Results on Discrete-Time Control-Lyapunov Functions - CiteSeerX
Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
FrP11-1
Results on Discrete-Time Control-Lyapunov Functions Christopher M. Kelletta a
1
and Andrew R. Teelb
2
Department of Electrical and Electronic Engineering, University of Melbourne, VIC 3052, Australia; [email protected] b Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106; [email protected]
Abstract We demonstrate the existence of a smooth controlLyapunov function (CLF) for difference equations asymptotically controllable to closed sets. This follows from a more general result on the existence of a weak Lyapunov function under the assumption of weak asymptotic stability of a closed (not necessarily compact) set for a difference inclusion. We use the derived CLF to construct a robust stabilizer. 1 Introduction The utility of control-Lyapunov functions (CLFs) in continuous-time nonlinear control design is well-known. A well-studied question is: what properties guarantee the existence of a CLF? An early result by Roxin [22] essentially generated a lower semicontinuous CLF given asymptotic controllability to the origin. In a seminal paper, Sontag [23] showed that, without limiting the controls a priori, asymptotic controllability to the origin implies the existence of a continuous CLF. Further refinements were made by Albertini and Sontag [1], Clarke, Ledyaev, Sontag, and Subbotin [4], Sontag [25], Clarke, Ledyaev, Rifford, and Stern [3], Rifford [20], and the authors [12] (see also [14] or [10]). We set out to solve the same problem in discrete-time as was solved in [1] and [12] for continuous-time; that asymptotic controllability to a closed (possibly noncompact) set A ⊆ Rn for the controlled difference equation x+ = f (x, u), x ∈ Rn , u ∈ U (1) implies the existence of a discrete-time CLF. Keerthi and Gilbert [9] demonstrated the existence of a solution to a generic optimal control problem where the cost function is assumed lower semicontinuous, implicitly deriving a CLF with no regularity. Gr¨ une and Neˇsi´c [8] generated a continuous CLF as a solution to an optimization problem given asymptotic controllability to the origin. This result required assumptions on the cost function defining the optimization problem as well as continuity of (1) in u and locally Lipschitz in x. While our CLF is also the value function of an optimization problem, we do not make assumptions on its structure a priori. Furthermore, our regularity requirement is merely that the set-valued map obtained by allowing the controls to range over the input space be continuous. With a discrete-time CLF in hand, designing a stabilizer robust to measurement noise and additive perturbations is straightforward. In the continuous-time case, the construction of a feedback stabilizer from a generic CLF is 1 This
work was done while C. M. Kellett was at UCSB. partially supported by AFOSR under grants F4962000-1-0106 and F49620-03-1-0203 and by NSF under grant ECS9988813. 2 Authors
0-7803-7924-1/03/$17.00 ©2003 IEEE
a nontrivial task. This comes from the fact that, in the general case, continuously differentiable CLFs fail to exist. The decrease condition thus obtained involves the Dini subderivate (alternatively one may use a characterization in terms of proximal subgradients) and the minimum involves the closed convex hull of the vector field defining the controlled differential equation; min DV (x; w) ≤ −V (x), w∈cof (x,U )
where f (x, U) is the set-valued map obtained by allowing the controls to range over the entire control space. Unfortunately, there is no guarantee that a feasible control generates the minimum in this statement and something more clever is required (see [4], [3], [14], or [10] for various constructions). However, in the discrete-time case we do not have a convexity requirement on the set-valued map and the decrease condition becomes merely min V (w) ≤ V (x)e−1 , (2) w∈f (x,U )
where e denotes the exponential. The construction of a stabilizer then follows directly from the CLF. When A is compact, the fact that a stabilizer thus designed is robust to measurement noise and additive perturbations follows from the continuous (or smooth) CLF which acts as a Lyapunov function for the closed loop system. This result is not surprising since robustness in continuoustime is induced via sample-and-hold control (see [11]). In order to demonstrate the existence of a discrete-time CLF for (1), we will consider the more general problem of weak uniform global asymptotic stability (weak-UGAS) of a closed set A for the difference inclusion x+ ∈ F (x). (3) By weak-UGAS of a set A we mean that, for every initial condition, there exists a solution of (3) which does not wander too far from, and is attracted to, the set A. Thus we see that asymptotic controllability to A of (1) can be cast as weak-UGAS of A for the difference inclusion x+ ∈ f (x, U). In continuous-time, Roxin [21] generated a lower semicontinuous weak Lyapunov function given weak stability (not asymptotic) of a closed set A. Deimling [6] generated a lower semicontinuous weak Lyapunov function given weak asymptotic stability of the origin for a differential inclusion. The authors [12] (see also [10] or [13]) generated a locally Lipschitz weak Lyapunov function given weak asymptotic stability of a closed (not necessarily compact) set A. We pause to point out a crucial difference in the discretetime case; namely that the Lyapunov function we obtain is smooth. In the continuous-time case, in order to get a smooth weak Lyapunov function under the assumption of weak asymptotic stability, it was necessary that the
