stabilization of nonlinear time-varying systems: a ...

Jrl Syst Sci & Complexity (2009) 22: 683–696

STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS: A CONTROL LYAPUNOV FUNCTION APPROACH∗ Zhongping JIANG · Yuandan LIN · Yuan WANG

Received: 18 June 2009 c

2009 Springer Science + Business Media, LLC Abstract This paper presents a control Lyapunov function approach to the global stabilization problem for general nonlinear and time-varying systems. Explicit stabilizing feedback control laws are proposed based on the method of control Lyapunov functions and Sontag’s universal formula. Key words Control Lyapunov functions (clf), global stabilization, nonlinear time-varying systems.

1 Introduction The last two decades have witnessed tremendous progress in the field of nonlinear control. One of the most powerful tools is the approach of control Lyapunov functions (clf)[1−2] which has been employed to address various issues tied to nonlinear control systems, such as nonlinear stabilization[3] and adaptive control[4−5]. However, most of the past work by the clf approach focused on time-invariant systems. The main purpose of this paper is to develop some tools based on control Lyapunov functions for stabilizing nonlinear time varying systems. The problem of stabilization of time varying systems has attracted much attention; see for instance, [6–10], and other work cited therein, where some recursive designs such as backstepping approach were adopted. Our work will be based on the clf approach. Essentially, we will generalize the well-known universal formula[2] to provide explicit feedback control laws that stabilize a general time-varying nonlinear system, under the assumption that a control Lyapunov function is known. It should be mentioned that, when applying the control Lyapunov function approach to time-varying systems, new challenging issues arise, mainly due to the presence of the time parameter t in the Lyapunov functions. In contrast to the time invariant case, a feedback law resulted from the universal formula may fail to stabilize the system even if a uniform control Lyapunov function is given. In this preliminary work, we will define uniform control Lyapunov functions and the associated concept of small control property for time varying systems. We will then provide several sufficient conditions under which the feedback laws given by the universal formula render the closed-loop system uniformly globally asymptotically stable. Zhongping JIANG Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA. Email: [email protected]. Yuandan LIN · Yuan WANG Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA. Email: [email protected]; [email protected]. ∗ This work has been supported in part by National Science Foundation under Grants Nos. ECS-0093176, DMS0906659, and DMS-0504296, and in part by National Natural Science Foundation of China under Grant Nos. 60228003 and 60628302.

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ZHONGPING JIANG · YUANDAN LIN · YUAN WANG

By the control Lyapunov function approach, we also consider a problem related to some open questions raised in [8]. The general form of the problems proposed in [8] was that if a system x˙ = f (x) + g(x)u is stabilizable by a feedback law k(x), can the system x˙ = f (x) + p(t)g(x)u be stabilized when p(·) possesses some persistent excitation property? In [8], the problem was discussed and solved for several classes of systems, which may lead to a solution to the more general type of systems in strict feedback forms. For some related work based on the condition of persistent excitation, please also see [7] and [11]. We will consider in this work the problem from the clf point of view. We first show by a simple example that not every stabilizing feedback law for a system x˙ = f (x) + g(x)u will stabilize the system x˙ = f (x) + p(t)g(x)u when a persistently exciting p(t) is presented. Our main concern is then whether the feedback law derived from the universal formula for a system is still stabilizing when p(t) multiplicatively appears in the control channel. A natural tool in our method is the so called weakly persistently exciting functions introduced in [12]. A weakly persistently exciting function still has the main feature that the energy of the function over any time interval of a given length maintains at least a certain level, but the function is allowed to take negative values. Because of the uncertainty of the sign of such functions involved, the proofs of the results are more complicated than one would expect, especially when it involves with manipulations of inequalities. The results we obtained are still preliminary. On the other hand, the method by the combination of universal formula and weakly persistently exciting functions may lead to more applications. We refer the interested reader to some recent contributions[6−8,12−17] for additional tools and results on stability and stabilization for nonlinear time-varying systems. The rest of the paper is organized as follows. In Section 2 we review the notion and properties of uniform global asymptotic stability and Sontag’s universal formula. In Section 3 we present some extension of control Lyapunov functions to deal with nonlinear time-varying systems. Several new results on time-varying stabilization will be developed in this section. In Section 4 we close the paper with brief concluding remarks.

