Further Results on Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks
|
|
- David Walters
- 5 years ago
- Views:
Transcription
1 Further Results on Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks Frédéric Mazenc a, Michael Malisoff b,, Zongli Lin c a Projet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, pl. Viala, Montpellier, France b Department of Mathematics, Louisiana State University, Baton Rouge, LA c Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, PO Box , Charlottesville, VA Abstract We consider a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions for the closed loop systems are available. In the presence of feedback delays and actuator errors, we explicitly construct input-tostate stability (ISS) Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the available Lyapunov functions for the original undelayed dynamics, which establishes that the closed loop systems are input-to-state stable (ISS) with respect to actuator errors. We illustrate our results using a generalized system from identification theory and other examples. Key words: Delayed systems, Input-to-state stability, Lyapunov function constructions Introduction Input-to-state stability (ISS) plays an important role in modern stability analysis and controller design (see Sontag (989, 006) and Teel (998)). The ISS paradigm was introduced in Sontag (989) and extended to delay systems in the seminal paper Teel (998), which gave sufficient conditions for ISS using control ISS Lyapunov- Razumikhin functions (ISS-LRFs). By contrast, Pepe & Jiang (006) gave sufficient conditions for ISS of delay systems via Lyapunov-Krasovskii functionals (see Section. for the relevant definitions and Section 3 for details on the known results). However, none of these works provide general methods for explicitly constructing Lyapunov functionals for feedback delayed dynamics. In applications, it is often essential to explicitly construct Lyapunov functions. For example, Lyapunov functions for delayless systems ẋ = f(x) can be used to construct stabilizing feedbacks K(x) for which ẋ(t) = f(x(t)) + g(x(t))[k(x(t)) + d(t)] is ISS with respect to the actuator error d(t), under standard assumptions (see Sontag (989)). In Karafyllis (006), Lyapunov functionals were used to characterize robust global asymptotic stability of nonlinear time-varying retarded systems. For Corresponding author: Tel.: (Zongli Lin). addresses: Frederic.Mazenc@supagro.inra.fr (Frédéric Mazenc), malisoff@lsu.edu (Michael Malisoff), zl5y@virginia.edu (Zongli Lin). Supported by NSF/DMS Grant certain classes of delayed systems, LRFs can be used to construct stabilizing feedbacks as well. For example, see Jankovic (00), which also constructs the needed LRFs in some cases. With these observations in mind, we pursue a different objective. We assume that we are given a time-varying control affine system for which a stabilizing feedback u = u s (x, t) and a Lyapunov function for the corresponding closed loop system ẋ = f(x, t) + g(x, t)u s (x, t) () are known. We do not require () to be exponentially stable. We introduce a vector of constant delays τ = (τ, τ,, τ n ) with τ i 0, in the feedback signal. We then explicitly construct an ISS Lyapunov-Krasovskii functional (ISS-LKF) for ẋ(t) = f(x(t), t) + g(x(t), t)[u s (ξ τ (t), t) + d(t)] () with ξ τ (t) = (x (t τ ), x (t τ ),, x n (t τ n )) and 0 τ i τ for some bound τ 0 (see, e.g., Definition and Nesic & Teel (004) for applications where general delays of this type such as different delays in the different components of x are used). Our ISS-LKF construction implies that () is ISS with respect to the actuator error d(t). The ISS-LKF we construct is an explicit formula involving the known Lyapunov function for (). Therefore, while the known results largely concern constructing feedbacks that achieve stability and with building LRFs (or sufficient conditions for stability), here we address the complementary problems of (a) quantifying Preprint submitted to Automatica 6 November 007
2 the effects of introducing feedback delays and actuator errors on the stability performance of given feedbacks and (b) explicitly building corresponding ISS-LKFs (but see Section 3 for a comparison of our work with Fan & Arcak (006), Pepe & Jiang (006), and Teel (998)). In Section, we provide the relevant definitions and notation and state our main result. We provide a detailed comparison of our work with the known stability results in Section 3. We prove our main theorem in Section 4. We extend our results to cascades in Section 5. In Section 6, we apply our results to dynamics from identification theory and to other examples. We close in Section 7 with a brief concluding remark. We include some of the technical details and extensions in the appendices. Preliminaries and statement of main result. Definitions and assumptions Let K denote the set of all continuous, strictly increasing and unbounded functions ρ : [0, ) [0, ) with ρ(0) = 0. Let KL denote the set of all continuous functions β : [0, ) [0, ) [0, ) satisfying () β(, t) K for each t 0, () β(s, ) is non-increasing for each s 0, and (3) β(s, t) 0 as t + for each s 0. Given φ : I R p, defined on an interval I, let φ I denote its (essential) supremum. Let denote the usual Euclidean norm (or the induced matrix norm). A function V : R n [0, ) [0, ) is called (uniformly) proper and positive definite provided there are α, ᾱ K such that α( x ) V (x, t) ᾱ( x ) for all x R n and t 0. When we say V is C, we understand its partial derivatives at t = 0 to be one-sided derivatives. Let C n (I) denote the set of all continuous R n -valued functions on any interval I, with the locally uniform topology. Consider a general delayed control system () with constant τ i (0, τ], f(0, t) 0, u(0, t) 0, and d L m ([0, ))(=all measurable essentially bounded functions [0, ) R m ) being an actuator error. We refer to the constant vector τ (0, τ] n as a vector feedback (time) delay. We stipulate τ later, although in many cases of interest, the bound τ can be raised to an arbitrary large positive constant (see Remarks 4 and A.4 below). Given constants t o 0, τ (0, τ] n, x o C n ([t o τ, t o ]), and d L m ([0, )), consider the initial value problem ẋ(t) = f(x(t), t)+g(x(t), t)[u(ξ τ (t), t)+d(t)] t t o a.e. and x(r) = x o (r) r [t o τ, t o ]. (IP) The assumptions we make below imply that () has globally well defined solutions, i.e., for all choices of t o 0, τ, x o C n ([t o τ, t o ]), and d L m ([0, )), (IP) has a unique solution t x(t; t o, x o, d, τ) defined on [t o τ, + ) (see Assumption H and Appendix A. below). We then call x o the initial function for t x(t; t o, x o, d, τ). When () has globally well defined solutions, we denote x(t; t o, x o, d, τ) simply by x(t) when t o, x o, d, and τ are clear. We extend the functions x(t) to R by setting x o (t) = x o (t o τ) for t t o τ. The following generalize the ISS notions from Sontag (989) and Sontag & Wang (995) to delayed systems (see also, e.g., Pepe & Jiang (006)). The first definition is unchanged if d [to,t] is replaced by d. Definition Assume that () has globally well defined solutions. Given a vector delay τ (0, τ] n, we call () (uniformly) input-to-state stable (ISS) provided that there exist β KL, γ K such that for all d L m ([0, )), x(t; t o, x o, d, τ) β( x o [to τ,t o], t t o ) + γ( d [to,t]) for all t o 0, x o C n ([t o τ, t o ]), and t t o. Let τ be a positive real number and κ be a positive integer. For a given t 0, x t ( ) denotes the restriction of x( ) to the interval [t κ τ, t] translated to [ κ τ, 0], i.e., x t (θ) = x(t + θ) for all θ [ κ τ, 0]. Definition A continuous functional U : C n (R) [0, ) [0, ) is called an ISS Lyapunov-Krasovskii functional (ISS-LKF) for () provided for all τ (0, τ] n and all trajectories x(t) := x(t; t o, x o, d, τ) of (), the function t U(x t, t) is locally absolutely continuous and there exist functions α i K for i =,, 3, 4 and κ N such that for all φ C n ([ κ τ, 0]), all trajectories x(t) of (), and all t t o + κ τ, we have (i) α ( φ(0) ) U(φ, t) α ( φ [ κ τ,0] ) and (ii) the time derivative D t U(x t, t) of U(x t, t) satisfies D t U(x t, t) α 3 (U(x t, t)) + α 4 ( d [to,t]) a.e.. Our assumptions will guarantee that if () admits an ISS-LKF, then it is ISS (see Appendix A. below).. Statement of main result Set F (x, t, u s ) = f(x, t) + g(x, t)u s (x, t). Consider () with d L m ([0, )) an unknown disturbance, τ (0, τ] n, and f : R n [0, ) R n, g : R n [0, ) R n m and u s : R n R R m satisfying: H The function u s is C and u s (0, t) 0, and f and g are locally Lipschitz. Also, there exist a σ K for which σ(r) r for all r 0, a C uniformly proper and positive definite V : R n [0, ) [0, ); positive constants L and K, and K i 0 (i =, 3, 4) such that for all x R n, q R n, and l 0, we have: H V t (x, l) + V x (x, l)f (x, l, u s ) σ ( n x ). H V x (x, l)g(x, l) K σ( x ), u s x (x, l) L. H3 f(x, l) K σ ( x ), g(x, l) K 3 (σ( x ) + ). H4 [ g(x, l) u s (q, l) ] K 4 [σ ( x ) + σ ( q )]. Assumption H allows many cases where () is a stable linear system with bounded g (using a quadratic Lyapunov function), as well as cases where the closed loop system is not exponentially stable or g is unbounded (see Section 6. below). In what follows, 0.49 τ = LK K + 8K , (3) but see Remarks 4 and A.4 for larger delay bounds.