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set-valued map defining the differential inclusion satisfy a certain covering condition (see Clarke, Ledyaev, and Stern [5, Theorem 2]). However, in the discrete-time case, certain nonrestrictive basic conditions and continuity of the set-valued map on Rn \A (coupled with weak UGAS of A) are enough to assert the existence of a smooth weak Lyapunov function. This follows by constructing a continuous Lyapunov function, and then using certain smoothing techniques that a negative finite difference can withstand. We first present our converse Lyapunov theorem for weak-UGAS of a closed set in Section 2 with the proof in Section 5. Our result on the existence of a CLF given asymptotic controllability to a closed set A is contained in Section 3. Finally, we construct a robust stabilizing feedback in Section 4. 2 A Weak Converse Lyapunov Theorem In what follows, Z≥0 represents the nonnegative integers and | · | denotes the usual Euclidean norm on Rn . For a closed set A we write the distance from a point x ∈ Rn to the set A as |x|A := inf a∈A |x − a|. We will use B (or B) to denote the open (or closed) unit ball in Rn . Definition 1 The set-valued map F (·) is said to be upper semicontinuous on the open set O if for each x ∈ O and ε > 0 there exists δ > 0 such that, for all ξ ∈ O satisfying |x − ξ| < δ, we have F (ξ) ⊆ F (x) + εB. Definition 2 The set-valued map F (·) is said to satisfy the basic conditions if it is upper semicontinuous on Rn and, for each x ∈ Rn , F (x) is nonempty and compact. Definition 3 We say the set-valued map F (·) is continuous on the open set O if, in addition to being upper semicontinuous on O, for each x ∈ O and ε > 0 there exists δ > 0 such that, for z ∈ O satisfying |z − x| < δ we have F (x) ⊆ F (z) + εB . Recall that a function α : R≥0 → R≥0 is said to belong to class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-K∞ if, in addition, it is unbounded. A function β : R≥0 × R≥0 → R≥0 is said to belong to class-KL if, for each t ≥ 0, β(·, t) is nondecreasing and lims→0+ β(s, t) = 0, and, for each s ≥ 0, β(s, ·) is nonincreasing and limt→∞ β(s, t) = 0. In what follows, F (·) maps Rn to subsets of Rn . We denote the set of solutions of the difference inclusion x+ ∈ F (x) for an initial condition x ∈ Rn by S(x). Definition 4 We say the closed set A ⊂ Rn is weakly uniformly globally asymptotically stable (weakly UGAS) for the difference inclusion x+ ∈ F (x) if there exists β ∈ KL such that for every x ∈ Rn there exists φ ∈ S(x) such that |φ(k, x)|A ≤ β(|x|A , k), for all k ∈ Z≥0 . Definition 5 A function V : Rn → R≥0 is said to be a weak discrete-time Lyapunov function for the difference inclusion x+ ∈ F (x) if the function V (·) is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (4) inf V (f ) ≤ V (x)e−1 .
f ∈F (x)
(5)
We note that the decrease condition (5) is equivalent to inf f ∈F (x) V (f ) ≤ V (x)−α(|x|A ) where α : R≥0 → R≥0 is continuous and positive definite. This equivalence can be seen in the arguments showing the decrease condition of a derived Lyapunov function in Section 5.2. We use (5) due to its resemblance to the continuous-time condition inf w∈F (x) h∇V (x), wi ≤ −V (x) which gives an exponential decrease of the Lyapunov function along solutions. With the necessary definitions in hand, we may now state the following discrete-time analog of [13] (see also [12]) which appeared as [10, Theorem 8.1]. The proof is contained in Section 5. Theorem 1 Suppose F (·) is a set-valued mapping from Rn to subsets of Rn satisfying the basic conditions, continuous on Rn \A, and that the set A is weakly UGAS for the inclusion (3). Then there exists a weak discrete-time Lyapunov function. 3 Control Lyapunov Functions In this section we formally present the result that asymptotic controllability to a closed (not necessarily compact) set A implies the existence of a control-Lyapunov function. In what follows we take U to be a metric space with a unique zero element, “0”, and, by abuse of notation, |u| := d(u, 0). We define the closed unit ball in the metric space U as B U := {ζ ∈ U : d(ζ, 0) ≤ 1}. Definition 6 Let A ⊆ Rn be a closed, nonempty set and let σ : R≥0 → R≥0 be continuous, nondecreasing. We say that (1) is uniformly globally asymptotically controllable (UGAC) to A with U ∩σ controls if there exists a function β ∈ KL such that: for each x ∈ Rn there exist a sequence u : Z≥0 → U and a solution φ(·, x, u) of x+ = f (x, u(k)) satisfying, for all k ∈ Z≥0 , |φ(k, x, u(k))|A ≤ β(|x|A , k), |u(k)| ≤ σ(|φ(k, x, u)|A ).
(6)
When the function σ ∈ K, we say that (1) is UGAC to A with vanishing controls. Messina, Tuna, and Teel [18, Corollary 4] use the notion of UGAC to A with vanishing controls, coupled with uniform complete observability, to show that, for a controlled difference equation, the set A can be robustly stabilized via dynamic output feedback Definition 7 A function V : Rn → R≥0 is said to be a control Lyapunov function with U ∩ σ controls for the system (1) if the function V (·) is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (7) min
V (f (x, u)) ≤ V (x)e−1 .
(8)
u∈U∩σ(|x|A )BU
The following theorem is the discrete-time equivalent of the result in [14] (see also [12]). Theorem 2 Suppose (1) is UGAC to A with U ∩ σ controls and that the set-valued map F (x) := f (x, U ∩ σ(|x|A )B U ) satisfies the basic conditions and is continuous on Rn \A. Then there exists a control Lyapunov function with U ∩ σ controls for (1).
