M 2M be defined by = W = 0 0r G  G0  d () () (2.6) where 0 denotes matrix transposition. Then, W is positive–definite W > . 0018-9286/03$17.00 © 2003 IEEE ( 0) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003 Assumption H1) means that Sd has n linearly independent left eigenvectors/generalized left eigenvectors in the sense of [4], [5]; it is satisfied by most practical systems. Assumption H2) is an expression of the controllability of the pair (J; QA1 ) on [0; r]. 1987 for x(1;  ) and p(1;  ), "(t) = 0 8 t  0. Employ (3.6) and (2.2) to write out "t fully and then break it up on [t 0 r; 0] and [0; t] as T t 0Q"(t) = T"t = = G()xt (;  )d: Consider t becomes 0 0 0r t>0 (3.4) = 0r 1 G() ()d: for some (t) obtain 2 M 0 Qp(t)] = Tpt = (3.7) 0 0 pt () = G ()W Txt (1;  ) = z(t;  ) 0 Qx(t;  ): (3.9) Since (p ; p(0)) = (x (1;  ); x(0;  )), it follows that x(1;  ) is a solution of the pair (3.7) and (3.8). We claim that x(1;  ) is the only solution. Otherwise, the pair (3.7) and (3.8) will have another solution, p(1;  ). Define "(t) = x(t;  ) 0 p(t;  ) and subtract (3.9) from (3.7) to obtain Q"(t) = 0T"t . Since the initial functions are identical 0 1 xt (;  ) = 0 G ()W 01 01 [z(t;  ) 0 xt (;  ) = G ()W 0r   < 0: G()pt ()d 0 G()G () (t)d = W (t) 0 Qp(t)] ; 0r   < 0: (3.14) 01 [z(t;  ) 0 Qx(t;  )] (3.15) Finally, append (3.15) to xt (0;  ) = x(t;  ) to arrive at (3.5). 0 1 0r G( ) 01 [z(t;  x(t;  ) 0r Since pt (1) = xt (1;  ) 8 t  0, we conclude from (3.14) that 0  () = G ()W 0 0  =  ; 0r 0 where (t) = W 01 [z(t;  ) 0 Qp(t)]. Returning this result to (3.13) gives (3.8)  = (3.13) . Substitute (3.13) into (3.7) and exploit (2.6) to = Observe that (3.2) can be put in the form 0 0 0 1 (3.12) p(t + ) = G () (t) pt () Consider the system (p0 ; p(0)) = ( ;  ): 1 repeat the process. By this method of steps [3, p. 6], it is found that 8 t 2 [0r; 1). That is, p(1;  ) = x(1;  ) 8 t 2 [0r; 1) to prove the uniqueness of the solution of the pair (3.7) and (3.8). To solve this pair, apply the separation of variables technique to write (3.6) t>0 ) 2 "(t) = 0 [z(t;  ) Tpt = z(t;  ) 0 Qp(t); (3.11) 8 t 2 [0; r]. Application of Gronwall inequality [8, Lemma 3.1, p. 15] to (3.12) yields k"(t)k = 0 8 t 2 [0; r]. With "(t) = 0 on [0; r] to play the role of initial function, replace t 2 [0; r] by t 2 [r; 2r] and T 1 0r0)J QA1 "()d: (t e 0 0 With this background material, we prove the following result. Theorem 3.1: For t +  > 0, the state of (Sd ;  ) admits the representation shown in (3.5) at the bottom of the page. Proof: Define the linear functional : L2 ([0r; 0]; n ) ! M by T t k(t)k = kQL Q(t)k t  kQL k e t0 kJ k ke0rJ QA k 2 k()kd (3.3) 1 (3.10) Since rankQ = n by assumption H1), there exists a left inverse QL n2M such that QL Q = In . Therefore (3.2) G() ()d: 0 0 ( z(t; _  ) = J z(t;  ) + QB0 u(t); 0r0)J QA1 "()d: (t e 2 [0; r]. Then, "() = 0 for t 0 r   < 0 and (3.10) Q"(t) = Again, using the transformation method [4], [5], (3.2) reduces (Sd ;  ) to z(0;  ) = Q + t 0 0 0r e + Definition 3.1: By the auxiliary initial function, it is meant n 0 n 1 0 1  = ( ;  ) where  2 L2 ([0r; 0]; ),  2 are given by (3.1) as shown at the bottom of the page. The pair (Sd ;  ) is the auxiliary system. It is observed from (3.1) that 1 is finite-dimensional as it belongs to the span of the M columns of G0 (1). Our goal is to exploit this observation to obtain a finite-dimensional