2 Preliminaries Before tackling the stabilization problem for general nonlinear time-varying systems, we first review the notion and properties of uniform global asymptotic stability that will be needed for the development of our stabilizing control schemes. Then, in Subsection 2.2, we recall the concept of control Lyapunov function and Sontag’s universal formula for time-invariant control systems. Throughout the paper, K is the class of continuous functions R+ → R+ which is zero at zero and strictly increasing; K∞ is a subset of the class-K functions which are unbounded; KL stands for the class of functions R+ × R+ → R+ which are of class K on the first argument and decreases to zero on the second argument. 2.1 Uniform Global Asymptotic Stability Consider a nonlinear time-varying system of the form: x(t) ˙ = f (t, x(t)),

t ∈ R≥0 ,

x ∈ Rn ,

(1)

where f : R≥0 × Rn → Rn is locally Lipschitz. Such a system is uniformly globally asymptotically stable (UGAS) at the origin if there exists some β ∈ KL such that for every solution x(·, t0 , x0 ) of (1) with the initial condition x(t0 ) = x0 , it holds that |x(t + t0 , t0 , x0 )| ≤ β(|x0 | , t),

∀ t ≥ 0.

STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS

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The following result provides an equivalent Lyapunov characterization of UGAS; see, for instance, [18, Theorem 49.3] and [19, Proposition 16]. Proposition 2.1 A system as in (1) is UGAS if and only if there exists a C ∞ Lyapunov function V : R≥0 × Rn → R≥0 such that • for some α, α ∈ K∞ , it holds that ∀ t ≥ 0, ∀ ξ ∈ Rn ;

α(|ξ|) ≤ V (t, ξ) ≤ α(|ξ|),

(2)

• for some positive definite function α, it holds that ∂V ∂V (t, ξ) + f (t, ξ) ≤ −α(|ξ|), ∂t ∂ξ

∀ t ≥ 0, ∀ ξ ∈ Rn .

(3)

Note that to get the existence of a C ∞ function for system (1), one may relax the Lipschitz condition of f by assuming that f (t, ξ) is locally Lipschitz on R≥0 ×(Rn \ {0}) and is continuous everywhere[19]. Observe that for a function V satisfying (2), property (3) is equivalent to the existence of a positive definite function α e(·) such that ∂V ∂V (t, ξ) + f (t, ξ) ≤ −e α(V (t, ξ)), ∂t ∂ξ

∀ t ≥ 0, ∀ ξ ∈ Rn .

In recent work [12] and [16], it was shown that, for the sufficiency part, condition (3) can be replaced by a weaker condition, as shown below. For τ > 0 and σ > 0, let Pτ,σ denote the collection of measurable, locally essentially bounded functions p : R → R≥0 such that Z

t

t+τ

p(s) ds ≥ σ,

∀t ≥ 0.

(4)

Proposition 2.2[16] A system as in (1) is UGAS if there exists a C 1 Lyapunov function satisfying (2) such that for some p ∈ Pτ,σ , the following holds: ∂V ∂V (t, ξ) + f (t, ξ) ≤ −p(t)α(|ξ|) ∂t ∂ξ

(5)

for all t ≥ 0 and all ξ ∈ Rn . In [16], this result was stated for periodic systems and was proved by establishing equivalence among different types of Lyapunov functions. Below we provide a more direct proof. Suppose there is some Lyapunov function V satisfying properties (2) and (5). By Proposition 13 in [20], one sees that there exists some C 1 , proper, and positive definite function ρ such that ρ′ (V )α ◦α−1 (V ) ≥ ρ(V ), and hence, for the Lyapunov function W given by W := ρ ◦ V , it holds that ∂W ∂W (t, ξ) + f (t, ξ) ≤ −p(t)W (t, ξ), ∀ t ≥ 0, ∀ ξ. (6) ∂t ∂ξ It then follows that (see also [7]) W (t + t0 , x(t + t0 , t0 , x0 )) ≤ W (t0 , x0 )e−

R

t+t0 t0

p(s) ds

,

∀ t ≥ 0, ∀ t0 , ∀ x0 .