3 Theorem 3 Under Assumption H and with τ in (3), the feedback delayed system () with any τ (0, τ] n, in closed loop with the feedback u s, admits the ISS-LKF U(x t, t) = V (x(t), t) + 4 τ and therefore is ISS. t τ ( r σ ( ) n x(l) )dl dr (4) Remark 4 Often, τ can be raised to an arbitrarily large positive constant. For example, if () satisfies Assumption H with V t 0 and f 0, then for any constant η (0, ), Assumption H also holds if we replace u s, L, σ, K, and K 3 with ηu s, η L, η / σ, K /η /, and K 3 /η /, respectively. Plugging into (3) and taking η arbitrarily small makes (3) arbitrarily large. Our control affineness assumption can be relaxed as well (see Remark 6 below). Remark 5 Growth restrictions on f and g are required. Indeed, the following system, evolving on R, ẋ (t) = x (t) + x 4 (t)x (t), ẋ (t) = u(x(t τ)), with τ being a nonnegative scalar delay, is globally asymptotically stabilized by the feedback u(x) = x x 5 when τ = 0, and V (x) = x is a Lyapunov function for the corresponding closed-loop system. However, this system is not globally asymptotically stabilizable by any continuous u(x(t τ)) for any constant τ > 0 (see (Mazenc & Bliman, 006, Appendix )). 3 Comparison with the literature Much of the delay systems literature uses (control) Lyapunov-Razumikhin functions (LRFs) (see Jankovic (00)). ISS-LRFs were used by Teel (998) to give sufficient conditions for nonlinear delayed systems to be ISS. Using LRFs, one can build stabilizing feedbacks for some control affine systems, and the stability of the closed loop dynamics enjoys some robustness relative to unmodeled dynamics (see Jankovic (00)). LKF methods can be viewed as generalizations of Razumikhin methods and have been used to give sufficient conditions for ISS as well (see Pepe & Jiang (006)). Michiels et al. (00) uses Lyapunov techniques to study delayed partially linear systems. For delayed linear systems, stabilizing feedbacks can often be built using linear matrix inequalities (see, e.g., Fattouh et al. (000), Niculescu (998), and Tarbouriech & da Silva, Jr. (000)). Our approach differs from these earlier works in our explicit construction of LKFs and our use of ISS to quantify the effect of introducing feedback delays and actuator errors into a priori stable closed loop nonlinear dynamics. Moreover, we allow undelayed closed loop dynamics that are not necessarily exponentially stable, and our results lead to new stabilizing feedbacks that guarantee ISS of cascades (see Section 5 below). The results Fan & Arcak (006) and Teel (998) lead to an ISS small gain approach for proving robustness to feedback delays. However, Fan & Arcak (006) and Teel (998) do not include our results. For example, they do not explicitly construct ISS-LKFs. See also Section 6. for an example covered by our results which is beyond the scope of Fan & Arcak (006). The work Pepe & Jiang (006) assumes the availability of a Lyapunov-like functional for the delayed dynamics, which we do not require here. The ISS-LKFs we construct are explicit expressions involving the available Lyapunov functions for the original undelayed dynamics. Our hypotheses correspond to cases where a feedback is known to stabilize an undelayed system but the state that is observed by the feedback in the implementation has a small unknown delay. It is desirable to find conditions under which the feedback continues to stabilize the system when feedback delays or actuator errors are introduced, which our work provides. 4 Proof of Theorem 3 In what follows, all inequalities and equalities should be understood to hold globally unless otherwise indicated. Consider the dynamics (). From Assumption H, V = V t (x(t), t) + V x F (x(t), t, u s ) + V x g(x(t), t)d(t) +V x g(x(t), t)[u s (ξ τ (t), t) u s (x(t), t)] σ ( n x(t) ) + V x g(x(t), t) d(t) + V x g(x(t), t) u s (ξ τ (t), t) u s (x(t), t) along any trajectory x(t) := x(t; t o, x o, d, τ) of (), where we omitted the argument (x(t), t) of V x. From Assumption H and the relation wz w /(δ) + δz / with w = σ( n x ), and with δ = 8 and then δ =, we get V σ ( n x(t) ) + K σ( x(t) ) d(t) +K Lσ( x(t) ) ξτ (t) x(t) 5 8 σ ( n x(t) ) + 4K d(t) +K L ξ τ (t) x(t). (5) Jensen s Inequality and the fact that τ i τ for all i give ξ τ (t) x(t) = ( n ) t i= i ẋ i (r)dr n i= τ i i ẋ i (r)dr τ (6) t t τ ẋ(r) dr for each t t o + τ. From Assumptions H3-H4, ẋ(r) f + g[u s (ξ τ (r), r) + d(r)] K σ ( x(r) ) + 4K 4 (σ ( x(r) ) + σ ( ξ τ (r) )) (7) +4K 3 (σ( x(r) ) + ) d(r) where we omitted the arguments (x(r), r) from f and g. By the relation wz w /4 + z with w = σ( x(r) ) and z = 4K 3 d(r), we conclude from (7) that ẋ(r) (K + 4K )σ ( x(r) ) +4K 4 σ ( ξ τ (r) ) + 6K 3 d(r) 4 + 4K 3 d(r) (K + 4K )σ ( x(r) ) + 6K 3 d(r) 4 +4K 4 σ ( n max i x i (r τ i ) ) + 4K 3 d(r). 3
4 Set Θ(r) = 4K 3 (4K 3 + )(r 4 + r ). Since τ i τ for all i, it follows that for all t t o + τ, we get t τ ẋ(r) dr t τ (K + 4K )σ ( x(r) )dr + t τ 4K 4σ ( n max i x i (r τ i ) )dr + τθ(d [to,t]) t τ (K + 4K )σ ( x(r) )dr +4K 4 t τ σ ( n x(r) ) dr + τθ(d [to,t]) t τ (K +8K )σ ( n x(r) )dr+ τθ(d [to,t]). Combining this inequality, (5), and (6) yields V 5 8 σ ( n x(t) ) +4K d(t) + L K τ Θ(d [to,t]) + L K τ t τ (K +8K )σ ( n x(r) )dr. Set R( τ) = L K τ(k + 8K ) /(4 τ) and α 4 (r) = L K τ Θ(r)+4Kr. Our bound (3) on τ gives R( τ) /(8 τ). By our choice of α 4, our bound on R( τ), and the fact (cf. Mazenc et al. (006)) that d dt t τ r σ ( n x(l) ) dl dr = τσ ( n x(t) ) t τ σ ( n x(l) )dl, we deduce that when t t o + τ, U from (4) gives U 5 8 σ ( n x ) + 4K d + σ ( n x ) + L K τ t τ (K +8K )σ ( n x(r) )dr + L K τ Θ(d [to,t]) t 4 τ t τ σ ( n x(l) )dl 8 σ ( n x ) t 8 τ t τ σ( n x(r) ) dr +α 4 ( d [to,t]) t t o + τ, where we omitted the argument t from x and d. (8) (9) Since σ K and V is proper and positive definite, we can easily find γ K such that γ(s) s for all s 0 and, 8 σ ( n x ) γ(v (x, t)) for all x and t. Since γ(s) s for all s 0, the relation γ(a + b) γ(a) + b (for a, b 0) with a = V (x(t), t)/ and (9) give U γ ( +α 4 ( d [to,t]) { V (x(t), t) + 8 τ t τ σ( x(l) ) dl α 3 (U(x t, t)) + α 4 ( d [to,t]), t t o + τ, }) where α 3 (s) := γ(s/{8 τ +}). Hence (ii) from the ISS- LKF definition holds with κ =. To check (i) from the ISS-LKF definition with κ =, we first choose α, ᾱ K so that α( x ) V (x, t) ᾱ( x ) everywhere. Given φ C n ([ τ, 0]) and t 0, the function s x(s) := φ(s t) satisfies x t = φ. Hence, α ( φ(0) ) V (x(t), t) U(φ, t) V (x(t), t) + t τ σ ( n x(l) )dl α ( φ [ τ,0] ) where α = α and α (s) = ᾱ(s) + τσ ( ns). Thus, U is an ISS-LKF for (). Hence, () is ISS by Remark A.. Remark 6 We can extend Theorem 3 to cover systems d that are not control affine, e.g., dtx(t) = f(x(t), t) + g(x(t), t)σ(u s (ξ τ (t), t) + d(t)) where Σ : R m R m is globally Lipschitz (e.g., saturation) if we change Assumption H to V t (x, l)+v x (x, l)[f(x, l)+g(x, l)σ(u s (x, l))] σ( n x ). The proof is as before except we add and subtract V x g(x(t), t)σ(u s (x(t), t)) at the start to get D t V σ ( n x(t) ) + B V x (x, t)g(x(t), t) { d(t) + u s (ξ τ (t), t) u s (x(t), t) }, where B is a Lipschitz constant for Σ. 5 Extension to cascades We extend Theorem 3 to cascades of the form ẋ(t) = f(x(t)) + g(x(t))z(t), ż(t) = u(x(t τ), x(t τ), z(t τ)) + d(t), (0) evolving on R n R with d L ([0, ) and constant delay τ > 0. We assume that the x-dynamics (with fictitious input z) satisfies Assumption H for the case where the vector fields, feedback, and Lyapunov function are time independent. Let u s : R n R m and the constants L, K, K,..., K 4 be as in Assumption H. Mazenc & Bliman (006) designed stabilizing controls for a family of delayed systems that includes (0), but those control laws do not yield ISS. Here we show that a suitable u renders (0) ISS, i.e., there are β KL and γ K such that for all t t o 0, all sufficiently small delays τ > 0, all d L ([0, )), and all trajectories (x, z)(t) := (x(t; t o, (x o, z o ), d, τ), z(t; t o, (x o, z o ), d, τ)) of (0), (x, z)(t) β( (x, z) [to τ,t o], t t o ) + γ( d [to,t]) () with the convention that the initial functions (x o, z o ) C n+ ([t o τ, t o ]) are constant on [t o τ, t o τ]. Corollary 7 Let Assumption H (with time invariant f, g, and u s ) hold for ẋ = f(x) + g(x)u s (x). Set τ c := min{/ 8, τ}, Z(t) := z(t) u s (x(t τ)), () where τ is defined in (3). Then for each constant delay τ (0, τ c ], the dynamics (0) is ISS when we choose u(x(t τ), x(t τ), z(t τ)) = Z(t τ)+ u s x (x()) [f(x())+g(x())z()]. (3) Let us sketch the proof of the preceding result. Set q = (x, Z). Note that (3) transforms (0) into q(t) = F (q(t)) + G(q(t))[U s (q(t τ)) + D(t)] (4) where U s (q)=(u s (x), Z) T, D(t)=(0, d(t)) T, ( ) f(x) + g(x)z F (q) =, and G(q) = 0 ( g(x) 0 0 ). The proof of Theorem 3 provides an ISS-LKF U(x t ) and α 3 K such that along the trajectories (4), U α 3 (U(x t )) + 4KZ (t) +6 L KK t [ 3 τ K3 Z 4 (l) + Z (l) ] (5) dl for all t t 0 + τ and all τ (0, τ]. The relation ab a + b and Jensen s Inequality give Z(t)[Z(t) Z(t τ)] = Z(t) Z(r τ)dr + Z(t) d(r)dr Z (t) + τ t Z (r)dr + Z(t) d(r)dr, (6) 4
5 since Ż(t) = Z(t τ) + d(t). Consider the functionals r Z(l) dldr, M(Z t ) = Z(t) + τ Γ(d, t) = 4(+τ) d [t, Q o,t] (Z t ) = 4M(Z t )+4M (Z t ) and Q (Z t ) = Q (Z t )+ 6 r (Z(l) +Z(l) 4 )dldr. One can easily prove that the derivative of M along the trajectories of (4) for all t t 0 + τ satisfies Ṁ = Z(t)[Z(t) Z(t τ)] [ τ ]Z (t) τ Z (l)dl + Z(t)d(t) 4 Z(t) τ t (7) Z(r) dr + Z(t)d(t) +Z(t) d(r)dr M/4 + Γ(d, t), where we applied ab a /6 + 4b with a = Z(t) twice, (6), τ / 8, and Jensen s Inequality. From this and ab a /4+b with b = M, one easily gets Q Z(t) 4 Z(t)4 + 4Γ(d, t) + 6Γ (d, t) (8) for all t t 0 + τ. The estimate (8) easily gives Q 8 Z(t) 8 Z(t)4 t 6 (Z(l) + Z(l) 4 )dl +α u ( d [t0,t]), t t 0 + τ and τ τ c, where α u (r) = 6( + τ ) (r + r 4 ). From (5), we can then determine a constant C > 0 and α a K such that the time derivative of U f (x t, Z t ) = U(x t ) + CQ (Z t ) satisfies U f α a (U f (x t, Z t )) + Cα u ( d [t0,t]) t t 0 + τ. By viewing Z as a disturbance in the x-dynamics, one can find Ω K such that q(t) Ω( q [t0 τ,t 0 ]) + Ω( d [t0,t 0 +τ]) for all t [t 0 τ, t 0 + τ] and trajectories q(t) of (4); see Appendix A. for similar arguments. This gives (cf. Appendix A.) an ISS estimate of the form q(t) β ( ) q [t, t t o τ,t o] o + γ( d [to,t]) (9) for all t t o. Moreover, z(t) Z(t) + L x(t τ), by our choice () of Z. Therefore, for all t t o, (9) gives (x, z)(t) β ( ) q [t, t t o τ,t o] o + L β ) ( q c [t, max{0, t τ t o τ,t o] o} + γ( d [to,t])+ L γ( d [to,t]), βc (s, r) := β(s, r)+se ṛ Substituting q [to τ,to] (+ L ) (x, z) [to τ,to] into the preceding estimate gives () with γ(s) := {4( + L ) γ(s)} / β(s, r) := {4 β({ + L }s, r) 6 Illustrations +4 L β c ({ + L }s, max{0, r τ})} /. 6. An example from identification theory Consider a dynamics (cf. Mazenc et al. (006) and several references contained therein) ẋ = m(t)m T (t)u (0) in which m : R R n is continuous and satisfies m(t) = for all t R and admits constants α (0, ) and β, c > 0 such that α I + c t m(τ)m T (τ)dτ β I, t R. () A Lyapunov function whose derivative along the trajectories of (0) is negative definite is not obvious; no time independent Lyapunov function has this property, because m T (t)x may be zero when x 0. Fortunately, (Mazenc et al., 006, Lemma ) gives: Lemma 8 Let the function m(t) satisfy the preceding conditions. Then, with κ = + c + 4α c 4 and P (t) = κi + t c s m(l)mt (l) dl ds, () the function V (x, t) := x T P (t)x satisfies V α x / along all trajectories of ẋ(t) = m(t)m T (t)x(t). Moreover, P (t) κ + c everywhere. Using Lemma 8, one checks that Assumption H holds for ẋ(t) = m(t)m T (t)[u(ξ τ (t)) + d(t)] with u(x) = x, σ(r) = r(α /{n}) /, K = (n/α ) / (κ + c ), K 4 = n/α, L =, K 3 =, and K = 0, so Theorem 3 gives: Corollary 9 Let the preceding hypotheses hold. Choose P as in () and τ from (3). Then for all constants τ i (0, τ], ẋ(t) = m(t)m T (t)[ξ τ (t) + d(t)] has the ISS-LKF U(x t, t) = x T (t)p (t)x(t) + α 8 τ and therefore is ISS. 6. Further illustrations t τ ( r x(l) dl ) dr Example 0 One easily checks that ẋ = ux /( + x ) with x R is not locally exponentially stabilizable by any C feedback u(x) (using e.g. (Khalil, 00, Theorem 4.4, p.6)). However, it satisfies Assumption H with V (x) = x, σ(r) = r / + r, u s (x) = x, L = K =, K = 0 and K 4 =. Hence, by Theorem 3, ẋ(t) = x (t) +x (t)[x(t τ) d(t)] (3) is ISS when 0 < τ τ = 0.98/ 65. Moreover, Theorem 3 provides an explicit ISS-LKF. In fact, by scaling u s, we can allow an arbitrarily large bound τ (see Remark 4). Example The system ẋ = u( + x ) / on R has an unbounded function g and satisfies Assumption H from Appendix A. below with K = K 4 =, u s (x) = x 0 dl, σ(r) = u +l s(r), V (x) = u s(x) and K = 0 and so is tractable by Lemma A.3: ẋ(t) = + x (t) [ x() 0 ] dl + d(t) +l is ISS when 0 < τ /8. Moreover, we can again allow an arbitrarily large delay bound (see Remark A.4 below for details). Since g is unbounded, this example is not covered by Fan & Arcak (006). 5
