Global stabilisation of nonlinear delay systems with a compact absorbing set

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.


Introduction
Remarkable progress has been made in recent years on the design of predictor feedback laws for nonlinear delay systems (Bekiaris-Liberis & Krstic, 2012;Bekiaris-Liberis & Krstic, 2013aKarafyllis, 2011;Karafyllis & Jiang, 2011;Karafyllis & Krstic, 2012aKrstic, 2004Krstic, , 2008Krstic, , 2009Krstic, , 2010. The main challenge to the implementation and design of predictor feedback for nonlinear delay systems is that, except for rare special cases, the solution mapping (used for the prediction) is not available explicitly.
The current status in the literature on input delay compensation is that when, in addition to input delays, • the full state is not measured, • the measurement is sampled and possibly delayed, and when, in addition to global stability, the following properties are required in closed loop, • exponential convergence for the disturbance-free case, • robustness with respect to perturbations of the sampling schedule, and • robustness with respect to measurement errors, predictor feedback designs are available only for two classes of systems: linear detectable and stabilisable * Corresponding author. Email: iasonkar@central.ntua.gr systems and globally Lipschitz systems in strict feedback form (Karafyllis & Krstic, 2013a).
In this paper, we present a result that removes the global Lipschitz restriction (an algebraic condition on the system's right-hand side), but imposes an assumption that the system has a compact absorbing set (a condition on the system's dynamic behaviour in open loop). Specifically, we consider general nonlinear systems of the forṁ (1.1) where U ⊆ m is a non-empty compact set with 0 ∈ U , τ ≥ 0 is the input delay and f : n × m → n is a smooth vector field with f (0, 0) = 0. The measurements are sampled and the output is given by where h : n → k is a smooth mapping with h(0) = 0, r ≥ 0 is the measurement delay, {τ i } ∞ i=0 is a partition of + (the set of sampling times) and the input e : + → is the measurement error. We focus on a class of nonlinear systems that is different from the class of globally Lipschitz systems: the systems with a compact absorbing set. A nonlinear system with a compact absorbing set is a system for which all solutions enter a specific compact set after an initial transient period (for systems without inputs C 2013 Taylor & Francis the name 'global uniform ultimate boundedness' is used in Khalil (1996); the term 'dissipative system' is used in the literature of finite-dimensional dynamical systems; see Stuart and Humphries (1998) and the discussion on page 22 of the book Temam (1997)).
Though it may appear that we merely trade one major restriction (global Lipschitzness) for another (compact absorbing set), which imposes a strong requirement on the system's open-loop behaviour, the latter restriction is less frequently violated in applications. Many engineering systems belong to the class of systems with a compact absorbing set because finite escape is rare in physical processes, control inputs usually saturate, and limit cycles are a frequent outcome of local instabilities.
The contribution of our paper is twofold: (1) Predictor feedback is designed and stability is proved for the class of nonlinear delay systems with a compact absorbing set under appropriate assumptions (Theorem 2.2).
(2) The result is then extended to nonlinear delay systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback (Theorem 2.4).
In both cases, we provide explicit formulae for the predictor feedback and explicit inequalities for the parameters of the applied control scheme and the upper diameter of the sampling partition. The proposed predictor feedback guarantees all properties listed at the beginning of the section for the class of nonlinear delay systems with a compact absorbing set: global asymptotic stability and global exponential attractivity in the absence of measurement error, robustness with respect to perturbations of the sampling schedule and robustness with respect to measurement errors.
Our predictor feedback design consists of the following elements: (1) An Inter-sample predictor (ISP), which uses the sampled, delayed and corrupted measurements of the output and provides an estimate of the (unavailable) delayed continuous output signal.
(2) A global observer (O), which uses the estimate of the delayed continuous output signal and provides an estimate of the delayed state vector. (3) An approximate or exact predictor (P), which uses the estimate of the delayed state vector in order to provide an estimate of the future state vector. (4) A delay-free controller (DFC), that is, a baseline feedback law that works for the delay-free version of the system, which in the presence of delay uses the estimate of the future state vector in order to provide the control action.
We refer to the above control scheme as the ISP-O-P-DFC control scheme. In Karafyllis and Krstic (2013a), the ISP-O-P-DFC control scheme was shown to achieve all the objectives mentioned at the beginning of this section by using approximate predictors that are based on successive approximations of the solution map for linear detectable and stabilisable systems and globally Lipschitz systems in strict feedback form. Here, we show that the ISP-O-P-DFC control scheme guarantees all the objectives listed at the beginning of this section using dynamic approximate predictors for systems with a compact absorbing set.
This methodological difference relative to Karafyllis and Krstic (2013a) merits further emphasis. We employ here a class of approximate predictors that are implemented by means of a dynamical system: the approximate predictor is a system described by integral delay equations (IDEs; see Karafyllis & Krstic, 2013b) and consists of a series connection of N approximate predictors (each making a prediction for the state vector δ = r+τ N time units ahead). Such dynamic predictors were introduced in Ahmed-Ali, Karafyllis, and Lamnabhi-Lagarrigue (2013) and Germani, Manes, and Pepe (2002), but here the predictor is designed in a novel way so that the prediction takes values in an appropriate compact set after an initial transient period. The dynamic predictor is different from other predictors proposed in the literature (e.g., exact predictors in Karafyllis and Krstic (2012a) and Krstic (2010); approximate predictors based on successive approximations in Karafyllis (2011) and Karafyllis and Krstic (2013a); and approximate predictors based on numerical schemes in Karafyllis and Krstic (2012b)). Theorem 2.4 employs a novel combination of approximate predictors and exact predictors in the control scheme, which can be used for other classes of nonlinear delay systems.
The main advantage of the dynamic predictor employed here over other predictor approximations (numerical in Karafyllis and Krstic (2012b) or successive approximations in Karafyllis (2011) and Karafyllis and Krstic (2013a)) is the existence of simple formulas (provided in Ahmed-Ali et al. 2013), for the estimation of the asymptotic gain of the measurement error for certain classes of systems. In contrast, the predictor for which the effect of measurement errors is most difficult to quantify is the numerical predictor (Karafyllis & Krstic, 2012b).
On the other hand, the disadvantages of the dynamic predictor are the difficulty of implementation (one has to approximate numerically the solution of the IDEs or the equivalent distributed delay differential equations) and that it works only for certain classes of nonlinear systems (globally Lipschitz systems and systems with a compact absorbing set). In contrast, the most easily programmable predictor is the numerical predictor (Karafyllis & Krstic, 2012b), which is the crudest version of the predictor based on successive approximations (Karafyllis, 2011;Karafyllis & Krstic, 2013a) -when only one successive approximation is used (and many grid points), then the predictor based on successive approximations coincides with the numerical predictor.
Though our approach to stabilisation of nonlinear systems with actuation and measurement delays is based on delay compensation via predictor design -an approach known for its ability to recover nominal performance in the absence of delay and after finite time in the presence of delay -this is not the only option for stabilisation of nonlinear systems with large dead times. For certain classes of nonlinear systems, other approaches exist that are capable of guaranteeing stability and robustness (Mazenc, Malisoff, & Lin, 2008;Mazenc, Mondie, & Francisco, 2004).
The paper is structured as follows. Section 2 contains the assumptions and the statements of the main results. The proofs are given in Section 3. Section 4 presents two illustrative examples. Concluding remarks are provided in Section 5.
Notation. Throughout this paper, we adopt the following notation: with τ 0 = 0 and lim i→∞ τ i = +∞. • By C 0 (A; ), we denote the class of continuous functions on A ⊆ n , which take values in ⊆ m . By C k (A; ), where k ≥ 1 is an integer, we denote the class of functions on A ⊆ n with continuous derivatives of order k, which take values in ⊆ m . • By int(A), we denote the interior of the set A ⊆ n .
• For a vector x ∈ n , we denote by x its transpose and by |x| its Euclidean norm. A ∈ n×m denotes the transpose of the matrix A ∈ m×n and |A| denotes the induced norm of the matrix A ∈ m×n , i.e., |A| = sup {|Ax| : x ∈ m , |x| = 1}. • A function V : n → + is called positive definite if V (0) = 0 and V (x) > 0 for all x = 0. A function V : n → + is called radially unbounded if the sets {x ∈ n : V (x) ≤ M} are either empty or bounded for all M ≥ 0. • For a function V ∈ C 1 (A; ), the gradient of V at x ∈ A ⊆ n , denoted by ∇V (x), is the row vector ∇V (x) = ∂V ∂x 1 (x) . . . ∂V ∂x n (x) . • The class of functions K ∞ is the class of strictly increasing, continuous functions a : + → + with a(0) = 0 and lim s→+∞ a(s) = +∞.

