Separation Theorems for a Class of Retarded Nonlinear Systems

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1 Separation Theorems or a Class o Retarded Nonlinear Systems A. Germani C. Manes P. Pepe Dipartimento di Ingegneria Elettrica e dell Inormazione 674 Poggio di Roio L Aquila Italy alredo.germani@univaq.it Dipartimento di Ingegneria Elettrica e dell Inormazione 674 Poggio di Roio L Aquila Italy costanzo.manes@univaq.it Dipartimento di Ingegneria Elettrica e dell Inormazione 674 Poggio di Roio L Aquila Italy pierdomenico.pepe@univaq.it Abstract: In this paper global and local separation theorems or a class o retarded nonlinear systems are investigated. IFAC Keywords: Time-Delay Systems Nonlinear Observers Separation Theorem. 1. INTRODUCTION Many results are available in the literature or the input/output exact linearization and or the stabilization o retarded (with time-delay in the state) nonlinear systems (see or instance Jankovic 21 Germani Manes & Pepe Oguchi Watanabe & Nakamizo 22 Hua Guan & Shi 24 Lien 24 Marquez-Martinez & Moog 24 Zhang & Cheng 25). All the above papers assume the ull knowledge o the system state. On the other hand ew works dealing with the observer problem o retarded nonlinear systems are available in the literature. Contributions can be ound in Germani Manes & Pepe 21 Germani & Pepe 25. Observers or nonlinear systems with delayed output have been built up in Germani Manes & Pepe 22 Kazantzis & Wright 25 Koshkouei & Burnham 29 and an output eedback stabilizing control law is ound in Zhang Zhang & Cheng 26. An observer based control law or the human glucose-insulin system described by retarded nonlinear equations (see Palumbo Panunzi & De Gaetano 27) is investigated in Palumbo Pepe Panunzi & De Gaetano 29. The input/output linearization method and the nonlinear observer developed in Germani Manes & Pepe 2 23 and in Germani & Pepe 25 respectively are used. The local convergence o the glucose evolution to the reerence signal is proved theoretically and shown by simulations. In this paper we investigate an observer based control law or the class o retarded systems which admit ull relative degree and time-delay matched with the control input (see Pepe 23 Germani & Pepe 25). Many practical systems belong to this class such as the above human glucose-insulin system or the two interacting species system with control acting on predators (see Kolmanovskii & Myshkis 1999). Any inite number o discrete timedelays o arbitrary size is allowed. It is assumed that the time-delays are constant and known. The input/ouptut exact linearization method and the observer developed This work is supported by the Italian MIUR Atheneum project RIA 28. *** Corresponding Author. in Germani Manes & Pepe 2 23 and in Germani & Pepe 25 respectively are used. The state eedback control law ound by the exact input/output linearization method is applied to the system using the estimated state instead o the true state. The asymptotic stability o the trivial solution o the closed-loop system is proved. In the case that the unctionals describing the equations are globally Lipschitz it is proved that the trivial solution