6 7 Conclusions We gave new explicit constructions of ISS Lyapunov- Krasovskii functionals for delayed dynamics, in terms of given Lyapunov functions for the corresponding undelayed dynamics. This led to general conditions under which a time-varying delayless system with a stabilizing feedback remains ISS with respect to actuator errors when time delays are introduced into the feedback. APPENDICES A. Explicit ISS estimate Throughout the appendices, we assume the following which automatically holds under our Assumption H: A The functions f, g, and u are locally Lipschitz and there exists a constant L > 0 such that for all x R n and t 0, we have (A) f(x, t) L x, (A) g(x, t) L( x + ), and (A3) u s / x(x, t) L. Lemma A. Let τ > 0 be a given constant. Then, for each κ N and τ (0, τ] n, there exists a γ κ,τ K (depending on κ and τ) such that for all t o 0, x o C n ([t o τ, t o ]), and d L m ([0, )), the corresponding solution t x(t; t o, x o, d, τ) of (IP) above satisfies x(t; t o, x o, d, τ) γ κ,τ ( x o [to τ,t o]) + γ κ,τ ( d [to,t]) for all t [t o, t o + κ τ]. In particular, we have globally well defined solutions. To prove Lemma A., let t o 0, τ (0, τ] n, x o C n ([t o τ, t o ]), and d L m ([0, )) be given. Set ˆτ = min i τ i. The existence of a unique maximal solution x(t) of (IP) on some interval [t o τ, t o + ε) for some ε > 0 follows from the classical theory applied with x o (t) and d(t) viewed as the inputs. Moreover, Assumption A gives ẋ(t) x(t) { L+ L ξ τ (t) + L d(t) }+ L ξ τ (t) + L d(t) for all t t o for which x(t) is defined. Integrating up to any such t [t o, t o + ˆτ], applying Gronwall s Inequality, and setting D := L x o [to τ,to] + L d [to,t] gives x(t) { x(t o ) + td} e t( L+ L x o [to τ,to] + L d [to,t] ) on all intervals [t o, t o + t] [t o, t o + ˆτ] for which the solution is defined. A standard maximality and local existence argument now easily allows us to conclude that x(t) is uniquely defined at least on [t o, t o + ˆτ]. Taking t = ˆτ in the above estimate and using e a+b e a +e b and ab a + b for a, b 0 provides a γ K such that x(t) [ L xo [to τ,t o] + ˆτ [to,t]] L d e Lˆτ } {e L ˆτ x o [to τ,to] + e Lˆτ d [to,t] e L τ { L xo [to τ,t o]e L τ x o [to τ,to] + ( L x o [to τ,t o]) +(e L τ x o [to τ,to] ) ( ) + e Lτ d [to,t] + ( τ L d [to,t]) + τ L d } [to,t]e L τ d [to,t] γ ( x o [to τ,t o]) + γ ( d [to,t]) (A.) for all t [t o, t o + ˆτ], where L = + τ L. Repeating this argument on [t o + ˆτ, t o + ˆτ] gives a γ K such that x(t) γ ( x [to,t o+ˆτ]) + γ ( d [to,t]) (A.) for all t [t o + ˆτ, t o + ˆτ]. Taking the supremum on the left side of (A.) over [t o, t o + ˆτ], substituting into (A.), and using the relation γ (a + b) γ (a) + γ (b) for a, b 0 gives x(t) γ 3 ( x o [to τ,to] )+γ 3 ( d [to,t]) for all t [t o, t o +ˆτ], where γ 3 (s) = γ (s)+γ (γ (s)). Lemma A. now follows by the obvious inductive argument. Corollary A. If () satisfies Assumption A and admits an ISS-LKF, then it is ISS. To prove this corollary, let U and the α i s be as in Definition. Take any trajectory x(t) of (). Then s χ(s) := α3 (α 4(s)) satisfies U(x t, t) χ( d ) D t U(x t, t) α 3 (U(x t, t))/ for all t t o + κ τ. Also, S := {t t o + κ τ : U(x t, t) χ( d )} is a (possibly empty) interval which we denote by I. Hence, either (a) the left endpoint of I is t o + κ τ or (b) U(x t, t) χ( d ) for all t t o + κ τ (by the argument from Sontag (989)). If case (a) occurs, then a standard comparison argument applied to I t U(x t, t) and (i) in the ISS-LKF definition (with φ(r) = x(t + r) and then φ(r) = x(t o + κ τ + r) to get an upper bound) provide a β a KL so that U(x t, t) χ( d ) x(t) β a ( x [to,to+κ τ], t t o ) for all t t o + κ τ, which gives x(t) β a ( x [to,to+κ τ], t t o ) + γ c ( d [to,t] ) for all t t o + κ τ, where γ c := α χ. Hence, for all trajectories x(t), x(t) β c ( x [to,to+κ τ], t t o ) + γ c ( d [to,t] ) for all t t o, where β c (s, r) = β a (s, r) + se κ τ r. Lemma A. provides a γ κ,τ K so that x(t) γ κ,τ ( x o [to τ,to] ) + γ κ,τ ( d [to,t]) for all t [t o, t o + κ τ]. Since β c (a+b, r) β c (a, r)+β c (b, 0) for all a, b, r 0, we can satisfy the ISS requirements with β(s, r) := β c (γ κ,τ (s), r) and γ(r) := γ c (r) + β c (γ κ,τ (r), 0). A. The case of equal delays When τ = τ =... = τ n, we can relax Assumption H to the following, which we call Assumption H : The function u s C. Also, there exist a σ K for which σ(r) r for all r 0, a C uniformly proper and positive definite V : R n [0, ) [0, ), and constants K > 0 and K i 0 (i =, 3, 4) such that for all x R n, q R n, l 0, and t 0: (H ) V t (x, t) + V x (x, t)[f(x, t) + g(x, t)u s (x, t)] σ ( x ); (H ) V x (x, t)g(x, t) K σ( x ); (H3 ) u s / x(x, t)f(x, l) K σ ( x ) and u s / x(x, t)g(x, l) K 3 (σ( x ) + ); and (H4 ) [ u s / x(x, t)g(x, l) u s (q, l) ] K 4 [σ ( x ) + σ ( q )]. Notice the use of both l and t in Assumptions (H3 )- (H4 ). Example above satisfies H but violates Assumption H4. We now use the delay bound τ = 4K 3K + 3K 4 +, (A.3) but see Remark A.4 for results with larger bounds. 6
7 Lemma A.3 Under Assumptions A and H with τ := τ = τ =... = τ n (0, τ] constant, the delayed system () in closed loop with the feedback u = u s admits the ISS-LKF U(x t, t) = V (x(t), t) + 8 τ and therefore is ISS. ( t ) t r σ ( x(l) )dl dr Let us prove Lemma A.3. Our dynamics have globally well defined solutions, by Lemma A.. By adding and subtracting V x g(x(t), t)u s (x(t), t), (H ) gives V σ( x(t) ) + V x g(x(t), t) d(t) + V x g(x(t), t) u s (x(t τ), t) u s (x(t), t) along any trajectory x(t) := x(t; t o, x o, d, τ) of the dynamics, where we omitted the argument (x(t), t) of V x. From (H ) and wz 4 w + z with w = σ( x ), V σ( x(t) ) + K σ( x(t) ) d(t) +K σ( x(t) ) u s (x(t τ), t) u s (x(t), t) σ ( x(t) ) + K d(t) +K u s (x(t τ), t) u s (x(t), t). (A.4) By the Fundamental Theorem of Calculus applied to [t τ, t] l u s (x(l), t) and Jensen s inequality, u s (x(t τ), t) u s (x(t), t) τ T t(x(l), x(l τ), d(l), l)dl, where (A.5) T t (a, b, d, l)= [ u s x (a, t) {f(a, l)+g(a, l)(u s(b, l)+d)} ]. By the Cauchy Inequality, we get T t (a, b, d, l) 3 [ u s x (a, t)f(a, l) + u s x (a, t)g(a, l) u s (b, l) + u s x (a, t)g(a, l) d ]. From (H3 )-(H4 ) and the relation wz w + z /4, we get T t (x(l), x(l τ), d(l), l) 3K 3 d(l) 4 + 3K 3 d(l) + (3K + 3K 4 + )σ( x(l) ) + 3K 4 σ( x(l τ) ), where we took z = 3K 3 d(l). Hence, when t t o + τ, u s (x(t τ), t) u s (x(t), t) τ [ (3K + 3K 4 + )σ( x(l) ) +3K 4 σ( x(l τ) ) ] dl +τ [ 3K 3 d(l) 4 + 3K 3 d(l) ] (A.6) dl τ(3k + 3K 4 + ) σ( x(l) ) dl [ +3τK 3 K3 d(l) 4 + d(l) ] dl. Substituting (A.6) into (A.4), and arguing analogously to the proof of (8), we deduce that when t t o + τ, U 4 τ τσ ( x(t) ) t 8 τ σ ( x(l) )dl σ( x ) +τk(3k +3K 4 +) σ( x(l) ) dl + α 4 ( d [to,t]) 4 σ( x(t) ) + ( 8 τ + τk (3K + 3K 4 + ) ) σ( x(l) ) dl + α 4 ( d [to,t]) along all of the trajectories of (), where α 4 (r) = 3 τ KK 3 (K 3 +)(r 4 +r )+Kr. The rest of the proof is essentially the same as in the proof of Theorem 3. Remark A.4 Our bound (A.3) can sometimes be raised to an arbitrarily large positive constant. To show why, first note that if Assumption H holds with V t 0 and f 0, and if η (0, ) is any fixed constant, then Assumption H also holds with u s, σ, K, and K 4 replaced by u η s, η / σ, K /η /, and η 3 K 4 respectively, and K = 0, with the feedback u η s(x, t) := ηu s (x, t). Next, note that applying wz K 4 w + z /(4K 4 ) with z = 3K 3 d(l) and arguing in almost the same way as in the proof of Lemma A.3 gives T t (x(l), x(l τ), d(l), l) (3K + 4K 4 )σ ( x(l) ) + 3K 4 σ ( x() )+3K 3 d(l) 4 /K 4 +3K 3 d(l). A slight variant of the rest of the proof of Lemma A.3 then gives ISS of () for all 0 < τ τ c := /{4K 3K + 4K 4 }. Combining the preceding observations (with K = 0), we conclude that if () satisfies H with V t 0, f 0, and equal delays, then u = u η s := ηu s renders the system ẋ(t) = g(x(t), t)[u(x(t τ), t) + d(t)] ISS as long as 0 < τ = 4 K 4η3 K η / 4 8ηK K4 =: τ η. Since τ η + as η 0, we get ISS with arbitrarily large delay bounds τ = τ η, if u s is properly selected. Moreover, we can find explicit ISS-LKFs for all τ > 0. References Fan, X., & Arcak, M. (006). Delay robustness of a class of nonlinear systems and applications to communication networks. IEEE Transactions on Automatic Control, 5(), Fattouh, A., Sename, O., & Dion, J.-M. (000). A LMI approach to robust observer design for linear timedelay systems. In Proceedings of the 39th IEEE Conference on Decision and Control (pp ), Sydney, Australia. Jankovic, M. (00). Control-Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Transactions on Automatic Control, 46(7), Karafyllis, I. (006). Lyapunov theorems for systems described by retarded functional differential equations. Nonlinear Analysis: Theory Methods and Applications, 64(3), Khalil, H. (00). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. Mazenc, F., & Bliman, P. (006). Backstepping design for time-delay nonlinear systems. IEEE Transactions on Automatic Control, 5(), Mazenc, F., Malisoff, M., & de Queiroz, M. (006). Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems. Automatica, 4(0), Michiels, W., Sepulchre, R., Roose, D., & Moreau, L. (00). A perturbation approach to the stabilization of nonlinear cascade systems with time-delay. In Proceedings of the 4st IEEE Conference on Decision and Control (pp ), Las Vegas, NV. 7
8 Nesic, D., & Teel, A. (004). Input-to-state stability of networked control systems. Automatica, 40, - 8. Niculescu, S.-I. (998). H memoryless control and an α-stability constraint for time-delay systems: An LMI approach. IEEE Transactions on Automatic Control, 43(5), Pepe, P., & Jiang, Z.-P. (006). A Lyapunov-Krasovskii methodology for ISS and iiss of time-delay systems. Systems and Control Letters, 55(), Sontag, E.D. (989). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 34(4), Sontag, E.D. (006). Input-to-State Stability: Basic concepts and results. Nonlinear and Optimal Control Theory (pp. 63-0). Berlin: Springer. Sontag, E.D., & Wang, Y. (995). On characterizations of the Input-to-State Stability property. Systems and Control Letters, 4(5), Tarbouriech, S., & da Silva, Jr., J. (000). Synthesis of controllers for continuous time delay systems with saturating controls via LMI s. IEEE Transactions on Automatic Control, 45(), 05-. Teel, A. (998). Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Transactions on Automatic Control, 43(7),
Backstepping Design for Time-Delay Nonlinear Systems
Backstepping Design for Time-Delay Nonlinear Systems Frédéric Mazenc, Projet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie, INRA, pl. Viala, 346 Montpellier, France, e-mail: mazenc@helios.ensam.inra.fr
More informationOn Sontag s Formula for the Input-to-State Practical Stabilization of Retarded Control-Affine Systems
On Sontag s Formula for the Input-to-State Practical Stabilization of Retarded Control-Affine Systems arxiv:1206.4240v1 [math.oc] 19 Jun 2012 P. Pepe Abstract In this paper input-to-state practically stabilizing
More informationAn asymptotic ratio characterization of input-to-state stability
1 An asymptotic ratio characterization of input-to-state stability Daniel Liberzon and Hyungbo Shim Abstract For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic
More informationConverse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form
Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form arxiv:1206.3504v1 [math.ds] 15 Jun 2012 P. Pepe I. Karafyllis Abstract In this paper
More informationSmall Gain Theorems on Input-to-Output Stability
Small Gain Theorems on Input-to-Output Stability Zhong-Ping Jiang Yuan Wang. Dept. of Electrical & Computer Engineering Polytechnic University Brooklyn, NY 11201, U.S.A. zjiang@control.poly.edu Dept. of
More informationL -Bounded Robust Control of Nonlinear Cascade Systems
L -Bounded Robust Control of Nonlinear Cascade Systems Shoudong Huang M.R. James Z.P. Jiang August 19, 2004 Accepted by Systems & Control Letters Abstract In this paper, we consider the L -bounded robust
More informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationarxiv: v3 [math.ds] 22 Feb 2012
Stability of interconnected impulsive systems with and without time-delays using Lyapunov methods arxiv:1011.2865v3 [math.ds] 22 Feb 2012 Sergey Dashkovskiy a, Michael Kosmykov b, Andrii Mironchenko b,
More informationObserver-based quantized output feedback control of nonlinear systems
Proceedings of the 17th World Congress The International Federation of Automatic Control Observer-based quantized output feedback control of nonlinear systems Daniel Liberzon Coordinated Science Laboratory,
More informationBACKSTEPPING FOR NONLINEAR SYSTEMS WITH DELAY IN THE INPUT REVISITED
BACKSTEPPING FOR NONLINEAR SYSTEMS WITH DELAY IN THE INPUT REVISITED FRÉDÉRIC MAZENC, SILVIU-IULIAN NICULESCU, AND MOUNIR BEKAIK Abstract. In this paper, a new solution to the problem of globally asymptotically
More informationOn Input-to-State Stability of Impulsive Systems
