The first subject of study in this report is to establish sufficient (and for some cases, necessary) conditions to guaranteeuniformglobal asymptotic stability (UGAS) (cf. Definition 2.5) of the origin, for nonlinear ordinary differential equations (ODE)
˙
x=f(t, x) x(t◦) =:x◦ . (2.1) Most of the literature for nonlinear systems in the last decades has been devoted to time-invariant systems nonetheless, the importance of nonau- tonomous systems should not be underestimated; these arise for instance as closed-loop systems in nonlinear trajectory tracking control problems, that is, where the goal is to design a control inputu(t, x) for the system
˙
x=f(x, u) x(t◦) =:x◦ (2.2a)
y=h(x) (2.2b)
such that the output y follows asymptotically a desired time-varying refer- ence yd(t). For a “feasible” trajectory yd(t) =h(xd(t)) some “desired” state trajectoryxd(t), satisfying ˙xd=f(xd, u), system (2.2) in closed loop with the control inputu=u(x, xd(t), yd(t)) may be written as
˙˜
x= ˜f(t,x)˜ , x(t˜ ◦) = ˜x◦, (2.3a)
˜
y = ˜h(t,x)˜ (2.3b)
where ˜x=x−xd, ˜y=y−ydand ˜f(t, x) :=f(˜x+xd(t), u(˜x+xd(t), xd(t), yd(t)).
The so stated tracking control problem, applies to many physical systems, e.g.
mechanical and electromechanical, for which there is a large body of literature (see [39] and references therein).
Another typical situation in which closed-loop systems of the form (2.3) arise, is in regulation problems (that is, when the desired set-point (xd, yd) is constant) such that the open-loop plant is not stabilisable by continuous time- invariant feedbacksu=u(x). This is the case, for instance, of some driftless (e.g. nonholonomic) systems, ˙x=g(x)u. See e.g. [4, 22].
A classical approach to analyse thestabilityof thenonautonomoussystem (2.1) is to search a so-called Lyapunov function with certain properties (see e.g. [66, 20]). Consequently, for the tracking control design problem previously described one looks for so-called Control Lyapunov Function (CLF) for system (2.2) so that the control lawuis derived from the CLF (see e.g. [23, 47, 57]).
In general, finding an adequate LF or CLF is very hard and one has to focus on systems with specific structural properties. This gave rise to approaches such as the so-called integrator backstepping [23] and feedforwarding [30, 57].
The second subject of study in this chapter, is a specific structure of sys- tems, which are wide enough to cover a large number of applications while simple enough to allow criteria for stability which are easier to verify than finding an LF for the closed-loop system. These arecascadedsystems. We dis- tinguish between two problems: stabilityanalysisand controldesign. For the design problem, we do not offer a general methodology as in [23, 57] however, we show through different applications, that simple (from a mathematical viewpoint) controllers can be obtained by aiming at giving the closed-loop system a cascaded structure.
2.1.1 Stability Definitions
There are various types of asymptotic stability that can be pursued for time- varying nonlinear systems. As we shall see in this section, from arobustness viewpoint, the most useful areuniform(global) asymptotic stability anduni- form(local) exponential stability (ULES). For clarity of exposition and self- containedness let us recall a few definitions (cf. e.g. [19, 66, 10]) as we use them throughout the chapter.
Definition 2.1.A continuous function α :R≥0 →R≥0 is said to belong to classK if it is strictly increasing andα(0) = 0. It is said to be ofclassK∞ if moreover α(s)→ ∞ass→ ∞.
Definition 2.2.A continuous functionβ :R≥0×R≥0→R≥0is said to belong toclassKLif, for each fixeds β(ã, s)is of class Kand for each fixedr,β(r,ã) is strictly decreasing and β(r, s)→0as s→ ∞.
For system (2.1), we define the following.