representation for the state of (Sd ;  ). Thus, replacing  with  gives the state of (Sd ;  ) as M (xt (1;  ); x(t;  )). Define a new variable z(t; ) 2 by (2.3) 0r0)J QA1 "()d (t 0 t r A. Auxiliary System State Representation z(t;  ) = Qx(t;  ) + e 0 III. APPROXIMATION OF DELAY SYSTEM 0r0)J QA1 "()d (t 0 t r ) ( )d; 0 Qx(t;  0r   < 0  = 0: )] if0r   < 0 if  = 0: (3.1) (3.5) 1988 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003 B. Approximation The following lemma plays a key role in relating Sd ;  . Lemma 3.1: ( ) (Sd ;  ) to ( ; ) = z(t; ) 8 t  0 (3.16) where z (t; ), z (t;  ) are defined by (2.3) and (3.2), respectively. Proof: From (3.1), premultiply 1 () by G(), integrate with z t  respect to  and apply (2.6) to verify that 0 0r 0 ( ) 1 ()d = G   0r ( ) 1( ) G    d: (3.17) Making use of (2.5), (3.1), (3.17), and (3.4), we have 0 0 z (0; ) = Q + 0r = Q0 + G()1 ()d 0r =z(0;  ): (3.18) (0; ) = Since (2.4) and (3.3) are the same differential equation, z  z  implies that z t  z t  8 t  . An immediate consequence of Lemma 3.1 is the following approximation result. Theorem 3.2: Let x 1  denote the solution of Sd ;  and x 1  the solution of Sd ;  . Then, 9 k > , 9  2 ; 0 such that 0 (; ) (; ) ( ) 0 0r ( ) (0 ) kx(t; ) 0 x(t;  )k < ke0t ; t  0: (3.19) Proof: Define (t;  ) = x(t; ) 0 x(t;  ). The fact that both x(1; ) and x(1;  ) satisfy (1.1) leads to _(t;  ) = A0 (t;  ) + A1 (t 0 r;  ) with initial function 0 (;  ) = 1 () 0 1 () for 0r   < 0. Also, as a result (0 ) = = = x(0;  ), it follows that (0;  ) = 0. 0 0 = z(t; ) 0 z(t;  ) 0 = Qx(t; ) + G()xt (; )d 0 ( ; )+ Qx t  = Q(t;  ) + (; ) 0 0r 0 0r : det[ 1( ) ] det( ( )= ( ) + 0 0r () s e G  d =0 (; ) 0 ( ) ( ; )d ( ) ( ; )d: G  t   (3.20) 0r ( ) s  0 0s e 0 ( ;  )d d (; ) (3.21) where E s  represents the Laplace transform of  t  . First, as a direct consequence of the Paley–Wiener Theorem (see, for example, [16, p. 272], the right hand side of (3.21) is an entire function. Taking advantage of the glcme and (2.2), compute 0 0r s ( ) = (sI 0 J )01 e G  d d: (3.22) (; ) ( ) () )=0 (0 ) (; ) = (; ) (; ) It is evident from (3.19) that to achieve the goal of obtaining an approximate finite-dimensional description for Sd ;  , it is enough to obtain a finite-dimensional representation for Sd ;  . (; ) (; ) ( ( ) ) (; ) where x(t; ) is the solu(Sd ; ) and z(t; ) is the corresponding solution of (2.4). Sim4 x(t;  ) where x(t;  ) is the solution of ilarly, set x(t;  )=  z (t;  ) (Sd ;  ) and z(t;  ) is the corresponding solution of (3.3). Then 0t ); t  0: (3.23) x(t; ) = x(t;  ) + O(e Furthermore, for t > r , x(t;  ) is the solution of the initial value Theorem 3.3: Set x t  tion of x t  z t  problem B0 _ ( ; ) = A00 FJ0 x(t;  ) + QB 0 t 0 (0) x(0;  ) = 0 Q(0) + 0r G()()d x t  () u t (3.24) (3.25) where = A0 0 A1 G0 (0r)W 01 Q 0 01 F0 = A1 G (0r)W : A0 (3.26) (3.27) Proof: Note that (3.23) is simply a restatement of (3.19). Recall that the validity of (3.19) rests on (3.18), namely, z  z  . However, in view of (3.17), the satisfaction of this initial condition implies that x  0 0  . With these initial conditions to ensure that (3.19) holds, we apply (3.5) to Sd ;  for t > r . The result is (0; ) = (0; ) (0; ) = = = (0) ( ) _ ( ; ) = A0 x(t;  ) + A1 xt(0r;  ) + B0 u(t) = A0 x(t;  ) + A1 G0 (0r)W 01 2 [z(t;  ) 0 Qx(t;  )] + B0 u(t) = A0 x(t;  ) + F0 z(t;  ) + B0 u(t): x t  E (s;  ) G  e 0 0 