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Let k denote the largest integer such that kτ ≤ t. By (4), one sees that Z t0 +t Z t0 +kτ t p(s) ds ≥ kσ ≥ − 1 σ. p(s) ds ≥ τ t0 t0

(7)

Therefore, t

W (t + t0 , x(t + t0 , t0 , x0 )) ≤ W (t0 , x0 )e−( τ −1)σ ≤ W (t0 , x0 )M e−σt/τ ,

(8)

where M = eσ . The desired UGAS property follows directly readily. In the above discussions, one has used the linear property of W in the decay estimation (6). For a Lyapunov function V satisfying such an exponential decay estimation, the condition of p being nonnegative can be further relaxed as shown in [12]. For τ > 0 and σ > 0, let wPτ,σ denote the set of all measurable, locally essentially bounded functions p such that • for some c > 0, it holds that p(t) ≥ −c, a.e.; Z t+τ • p(s) ds ≥ σ for all t ≥ 0. t

Observe that wP is a larger class than P since a function in wP is not required to be nonnegative. Also notice that any p ∈ Pτ,σ yields a function p(t) − c in wPτ,σ′ for any given 0 ≤ c < σ/τ and σ ′ = σ − cτ . For instance, sin2 t − 14 ∈ wPτ,σ with τ = π and σ = π/4. Since the function takes negative values in some intervals, sin2 t − 14 6∈ Pτ,σ for any τ, σ > 0. The next result is a variant of [12, Theorem 10]. Proposition 2.3 A system as in (1) is UGAS if there exists a C 1 Lyapunov function V satisfying (2) such that for some p ∈ wPτ,σ , the following holds: ∂V ∂V (t, ξ) + f (t, ξ) ≤ −p(t)V (t, ξ) ∂t ∂ξ

(9)

for all t ≥ 0 and all ξ ∈ Rn . The proof of Proposition 2.3 follows essentially the same lines as the proof from (6) to (8), except one needs to note that for p ∈ wPτ,σ , Z

t0 +t

p(s) ds =

t0

Z

t0 +kτ

p(s) ds +

t0

Z

t0 +t

t0 +kτ

p(s) ds ≥ kσ − cτ ≥

t τ

− 1 σ − cτ,

where k still has the same meaning as in (7). Hence, the decay property (9) implies that V (t + t0 , x(t + t0 , t0 , xo )) ≤ V (t0 , xo )M1 e−σt/τ , where M1 = eσ+cτ . 2.2 Control Lyapunov Functions Recall that a C ∞ function V (x) is said to be a (global) clf for a time-invariant affine-incontrol system x˙ = f (x) + g(x)u, if V (x) is positive definite and proper, and satisfies the following implication inf a(ξ) + b(ξ)u < 0, ∀ξ 6= 0 u


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with a(ξ) = ∂V∂ξ(ξ) f (ξ) and b(ξ) = ∂V∂ξ(ξ) g(ξ). We refer the interested reader to [3] for a tutorial on control Lyapunov functions. In the seminal work [2], a universal formula was established for the design of feedback stabilizers when a clf is given. More precisely, it is shown that the feedback law u = κ(a(ξ), b(ξ)) given by the universal formula is of class C ∞ on Rn \ {0} and stabilizes the system, where the function κ is defined by  √  a + a2 + b 4 − , if b 6= 0, κ(a, b) = (10) b  0, if b = 0.

Furthermore, if the clf function V satisfies the small control property, i.e., for any ε > 0, there exists some δ > 0 such that whenever 0 < |ξ| < δ, there exists some |u| < ε such that a(ξ) + b(ξ)u < 0, then the feedback given by u = κ(a(ξ), b(ξ)) is almost smooth, where an almost smooth function means a function defined on Rn which is smooth away from 0 and continuous everywhere. In the next section, we generalize this method to general time-varying systems. For a system given by x(t) ˙ = F (t, x(t)) and a C 1 function V : R≥0 × Rn → Rn , we let d V (t, x(t)) along the solution x(t), that is, V˙ (t, x(t)) denote dt ∂V ∂V V˙ (t, x(t)) = (t, x(t)) + (t, x(t))F (t, x(t)). ∂t ∂x

3 Control Lyapunov Functions for Time-Varying Systems To simplify the presentation, we only consider a single-input time-varying control system: x(t) ˙ = f (t, x(t)) + g(t, x(t))u(t),

(11)

where f and g are smooth maps from R+ × Rn to Rn . The admissible class of input functions is composed of measurable, locally essentially bounded functions. A function ψ : R+ × Rn → R is said to be almost smooth if it is continuous everywhere and is smooth on R+ × (Rn \ {0}). Definiton 3.1 System (11) is said to be uniformly stabilizable if there exists some feedback law k : R+ × Rn → R which is smooth on R+ × (Rn \ {0}) such that the resulted closed-loop system x(t) ˙ = f (t, x(t)) + g(t, x(t))k(t, x(t)) is UGAS at the origin. Definiton 3.2 A C ∞ function V is said to be a clf for (11), if it satisfies property (2) and the following: inf {a(t, ξ) + b(t, ξ)u} < 0, ∀ ξ 6= 0, (12) u

∂V ∂t

∂V ∂ξ

where a(t, ξ) = (t, ξ) + f (t, ξ) and b(t, ξ) = ∂V ∂ξ g(t, ξ). Moreover, V is said to be a uniform control Lyapunov function (uniform-clf) for (11), if, instead of (12), it satisfies a stronger property inf a(t, ξ) + b(t, ξ)u ≤ −α(|ξ|), (13) u

where α is a positive definite function.