is globally uniormly asymptotically stable. In the case that the unctionals describing the equations are only locally Lipschitz it is proved that the the trivial solution o the closed loop system is locally asymptotically stable. In the global case distributed delay terms are also allowed. A numerical example concerning the global case is reported with simulations. Notations R denotes the set o real numbers R denotes the set o non negative reals ). The symbol stands or the Euclidean norm o a real vector or the induced Euclidean norm o a matrix. The essential supremum norm o an essentially bounded unction is indicated with the symbol. A unction v : R R m m positive integer is said to be essentially bounded i ess sup t v(t) <. For given times T 1 < T 2 we indicate with v T1 T 2 ) : R R m the unction given by v T1T 2)(t) = v(t) or all t T 1 T 2 ) and = elsewhere. An input v is said to be locally essentially bounded i or any T > v T ) is essentially bounded. For a positive integer n or a positive real (maximum involved time-delay) C denotes the space o the continuous unctions mapping into R n. For a unction x : c) R n with < c or any positive real t x t is the unction in C deined as x t (τ) = x(t τ) τ. For a positive real H C H C denotes the subset o the unctions x C such that x < H B H () R n denotes the subset o the vectors x R n such that x < H. For given smooth unctions g : R n R n h : R n R L i h(x) L gl i h(x) x Rn i = 1... denote the Lie derivatives L h(x) = h(x) L i h(x) = Li 1 h(x) (x) i = x

2 L g L j h(x) = Lj h(x) g(x) j = 1... (1) x A control aine single-input single-output nonlinear system described by unctions g h admits (uniorm) ull relative degree in a ball B H () (in R n ) i x B H () L g L i h(x) = i = 1... n 2 L g L n 1 h(x) For positive integers n m n m denotes a matrix o zeros in R n m I n denotes the identity matrix in R n n. For a positive integer n A B C are a Brunovskii triple o matrices i.e. A = n 1 1 I n 1 B = 1 n 1 n C = 1 1 n 1 (2) Let us here recall that a unction γ : R R is: o class P i it is continuous zero at zero and positive at any positive real; o class K i it is o class P and strictly increasing; o class K i it is o class K and it is unbounded; o class L i it is continuous and it monotonically decreases to zero as its argument tends to. A unction β : R R R is o class KL i β( t) is o class K or each t and β(s ) is o class L or each s. 2. RETARDED SYSTEMS WITH MATCHED TIME-DELAY AND CONTROL INPUT In this paper we consider the ollowing retarded nonlinear system (see Pepe 23 Germani & Pepe 25) ẋ(t) = (x(t)) g(x(t)) (p 1 (x t )u(t) p 2 (x t )) t y(t) = h(x(t)) t (3) where: x(t) R n ; n is positive integer; u(t) y(t) R are the control input and measured output respectively; : R n R n g : R n R n h : R n R are smooth unctions (admit continuous partial derivatives o any order); p 1 : C R p 2 : C R are continuously Frechét dierentiable unctionals; () = p 2 () =. 