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Proof: By assumption, the set-valued map satisfies all the conditions of Theorem 1. Let β ∈ KL, σ(·) nondecreasing, and φ, a solution of x+ = f (x, u) satisfying (6), come from the assumption of UGAC to A with U ∩ σ controls. By construction, for all k ∈ Z≥0 , φ(k + 1, x, u) ∈ F (φ(k, x, u(k))). Therefore, A is weakly UGAS for x+ ∈ F (x). Appealing to Theorem 1, there exist functions α1 , α2 ∈ K∞ , and V : Rn → R≥0 such that, for all x ∈ Rn α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and min V (f ) ≤ V (x)e−1 , f ∈F (x)
where we may replace the infimum with a minimum since F (x) is compact for every x ∈ Rn . Therefore, from the definition of F (·), we have min
V (f ) ≤ V (x)e−1 ,
f ∈f (x,U ∩σ(|x|A )BU )
and so V (·) is a control Lyapunov function with U ∩ σ controls for (1). 4 Robust Stabilization We now turn to the design of a feedback stabilizer that yields robustness to measurement noise and additive perturbations. For this purpose, we restrict our attention to compact sets A. It is possible to discuss robustness for noncompact sets, but it is more involved (see [10]). We require the notion of strong global asymptotic stability (strong GAS) that governs how all solutions to a difference inclusion behave (as opposed to the previous concept of weak GAS that constrains at least one solution but that says nothing about how all solutions behave). Definition 8 We say that the compact set A ⊂ Rn is strongly GAS for the difference inclusion x+ ∈ F (x) if there exists a function β ∈ KL such that for every initial condition x ∈ Rn all solutions φ ∈ S(x) satisfy |φ(k, x)|A ≤ β(|x|A , k), for all k ∈ Z≥0 . For a continuous function δ : Rn → R≥0 we define the δ-perturbation of F (·) as Fδ (x) := F {x} + δ(x)B + δ(x)B. (9) We may clearly interpret the “inner perturbation” above as measurement noise and the “outer perturbation” as an additive disturbance. Definition 9 We say that the compact set A is robustly strongly GAS for the difference inclusion x+ ∈ F (x) if there exists a continuous function δ : Rn → R≥0 such that δ(x) > 0 for x ∈ Rn \A and A is strongly GAS for the difference inclusion x+ ∈ Fδ (x). Definition 10 A function V : Rn → R≥0 is said to be a strong discrete-time Lyapunov function for the difference inclusion x+ ∈ F (x) if it is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (10)
Theorem 3 Let F (·) mapping Rn to subsets of Rn be compact and nonempty. If there exists a strong discretetime Lyapunov function for the difference inclusion x+ ∈ F (x) then A is robustly strongly GAS for the inclusion. Theorem 4 Suppose the set-valued map F (x) := f (x, U ∩ σ(|x|A )B U ) satisfies the basic conditions and is continuous on Rn . If the system x+ = f (x, u) is UGAC to A with U ∩ σ controls then there exists a feedback function κ : Rn → U such that A is robustly strongly GAS. If the system is asymptotically controllable to A with vanishing controls then, moreover, κ(·) can be taken to satisfy lim supx→0 |κ(x)| = 0. Proof: Since x+ = f (x, u) is UGAC to A and F (·) satisfies the basic conditions and is continuous on Rn , we appeal to Theorem 2 to obtain functions α1 , α2 ∈ K∞ and a smooth function V : Rn → R≥0 such that, for each x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and min V (f (x, u)) ≤ V (x)e−1 . u∈U∩σ(|x|A )BU
Let λ ∈ (0, e−1 ] and then, for each x ∈ Rn , we let κ(x) ∈ U ∩ σ(|x|A )B U satisfy V (f (x, κ(x))) ≤ λV (x). We see that V (·) is then a strong discrete-time Lyapunov function for the closed-loop system x+ = f (x, κ(x)). Therefore, appealing to Theorem 3 yields robust strong UGAS of the set A for the closed loop system. Furthermore, when σ ∈ K we obtain lim supx→0 |κ(x)| = 0. 5 Proof of Theorem 1 We now demonstrate that weak-UGAS of a closed set A implies the existence of a weak discrete-time Lyapunov function. We require several preliminary results. The proofs of Lemmas 1-4 can be found in [10, Chapter 2]. 5.1 Preliminary Lemmas In what follows, we will make use of the perturbed difference inclusion x+ ∈ Fδ (x) where Fδ (·) is defined by (9). We denote the solution set of this inclusion from the point x ∈ Rn by Sδ (x). This lemma resembles continuous-time results [26, Lemmas 4 & 5] which derived from [7, §7, Theorem 3]. Lemma 1 Let F (·) mapping Rn to subsets of Rn satisfy the basic conditions. Let x ∈ Rn be given and suppose that all solutions φ ∈ S(x) are defined for all k ≥ 0. Then each sequence {φn }∞ n=1 of solutions in S(x) has a subsequence converging to a function φ ∈ S(x) and the convergence is uniform on each finite time interval. The following result is similar to a result on closeness of solutions for differential inclusions (see, for example, [7, §8 Cor. 2]) and is a simplification of [10, Lemma 2.8].
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Lemma 2 Let F : Rn → subsets of Rn satisfy the basic conditions. Let the triple (K, ε, x) be such that K ∈ Z>0 , ε > 0, x ∈ Rn . Then there exists δ > 0 such that for every xδ ∈ {x} + δB and every φδ ∈ Sδ (xδ ) there exists φ ∈ S(x) such that for all k ∈ {0, . . . , K} we have ||φδ (k, xδ )|A − |φ(k, x)|A | ≤ ε . (12)
The following theorem is a consequence of [10, Theorem 7.1] and [10, Remark 7.3] (see also [15, Theorem 1]):
Lemma 3 Suppose F (·) is a set-valued map from Rn to subsets of Rn continuous on an open set O ⊆ Rn and that, for each x ∈ Rn , F (x) is compact and nonempty.