Systems with an absorbing compact set
Consider the system (1.1) and (1.2). Our main assumption guarantees that there exists a compact set which is robustly globally asymptotically stable (the adjective robust means uniformity to all measurable and essentially bounded inputs u : + → U ). We call the compact set "absorbing" because the solution "is absorbed" in the set after an initial transient period.
H1: There exist a radially unbounded (but not necessarily positive definite) function V ∈ C 2 ( n ; + ), a positive definite function W ∈ C 1 ( n ; + ) and a constant R > 0 such that the following inequality holds for all (x, u) (2.1) Indeed, assumption H1 guarantees that for every initial condition x(0) ∈ n and for every measurable and essentially bounded input u : + → U the solution x(t) of (1.1) enters the compact set S = {x ∈ n : V (x) ≤ R} after a finite transient period, i.e., there exists T ∈ C 0 ( n ; + ) such that x(t) ∈ S, for all t ≥ T (x(0)). Moreover, notice that the compact set S = {x ∈ n : V (x) ≤ R} is positively invariant. This fact is guaranteed by the following lemma that is an extension of Theorem 5.1 in Khalil (1996, p. 211).
Lemma 2.1: Consider system (1.1) under hypothesis H1. Then there exists T ∈ C 0 ( n ; + ) such that for every x 0 ∈ n and for every measurable and essentially bounded input u : [−τ, +∞) → U the solution x(t) ∈ n of (1.1) with initial condition x(0) = x 0 and corresponding to input u : Our second assumption guarantees that we are in a position to construct an appropriate local exponential stabiliser for the delay-free version system (1.1), i.e. system (1.1) with τ = 0.
H2: There exist a positive definite function P ∈ C 2 ( n ; + ), constants μ, K 1 > 0 with K 1 |x| 2 ≤ P (x) for all x ∈ n with V (x) ≤ R and a globally Lipschitz mapping k : n → U with k(0) = 0 such that the following inequality holds (2. 2) The requirement that the mapping k : n → U with k(0) = 0 is globally Lipschitz is not essential. Note that if assumption H2 holds for certain locally Lipschitz mapping k : n → U with k(0) = 0 and if U is a star-shaped set (i.e. (λu ∈ U ) for every λ ∈ U [0,1] and u ∈ U ), then we are in a position to definek(x) = (R + 1 − min (R + 1, max(R, V (x)))) k(x) and we note that assumption H2 holds for the globally Lipschitz functionk : n → U . Our third assumption guarantees that we are in a position to construct an appropriate local exponential observer for the delay-free version of system (1.1), (1.2), i.e. systems (1.1) and (1.2) with r = τ = 0.
H3: There exist a symmetric and positive definite matrix Q ∈ n×n , constants ω > 0, b > R and a matrix L ∈ n×k such that the following inequality holds: Indeed, assumption H3 in conjunction with assumption H1 guarantees that for every x(0) ∈ S = {x ∈ n : V (x) ≤ R} and for every measurable and essentially bounded input u : + → U the solution of system (1.1), (1.2) with r = τ = 0, e ≡ 0 anḋ (0)| is sufficiently small. This is why system (2.4) is termed 'a local exponential observer'. The reader should note that assumption H3 holds automatically for nonlinear systems of the forṁ for every b > R > 0 and for every non-empty set U ⊆ m , where f i : i → (i = 1, . . . , n) are smooth mappings. In order to be able to construct a feedback stabiliser for system (1.1) and (1.2), we need an additional technical assumption.
H4: There exist constants c ∈ (0, 1), R ≤ a < b such that the following inequality holds: (2.6) Assumption H4 imposes constraints for the evolution of the trajectories of the local observer (2.4). Indeed, inequality (2.6) imposes a bound on the derivative of the Lyapunov function V ∈ C 1 ( n ; + ) along the trajectories of the local observer (2.4) for certain regions of the state space. Assumption H4, in conjunction with assumptions H1 and H3, allows us to construct a global exponential sampled-data observer for system (1.1) and (1.2) in the same spirit as in Ahmed-Ali et al. (2013): first, we exploit assumption H4 to construct an observer for which the observer states enter the compact setS = {z ∈ n : V (z) ≤ b} after some finite time. Since assumption H1 guarantees that the states of the system enter the compact set S = {x ∈ n : V (x) ≤ R} after some finite time, we can guarantee that the system operates on the set for which assumption H3 holds after an initial transient period. Therefore, we can exploit the local exponential observer of assumption H3 and guarantee exponential convergence of the error after an initial transient period.