G  xt   Since  t  is the solution of a differential-difference equation, it is of exponential order (see, for example, [8, p. 16]) and, hence, Laplace transformable. Application of the Laplace transform to (3.20) gives Q )d In displaying (3.22), we have introduced QL 2 n2M , the left inverse of Q, so as to make QE s  the subject of (3.22). Thus, the system poles are given by fs 2 Q s QL = sI 0 J g   A n  J . Since +  J  A \ 0 , it follows that the system poles are located in 0 where 0 is the complement of +0 in . By [8, Cor. 6.1, 0 0 p. 215], 9 k0 > , 9  2 ; 0 such that kQ t  k < k0 e0t , t  . That is, k t  k  kQL Q t  k  kQL kk0 e0t , 4 t  . The stated result follows with k kQL kk0 . of x ;  0 0 For t  , (3.16) gives 0r  0 0s e 0 ( ;  C. Finite-Dimensional Representation 0 ( ; )= ( ; ) (sI 0 J )01 Q1(s)QL QE (s) 0 = 0 G()es 0 1 G() ()d (0; ) Then, substitute this result into the left-hand side of (3.21) to get 0rJ 0 e0rs I e QA1 : (3.28) Appending the previously described initial conditions to (3.28) leads to the IVP described by (3.24) and (3.25). IV. FINITE-DIMENSIONAL OBSERVER THEORY The foregoing approximation theory points to the possibility of a finite-dimensional observer theory for the delay system (1.1) and (1.2) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003 1989 Fig. 1. Evolution in time of the actual and observed state. Re Theorem 4.1: Let A0 F0 0 J A= where let n A; C B= B0 C = [C0 j 0p2M ] QB0 p 2 M zero matrix. where x^(t;  ) 0p2M denotes the x^(t;  ) x^(t;  ) = z^(t;  ) and z^(t;  ) 2 M n2M n p2 M 2 ( + ) ( + ) 2 ( +n) observable. Then, there exists a gain L 2 d x^(t;   dt ( 1 Also 2 d ^(t;  ) = A^(t;  ) 0 L y(t) 0 C x^(t;  ) : (4.2)    dt Define (t;  ) = x(t; ) 0 x(t;  ) so that x(t; ) = x(t;  )+ (t;  ): Then 1 d ^(t;   dt )= A 0 LC ^(t;  ) 0 LC (t;  ) 0 Substi- (4.3) 0 e A0LC t0 2 LC ( ;  )d: ( )( ) 0 0 Choose A; C  (4.4)   where  is described in (3.19). Since the pair is assumed observable, 9L 2 (M +n)2p such that (4.5) A model of the combustion stability in a liquid propellant rocket motor is given in [6]. Here, we employ the simplified version given in [9, p. 39]. In our notation, x x1 ; x2 0 where x1 is the pressure in the combustion chamber, x2 is fuel consumption =( A0 = 01 0 0 0 k = 10 01 ; ) A1 = r=1 0 1 0 0 =1 =1 =1 = (; ) (; ) =1 ()=1 ( ) = (1 1) [ 1 0] det1( ) = det[ ]= + 1) + 2 = 1 and , , k are positive constants. Take , ,k , C0 ; . The problem is to construct a finite-dimensional observer which estimates both x 1  and z 1  . Take 0 ,u t , and the initial function   ; 0,  2 0 ; . Solution: Verify that s sI 0 A0 0 e0rs A1 s2 e0s s e0s . For 0 , use the code in + [13] to compute the system open-loop poles in 0 as [1 1] s 2 +0 : det1(s) = 0 = fs1 ; s31 g p where s1 = 0:0880 + 1:3197i, s13 = 0:0880 0 1:3197i, i = 01. With MATLAB, the left eigenvector of 1(s1 ) is computed as q1 = (0:8816; 00:2782 0 0:3814i). Then, following [5] (A) ) 0 A. Example +( ^(t;  ) =e A0LC t^(0;  ) t 0 as t ! 