sup V (f ) ≤ V (x)e−1 . f ∈F (x)
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For each triple (K, ε, x0 ) such that K ∈ Z>0 , ε > 0, and x0 ∈ Rn , and for each solution φ ∈ S(x0 ) such that φ(k, x0 ) ∈ O for all k ∈ {0, . . . , K} there exists a δ > 0 such that, for every xδ ∈ {x0 }+δB there exists a solution ψ ∈ S(xδ ) such that, for all k ∈ {0, . . . , K + 1}, ||φ(k, x0 )|A − |ψ(k, xδ )|A | ≤ ε. The following lemma appeared as [10, Lemma 2.4]. Lemma 4 If γ > 1 and µ ∈ K∞ satisfies (µ − Id) ∈ K∞ then there exists ρ ∈ K∞ such that ρ ◦ µ(s) = γ · ρ(s) for all s ≥ 0. The following lemma is [24, Proposition 7]. Lemma 5 For each β ∈ KL and λ > 0, there exist α1 , α2 ∈ K∞ such that, for all (s, t) ∈ R≥0 × R≥0 , α1 (β(s, t)) ≤ α2 (s)e−λt . The following appeared as [19, Theorem 8]. Lemma 6 Let V : Rn → R≥0 and suppose that, for α ∈ K∞ , V (φ(k +1, x))−V (φ(k, x)) ≤ −α(V (φ(k, x))). Then there exists β ∈ KL such that V (φ(k, x)) ≤ β(V (x), k), ∀k ∈ Z≥0 . (13) The following is a standard approximation result. Lemma 7 Let O ⊂ Rn be open and let µ : O → (0, ∞) be continuous. Suppose V : O → R is continuous. Then there exists a smooth function Vs : O → R such that, for all x ∈ O, |V (x) − Vs (x)| ≤ µ(x) . The following result appeared as [26, Lemma 17] which derives from [17, Lemma 4.3] and [16, Theorem 6]. Lemma 8 Let A ⊂ Rn be a closed set and assume that Vs : Rn → R≥0 is continuous, the restriction of Vs to Rn \A is smooth, Vs (x) = 0 for all x ∈ A, and Vs (x) > 0 for all x ∈ Rn \A. Then there exists a function ρ ∈ K∞ , strictly convex, and smooth on (0, ∞) such that V := ρ ◦ Vs is smooth on Rn . 5.2 Core of Proof of Theorem 1 Let β ∈ KL and φˆ ∈ S(x) come from the assumption of weak UGAS; i.e., ˆ x)|A ≤ β(|x|A , k), ∀k ∈ Z≥0 . |φ(k, (14) Let α ¯1, α ˆ 2 ∈ K∞ come from Lemma 5 such that α ¯ 1 (β(s, k)) ≤ α ˆ 2 (s)e−k . (15) n We define a function V2 : R → R≥0 by ∞ X V2 (x) := inf α ¯ 1 (|φ(k, x)|A ) , ∀x ∈ Rn . (16) φ∈S(x)
k=0
Upper and Lower bounds: We see that V2 (x) ≥ inf α ¯ 1 (|φ(0, x)|A ) = α ¯ 1 (|x|A ).
(17)
The upper bound follows from (14) and (15): ∞ ∞ X X V2 (x) ≤ α ¯ 1 (β(|x|A , k)) ≤ α ˆ 2 (|x|A ) e−k e α ˆ 2 (|x|A ) =: α ¯ 2 (|x|A ). = e−1 Preliminary Claims:
k=0
Proof: From the definition P∞ of V2∗ we have that for any φ∗ ∈ S(x), V2 (x) ≤ ¯ 1 (|φ (k, x)|A ). So we just k=0 α need to construct a solution φ∗ ∈ S(x) such that ∞ X V2 (x) ≥ α ¯ 1 (|φ∗ (k, x)|A ) . (20) k=0
∞
We let {φi }i=1 be such that ∞ X V2 (x) ≥ α ¯ 1 (|φi (k, x)|A ) − 1/i . k=0
(21)
∞
From Lemma 1, the sequence {φi }i=1 has a converging subsequence (which we won’t relabel) converging to a solution φ∗ ∈ S(x) and the convergence is uniform on finite intervals. We first claim that ∞ X V ∗ := α ¯ 1 (|φ∗ (k, x)|A ) < ∞ . (22) k=0
Suppose not. Then there exists κ ∈ Z>0 such that κ X α ¯ 1 (|φ∗ (k, x)|A ) ≥ α ¯ 2 (|x|A ) + 1 . (23) k=0
Then, using the continuity of α ¯ 1 , the uniform convergence of φi to φ∗ on {0, . . . , κ}, (21), and (23), for each ε > 0 there exists i∗ such that for all i ≥ i∗ , κ X α ¯ 1 (|φi (k, x)|A ) − 1/i V2 (x) ≥ ≥
k=0 κ X
α ¯ 1 (|φ∗ (k, x)|A ) − ε − 1/i
k=0
≥ α ¯ 2 (|x|A ) + 1 − ε − 1/i . (24) Taking the limit as ε → 0+ and i → ∞ establishes a contradiction with the upper bound (18). Using each ε > 0 there exists κε ∈ Z>0 such that Pκε (22), for ¯ 1 (|φ∗ (k, x)|A ) ≥ V ∗ −ε . Then, as above, for each k=0 α ε > 0 there exists i∗ ∈ Z>0 such that for all i ≥ i∗ , we have κε X α ¯ 1 (|φi (k, x)|A ) − 1/i V2 (x) ≥ ≥
k=0 κε X
α ¯ 1 (|φ∗ (k, x)|A ) − ε − 1/i
k=0
≥ V ∗ − 2ε − 1/i . (25) Taking the limit as ε → 0+ and i → ∞ gives V2 (x) ≥ V ∗ ; i.e., gives (20), and thus establishes the result. Claim 2 For an optimal solution φ∗ ∈ S(x), k−1 X V2 (φ∗ (k, x)) = V2 (x) − α ¯ 1 (|φ∗ (j, x)|A ).
(26)
j=0
φ∈S(x)
k=0
Claim 1 There exists a solution φ∗ ∈ S(x) (which we will call optimal) such that ∞ X V2 (x) = α ¯ 1 (|φ∗ (k, x)|A ) . (19)
Proof: Let ψ ∗ ∈ S(φ∗ (k, x)) be optimal from the point φ∗ (k, x). Using the definition of V2 (φ∗ (k, x)), the optimality of ψ ∗ , and the definition of V2 (x) we may write ∞ X V2 (φ∗ (k, x)) = α ¯ 1 (|ψ ∗ (j, φ∗ (k, x))|A )
k=0
j=0
(18) ≤ V2 (x) −
k−1 X i=0
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α ¯ 1 (|φ∗ (i, x)|A ) .