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Remark 3.3 It is of interest to note that the condition (12) is equivalent to requiring b(t, ξ) = 0

⇒

a(t, ξ) < 0,

∀ ξ 6= 0.

(14)

Clearly, by means of Proposition 2.1, the existence of a uniform-clf is a necessary condition for a system as in (11) to be uniformly stabilizable. Namely, Lemma 3.4 If a system as in (11) is uniformly stabilizable, then the system admits a uniform-clf. It is interesting to point out that the existence of a clf does not necessarily imply that the system can be uniformly stabilized. For instance, for the planar system x˙ 1 = −

x1 + x2 , 1 + t2

x˙ 2 = x2 u,

V (x1 , x2 ) = x21 + x22 is a clf. But the system cannot be uniformly stabilized. Indeed, for any feedback law u = k(t, ξ), the x1 -component of the trajectory with the initial condition x(t0 ) = (x1o , 0) is a solution of the one-dimensional system x˙ 1 = −x1 /(1 + t2 ), x1 (t0 ) = x1o which is not a UGAS system. Even if a system admits a uniform-clf, the universal formula does not necessarily yield a feedback law that guarantees the UGAS property of the closed-loop system. Indeed, for a general system with the feedback given by the universal formula, the following holds for the closed-loop solutions: p V˙ (t, x(t)) = − a2 (t, x(t)) + b4 (t, x(t)). (15)

Without knowing if the function a2 (t, ξ) + b4 (t, ξ) dominates a positive definite function of V , one cannot determine if the closed-loop system is uniformly asymptotically stable. For instance, for the simple system x+u , x˙ = √ 1 + t2 2

√ the function V (x) = x2 /2 is a uniform-clf satisfying that V˙ (x(t)) = x (t)+x(t)u(t) . The universal 1+t2 formula leads to r r 2 1 2V (x(t)) 1 x (t) V˙ (x(t)) = − √ 1+ =−√ 1+ . 1 + t2 1 + t2 1 + t2 1 + t2

It can be verified that the closed-loop system is not UGAS. In this work we will investigate sufficient conditions under which the feedback laws given by the universal formula render the closed-loop system the UGAS property. As a preparation, we first extend the result regarding the small control property to time varying systems. Definition 3.5 A clf V is said to satisfy the small control property (SCP) if for every ε > 0, every t0 > 0, there exists some δ > 0 such that for all 0 < |ξ| < δ and all t ∈ (t0 − δ, t0 + δ), there exists some |µ| < ε such that a(t, ξ) + b(t, ξ)µ < 0. Note that in Definition 3.5, the choice of δ is allowed to be dependent on t0 (i.e., the choice of δ is not required to be uniform in t0 ). Instead of saying V satisfies the small control property, we will sometimes say that the pair (a(t, ξ), b(t, ξ)) satisfies the small control property. Lemma 3.6 Let V be a clf for a system as in (11). If V satisfies the small control property, then the feedback function given by the universal formula u = k(t, ξ) := κ(a(t, ξ), b(t, ξ)) is almost smooth. The proof of Lemma 3.6 will be given in Appendix A. The next result follows directly from Lemma 3.6, Proposition 2.2, and Equation (15).