On Input-to-State Stability of Impulsive Systems João P. Hespanha Electrical and Comp. Eng. Dept. Univ. California, Santa Barbara Daniel Liberzon Coordinated Science Lab. Univ. of Illinois, Urbana-Champaign
More informationInput to state Stability
Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part III: Lyapunov functions and quantitative aspects ISS Consider with
More informationAdaptive Tracking and Estimation for Nonlinear Control Systems
Adaptive Tracking and Estimation for Nonlinear Control Systems Michael Malisoff, Louisiana State University Joint with Frédéric Mazenc and Marcio de Queiroz Sponsored by NSF/DMS Grant 0708084 AMS-SIAM
More informationSINCE the 1959 publication of Otto J. M. Smith s Smith
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 287 Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems Miroslav Krstic, Fellow, IEEE Abstract We present an approach for compensating
More informationStrong Lyapunov Functions for Systems Satisfying the Conditions of La Salle
06 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 Strong Lyapunov Functions or Systems Satisying the Conditions o La Salle Frédéric Mazenc and Dragan Ne sić Abstract We present a construction
More informationOn integral-input-to-state stabilization
On integral-input-to-state stabilization Daniel Liberzon Dept. of Electrical Eng. Yale University New Haven, CT 652 liberzon@@sysc.eng.yale.edu Yuan Wang Dept. of Mathematics Florida Atlantic University
More informationStability Criteria for Interconnected iiss Systems and ISS Systems Using Scaling of Supply Rates
Stability Criteria for Interconnected iiss Systems and ISS Systems Using Scaling of Supply Rates Hiroshi Ito Abstract This paper deals with problems of stability analysis of feedback and cascade interconnection
More informationDelay-independent stability via a reset loop
Delay-independent stability via a reset loop S. Tarbouriech & L. Zaccarian (LAAS-CNRS) Joint work with F. Perez Rubio & A. Banos (Universidad de Murcia) L2S Paris, 20-22 November 2012 L2S Paris, 20-22
More informationOn Characterizations of Input-to-State Stability with Respect to Compact Sets
On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag and Yuan Wang Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Mathematics,
More informationGlobal stabilization of feedforward systems with exponentially unstable Jacobian linearization
Global stabilization of feedforward systems with exponentially unstable Jacobian linearization F Grognard, R Sepulchre, G Bastin Center for Systems Engineering and Applied Mechanics Université catholique
More informationNonlinear L 2 -gain analysis via a cascade
9th IEEE Conference on Decision and Control December -7, Hilton Atlanta Hotel, Atlanta, GA, USA Nonlinear L -gain analysis via a cascade Peter M Dower, Huan Zhang and Christopher M Kellett Abstract A nonlinear
More informationMinimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality
Minimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality Christian Ebenbauer Institute for Systems Theory in Engineering, University of Stuttgart, 70550 Stuttgart, Germany ce@ist.uni-stuttgart.de
More informationA small-gain type stability criterion for large scale networks of ISS systems
A small-gain type stability criterion for large scale networks of ISS systems Sergey Dashkovskiy Björn Sebastian Rüffer Fabian R. Wirth Abstract We provide a generalized version of the nonlinear small-gain
More informationEvent-based Stabilization of Nonlinear Time-Delay Systems
Preprints of the 19th World Congress The International Federation of Automatic Control Event-based Stabilization of Nonlinear Time-Delay Systems Sylvain Durand Nicolas Marchand J. Fermi Guerrero-Castellanos
More information1 The Observability Canonical Form
NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)
More informationRobust Control for Nonlinear Discrete-Time Systems with Quantitative Input to State Stability Requirement
Proceedings of the 7th World Congress The International Federation of Automatic Control Robust Control for Nonlinear Discrete-Time Systems Quantitative Input to State Stability Requirement Shoudong Huang
More informationON INPUT-TO-STATE STABILITY OF IMPULSIVE SYSTEMS
ON INPUT-TO-STATE STABILITY OF IMPULSIVE SYSTEMS EXTENDED VERSION) João P. Hespanha Electrical and Comp. Eng. Dept. Univ. California, Santa Barbara Daniel Liberzon Coordinated Science Lab. Univ. of Illinois,
More informationPassivity-based Stabilization of Non-Compact Sets
Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained
More informationc 2002 Society for Industrial and Applied Mathematics
SIAM J. CONTROL OPTIM. Vol. 4, No. 6, pp. 888 94 c 22 Society for Industrial and Applied Mathematics A UNIFYING INTEGRAL ISS FRAMEWORK FOR STABILITY OF NONLINEAR CASCADES MURAT ARCAK, DAVID ANGELI, AND
More informationStabilization in spite of matched unmodelled dynamics and An equivalent definition of input-to-state stability
Stabilization in spite of matched unmodelled dynamics and An equivalent definition of input-to-state stability Laurent Praly Centre Automatique et Systèmes École des Mines de Paris 35 rue St Honoré 7735
More informationViscosity Solutions of the Bellman Equation for Perturbed Optimal Control Problems with Exit Times 0
Viscosity Solutions of the Bellman Equation for Perturbed Optimal Control Problems with Exit Times Michael Malisoff Department of Mathematics Louisiana State University Baton Rouge, LA 783-4918 USA malisoff@mathlsuedu
More informationStability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach
Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach Michael Malisoff (LSU) Joint with Frederic Mazenc and Silviu-Iulian Niculescu 0/6 Motivation
More informationInput to state Stability
Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part IV: Applications ISS Consider with solutions ϕ(t, x, w) ẋ(t) =
More informationAdaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification
Adaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification Michael Malisoff, Louisiana State University Joint with Frédéric Mazenc and Marcio
More informationA new method to obtain ultimate bounds and convergence rates for perturbed time-delay systems
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR ONTROL Int. J. Robust. Nonlinear ontrol 212; 22:187 188 ublished online 21 September 211 in Wiley Online Library (wileyonlinelibrary.com)..179 SHORT OMMUNIATION
More informationLocal ISS of large-scale interconnections and estimates for stability regions