We are now ready to state the first main result of the paper. Note that the dynamic feedback stabiliser is explicitly given and that all parameters included in the feedback stabiliser are required to satisfy explicit inequalities that can be verified easily in practice.
and p : + → [0, 1] is an arbitrary locally Lipschitz function that satisfies p(s) = 1 for all s ≥ b and p(s) = 0 for all s ≤ a. Let q : → + be a continuously differentiable function with q(s) = 1 for s ≤ 1 and sq(s) ≤ K for s ≥ 1, where K ≥ 1 is a constant. Let ψ : n → [1, +∞) be a smooth function that satisfies the following implication: Let N > 0 be an integer and T s > 0, σ > 0 be constants so that Then there exist a constant > 0 and a locally Lips- , exists and satisfies the following estimate for all t ≥ 0: (2.17)

Remark 2.3:
(a) As noted in the Introduction, Theorem 2.2 shows that the control scheme that consists of the series connection of (i) the sampled-data hybrid observer (2.12)-(2.14), which provides an estimate of the delayed state vector x(t − r), (ii) the dynamic predictor (2.15) which provides an estimate of the future value of the state vector x(t + τ ), and (iii) the control law (2.16)  with upper diameter less or equal to T s > 0). (b) The sampled-data hybrid observer (2.12)-(2.14) uses the local exponential observer involved in assumptions H3 and H4 with some modifications. The first modification involves the replacement of the unavailable output signal y(t − r) with the signal w(t), which is generated by the ISP (2.13) and (2.14) (see also Ahmed-Ali et al., 2013). The second modification is the addition of a 'correction term' of the form − ϕ(z,y,u) |∇V (z)| 2 (∇V (z)) which has the task to guarantee the validity of the differential in- (z,y,u) |∇V (z)| 2 (∇V (z)) was used in Ahmed- Ali et al. (2013) in order to guarantee that the solution enters an appropriate compact set in finite time and in this appropriate compact set the local exponential observer works. (c) The input e : + → quantifies the effect of measurement errors. Inequality 2.17 shows that the 'asymptotic gain' of the closed-loop system with respect to the measurement error is lin- However, the ISS-like inequality (2.17) does not guarantee the ISS property with linear gain. In general, the locally Lipschitz function C ∈ K ∞ is nonlinear and the gain function with respect to the measurement error is nonlinear. (d) The predictor (2.15) is a system described by IDEs (see Karafyllis & Krstic, 2013b)  ξ j −1 (t) instead of ξ j −1 (t) guarantees that the prediction will take values in an appropriate compact set. The dynamic predictor (2.15) is different from other predictors proposed in the literature (e.g., exact predictors in Karafyllis & Krstic, 2012a;Krstic, 2010; approximate predictors based on successive approximations in Karafyllis, 2011;Karafyllis & Krstic, 2013a; approximate predictors based on numerical schemes in Karafyllis & Krstic, 2012b). (e) An example of a function q : → + that satisfies the requirements of Theorem 2.2 is the function q(s) := 2s −1 − s −2 for s>1 and q(s) = 1 for s ≤ 1. (f) Since the function C ∈ K ∞ is a locally Lipschitz function, it follows from estimate (2.17) that the dynamic hybrid controller (2.12)-(2.16) guarantees not only global asymptotic stability but local exponential stability as well in the absence of measurement error. Note that the stability properties of the closed-loop system are robust with respect to perturbations of the sampling schedule.
The following result uses a preliminary predictor feedback in order to transform the given system to a system with a compact absorbing set. However, the result of Theorem 2.4 does not allow us to conclude robust global asymptotic stability for the closed-loop system: only exponential attractivity holds for the closed-loop system. The notion of forward completeness used in the statement of Theorem 2.4 is the standard notion used in Angeli and Sontag (1999): the solution exists for times, all initial conditions and all measurable and locally essentially bounded inputs.