Definition 2.3 (Uniform boundedness). We say that the solutions of (2.1) are uniformly (resp. globally) bounded (UB, resp., UGB) if there ex- ist a class K (resp. K∞) function α and a numberc > 0 such that
|x(t; t◦, x◦)| ≤α(|x◦|) + c ∀t ≥t◦. (2.4) Definition 2.4 (Uniform stability). The origin of system (2.1) is said to be uniformly stable (US) if there exist a constant r > 0 and γ ∈ K such that, for each(t◦, x◦) ∈R≥0 ×Br,
|x(t; t◦, x◦)| ≤γ(|x◦|) ∀t ≥t◦ . (2.5) If the bound (2.5) holds for all (t◦, x◦) ∈R≥0 ×Rn and with γ ∈ K∞, then the origin is uniformly globally stable (UGS).
Remark 2.1.
1. The formulation of uniform stability given above is equivalent to the clas- sical one, i.e., “the system (2.1) is uniformly stable in the sense defined above if and only if for each ε there exists δ(ε) such that |x◦| ≤ δ im- plies that |x(t)| ≤ ε for all t ≥t◦. Necessity becomes evident if we take δ(s) = γ−1(s) for sufficiently small1.
2. It is clear from the above that uniform global boundedness is a necessary condition for uniform global stability, that is, in the case that γ ∈ K∞we have that (2.5) implies (2.4). However, a system may be (locally) uniformly stable and yet, have unbounded solutions.
3. Another common characterization of UGS and which we use in some proofs is the following (see e.g. [23]): “the system is UGS if it is US and uniformly globally bounded (UGB)”. Indeed, observe that US im- plies that there exists γ ∈ K such that (2.5) holds. Then, using (2.4) and letting b ≤r, one can construct ¯γ∈ K∞ such that ¯γ(s)≥α(s) +cfor all s≥b >0 and ¯γ(s)≥α(s) for alls≤b≤r. Hence (2.5) holds with ¯γ.
Definition 2.5 (Uniform asymptotic stability). The origin of system (2.1) is said to be uniformly asymptotically stable (UAS) if it is uniformly stable and uniformly attractive, i.e., there exists r > 0 and for each σ > 0 there existsT >0, such that
|x◦| ≤r =⇒ |x(t;t◦, x◦)| ≤σ ∀t≥t◦+T . (2.6) If moreover the origin of system is UGS and the bound (2.6) holds for each r >0 then the origin is uniformly globally asymptotically stable (UGAS).
1 “Sufficiently small” here means thatsis taken to belong to the domain ofγ−1. Recall that in general,classKfunctions are not invertible onR≥0. For instance, tanh(|ã|)∈ Kbut tanh−1: (−1,1)→R≥0.
Remark 2.2 (class-KLcharacterization of UGAS).It is known (see, e.g., [10, Section 35] and [26, Proposition 2.5]) that the two properties characterizing uniform global asymptotic stability hold if and only if there exists a function β∈ KLsuch that, for all (t◦, x◦)∈R≥0×Rn, all solutions satisfy
|x(t;t◦, x◦)| ≤β(|x◦|, t−t◦) ∀t≥t◦. (2.7) The local counterpart is that the origin of system (2.1) is UAS if there exist a constantr >0 andβ ∈ KLsuch that for all (t◦, x◦)∈R≥0×Br.
Definition 2.6 (Exponential convergence). The trajectories of system (2.1) are said to be exponentially convergent if there existsr >0such thatfor eachpair of initial conditions (t◦, x◦)∈R≥0×Br, there exist γ1 andγ2>0 such that each solution x(t;t◦, x◦) of (2.1), satisfies
|x(t;t◦, x◦)| ≤γ1|x◦|e−γ2(t−t◦) . (2.8) The trajectories of the system are said to be globally exponentially convergent if the constantsγiexist for each pair of initial conditions(t◦, x◦)∈R≥0×Rn. Remark 2.3.Sometimes, in the literature, the dependence of γ1 and γ2 on the initial conditions is overlooked and, inappropriately, (uniform) global ex- ponential stability is concluded. Motivated by such abuse of terminology we introduce below the following definition of exponential stability in which we emphasize the uniformity with respect to initial conditions.