3. GLOBAL SEPARATION THEOREM Let us introduce in this section the ollowing hypotheses or (3): H1) the triple g h has ull uniorm relative degree; H2) the nonlinear delay-ree system described by the triple ( g h) is Globally Uniormly Lipschitz Drit Observable (GULDO see Dalla Mora Germani Manes 2) i.e. the unction Φ : R n R n deined or x R n by Φ ( x) = h(x) L h(x)... L n 1 h(x) T is a dieomorphism in R n and there exist positive reals γ φ and γ φ 1 such that v 1 v 2 R n Φ(v 1 ) Φ(v 2 ) γ Φ v 1 v 2 Φ 1 (v 1 ) Φ 1 (v 2 ) γ Φ 1 v 1 v 2 ; (4) H3) there exist a positive real γ 1 such that v 1 v 2 R n L n h(φ 1 (v 1 )) L n h(φ 1 (v 2 )) γ 1 v 1 v 2 ; (5) H4) there exist a positive real γ 2 such that v 1 v 2 C L g L n 1 h(φ 1 (v 1 ()))p 2 (Ψ 1 (v 1 )) L g L n 1 h(φ 1 (v 2 ()))p 2 (Ψ 1 (v 2 )) γ 2 v 1 v 2 (6) where Ψ 1 (v i )(τ) = Φ 1 (v i (τ)) τ i = 1 2; H5) there exists a unction G : C R such that φ C L g L n 1 h(φ())p 1 (φ) = G(H(φ)) where H : C C is deined as H(φ)(τ) = h(φ(τ)) τ ; H6) the unctional φ p 1 (φ) is not equal to zero φ C. Lemma 1. Let n be a positive integer. Let L : C C R n and M : C C R n be smooth unctions with L( ) = M( ) =. Let there exist a positive constant l such that the inequality holds L(x x y) L(x x) l y x y C (7) Let there exist a Hurwitz matrix H such that L(v v) = Hv() v C (8) Consider the system in the unknown variables v(t) w(t) R n v(t) = L(v t v t w t ) ẇ(t) = M(w t v t ) (9) Let system (9) be such that or a suitable unction β o class KL the inequality holds w(t) β( w t) t (1) Then the trivial solution o the system (9) is globally uniormly asymptotically stable. Proo. Let P be a symmetric positive deinite matrix such that H T P P H = I. Let us consider the irst equation in system (9) and let us consider w t as a disturbance. Let us choose the Liapunov-Krasovskii unctional V (φ) = φ T ()P φ() φ C. We have or all φ ψ C D V (φ()) V (φ ψ) = L(φ φ ψ) = V (φ()) V (φ()) L(φ φ) (L(φ φ ψ) L(φ φ)) (11) Thereore φ ψ C the ollowing inequality holds V (φ()) L(φ φ ψ) φ T ()φ() 2 P φ() L(φ φ ψ) L(φ φ) (12) From condition (7) we get D V (φ ψ) φ T ()φ() 2 P φ() l ψ (13) From results in Pepe & Jiang 26 the inequality ollows v(t) β 1 ( v t t t ) β 2 ( w t t) ) t t (14)

3 where β 1 is a suitable unction o class KL satisying β 1 (s ) s s and β 2 is a suitable unction o class K. From the inequalities (1) (14) it ollows that the system (9) is stable. As ar as the global attractivity is concerned it is proved by the inequality v(t) β 1 (β 1 ( v ) β 2 ( w t ) ) t t ) β 2 ( w t t) ) t t (15) For take ɛ >. There exists a time t such that by the inequality (1) the inequality holds β 2 ( w t t) ) < ɛ 2 t t There exists a time t t such that β 1 (β 1 ( v ) β 2 ( w t) ) t t ) < ɛ 2 Then t t v(t) β 1 (β 1 ( v ) β 2 ( w t ) ) t t ) β 2 ( w t t) ) < ɛ (16) Thereore taking into account the inequality (1) the global attractivity property o the system (9) is proved. The uniormity ollows rom the act that the involved unctions β β 1 and β 2 are independent o the initial conditions. Theorem 2. Consider the system (3) and let the hypotheses H1 H6 be satisied. Then there exist a row vector Γ and a column vector K in R n such that the trivial solution o the closed loop system ẋ(t) = (x(t)) g(x(t))(p 1 (x t )u(t) p 2 (x t )) y(t) = h(x(t)) Φ(ˆx(t)) ˆx(t) = (ˆx(t)) g(ˆx(t))p 2 (ˆx t ) ˆx(t) BG(y t )u(t) K(y(t) h(ˆx(t))) 1 u(t) = Ln h(ˆx(t)) L gl n 1 h(ˆx(t))p 2 (ˆx t ) ΓΦ(ˆx(t)) G(y t ) (17) is globally uniormly asymptotically stable. Proo. Let us introduce the variables z(t) = Φ(x(t)) e(t) = Φ(ˆx(t)) Φ(x(t)) t. Then the couple o variables z(t) e(t) satisy the equations ż(t) = L(z t z t e t ) ė(t) = M(e t z t ) (18) where L is the unctional deined as or v w C L(v w) = Av() B L n h(φ 1 (v())) L g L n 1 h(φ 1 (v()))p 2 (Ψ 1 (v)) L n h(φ 1 (w())) L g L n 1 h(φ 1 (w()))p 2 (Ψ 1 (w)) Γw() (19) and M is the unctional deined as or v w C M(v w) = L(w v w) (A BΓ)(v() w()) KCv() (2) Taking into account the hypotheses H1 H6 by the results in Germani Manes & Pepe 21 Germani & Pepe 25 it ollows that there exist a gain vector K and positive reals a b such that e(t) a e bt e. Taking into account that L(v v) = (A BΓ)v() and that (A B) is a controllable pair we can choose Γ such that A BΓ is Hurwitz. Thus the hypotheses o Lemma 1 are satisied by the equations (18). By applying Lemma 1 the global uniorm asymptotic stability o the closed-loop system (17) is proved. Theorem 3. Consider the system (3) and let the hypotheses H1 H2 H5 H6 be satisied. Let (5) be satisied v 1 v 2 R n with h(v 1 ) = h(v 2 ) and let (6) be satisied v 1 v 2 C with h(v 1 (τ)) = h(v 2 (τ)) τ. Let h(x) = x i or some i { n}. Let the unction Φ 1 be known. Then there exist a row vector Γ and a column vector K in R n such that the trivial solution o the closed loop system ẋ(t) = (x(t)) g(x(t))(p 1 (x t )u(t) p 2 (x t )) y(t) = h(x(t)) 1 Φ(ˆx(t)) ˆx(t) = (ˆx(t)) g(ˆx(t))p 2 (ˆx t ) ˆx(t) B(G(y t )u(t) L n h( x(t)) L g L n 1 h( x(t))p 2 ( x t ) L n h(ˆx(t)) L g L n 1 h(ˆx(t))p 2 (ˆx t )) K(y(t) h(ˆx(t))) u(t) = Ln h( x(t)) L gl n 1 h( x(t))p 2 ( x t ) ΓΦ( x(t)) G(y t ) (21) with x(t) = Φ 1 ( y(t) L h(ˆx(t)) L n 1 h(ˆx(t)) T ) t is globally uniormly asymptotically stable. Proo. The proo is similar to the one given or Theorem 2 and or lack o space is here omitted. Remark 4. Γ and K must be chosen such that the A BΓ and A KC are Hurwitz. Hypotheses H1-H4 and H6 concern observability Lipschitz and ull relative degree conditions. Hypothesis H5 imposes that the term multiplying the input in the model (3) rewritten in normal orm depends on only the measured output. 4. LOCAL SEPARATION THEOREM Let us introduce in this section the ollowing hypotheses or (3): J1) there exists a positive real H such that the triple g h has ull relative degree in the ball B H (); J2) the nonlinear delay-ree system described by the triple ( g h) is uniormly Lipschitz drit observable (ULDO see Dalla Mora Germani Manes 2) i.e. the unction Φ : B H () Ω (where Ω is a suitable open bounded set in R n ) deined or x B H () by Φ ( x) = h(x) L h(x)... L n 1 h(x) T is a dieomorphism and there exist positive reals γ φ and γ φ 1 such that or all v 1 v 2 B H () w 1 w 2 Ω Φ(v 1 ) Φ(v 2 ) γ Φ v 1 v 2 Φ 1 (w 1 ) Φ 1 (w 2 ) γ Φ 1 w 1 w 2 ; (22)