(27)
Reversing directions, and this time using the optimality of φ∗ , it is easy to see that k−1 X V2 (x) ≤ α ¯ 1 (|φ∗ (i, x)|A ) + V2 (φ∗ (k, x)). i=0
Continuity: We define 1 α ˜ (s) := min s, α ¯1 ◦ α ¯ 2−1 (s) , ∀s ∈ R≥0 , (28) 2 and note that Id − α ˜ ∈ K∞ . We first address lower semicontinuity of V2 (·). Appealing to Claim 2 twice and using (18) we see that, along an optimal trajectory φ∗ ∈ S(x), V2 (φ∗ (k + 1, x)) − V2 (φ∗ (k, x)) = −¯ α1 (|φ∗ (k, x)|A ) ≤ −¯ α1 ◦ α ¯ 2−1 (V2 (φ∗ (k, x))) ≤ −˜ α (V2 (φ∗ (k, x))) . Using Lemma 6 we have the existence of a function βs ∈ KL such that, along an optimal trajectory φ∗ ∈ S(x), V2 (φ∗ (k, x)) ≤ βs (V2 (x), k), ∀k ∈ Z≥0 . (29) Therefore, using (17) and (18), ˜ A , k). (30) |φ∗ (k, x)|A ≤ α ¯ 1−1 (βs (¯ α2 (|x|A ), k)) =: β(|x| ∗ Let ε > 0. Let xn → x and φn ∈ S(xn ) be optimal for each n. Let ε˜ := min α ¯ 2−1 2ε , 1 and choose κ ∈ Z>0 such that ˜ A + 1, κ) ≤ ε˜. 2β(|x| (31) ε Since α ¯ 1 (·) is continuous, for 2κ > 0 on the compact ˜ A + 1, 0) + 1] there exists δ ∈ (0, ε˜ ] such interval [0, β(|x| 2 ˜ A + 1, 0) + 1] then that, if s1 , s2 ∈ [0, β(|x| ε |s1 − s2 | ≤ δ =⇒ |¯ α1 (s1 ) − α ¯ 1 (s2 )| ≤ . (32) 2κ With the triple (κ, δ, x) Lemma 2 yields a δ1 > 0 and a solution φˆn ∈ S(x) defined on {0, . . . , κ} such that, for each n such that |xn − x| < δ1 , ||φˆn (k, x)|A − |φ∗n (k, xn )|A | ≤ δ, ∀k ∈ {0, . . . , κ}. (33) Since | · |A is globally Lipschitz with Lipschitz constant 1, using (32) we may write that, for all k ∈ {0, . . . , κ}, ε . (34) |¯ α1 (|φˆn (k, x)|A ) − α ¯ 1 (|φ∗n (k, xn )|A )| ≤ 2κ Let φ¯n ∈ S(φˆn (κ, x)) be optimal. We define φˆn (k, x), k ∈ {0, . . . , κ} ˜ φn (k, x) := φ¯n (k − κ, φˆn (κ, x)), k ∈ Z>κ . (35) Then, by definition, ∞ X V2 (φ˜n (κ, x)) = α ¯ 1 (|φ˜n (k, x)|A ). (36)
Therefore, since ε > 0 was arbitrary, V2 (·) is lower semicontinuous on Rn . We now demonstrate upper semicontinuity of V2 (·) on A. Let x ∈ A and recall that this implies V2 (x) = 0. Using the upper bound (18), since α ¯ 2 (·) is continuous, for any ε > 0 there exists δ > 0 such that if |x − z| < δ we may write V2 (z) ≤ α ¯ 2 (|z|A ) = α ¯ 2 (|z|A ) + V2 (x) ≤ ε + V2 (x). Therefore, V2 (·) is upper semicontinuous on A. All that remains is to prove upper semicontinuity of V2 (·) on Rn \A. For x ∈ Rn \A, letφ∗ ∈ S(x) be optimal. Pick ε > 0 and define ε˜ := α ¯ 2−1 2ε . Let κ ∈ Z>0 be such that κ − 1 is the smallestP value such that φ∗ (κ − 1, x) ∈ Rn \A ∞ ∗ and V2 (φ (κ, x)) = k=κ α ¯ 1 (|φ∗ (k, x)|A ) < α ¯ 1 2ε˜ . Note that, with (17), this implies that ε˜ (40) |φ∗ (κ, x)|A ≤ . 2 Since α ¯ 1 (·) is continuous, with the modulus of continuity ε interval [0, α ¯ 2 (|x|A + 1) + 1] there 2κ , on the compact ε˜ exists δ ∈ 0, 2 such that for s1 , s2 ∈ [0, α ¯ 2 (|x|A +1)+1], ε |s1 − s2 | ≤ δ =⇒ |¯ α1 (s1 ) − α ¯ 1 (s2 )| ≤ . (41) 2κ Since F (·) is continuous on Rn \A, with the triple (κ − 1, δ, x) and the solution φ∗ ∈ S(x), Lemma 3 yields δ1 > 0 such that for all z ∈ {x} + δ1 B there exists ψ ∈ S(z) such that, for all k ∈ {0, . . . , κ}, ||φ∗ (k, x)|A − |ψ(k, z)|A | ≤ δ. With (41) this implies that κ−1 X
k=0
≥ V2 (x) − ε.
κ−1 ε X ≤ α ¯ 1 (|φ∗ (k, x)|A ) . 2
(42)
k=0
k=0
Let ψ ∗ ∈ S(ψ(κ, z)) be optimal; i.e., ∞ X V2 (ψ(κ, z)) = α ¯ 1 (|ψ ∗ (k − κ, ψ(κ, z))|A ) .