1. B0 whence, by the variation of constants formula ( ) =O(e0 t ) y(t) = C0 x(t; ) = C0 [x(t;  ) + (t;  )] = Cx(t;  ) + C0 (t;  ) = C [^x(t;  ) + ^(t;  )] + C0 (t;  ): 0 ( 1 0 1 (; ) ^( ; ) = C^(t;  ) + C (t;  ). ) 1 (4.1) is an exponential state observer for the delay system (1.1) with output (1.2). Proof: For t  , define the error function  t  x t  0 x t  where x t  is the state of (3.24). Subtracting (4.1) from (3.24) gives () ( 1 k^(t;  )k  k e0 t k^(0;  )k t + k e0 t0 kLC kke0 d = k e0 t k^(0;  )k k 0t 0 t + k(k0kLC 0 ) e 0 e Assume that is completely (M +n)2p such that Therefore, y t 0 C x t  tuting this into (4.2), one gets ) triangle inequality to (4.4) yields M. ) = Ax^(t;  ) + Bu(t) + L y(t) 0 C x^(t;  ) 0 ^( ; ) = ( ; ) ^( ; )  A 0 LC  (01; 0 ). Thus, for t sufficiently large, 9k > 0 such that ke A0LC t k < k e0 0" t where "0t> 0 is arbitrarily small. Also, by Theorem 3.2, k(t;  )k < ke ; t  0. 4 Let  0 =  0 " and define =min( 0 ;  ). Then, application of the \ 0 = J= + 0:0880 01:3197 1:3197 0:0880 Q= 0:8816 00:2782 0:0000 00:3814 1990 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003 where rankQ = 2 = n. From (2.6), W is computed and verified to be positive definite. F0 is then computed from (3.27). The results are W = F0 = 00:5008 0:8649 00:5008 0:5604 4:2238 3:7745 0:0000 0:0000 : From (3.26), A0 and thence the overall system matrix A can be computed as A0 = A= 03:7236 2:6144 01:0000 01:0000 03:7236 2:6144 01:0000 01:0000 4:2238 3:7745 0:0000 0:0000 0:0000 0:0000 0:0880 0:0000 0:0000 1:3197 01:3197 : 0:0880 The eigenvalues of A are given by = f02:361860:8717; 0:088061:3197g. From (A) the definition in Theorem 4.1, C = [1 1 0 0] and it is easily verified that (C; A) is completely observable. A gain L = [3:0241; 0:9282; 4:4226; 01:0053]0 is determined so that  (A 0 LC ) = f02:200060:5000i; 02:1000; 02:0000g. The overall plant-observer system is given by (1.1) and (1.2) and (4.1). Set the observer initial condition x ^(0;  ) = 0. Then, with the help of MATLAB, the plant-observer system was integrated by the method of steps. Fig. 1 shows the evolution of the components of the actual state x(t) and the observed state x ^(t;  ) with time, t. The reconstruction takes place in about 4 s. V. CONCLUDING REMARKS A finite-dimensional observer that exponentially reconstructs the state of the delay system has been given. Through z^(t;  ), it also constructs z (t; ) which, up to a constant gain, is the feedback controller given in [4]–[6] for this class of systems. It is pointed out that 0 reflects a desired margin of absolute stability. However, the optimal value for 0 represents an unsolved problem. For additional comments on the choice of 0 , see the concluding remarks in [4]. The application of z^(t;  ) to a finite-dimensional feedback stabilization theory for delay systems is left for future work. [8] J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. [9] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht, The Netherlands: Kluwer, 1999. [10] E. B. Lee, “Approximation of linear input/output delay-differential systems,” in Lecture Notes in Pure and Applied Mathematics: Differential Equations, Dynamical Systems and Control Science, K. D. Elworthy, W. N. Everitt, and E. B. Lee, Eds. New York: Marcel Dekker, 1994, vol. 152, pp. 659–682. [11] J. Leyva-Ramos and A. E. Pearson, “An asymptotic modal observer for linear autonomous time lag systems,” IEEE Trans. Automat. Contr., vol. AC 40, pp. 1291–1294, July 1995. [12] A. Manitius, “Completeness and F -completeness of eigenfunctions associated with retarded functional differential equations,” J. Diff. Equat., vol. 35, pp. 1–29, 1980. [13] A. Manitius, H. Tran, G. Payre, and R. Roy, “Computation of eigenvalues associated with functional differential equations,” SIAM J. Sci. Stat. Comp., vol. 