4 J3) there exist a positive real γ 1 such that v 1 v 2 Ω L n h(φ 1 (v 1 )) L n h(φ 1 (v 2 )) γ 1 v 1 v 2 ; J4) there exist a positive real γ 2 such that v 1 v 2 C with v i (τ) Ω τ i = 1 2 L g L n 1 h(φ 1 (v 1 ()))p 2 (Ψ 1 (v 1 )) L g L n 1 h(φ 1 (v 2 ()))p 2 (Ψ 1 (v 2 )) γ 2 v 1 v 2 (23) where Ψ 1 (v i )(τ) = Φ 1 (v i (τ)) τ i = 1 2; J5) there exist a positive integer m and positive reals k k = m such that v C p 1 (v) = p 1 (v() v( 1 )... v( m )) p 2 (v) = p 2 (v() v( 1 )... v( m )) where p 1 p 2 : R n m1 R are smooth unctions (admit continuous partial derivatives o any order); J6) the unctional φ p 1 (φ) is not equal to zero or all φ C H. Lemma 5. Let S = {(z 1 z 2 ) C C : z i (τ) Ω τ } Let L : S R be the unctional deined or (z 1 z 2 ) S as L(z 1 z 2 ) = L n h(φ 1 (z 2 ())) L g L n 1 (Φ 1 (z 2 ()))p 1 (Ψ 1 (z 2 )) 1 L g L n 1 (Φ 1 (z 1 ()))p 1 (Ψ 1 (z 1 )) L n h(φ 1 (z 1 ())) L g L n 1 (Φ 1 (z 1 ()))p 2 (Ψ 1 (z 1 )) W z 1 () L g L n 1 (Φ 1 (z 2 ()))p 2 (Ψ 1 (z 2 )) (24) with W any given row vector in R n. Then the unctional L admits continuous Frechét derivative at ( ) D F L and moreover there exist row vectors b k R n k = 1... m such that or all y 1 y 2 S satisying y 1 y 2 S D F L(y 1 y 1 y 2 ) W y 1 () = b y 2 (t) b k y 2 (t k ) (25) Proo. The Frechét dierentiability property ollows rom the act that g h p 1 p 2 (see the hypothesis J5) are smooth Φ is a dieomorphism. As ar as the equality (25) is concerned by the hypothesis J5 there exist row vectors a k b k R n k = 1... m such that D F L(z 1 z 2 ) = a z 1 () b z 2 () a k z 1 ( k ) b k z 2 ( k ) (26) Since L(z 1 z 1 ) = W z 1 () and (y 1 y 1 y 2 ) = (y 1 y 1 ) ( y 2 ) we get D F L(y 1 y 1 y 2 ) W y 1 () = D F L(y 1 y 1 y 2 ) D F L(y 1 y 1 ) = D F L( y 2 ) = b y 2 () b k y 2 ( k ) (27) k= Theorem 6. Consider the system (3) and let the hypotheses J1 J6 be satisied. Then there exist a row vector Γ and a column vector K in R n such that the trivial solution o the closed loop system ẋ(t) = (x(t)) g(x(t))(p 1 (x t )u(t) p 2 (x t )) y(t) = h(x(t)) ˆx(t) = (ˆx(t)) g(ˆx(t))(p 1 (ˆx t )u(t) p 2 (ˆx t )) 1 Φ(ˆx(t)) K(y(t) h(ˆx(t))) ˆx(t) u(t) = Ln h(ˆx(t)) L gl n 1 h(ˆx(t))p 2 (ˆx t ) ΓΦ(ˆx(t)) L g L n 1 (ˆx(t))p 1 (ˆx t ) is locally asymptotically stable. (28) Proo. Let us introduce the variables ẑ(t) = Φ(ˆx(t)) e(t) = Φ(x(t)) Φ(ˆx(t)) t. Then the couple o variables ẑ(t) e(t) satisies the equations ẑ(t) = (A BΓ)ẑ(t) KCe(t) ė(t) = (A KC)e(t) B(L(ẑ t ẑ t e t ) Γẑ(t)) (29) where L is the unctional deined in (24) with W = Γ. By Lemma 5 we get that the irst order approximation o (29) is given by the ollowing equations ẑ(t) = (A BΓ)ẑ(t) KCe(t) ė(t) = (A KC)e(t) Bb e(t) B b k e(t k ) (3) Since we can choose Γ such that ABΓ has any prescribed eigenvalues in the open let complex semi-plane the proo o the theorem is completed provided that the trivial solution o the second equation in (3) is exponentially stable or some K. We will use the same reasoning used in Palumbo Pepe Panunzi & De Gaetano 29 or the single-delay glucose-insulin system. We get e(t) = exp((a KC)t)e() exp((a KC)(t τ))b (b e(τ) b k e(τ k ))dτ (31) Let us choose an n-pla o negative real distinct eigenvalues in decreasing order λ = (λ 1 λ 2... λ n ). As in Ciccarella Dalla Mora & Germani 1993 let V (λ) be the diagonalizing (or A KC) Vandermonde matrix let ψ(t) = V (λ)e(t) t. Then rom (31) we get ψ(t) exp(λ 1 t) ψ() b V 1 (λ) ψ(τ) exp(λ 1 (t τ)) n b k V 1 (λ) ψ(τ k ) dτ