Local ISS of large-scale interconnections and estimates for stability regions Sergey N. Dashkovskiy,1,a,2, Björn S. Rüffer 1,b a Zentrum für Technomathematik, Universität Bremen, Postfach 330440, 28334
More informationNavigation and Obstacle Avoidance via Backstepping for Mechanical Systems with Drift in the Closed Loop
Navigation and Obstacle Avoidance via Backstepping for Mechanical Systems with Drift in the Closed Loop Jan Maximilian Montenbruck, Mathias Bürger, Frank Allgöwer Abstract We study backstepping controllers
More informationAsymptotic Controllability and Input-to-State Stabilization: The Effect of Actuator Errors
Asymptotic Controllability and Input-to-State Stabilization: The Effect of Actuator Errors Michael Malisoff 1 and Eduardo Sontag 2 1 Department of Mathematics, Louisiana State University and A. & M. College,
More informationSTABILITY ANALYSIS FOR NONLINEAR SYSTEMS WITH TIME-DELAYS. Shanaz Tiwari. A Dissertation Submitted to the Faculty of
STABILITY ANALYSIS FOR NONLINEAR SYSTEMS WITH TIME-DELAYS by Shanaz Tiwari A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements
More informationAbstract. Previous characterizations of iss-stability are shown to generalize without change to the
On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag and Yuan Wang Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Mathematics,
More informationOn finite gain L p stability of nonlinear sampled-data systems
Submitted for publication in Systems and Control Letters, November 6, 21 On finite gain L p stability of nonlinear sampled-data systems Luca Zaccarian Dipartimento di Informatica, Sistemi e Produzione
More informationTracking Control and Robustness Analysis for a Nonlinear Model of Human Heart Rate During Exercise
Tracking Control and Robustness Analysis for a Nonlinear Model of Human Heart Rate During Exercise Frédéric Mazenc a Michael Malisoff b,1 Marcio de Querioz c, a Projet INRIA DISCO, CNRS-Supélec, 3 rue
More informationPredictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions
Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions M. Lazar, W.P.M.H. Heemels a a Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
More informationAnalysis of Input to State Stability for Discrete Time Nonlinear Systems via Dynamic Programming
Analysis of Input to State Stability for Discrete Time Nonlinear Systems via Dynamic Programming Shoudong Huang Matthew R. James Dragan Nešić Peter M. Dower April 8, 4 Abstract The Input-to-state stability
More informationNetworked Control Systems, Event-Triggering, Small-Gain Theorem, Nonlinear
EVENT-TRIGGERING OF LARGE-SCALE SYSTEMS WITHOUT ZENO BEHAVIOR C. DE PERSIS, R. SAILER, AND F. WIRTH Abstract. We present a Lyapunov based approach to event-triggering for large-scale systems using a small
More informationAdaptive and Robust Controls of Uncertain Systems With Nonlinear Parameterization
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 0, OCTOBER 003 87 Adaptive and Robust Controls of Uncertain Systems With Nonlinear Parameterization Zhihua Qu Abstract Two classes of partially known
More informationLecture Note 7: Switching Stabilization via Control-Lyapunov Function
ECE7850: Hybrid Systems:Theory and Applications Lecture Note 7: Switching Stabilization via Control-Lyapunov Function Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio
More informationOn the construction of ISS Lyapunov functions for networks of ISS systems
Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24-28, 2006 MoA09.1 On the construction of ISS Lyapunov functions for networks of ISS
More informationConvergent systems: analysis and synthesis
Convergent systems: analysis and synthesis Alexey Pavlov, Nathan van de Wouw, and Henk Nijmeijer Eindhoven University of Technology, Department of Mechanical Engineering, P.O.Box. 513, 5600 MB, Eindhoven,
More informationFinite-time stability and input-to-state stability of stochastic nonlinear systems
Finite-time stability and input-to-state stability of stochastic nonlinear systems KANG Yu, ZHAO Ping,. Department of Automation, University of Science and Technology of China, Hefei 36, Anhui, P. R. China
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationFurther Equivalences and Semiglobal Versions of Integral Input to State Stability
Further Equivalences and Semiglobal Versions of Integral Input to State Stability David Angeli Dip. Sistemi e Informatica University of Florence 5139 Firenze, Italy angeli@dsi.unifi.it E. D. Sontag Department
More informationAn Asymmetric Small-Gain Technique to Construct Lyapunov-Krasovskii Functionals for Nonlinear Time-Delay Systems with Static Components
2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 An Asymmetric Small-Gain Technique to Construct Lyapunov-Krasovskii Functionals for Nonlinear Time-Delay
More informationtion. For example, we shall write _x = f(x x d ) instead of _x(t) = f(x(t) x d (t)) and x d () instead of x d (t)(). The notation jj is used to denote
Extension of control Lyapunov functions to time-delay systems Mrdjan Jankovic Ford Research Laboratory P.O. Box 53, MD 36 SRL Dearborn, MI 4811 e-mail: mjankov1@ford.com Abstract The concept of control
More informationA LaSalle version of Matrosov theorem
5th IEEE Conference on Decision Control European Control Conference (CDC-ECC) Orlo, FL, USA, December -5, A LaSalle version of Matrosov theorem Alessro Astolfi Laurent Praly Abstract A weak version of
More informationOn Input-to-State Stability of Impulsive Systems
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 TuC16.5 On Input-to-State Stability of Impulsive Systems João
More informationIntroduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
More informationFOR OVER 50 years, control engineers have appreciated
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 7, JULY 2004 1081 Further Results on Robustness of (Possibly Discontinuous) Sample Hold Feedback Christopher M. Kellett, Member, IEEE, Hyungbo Shim,
More informationAnalysis of different Lyapunov function constructions for interconnected hybrid systems
Analysis of different Lyapunov function constructions for interconnected hybrid systems Guosong Yang 1 Daniel Liberzon 1 Andrii Mironchenko 2 1 Coordinated Science Laboratory University of Illinois at
More informationOn Saturation, Discontinuities and Time-Delays in iiss and ISS Feedback Control Redesign
2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA05.5 On Saturation, Discontinuities and Time-Delays in iiss and ISS Feedback Control Redesign P. Pepe H.