(43)
k=κ
Since | · |A is globally Lipschitz with Lipschitz constant 1 and using (40) with the fact that δ ≤ 2ε˜ we have that |φ∗ (κ, x)|A + |φ∗ (κ, x) − ψ(κ, z)| ε˜ + δ ≤ ε˜. (44) ≤ 2 Therefore, using the upper bound (18) and the definition of ε˜ we see that ε V2 (ψ(κ, z)) ≤ α ¯ 2 (|ψ(κ, z)|A ) ≤ α ¯ 2 (˜ ε) = . (45) 2 Therefore, using Claim 2, (42), (43), the nonoptimality of ψ ∈ S(z), and (45) one may show |ψ(κ, z)|A
k=κ
With (30) and (31), we see that, for n sufficiently large ˜ A + 1, κ) ≤ ε˜ . |φ∗n (κ, xn )|A ≤ β(|x| (37) 2 Therefore, from (33) (with δ ≤ 2ε˜ ) and (35) we have |φ˜n (κ, x)|A ≤ ε˜ and, with the upper bound (18) ε V2 (φ˜n (κ, x)) ≤ α ¯ 2 (|φ˜n (κ, x)|A ) ≤ α ¯ 2 (˜ ε) = . (38) 2 Using Claim 2, (34), (35), (36), and (38) we have κ−1 X ε V2 (xn ) ≥ α ¯ 1 (|φ˜n (k, x)|A ) − + V2 (φ∗n (κ, xn )) 2
α ¯ 1 (|ψ(k, z)|A ) −
V2 (x) =
≤
κ−1 X
α ¯ 1 (|φ∗ (k, x)|A ) + V2 (φ∗ (κ, x))
k=0
ε − V2 (ψ(κ, z)) ≥ V2 (z) − ε, 2 which establishes upper semicontinuity on Rn \A. Decrease Condition: Since φ∗ (1, x) ∈ F (x) we may use Claim 2 and (28) to write min V2 (f ) ≤ V2 (x) − α ¯ 1 (|x|A ) f ∈F (x) ≤ V2 (x) − α ¯1 α ¯ 2−1 (V2 (x)) ≤ V2 (x) − α ˜ (V2 (x)). (46) ≥ V2 (z) −
Recall that (Id − α ˜ ) ∈ K∞ . Let µ ∈ K∞ be defined as
(39)
5965
µ(s) := (Id − α ˜ )−1 (s),
∀s ≥ 0.
(47)
We see then that µ−1 (s) = s − α ˜ (s) (48) −1 or s − µ (s) = α ˜ (s) or µ(s) − s = α ˜ ◦ µ(s) from which it follows that (µ − Id) ∈ K∞ . Let γ = e1 > 1 and let ρ ∈ K∞ come from Lemma 4 and define V1 := ρ(V2 ). Combining (46) and (48) we see that min V2 (f ) ≤ µ−1 (V2 (x)), (49) f ∈F (x)
which allows us to write min V1 (f ) = min ρ(V2 (f )) ≤ ρ ◦ µ−1 (V2 (x)) f ∈F (x)
f ∈F (x)
= e−1 ρ(V2 (x)) = e−1 V1 (x).
(50)
We note that V1 (·) is continuous by virtue of the continuity of V2 (·) and of ρ(·). Finally, we define K∞ functions α := ρ ◦ α ¯ 1 , and α := ρ◦α ¯ 2 so that α(|x|A ) ≤ V1 (x) ≤ α(|x|A ). 1/2
Smoothing: Let ε > 0 be such that ε ≤ ee1/2 −1 . We +1 appeal to Lemma 7 with µ(x) = εV1 (x) for all x ∈ Rn \A. Then we have that, for all x ∈ Rn \A, (1 − ε)V1 (x) ≤ Vs (x) ≤ (1 + ε)V1 (x). (51) We complete the definition of Vs by defining Vs (x) = 0 for x ∈ A. Using (50), (51), and the bound on ε we may write min Vs (f ) ≤ (1 + ε) min V1 (f ) ≤ (1 + ε)e−1 V1 (x)
f ∈F (x)
f ∈F (x)
≤
1 + ε −1 e Vs (x) ≤ e−1/2 Vs (x). 1−ε
Let V¯s = (Vs )2 . Then min V¯s (f ) = min Vs2 (f ) ≤ e−1 Vs2 (x) = e−1 V¯s (x). f ∈F (x)
f ∈F (x)
We note that V¯s satisfies all the conditions of Lemma 8. Let ρ ∈ K∞ come from Lemma 8. Then V := ρ ◦ V¯s is smooth on Rn and, since ρ(·) is convex, we may write ¯ min V (f ) = ρ min Vs (f ) f ∈F (x) f ∈F (x) ≤ ρ e−1 V¯s (x) ≤ e−1 ρ(V¯s (x)) = e−1 V (x). The upper and lower bounds on V (·) are then α1 (s) := ρ 21 α(s) , and α2 (s) := ρ(2α(s)), for all s ≥ 0. References [1] F. Albertini and E. D. Sontag. Continuous controlLyapunov functions for asymptotically controllable timevarying systems. Int. J. Control, 72:1630–1641, 1999. [2] J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkh¨ auser, 1990. [3] F. H. Clarke, Y. S. Ledyaev, L. Rifford, and R. J. Stern. Feedback stabilization and Lyapunov functions. SIAM J. Control Optim., 39(1):25–48, 2000. [4] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag, and A. I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control, 42:1394– 1407, 1997. [5] F. H. Clarke, Y. S. Ledyaev, and R. J. Stern. Asymptotic stability and smooth Lyapunov functions. J. Differential Equations, 149:69–114, 1998. [6] K. Deimling. Multivalued Differential Equations. Walter de Gruyter, 1992. [7] A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Pub., 1988.