5 (32) Let ξ(t) = exp( λ 1 t) ψ(t). From (32) we obtain by applying the Bellman-Gronwall inequality ξ(t) exp( n b V 1 (λ) t)ξ() exp( n b V 1 (λ) (t τ)) exp( λ 1 τ) b k V 1 (λ) ψ(τ k ) dτ Turning back to the variable ψ(t) we get rom (33) ψ(t) exp((λ 1 n b V 1 (λ) )t) ψ() exp((λ 1 n b V 1 (λ) )(t τ)) (33) b k V 1 (λ) ψ(τ k ) dτ (34) Let M = sup τ ψ(τ). Let p(t) = Mexp(αt) t α a suitable negative real which will be ound next. Let us see i there exists a negative α such that p(t) satisies (34) with the equality sign i.e. Mexp(αt) = exp((λ 1 n b V 1 (λ) )t)m exp((λ 1 n b V 1 (λ) )(t τ)) b k V 1 (λ) Mexp(α(τ k ))dτ (35) I (35) holds then rom the choice o M and a standard step procedure it ollows that ψ(t) Mexp(αt) t. Thereore i α can be chosen negative the proo o the exponential stability o the trivial solution o the second equation in (3) is complete. Solving the integral in (35) we get that the equality (35) holds i and only i the ollowing condition can be ulilled: m n b k V 1 (λ) exp( α k ) = α λ 1 n b V 1 (λ) (36) The equation (36) in the unknown variable α admits a unique negative solution i and only i the value at zero o the right-hand side unction o α is greater than the value at zero o the let-hand side unction o α i.e. i and only i λ 1 V 1 (λ) n b k < (37) k= It is proved in Ciccarella Dalla Mora & Germani 1993 that there exist an n-pla o real negative distinct eigenvalues λ such that the inequality (37) holds. Since or any choice o λ there exist K such that the set o eigenvalues o A KC is equal to λ the proo o the theorem is complete. 5. A NUMERICAL EXAMPLE Let us consider the unstable system described by the ollowing equations ẋ 1 (t) = x 2 (t) sech(x 1 (t)) 1 ẋ 2 (t) = tanh(x 2 (t )) x 1 (t) u(t) y(t) = x 1 (t) (38) 2 where as well known or a R sech(a) = exp(a)exp( a) tanh(a) = exp(a) exp( a) exp(a)exp( a). We get or x = x1 R x 2 2 w C φ(x) = x 1 x 2 sech(x 1 ) 1 L 2 h(x) = x 1 tanh(x 1 )sech(x 1 ) (x 2 sech(x 1 ) 1) L g L h(x) = 1 G(H(w))(τ) = 1 τ Φ(ˆx(t)) 1 = ˆx(t) tanh(x 1 )sech(x 1 ) 1 (39) All the hypotheses o Theorem 3 are satisied by the system (38). The observer based control law or the system (38) is given by (see (21)) ˆx1 (t) ˆx = 2 (t) sech(ˆx 1 (t)) 1 ˆx 2 (t) tanh(ˆx 2 (t )) ˆx 1 (t) u(t) 1 K(y(t) ˆx tanh(ˆx 1 (t))sech(ˆx 1 (t)) 1 1 (t)) B (y(t) ˆx 1 (t) (ˆx 2 (t) sech(ˆx 1 (t)) 1) (tanh(ˆx 1 (t))sech(ˆx 1 (t)) tanh(y(t))sech(y(t))) tanh(ˆx 2 (t ) sech(ˆx 1 (t )) sech(y(t ))) tanh(ˆx 2 (t ))) u(t) = y(t) tanh(y(t))sech(y(t)) (ˆx 2 (t) sech(ˆx 1 (t)) 1) tanh(ˆx 2 (t ) sech(ˆx 1 (t )) sech(y(t ))) y(t) Γ ˆx 2 (t) sech(ˆx 1 (t)) 1 (4) By Theorem 3 there exist a row vector Γ and a column vector K (chosen such that A BΓ and A KC are Hurwitz) such that the trivial solution o the closed loop system (38) (4) is globally uniormly asymptotically stable. The perormed simulations validate this result. In ig. 1 the evolutions o the true and estimated state are plotted. The initial state variables are set constant and equal to 1 5 set constant and equal to the initial estimated state variables are. The time-delay is set equal to 5. The gain vector Γ and the gain vector K are chosen such to have eigenvalues.1.2 and or A BΓ and A KC respectively. Both the true and estimated state converge to.