Theorem 2.4: Consider the forward complete systeṁ
where τ > 0 is the input delay,f : n × m → n is a smooth vector field withf (0, 0) = 0 and sampled measurements given by

19)
where h : n → k is a smooth mapping with h(0) = 0, r ≥ 0 is the measurement delay and{τ i } ∞ i=0 is a partition of + (the set of sampling times).
Suppose that there exist smooth functions a 1 : n → l , a 2 : l → m with a 1 (0) = 0, a 1 (0) = 0 and a nonempty compact set U ⊂ m with 0 ∈ U such that the vector field f (x, u) :=f (x, a 2 (a 1 (x)) + u) satisfies assumptions H1-H4. Moreover, suppose that there exists a locally Lipschitz vector field g : l × m → l with g(0, 0) = 0 such that the equation ∇a 1 (x)f (x, v) = g(a 1 (x), v) holds for all (x, v) ∈ n × m . Assume that the systeṁ is forward complete. Finally, suppose that there exists a mapping : Letk : n × k × m → n be the vector field defined by (2.7)-(2.9) for certain locally Lipschitz function p : + → [0, 1] that satisfies p(s) = 1 for all s ≥ b and p(s) = 0 for all s ≤ a. Let q : → + be a continuously differentiable function with q(s) = 1 for s ≤ 1 and sq(s) ≤ K for s ≥ 1, where K ≥ 1 is a constant. Let ψ : n → [1, +∞) be a smooth function that satisfies implication (2.10). Let N > 0 be an integer and T s > 0, σ > 0 be constants so that (2.11) holds.
Then for every partition , the solution of (2.18) and (2.19) with (2.12)-(2.16) and v(t) = u(t) + a 2 (θ (t)), for t ≥ 0, (2.22) where (2.21) is an exact prediction of a 1 (x(t + τ ))) and approximate predictors ((2.15) and (2.16) provides an approximate prediction of k(x(t + τ ))). Therefore, Theorem 2.4 generalises the results in Karafyllis and Krstic (2012a) and the result of Theorem 2.2. However, as remarked above the result of Theorem 2.4 is simple exponential attractivity for the closed-loop system. The existence of functions a 1 : n → l , a 2 : l → m satisfying the assumptions of Theorem 2.4 is a restrictive assumption, which can be verified in certain cases (see Example 4.2).

Proofs of main results
We start with the proof of Theorem 2.2. Proof of Theorem 2.2: We first notice that the following inequality holds for all (z, w, u (3.1) using definition (2.9) and noticing that p(V (z)) = 1, we conclude that (3.1) holds.

Concluding remarks
In this work, we have shown that the ISP-O-P-DFC control scheme can be applied to nonlinear systems with a compact absorbing set. The results guarantee the following properties for the closed-loop system even when the full state is not measured and when the measurement is sampled and possibly delayed: • Global asymptotic stability and exponential convergence for the disturbance-free case; • Robustness with respect to perturbations of the sampling schedule; • Robustness with respect to measurement errors.
The input and measurement delays can be arbitrarily large but must be both accurately known and constant.
To our knowledge, this is the first time that predictor feedback is proposed for systems with a compact absorbing set. While predictor feedback has been proposed and applied to nonlinear systems for which the solution mapping is known (see Karafyllis & Krstic, 2012a;Krstic, 2010), or to disturbance-free systems with state measurements (see Karafyllis & Krstic, 2012b), here we are in a position to guarantee important properties for the closed-loop system under the effect of various disturbances and without knowledge of the solution mapping. The results of the present work can be directly compared to the results in Karafyllis and Krstic (2013a) for globally Lipschitz systems. However, in this paper, we have not considered the effect of modelling errors as in Karafyllis and Krstic (2013a): this is a topic for future research.
More remains to be done for the class of systems that can be transformed to a nonlinear system with a compact absorbing set by means of a preliminary predictor feedback. Although Theorem 2.4 guarantees global exponential attractivity in the absence of measurement errors, additional assumptions must be employed for the global asymptotic stability and exponential convergence in the disturbancefree case.
An extension of the previous results to the case where the control is applied through a zero-order hold device is also an open problem and it is under investigation by the authors.