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Proposition 3.7 Suppose a system as in (11) admits a clf V that satisfies the small control property. Assume further that p a2 (t, ξ) + b4 (t, ξ) ≥ p(t)α(|ξ|), ∀ξ ∈ Rn , ∀ t ≥ 0 (16)

for some p ∈ Pτ,σ and some positive definite function α, then the closed-loop system under the almost smooth feedback u = κ(a(t, x), b(t, x)) is UGAS. As a side remark, we would like to point out that if the property p ∈ Pτ,σ is relaxed into the following weaker property Z ∞

0

p(s) ds = ∞,

(17)

then one will achieve a weaker stability. Namely, one has the following (see [14, Proposition 3.3]). Proposition 3.8 Suppose a system as in (11) admits a clf V that satisfies the small control property. Assume further that (16) holds for some positive definite functions α and p satisfying (17). Then, the closed-loop system under the almost smooth feedback u = κ(a(t, x), b(t, x)) is semi-uniformly GAS (see [14]), that is, for some β ∈ KL and some nonnegative function ρ, it holds that t |x(t + t0 )| ≤ β |x0 | , , ∀ t ≥ 0. 1 + ρ(t0 ) 3.1 Controller Design with p-CLF In applications, a clf satisfying condition (14) can be hard to find. Often, the following holds, instead of (14): b(t, ξ) = 0 ⇒ a(t, ξ) ≤ −p(t)e a(ξ),

∀ ξ 6= 0,

where e a is positive definite and p ∈ Pτ,σ . Such a clf is called a p-clf. Proposition 3.9 Consider a system as in (11). Suppose that there exists a C ∞ function V satisfying (2) and the following: a(t, ξ) = p(t)a0 (t, ξ) − a1 (t, ξ),

b(t, ξ) = p(t)b0 (t, ξ)

(18)

with the conditions (a) a1 (t, ξ) ≥ 0; (b) b0 (t, ξ) = 0 ⇒ a0 (t, ξ) < 0 for all ξ 6= 0; (c) p ∈ Pτ,σ for some τ > 0 and σ > 0; and (d) there is some continuous, positive definite function α3 such that |a0 (t, ξ)| + b20 (t, ξ) ≥ α3 (V (t, ξ)).

(19)

Assume further that (a0 (t, ξ), b0 (t, ξ)) satisfies the small control property. Then, the feedback law by the universal formula u = κ(a0 (t, x), b0 (t, x)) is almost smooth, and the corresponding closed-loop system is UGAS. Proof As it can be directly checked, the solutions of the closed-loop system satisfy q V˙ (t, x(t)) ≤ −p(t) a20 (t, x(t)) + b40 (t, x(t)) ≤ −p(t)α3 (V (t, x(t)))/2

690


With the help of Proposition 2.2 and Lemma 3.6, Proposition 3.9 follows readily. Remark 3.10 Suppose that the function p in property (c) of Proposition 3.9 is bounded, that is, 0 ≤ p(t) ≤ M for some constant M < ∞, then condition (19) can be relaxed into |a0 (t, ξ)| + b20 (t, ξ) + a1 (t, ξ) ≥ α3 (V (t, ξ)). To see this, note that with u = κ(a0 (t, x), b0 (t, x)), one has q V˙ (t, x(t)) ≤ −p(t) a20 (t, x(t)) + b40 (t, x(t)) − a1 (t, ξ) q p(t) ≤− a20 (t, x(t)) + b40 (t, x(t)) + a1 (t, ξ) M p(t) α3 (V (t, ξ)) (|a0 (t, ξ)| + b20 (t, ξ) + |a1 (t, ξ)|) ≤ −p(t) . ≤− 2M 2M The stability property then follows as in the proof of Proposition 3.9. In Proposition 3.9, a common function p(t) ∈ Pτ,σ is required for the pair (a(t, ξ), b(t, ξ)). This can be made slightly more flexible as in what follows. By using transformations on the input variable u of the form u = q0 (t)v, a system as in (11) is changed to x(t) ˙ = f (t, x(t)) + q0 (t)g(t, x(t))v(t). Proposition 3.9 can be modified as follows. Corollary 3.11 For a system as in (11). Suppose that there exists a C ∞ function V satisfying (2) and the following: a(t, ξ) = p(t)a0 (t, ξ) − a1 (t, ξ),

b(t, ξ) = p0 (t)b0 (t, ξ)