[8] L. Gr¨ une and D. Neˇsi´c. Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models. SIAM J. Control Optim., 42(1):98–122, 2003. [9] S. S. Keerthi and E. G. Gilbert. An existence theorem for discrete-time infinite-horizon optimal control problems. IEEE Trans. Automat. Control, 30(9):907–909, 1985. [10] C. M. Kellett. Advances in Converse and Control Lyapunov Functions. PhD thesis, University of California, Santa Barbara, June 2002. [11] C. M. Kellett, H. Shim, and A. R. Teel. Robustness of discontinuous feedback via sample and hold control. In Proc. Amer. Control Conf., Anchorage, Alaska, May 2002. [12] C. M. Kellett and A. R. Teel. Weak converse Lyapunov theorems and control Lyapunov functions. To appear SIAM J. Control Optim.. [13] C. M. Kellett and A. R. Teel. A converse Lyapunov theorem for weak uniform asymptotic stability of sets. In Proc. Mathematical Theory of Networks and Systems, Perpignan, France, June 2000. [14] C. M. Kellett and A. R. Teel. Uniform asymptotic controllability to a set implies locally Lipschitz controlLyapunov function. In Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, Dec. 2000. [15] C. M. Kellett and A. R. Teel. Results on converse Lyapunov theorems for difference inclusions. In Proc. 42nd IEEE Conf. on Decision and Control, Maui, Hawaii, Dec. 2003. [16] J. Kurzweil. On the inversion of Ljapunov’s second theorem on stability of motion. Amer. Math. Soc. Transl. Ser. 2, 24:19–77, 1956. [17] Y. Lin, E. D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim., 34:124–160, 1996. [18] M. Messina, S. E. Tuna, and A. R. Teel. Output feedback stabilization by certainty equivalence model predictive control. Submitted to Automatica. [19] D. Neˇsi´c, A. R. Teel, and E. D. Sontag. Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems Control Lett., 38:49–60, 1999. [20] L. Rifford. Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim., 39(4):1043–1064, 2000. [21] E. Roxin. Stability in general control systems. J. Differential Equations, 1:115–150, 1965. [22] E. Roxin. On asymptotic stability in control systems. Rend. Circ. Mat. Palermo (2), 15(2):193–208, 1966. [23] E. D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim., 21:462–471, 1983. [24] E. D. Sontag. Comments on integral variants of ISS. Systems Control Lett., 34:93–100, 1998. [25] E. D. Sontag. Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var., 4:537–557, October 1999. [26] A. R. Teel and L. Praly. A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions. ESAIM Control Optim. Calc. Var., 5:313–367, 2000.
5966
FrP11-1
Results on Discrete-Time Control-Lyapunov Functions Christopher M. Kelletta a
1
and Andrew R. Teelb
2
Department of Electrical and Electronic Engineering, University of Melbourne, VIC 3052, Australia; [email protected] b Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106; [email protected]
Abstract We demonstrate the existence of a smooth controlLyapunov function (CLF) for difference equations asymptotically controllable to closed sets. This follows from a more general result on the existence of a weak Lyapunov function under the assumption of weak asymptotic stability of a closed (not necessarily compact) set for a difference inclusion. We use the derived CLF to construct a robust stabilizer. 1 Introduction The utility of control-Lyapunov functions (CLFs) in continuous-time nonlinear control design is well-known. A well-studied question is: what properties guarantee the existence of a CLF? An early result by Roxin [22] essentially generated a lower semicontinuous CLF given asymptotic controllability to the origin. In a seminal paper, Sontag [23] showed that, without limiting the controls a priori, asymptotic controllability to the origin implies the existence of a continuous CLF. Further refinements were made by Albertini and Sontag [1], Clarke, Ledyaev, Sontag, and Subbotin [4], Sontag [25], Clarke, Ledyaev, Rifford, and Stern [3], Rifford [20], and the authors [12] (see also [14] or [10]). We set out to solve the same problem in discrete-time as was solved in [1] and [12] for continuous-time; that asymptotic controllability to a closed (possibly noncompact) set A ⊆ Rn for the controlled difference equation x+ = f (x, u), x ∈ Rn , u ∈ U (1) implies the existence of a discrete-time CLF. Keerthi and Gilbert [9] demonstrated the existence of a solution to a generic optimal control problem where the cost function is assumed lower semicontinuous, implicitly deriving a CLF with no regularity. Gr¨ une and Neˇsi´c [8] generated a continuous CLF as a solution to an optimization problem given asymptotic controllability to the origin. This result required assumptions on the cost function defining the optimization problem as well as continuity of (1) in u and locally Lipschitz in x. While our CLF is also the value function of an optimization problem, we do not make assumptions on its structure a priori. Furthermore, our regularity requirement is merely that the set-valued map obtained by allowing the controls to range over the input space be continuous. With a discrete-time CLF in hand, designing a stabilizer robust to measurement noise and additive perturbations is straightforward. In the continuous-time case, the construction of a feedback stabilizer from a generic CLF is 1 This
work was done while C. M. Kellett was at UCSB. partially supported by AFOSR under grants F4962000-1-0106 and F49620-03-1-0203 and by NSF under grant ECS9988813. 2 Authors
0-7803-7924-1/03/$17.00 ©2003 IEEE
a nontrivial task. This comes from the fact that, in the general case, continuously differentiable CLFs fail to exist. The decrease condition thus obtained involves the Dini subderivate (alternatively one may use a characterization in terms of proximal subgradients) and the minimum involves the closed convex hull of the vector field defining the controlled differential equation; min DV (x; w) ≤ −V (x), w∈cof (x,U )
where f (x, U) is the set-valued map obtained by allowing the controls to range over the entire control space. Unfortunately, there is no guarantee that a feasible control generates the minimum in this statement and something more clever is required (see [4], [3], [14], or [10] for various constructions). However, in the discrete-time case we do not have a convexity requirement on the set-valued map and the decrease condition becomes merely min V (w) ≤ V (x)e−1 , (2) w∈f (x,U )