More informationLyapunov Stability of Linear Predictor Feedback for Distributed Input Delays
IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL. 56 NO. 3 MARCH 2011 655 Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays Nikolaos Bekiaris-Liberis Miroslav Krstic In this case system
More informationComments on integral variants of ISS 1
Systems & Control Letters 34 (1998) 93 1 Comments on integral variants of ISS 1 Eduardo D. Sontag Department of Mathematics, Rutgers University, Piscataway, NJ 8854-819, USA Received 2 June 1997; received
More informationGlobal Stabilization of Oscillators With Bounded Delayed Input
Global Stabilization of Oscillators With Bounded Delayed Input Frédéric Mazenc, Sabine Mondié and Silviu-Iulian Niculescu Abstract. The problem of globally asymptotically stabilizing by bounded feedback
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationFrom convergent dynamics to incremental stability
51st IEEE Conference on Decision Control December 10-13, 01. Maui, Hawaii, USA From convergent dynamics to incremental stability Björn S. Rüffer 1, Nathan van de Wouw, Markus Mueller 3 Abstract This paper
More informationDecentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections
Decentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections Yi Guo Abstract The problem of decentralized disturbance attenuation is considered for a new
More informationControlling Human Heart Rate Response During Treadmill Exercise
Controlling Human Heart Rate Response During Treadmill Exercise Frédéric Mazenc (INRIA-DISCO), Michael Malisoff (LSU), and Marcio de Queiroz (LSU) Special Session: Advances in Biomedical Mathematics 2011
More informationA Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1
A Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1 Ali Jadbabaie, Claudio De Persis, and Tae-Woong Yoon 2 Department of Electrical Engineering
More informationRobust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC15.1 Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers Shahid
More informationA non-coercive Lyapunov framework for stability of distributed parameter systems
17 IEEE 56th Annual Conference on Decision and Control (CDC December 1-15, 17, Melbourne, Australia A non-coercive Lyapunov framework for stability of distributed parameter systems Andrii Mironchenko and
More informationnamics Conclusions are given in Section 4 2 Redesign by state feedback We refer the reader to Sontag [2, 3] for the denitions of class K, K and KL fun
Robust Global Stabilization with Input Unmodeled Dynamics: An ISS Small-Gain Approach Λ Zhong-Ping Jiang y Murat Arcak z Abstract: This paper addresses the global asymptotic stabilization of nonlinear
More informationControl under Quantization, Saturation and Delay: An LMI Approach
Proceedings of the 7th World Congress The International Federation of Automatic Control Control under Quantization, Saturation and Delay: An LMI Approach Emilia Fridman Michel Dambrine Department of Electrical
More informationDISCRETE-TIME TIME-VARYING ROBUST STABILIZATION FOR SYSTEMS IN POWER FORM. Dina Shona Laila and Alessandro Astolfi
DISCRETE-TIME TIME-VARYING ROBUST STABILIZATION FOR SYSTEMS IN POWER FORM Dina Shona Laila and Alessandro Astolfi Electrical and Electronic Engineering Department Imperial College, Exhibition Road, London
More informationA new robust delay-dependent stability criterion for a class of uncertain systems with delay
A new robust delay-dependent stability criterion for a class of uncertain systems with delay Fei Hao Long Wang and Tianguang Chu Abstract A new robust delay-dependent stability criterion for a class of
More informationRobust distributed linear programming
Robust distributed linear programming Dean Richert Jorge Cortés Abstract This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization
More informationExponential stability of families of linear delay systems
Exponential stability of families of linear delay systems F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany fabian@math.uni-bremen.de Keywords: Abstract Stability, delay systems,
More informationNonlinear Scaling of (i)iss-lyapunov Functions
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 4, APRIL 216 187 Nonlinear Scaling of (i)iss-lyapunov Functions Christopher M. Kellett and Fabian R. Wirth Abstract While nonlinear scalings of Lyapunov
More informationEvent-Triggered Decentralized Dynamic Output Feedback Control for LTI Systems
Event-Triggered Decentralized Dynamic Output Feedback Control for LTI Systems Pavankumar Tallapragada Nikhil Chopra Department of Mechanical Engineering, University of Maryland, College Park, 2742 MD,
More informationDisturbance Attenuation for a Class of Nonlinear Systems by Output Feedback
Disturbance Attenuation for a Class of Nonlinear Systems by Output Feedback Wei in Chunjiang Qian and Xianqing Huang Submitted to Systems & Control etters /5/ Abstract This paper studies the problem of
More informationState-norm estimators for switched nonlinear systems under average dwell-time
49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA State-norm estimators for switched nonlinear systems under average dwell-time Matthias A. Müller
More informationHybrid Systems Techniques for Convergence of Solutions to Switching Systems
Hybrid Systems Techniques for Convergence of Solutions to Switching Systems Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel Abstract Invariance principles for hybrid systems are used to derive invariance
More informationSTABILITY AND STABILIZATION OF A CLASS OF NONLINEAR SYSTEMS WITH SATURATING ACTUATORS. Eugênio B. Castelan,1 Sophie Tarbouriech Isabelle Queinnec
STABILITY AND STABILIZATION OF A CLASS OF NONLINEAR SYSTEMS WITH SATURATING ACTUATORS Eugênio B. Castelan,1 Sophie Tarbouriech Isabelle Queinnec DAS-CTC-UFSC P.O. Box 476, 88040-900 Florianópolis, SC,
More informationStability of Linear Distributed Parameter Systems with Time-Delays
Stability of Linear Distributed Parameter Systems with Time-Delays Emilia FRIDMAN* *Electrical Engineering, Tel Aviv University, Israel joint with Yury Orlov (CICESE Research Center, Ensenada, Mexico)
More informationResearch Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
Applied Mathematics Volume 202, Article ID 689820, 3 pages doi:0.55/202/689820 Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
More informationA Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA A Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems Seyed Hossein Mousavi 1,
More informationOn robustness of suboptimal min-max model predictive control *
Manuscript received June 5, 007; revised Sep., 007 On robustness of suboptimal min-max model predictive control * DE-FENG HE, HAI-BO JI, TAO ZHENG Department of Automation University of Science and Technology
More informationDeakin Research Online
Deakin Research Online This is the published version: Phat, V. N. and Trinh, H. 1, Exponential stabilization of neural networks with various activation functions and mixed time-varying delays, IEEE transactions
More informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationNorm-controllability, or how a nonlinear system responds to large inputs
Norm-controllability, or how a nonlinear system responds to large inputs Matthias A. Müller Daniel Liberzon Frank Allgöwer Institute for Systems Theory and Automatic Control IST), University of Stuttgart,
More informationL 2 -induced Gains of Switched Systems and Classes of Switching Signals
L 2 -induced Gains of Switched Systems and Classes of Switching Signals Kenji Hirata and João P. Hespanha Abstract This paper addresses the L 2-induced gain analysis for switched linear systems. We exploit
More informationExplicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems
Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems D. Nešić, A.R. Teel and D. Carnevale Abstract The purpose of this note is to apply recent results
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More information1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 5, MAY 2011
1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 L L 2 Low-Gain Feedback: Their Properties, Characterizations Applications in Constrained Control Bin Zhou, Member, IEEE, Zongli Lin,
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationOn a small gain theorem for ISS networks in dissipative Lyapunov form
On a small gain theorem for ISS networks in dissipative Lyapunov form Sergey Dashkovskiy, Hiroshi Ito and Fabian Wirth Abstract In this paper we consider several interconnected ISS systems supplied with
More informationNumerical schemes for nonlinear predictor feedback
Math. Control Signals Syst. 204 26:59 546 DOI 0.007/s00498-04-027-9 ORIGINAL ARTICLE Numerical schemes for nonlinear predictor feedback Iasson Karafyllis Miroslav Krstic Received: 3 October 202 / Accepted:
More informationResearch Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 21, Article ID 68572, 12 pages doi:1.1155/21/68572 Research Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary
More informationOutput Input Stability and Minimum-Phase Nonlinear Systems
422 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002 Output Input Stability and Minimum-Phase Nonlinear Systems Daniel Liberzon, Member, IEEE, A. Stephen Morse, Fellow, IEEE, and Eduardo
More informationResearch Article Mean Square Stability of Impulsive Stochastic Differential Systems
International Differential Equations Volume 011, Article ID 613695, 13 pages doi:10.1155/011/613695 Research Article Mean Square Stability of Impulsive Stochastic Differential Systems Shujie Yang, Bao
More informationA Simple Self-triggered Sampler for Nonlinear Systems
Proceedings of the 4th IFAC Conference on Analysis and Design of Hybrid Systems ADHS 12 June 6-8, 212. A Simple Self-triggered Sampler for Nonlinear Systems U. Tiberi, K.H. Johansson, ACCESS Linnaeus Center,
More information