with the conditions: (a) a1 (t, ξ) ≥ 0; (b) there exists some smooth function q0 (t) such that p0 (t)q0 (t) = p(t); (c) b0 (t, ξ) = 0 ⇒ a0 (t, ξ) < 0 for all ξ 6= 0; (d) p ∈ Pτ,σ for some τ > 0, σ > 0; and (e) there is some continuous, positive definite function α3 such that (19) holds for a0 (t, ξ) and b0 (t, ξ). Assume further that (a0 (t, ξ), b0 (t, ξ)) satisfies the small control property. Then, the feedback law by the modified universal formula u = q0 (t)κ(a0 (t, x), b0 (t, x)) is almost smooth, and the corresponding closed-loop system is UGAS. Example 3.12 The function V (ξ) = ξ 2 /2 is a clf for the system x˙ = u sin t as in Corollary 3.11 with p0 (t) = q0 (t) = sin t, a0 (t, ξ) = a1 (t, ξ) = 0 (Note that p0 (t) = sin t alone is not a class Pτ,σ function). A direct application of Corollary 3.11 yields the time-varying feedback law u = −x sin t for the closed-loop system to be UGAS. 3.2 Stabilization with Persistently Exciting Functions in Control Channels Suppose a system as in (11) is uniformly stabilizable. Let p ∈ Pτ,σ . An interesting question is whether or not there is a feedback law u = k1 (t, x) that uniformly stabilizes the system x˙ = f (t, x) + p(t)g(t, x)u.

(20)


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In this section, we consider the specific question that if a system as in (11) admits a clf, can the universal formula be used to find a stabilizing feedback law for (20)? Example 3.13 It is easily seen that u = −4x stabilizes both systems x˙ = x + u and x˙ = x3 + x2 u. However, the same feedback stabilizes uniformly the system x˙ = x + (sin2 t)u, but not the system x˙ = x3 + (sin2 t)x2 u. In fact, the latter system cannot be stabilized by any linear feedback u = −kx for any k > 0. We now assume that (11) admits a clf V satisfying (2) and the following: inf {a(t, ξ) + b(t, ξ)u} < 0, u

∀ ξ 6= 0,

∂ ∂ ∂ where a(t, ξ) = ∂t V (t, ξ) + ∂ξ V (t, ξ)f (t, ξ) and b(t, ξ) = ∂ξ V (t, ξ)g(t, ξ). Assume further that the feedback law given by the universal formula u = κ(a(t, x), b(t, x)) stabilizes the system (11) uniformly, that is, for some positive definite function α, p a2 (t, ξ) + b4 (t, ξ) ≥ α(|ξ|). (21)

Without loss of generality, because of property (2), (21) can be re-formulated as p a2 (t, ξ) + b4 (t, ξ) ≥ α b(V (t, ξ))

for some positive definite function α b. By Lemmas 11 and 12 in [20], one sees that, for each µ > 0, there exists a proper, positive definite function ρ which is smooth on R \ {0} and C 1 everywhere such that ρ′ (V (t, ξ))b α(V (t, ξ)) ≥ µρ(V (t, ξ)), and hence, with W = ρ ◦ V , it holds that inf {e a(t, ξ) + eb(t, ξ)u} ≤ −µW (t, ξ), u

(22)

∂ ∂ ∂ where e a(t, ξ) = ∂t W (t, ξ) + ∂ξ W (t, ξ)f (t, ξ) and eb(t, ξ) = ∂ξ W (t, ξ)g(t, ξ). Hence, we get the following lemma. Lemma 3.14 Suppose a system as in (11) admits a smooth clf V satisfying (2). Assume further that (21) holds for the function V . Then, for any µ > 0, there exists a C 1 clf W , smooth on R+ × (Rn \ {0}), for which (22) holds. Consider p ∈ Pτ,σ for a system as in (20). Replacing p by M p and u by v := u/M for some M > 0 if necessary, one may assume the following condition on the function p: Z t+τ (p(s) − 1) ds ≥ σ, ∀ t ≥ 0, t

that is, (p − 1) ∈ wPτ,σ . We are now ready to state the following theorem, whose proof is postponed to Appendix B. Theorem 1 Consider the problem of stabilizing a system as in (20). Assume the following: (a) the function p satisfies that 0 ≤ p(t) ≤ M for all t ≥ 0 and some M > 0 and that p − 1 ∈ wPτ,σ ; (b) the system (11) admits a C 1 clf V , smooth on R≥0 × (Rn \ {0}), satisfying (2) and p a2 (t, ξ) + b4 (t, ξ) ≥ cV (t, ξ), (23) √ √ where c = max 2 M, 2 + 2 ; and