Definition 2.7 (Uniform exponential stability). The origin of the sys- tem (2.1) is said to be uniformly (locally) exponentially stable (ULES) if there exist constantsγ1, γ2 andr >0 such that for all (t◦, x◦)∈R≥0×Br
|x(t;t◦, x◦)| ≤γ1|x◦|e−γ2(t−t◦) ∀t≥t◦. (2.9) If for each r > 0 there exist γ1, γ2 such that condition (2.9) holds for all (t◦, x◦) ∈ R≥0×Br then, the system is said to be uniformly semiglobally exponentially stable (USGES)2.
Finally, the origin of system (2.1) is said to be uniformly globally expo- nentially stable (UGES) if there existγ1, γ2>0 such that (2.9) holds for all (t◦, x◦)∈R≥0×Rn.
2.1.2 WhyUniform Stability?
One of the main interests of theuniformforms of stability, isrobustnesswith respect to bounded disturbances. If the time-varying system (2.1) withf(t,ã)
2 This property has been called also, uniform exponential stability in any ball –cf.
[25].
locally Lipschitz uniformly int, is ULAS or ULES then the system is locally Input-to-State Stable (ISS); that is, for this system, there exist β ∈ KL, γ ∈ K and a number δ such that, for all t ≥t◦≥0 (see e.g. [19, Definition 5.2])
max {|x◦|, u ∞} ≤δ =⇒ |x(t; t◦, x◦, u)| ≤β(|x◦|, t −t◦) + γ( u ∞) . (2.10) This fact can be verified invoking [19, Lemma 5.4], and the converse Lya- punov theorems in [24, 26]. The importance of this implication is that, in particular, local ISS implies total stability, which can be defined as follows.
Definition 2.8 (Total stability3). The origin of of x˙ =f(t, x,0), is said to be totally stable if for the systemx˙ =f(t, x, u)small bounded inputsu(t, x) and small initial conditions x◦ = x(t◦) yield small state trajectories for all t≥t◦, i.e., if for eachε >0 there existsδ >0 such that
max{|x◦|, u ∞} ≤δ =⇒ |x(t;t◦, x◦, u)| ≤ε ∀t≥t◦≥0 . (2.11) In contrast to this, weaker forms of asymptotic stability for time-varying sys- temsdo notimply total stability. More precisely:
Proposition 2.1.Consider the system (2.1) and assume thatf(t,ã)is locally Lipschitz uniformly int, and the origin is UGS. The following conditions are notsufficient for total stability:
1. The origin is globally attractive
2. The trajectories of the system are exponentially convergent andf(t, x)is globally Lipschitz in x, uniformly int.
Proof . We present an interesting example of an UGS nonlinear time-varying system which satisfies items 1 and 2 of the proposition above; yet, is not totally stable.
Example 2.1.[44] Consider system (2.1) with
f(t, x) = −a(t)sgn(x) if |x| ≥a(t)
−x if |x| ≤a(t) (2.12) anda(t) = 1
t+ 1. This system has the following properties:
3 The definition provided here is a modern version of the notion of total stability, originally introduced in [29] and which is more suitable for the purposes of these notes.
1. The function f(t, x) is globally Lipschitz in x, uniformly in t and the system is UGS with linear gain equal to one i.e., with γ(s) =s.
2. For eachr >0 andt◦≥0 there exist strictly positive constantsκandλ such that for allt≥t◦ and all|x(t◦)| ≤r
|x(t)| ≤κ|x(t◦)|e−λ(t−t◦) (2.13) 3. The origin isnottotally stable. Furthermore, there always exist a bounded (arbitrarily small) additive perturbation, and (t◦, x◦) ∈ R≥0×Rn such that the trajectories of the system grow unboundedlyas t→ ∞.
The proof of these properties is provided in [44]. See also [35] for examples of linear time-varying systems proving the claim in Proposition 2.1.
This lack of total stability for GAS (but not UGAS) and LES (but not ULES) nonautonomous systems, added to the unquestionable interest on total stability in time-varying systems arising in practical applications, is our main motivation to study sufficient conditions that guarantee UGAS and ULES for nonlinear nonautonomous systems.
As it has been mentioned before the stability analysis problem and hence, control design for time-varying systems is in general very hard to solve. By restricting the class of NLTV systems to cascades, we can establish simple- to-verify conditions for UGAS and UGES. The importance of these results is evident from their application to specific control design problems that we address.