6 2.5 x Fig. 1. True (solid line) and Estimated (dashed line) State 6. CONCLUSIONS In this paper the separation principle or a class o retarded nonlinear systems is investigated. Global and local convergence results are provided in the case o global Lipschitz property and local Lipschitz property o the unctionals describing the equations respectively. Full relative degree is assumed. Future work will concern the separation theorem or retarded nonlinear systems with un-matched timedelay and control input. One such a case has been studied in Di Ciccio Bottini Pepe & Foscolo where an observerbased control law is developed or a continuous stirred tank reactor with recycle time-delay using some results in Germani Manes & Pepe 21. Such application shows the possibility o extending the results o this paper to a larger class o retarded nonlinear systems and this will be the topic o the orthcoming research. The separation theorem or nonlinear systems with delayed output will be also topic o the orthcoming research. REFERENCES 1 Ciccarella G. M. Dalla Mora A. Germani A Luenberger-like observer or nonlinear systems International Journal o Control vol. 57 no. 3 pp Ciccarella G. M. Dalla Mora A. Germani Asymptotic linearization and stabilization or a class o nonlinear systems Journal o Optimization Theory and applications Vol. 84 N Dalla Mora M. A. Germani C. Manes Design o state observers rom a drit-observability property IEEE Transactions on Automatic Control vol. 45 no. 8 pp Di Ciccio M.P. M. Bottini P. Pepe P. U. Foscolo Observed-Based Control o a Continuous Stirred Tank Reactor with Recycle Time-Delay 14 th IFAC Conerence on Methods and Models in Automation and Robotics Miedzydroje Poland August Germani A. C. Manes P. Pepe Local Asymptotic Stability or Nonlinear State Feedback Delay Systems Kybernetika vol. 36 n. 1 pp Germani A. C. Manes P. Pepe Input-Output Linearization with Delay Cancellation or Nonlinear Delay Systems: the Problem o the Internal Stability International Journal o Robust and Nonlinear Control Vol. 13 No. 9 pp Germani A. C. Manes P. Pepe An Asymptotic State Observer or a Class o Nonlinear Delay Systems Kybernetika vol. 37 n.4 pp Germani A. C. Manes P. Pepe A New Approach to State Observation o Nonlinear Systems with Delayed Output IEEE Transactions on Automatic Control Vol. 47 No. 1 pp January Germani A. P. Pepe A State Observer or a Class o Nonlinear Systems with Multiple Discrete and Distributed Time Delays European Journal o Control Vol. 11 Issue 3 pp Hua C. X. Guan P. Shi Robust stabilization o a class o nonlinear time-delay systems Applied Mathematics and Computation Vol. 155 pp Jankovic M. Control Lyapunov-Razumikhin Functions and Robust Stabilization o Time Delay Systems IEEE Transactions on Automatic Control Vol. 46 No. 7 pp Kazantzis N. R. Wright Nonlinear observer design in the presence o delayed output measurements Systems & Control Letters Vol. 54 Issue 9 pp Koshkouei A. K. Burnham Discontinuous observers or non-linear time-delay systems International Journal o Systems Science Vol. 4 Issue 4 pp Lien C.-H. Global Exponential Stabilization or Several Classes o Uncertain Nonlinear Systems with Time-Varying Delay Nonlinear Dynamics and Systems Theory Vol. 4 No. 1 pp Marquez-Martinez L.A. C.H. Moog Input-output eedback linearization o time-delay systems IEEE Transactions on Automatic Control Vol. 49 N. 5 pp Oguchi T. A. Watanabe and T. Nakamizo Input- Output Linearization o Retarded Non-linear Systems by Using an Extension o Lie Derivative International Journal o Control Vol. 75 No Palumbo P. S. Panunzi and A. De Gaetano Qualitative behavior o a amily o delay dierential models o the glucose insulin system Discrete and Continuous Dynamical Systems - Series B Vol Palumbo P. P. Pepe S. Panunzi A. De Gaetano Observer-based closed-loop control o plasma glycemia 48 th IEEE Conerence on Decision and Control Shanghai China December Pepe P. Preservation o the Full Relative Degree or a Class o Delay Systems under Sampling ASME Journal o Dynamic Systems Measurement and Control Vol. 125 No. 2 pp Pepe P. Z.-P. Jiang A Lyapunov-Krasovskii Methodology or ISS and iiss o Time-Delay Systems Systems & Control Letters Vol. 55 Issue 12 pp Zhang X C. Zhang Z. Cheng Asymptotic stabilization via output eedback or nonlinear systems with delayed output International Journal o Systems Science Vol. 37 Issue 9 pp Zhang X. Z. Cheng Global stabilization o a class o time-delay nonlinear systems International Journal o Systems Science Vol. 36 No. 8 pp