8, pp. 222–247, 1987. [14] A. E. Pearson and Y. A. Fiagbedzi, “An observer for time lag systems,” IEEE Trans. Automat. Contr., vol. AC–34, pp. 775–777, July 1989. [15] K. Watanabe, “Finite spectrum assignment and observer for multivariable systems with commensurate delays,” IEEE Trans. Automat. Contr., vol. AC–31, pp. 543–550, Apr. 1986. [16] A. Zygmund, Trigonometric Series. Cambridge, U.K.: Cambridge Univ. Press, 1979. Probabilistic Enhancement of Classical Robustness Margins: A Class of Nonsymmetric Distributions Constantino M. Lagoa Abstract—In this note, we address the problem of risk assessment when the robustness margin is exceeded, without a priori knowledge of the distribution of the uncertainty. The only assumption is that the distribution belongs to a given class. In contrast to previous work, this class contains both symmetric and nonsymmetric distributions. We prove that the assessment of risk can be done using only a subset of the admissible distributions. Also, if the set of uncertainties that verify the specifications is convex, it is proven that risk assessment can be done using only a finite subset of the class. Finally, a way of estimating risk is provided for the nonconvex case. Index Terms—Distributional robustness, Monte Carlo simulation, risk assessment. I. INTRODUCTION REFERENCES [1] J. G. Borisovic and A. S. Turbabin, “On the Cauchy problem for linear non homogeneous differential equations with retarded argument,” Soviet Math. Dokl., vol. 10, no. 2, pp. 401–405, 1969. [2] M. Darouach, “Linear functional observers for systems with delays in state variables,” IEEE Trans. Automat. Contr., vol. AC 46, pp. 491–496, Mar. 2001. [3] L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations With Deviating Arguments. New York: Academic, 1973. [4] Y. A. Fiagbedzi, “Further generalization of the delay system transformation,” IMA J. Math. Control Inform., vol. 13, pp. 89–103, 1996. [5] Y. A. Fiagbedzi and A. E. Pearson, “Output feedback stabilization of delay systems via generalization of the transformation method,” Int. J. Control, vol. 51, pp. 801–822, 1990. , “Feedback stabilization of linear autonomous time lag systems,” [6] IEEE Trans. Automat. Contr., vol. AC 31, pp. 847–855, Sept. 1986. [7] G. Gu, P. P. Khargonekhar, E. B. Lee, and P. Misra, “Finite-dimensional approximation of unstable infinite-dimensional systems,” SIAM J. Control Optim., vol. 30, no. 3, pp. 704–716, 1992. The focal point of this note is a control system subjected to parametric uncertainty. We denote the uncertainty vector by q 2 ` and each defines a system q . Now, let  ` , with 0 2 , be a set defining the shape of the uncertainty. For  0, we define r = and obtain a family of systems indexed by 2 r . Let P denote the desired property to be satisfied for every system in the family; i.e., property P might represent a specification involving stability, settling time, overshoot, etc. Classical robustness theory is aimed at checking if the uncertain system satisfies property P for every q S Q R r q Q Q Q R rQ Manuscript received February 15, 2002; revised February 20, 2003. Recommended by Associate Editor E. Bai. This work was supported by the National Science Foundation under Grant ECS-9984260. The author is with the Electrical Engineering Department, the Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.819284 0018-9286/03$17.00 © 2003 IEEE