692


(c) a(t, ξ) ≤ V (t, ξ). Then, the feedback law given by the universal formula u = κ(a(t, ξ), b(t, ξ)) is smooth away from 0 and it globally and uniformly stabilizes the system (20). Furthermore, if the pair (a, b) satisfies the small control property, then the feedback law is almost smooth. Remark 3.15 For the conditions in Theorem 1, we note the following: 1) condition (a) amounts to requiring the function p appeared in (20) to be bounded; 2) condition (b) amounts to requiring system (11) to admit a clf satisfying (21) for some positive definite function α (which is always the case when f and g are time invariant); 3) one may always modify the clf V so that (23) holds (c.f. Lemma 3.14); and 4) condition (c) requires that for the new clf for which (23) holds, the term a(t, x) is bounded above by V . As one can see, condition (c) is the only real restriction required for Theorem 1. Notice that, as suggested by Example 3.13, some conditions on the upper bound of the term a(t, ξ) should be imposed to guarantee the stabilizability. In the special case when f and g are both time invariant, Theorem 1 states that under condition (a)–(c), the feedback law given by the universal formula for the system x˙ = f (x) + g(x)u also stabilizes the system x˙ = f (x) + p(t)g(x)u. One of the technical issues in the proof of Theorem 1 is that when working with a function q(·) of class wP, the uncertainty of the sign of q(·) makes it hard to manipulate with inequalities. In the special case when (11) admits a clf V for which a(t, ξ) does not change sign, the conditions of Theorem 1 can be significantly simplified, as stated below. The proof of the following proposition will be postponed to Appendix C. Proposition 3.16 Let p ∈ Pτ,σ for some τ > 0 and σ > 0. Assume that 0 ≤ p(t) ≤ M for some M > 0. If system (11) admits a smooth clf V such that • for some positive definite function α, (21) holds; and • a(t, ξ) ≤ 0 for all t, ξ, then the feedback law u = κ(a(t, ξ), b(t, ξ)) globally and uniformly stabilizes the corresponding system (20). Note that Proposition 3.16 means that if the term a(t, ξ) is nonpositive and if the persistently excited function p(·) is bounded, then the system (20) can be stabilized by the universal formula. The next result deals with the case when a(t, ξ) is always nonnegative. Its proof can be found in Appendix D. Proposition 3.17 Let p ∈ Pτ,σ for some τ > 0 and σ > 0. Assume that system (11) admits an almost smooth clf V such that p • a2 (t, ξ) + b4 (t, ξ) ≥ V (t, ξ); and • 0 ≤ a(t, ξ) ≤ cV (t, ξ) for some c ≥ 0.

Let M > 0 be such that (M p(·) − c) ∈ wPτ,σ . Then, the feedback law u = M κ(a(t, ξ), b(t, ξ)) globally and uniformly stabilizes the system. For the two systems in Example 3.13, let V (x) = x2 /2. It is readily seen that the first system x˙ = x + (sin2 t)u satisfies all conditions of Proposition 3.17; while the property a(t, ξ) ≤ cV (t, ξ) fails for the second system x˙ = x3 + (sin2 t)x2 u.


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Remark 3.18 In Theorem 1 and Propositions 3.16 and 3.17, the function p in (20) is required to be a class Pτ,σ function. As in the discussion for Corollary 3.11, one may relax this requirement slightly by converting the system to x˙ = f (t, x) + p1 (t)g(t, x)v

(24)

with p1 (t) = p(t)q0 (t) and u = q0 (t)v. Suppose system (24) satisfies the requirements of Theorem 1 or Proposition 3.16, then the modified feedback u = q0 (t)κ(a(t, ξ), b(t, ξ)) globally uniformly stabilizes the system (20).

4 Concluding Remarks In this paper, we have proposed a control Lyapunov function approach to the global stabilization problem for general nonlinear time-varying systems. Sufficient conditions are given under which explicit feedback laws generated by the universal formula can be found to stabilize globally and uniformly the system in question. Topics for future research include, among many others, feedback stabilization of time-varying systems with restricted inputs, and adaptive control of nonlinear time-varying systems with unknown parameters.

Appendix A Proof of Lemma 3.6 Let V be a clf for system (11) that satisfies the small control property. The proof of Theorem 1 in [2] shows that the function κ(a, b) given by (10) is analytic on the set S := {(a, b) ∈ R2 : b 6= 0 or a < 0}. Consequently, the feedback function k(t, ξ) := κ(a(t, ξ), b(t, ξ)) is smooth on the set R+ × (Rn \ {0}). Below we show that the function k(t, ξ) is continuous on the set R+ × {0}. Let t0 ≥ 0 and ε > 0 be given. We will show that there exists some δ > 0 such that |k(t, ξ)| < ε,

∀ |t − t0 | < δ, ∀ |ξ| < δ.

(25)

Let B1 = {(t, ξ) : b(t, ξ) 6= 0}. Since k(t, ξ) = 0 whenever b(t, ξ) = 0, it is enough to find δ such that (25) holds at the points (t, ξ) ∈ B1 . By the small control property, there is some δ > 0 such that for every |ξ| < δ and every t ∈ (t0 − δ, t0 + δ), there is some |µ| < ε/3 such that a(t, ξ) + b(t, ξ)µ < 0.

(26)

Without loss of generality, one may also assume (by continuity and the fact that b(t0 , 0) = 0) that δ > 0 is chosen so that |b(t, ξ)|
0 be a constant as stated in Proposition 3.17. With u = M κ(a(t, ξ), b(t, ξ)), one has p a(t, ξ) + p(t)b(t, ξ)κ(a(t, ξ), b(t, ξ)) = a(t, ξ) − M p(t) a(t, ξ) + a2 (t, ξ) + b4 (t, ξ) p ≤ cV (t, ξ) − M p(t) a2 (t, ξ) + b4 (t, ξ) ≤ −(M p(t) − c)V (t, ξ).

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Thus, the following holds for the closed-loop system of (20): ∂V ∂V (t, ξ) + (f (t, ξ) + M p(t)g(t, ξ)κ(t, ξ)) ≤ −(M p(t) − c)V (t, ξ). ∂t ∂ξ By assumption, (M p(·) − c) ∈ wPτ,σ . One concludes that the system is ugas. The proof of Proposition 3.17 is thus completed. References [1] Z. Artstein, Stabilization with relaxed controls, Nonlinear Analysis, Theory, Methods & Applications, 1983, 7: 1163–1173. [2] E. D. Sontag, A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization, Systems & Control Letters, 1989, 13: 117–123. [3] E. D. Sontag, Mathematical Control Theory (second edition), Springer-Verlag, New York, 1998. [4] M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. [5] L. Praly, G. Bastin, J. B. Pomet, and Z. P. Jiang, Adaptive stabilization of nonlinear systems, in P.V. Kokotović, editor, Foundations of Adaptive Control, Springer-Verlag, Berlin, 1991, 347–433. [6] I. Karafyllis and J. Tsinias, A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization, SIAM Journal on Control and Optimization, 2003, 42(4): 936–965. [7] A. Loria and E. Panteley, Uniform exponential stability of linear time-varying systems: Revisited, Systems & Control Letters, 2002, 47: 13–24. [8] A. Loria, A. Chaillet, G. Besancon, and Y. Chitour, On the PE stabilization of time-varying systems: Open questions and preliminary answers, in Proc. 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, 2005. [9] J. Tsinias, Backstepping design for time-varying nonlinear systems, Systems & Control Letters, 2000, 39: 219–227. [10] L. L. Xie and L. Guo, How much uncertainty can be dealt with by feedback, IEEE Trans. Automat. Contr., 2000, 45(12): 2203–2217. [11] F. Mazenc and D. Nesic, Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem, Math. Control, Signals Syst., 2007, 19: 151–182. [12] F. Mazenc and M. Malisoff, Further results on Lyapunov functions for slowly time-varying systems, Math. Control, Signals Syst., 2007, 19(1): 1–21. [13] D. Aeyels, R. Sepulchre, and J. Peuteman, Asymptotic stability conditions for time-variant systems and observability: Uniform and non-uniform criteria, Mathematics of Control, Signals, and Systems, 1998, 11: 1–27. [14] D. Z. Cheng, Y. Lin, and Y. Wang, On nonuniform and semi-uniform input-to-state stability for time varying systems, in Proc. 16th IFAC World Congress, Prague, 2005. [15] T. C. Lee and Z. P. Jiang, A generalization of Krasovskii-LaSalle theorem for nonlinear timevarying systems: Converse results and applications, IEEE Trans. Automatic Control, 2005, 50: 1147–1163. [16] F. Mazenc and M. Malisoff, Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems, Automatica, 2005, 41: 1973–1978. [17] A. R. Teel, E. Panteley, and A. Loria, Integral characterizations of uniform asymptotic and exponential stability with applications, Math. Control, Signals Syst., 2002, 15: 177–201. [18] W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967. [19] Y. Lin, Input-to-state stability with respect to noncompact sets, in Proc. 13th IFAC World Congress, volume E, pages 73–78, San Francisco, July 1996, IFAC Publications. [20] L. Praly and Y. Wang, Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability. Math. Control, Signals Syst., 1996, 9: 1–33.