where e denotes the exponential. The construction of a stabilizer then follows directly from the CLF. When A is compact, the fact that a stabilizer thus designed is robust to measurement noise and additive perturbations follows from the continuous (or smooth) CLF which acts as a Lyapunov function for the closed loop system. This result is not surprising since robustness in continuoustime is induced via sample-and-hold control (see [11]). In order to demonstrate the existence of a discrete-time CLF for (1), we will consider the more general problem of weak uniform global asymptotic stability (weak-UGAS) of a closed set A for the difference inclusion x+ ∈ F (x). (3) By weak-UGAS of a set A we mean that, for every initial condition, there exists a solution of (3) which does not wander too far from, and is attracted to, the set A. Thus we see that asymptotic controllability to A of (1) can be cast as weak-UGAS of A for the difference inclusion x+ ∈ f (x, U). In continuous-time, Roxin [21] generated a lower semicontinuous weak Lyapunov function given weak stability (not asymptotic) of a closed set A. Deimling [6] generated a lower semicontinuous weak Lyapunov function given weak asymptotic stability of the origin for a differential inclusion. The authors [12] (see also [10] or [13]) generated a locally Lipschitz weak Lyapunov function given weak asymptotic stability of a closed (not necessarily compact) set A. We pause to point out a crucial difference in the discretetime case; namely that the Lyapunov function we obtain is smooth. In the continuous-time case, in order to get a smooth weak Lyapunov function under the assumption of weak asymptotic stability, it was necessary that the
5961
set-valued map defining the differential inclusion satisfy a certain covering condition (see Clarke, Ledyaev, and Stern [5, Theorem 2]). However, in the discrete-time case, certain nonrestrictive basic conditions and continuity of the set-valued map on Rn \A (coupled with weak UGAS of A) are enough to assert the existence of a smooth weak Lyapunov function. This follows by constructing a continuous Lyapunov function, and then using certain smoothing techniques that a negative finite difference can withstand. We first present our converse Lyapunov theorem for weak-UGAS of a closed set in Section 2 with the proof in Section 5. Our result on the existence of a CLF given asymptotic controllability to a closed set A is contained in Section 3. Finally, we construct a robust stabilizing feedback in Section 4. 2 A Weak Converse Lyapunov Theorem In what follows, Z≥0 represents the nonnegative integers and | · | denotes the usual Euclidean norm on Rn . For a closed set A we write the distance from a point x ∈ Rn to the set A as |x|A := inf a∈A |x − a|. We will use B (or B) to denote the open (or closed) unit ball in Rn . Definition 1 The set-valued map F (·) is said to be upper semicontinuous on the open set O if for each x ∈ O and ε > 0 there exists δ > 0 such that, for all ξ ∈ O satisfying |x − ξ| < δ, we have F (ξ) ⊆ F (x) + εB. Definition 2 The set-valued map F (·) is said to satisfy the basic conditions if it is upper semicontinuous on Rn and, for each x ∈ Rn , F (x) is nonempty and compact. Definition 3 We say the set-valued map F (·) is continuous on the open set O if, in addition to being upper semicontinuous on O, for each x ∈ O and ε > 0 there exists δ > 0 such that, for z ∈ O satisfying |z − x| < δ we have F (x) ⊆ F (z) + εB . Recall that a function α : R≥0 → R≥0 is said to belong to class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-K∞ if, in addition, it is unbounded. A function β : R≥0 × R≥0 → R≥0 is said to belong to class-KL if, for each t ≥ 0, β(·, t) is nondecreasing and lims→0+ β(s, t) = 0, and, for each s ≥ 0, β(s, ·) is nonincreasing and limt→∞ β(s, t) = 0. In what follows, F (·) maps Rn to subsets of Rn . We denote the set of solutions of the difference inclusion x+ ∈ F (x) for an initial condition x ∈ Rn by S(x). Definition 4 We say the closed set A ⊂ Rn is weakly uniformly globally asymptotically stable (weakly UGAS) for the difference inclusion x+ ∈ F (x) if there exists β ∈ KL such that for every x ∈ Rn there exists φ ∈ S(x) such that |φ(k, x)|A ≤ β(|x|A , k), for all k ∈ Z≥0 . Definition 5 A function V : Rn → R≥0 is said to be a weak discrete-time Lyapunov function for the difference inclusion x+ ∈ F (x) if the function V (·) is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (4) inf V (f ) ≤ V (x)e−1 .
f ∈F (x)
(5)
We note that the decrease condition (5) is equivalent to inf f ∈F (x) V (f ) ≤ V (x)−α(|x|A ) where α : R≥0 → R≥0 is continuous and positive definite. This equivalence can be seen in the arguments showing the decrease condition of a derived Lyapunov function in Section 5.2. We use (5) due to its resemblance to the continuous-time condition inf w∈F (x) h∇V (x), wi ≤ −V (x) which gives an exponential decrease of the Lyapunov function along solutions. With the necessary definitions in hand, we may now state the following discrete-time analog of [13] (see also [12]) which appeared as [10, Theorem 8.1]. The proof is contained in Section 5. Theorem 1 Suppose F (·) is a set-valued mapping from Rn to subsets of Rn satisfying the basic conditions, continuous on Rn \A, and that the set A is weakly UGAS for the inclusion (3). Then there exists a weak discrete-time Lyapunov function. 3 Control Lyapunov Functions In this section we formally present the result that asymptotic controllability to a closed (not necessarily compact) set A implies the existence of a control-Lyapunov function. In what follows we take U to be a metric space with a unique zero element, “0”, and, by abuse of notation, |u| := d(u, 0). We define the closed unit ball in the metric space U as B U := {ζ ∈ U : d(ζ, 0) ≤ 1}. Definition 6 Let A ⊆ Rn be a closed, nonempty set and let σ : R≥0 → R≥0 be continuous, nondecreasing. We say that (1) is uniformly globally asymptotically controllable (UGAC) to A with U ∩σ controls if there exists a function β ∈ KL such that: for each x ∈ Rn there exist a sequence u : Z≥0 → U and a solution φ(·, x, u) of x+ = f (x, u(k)) satisfying, for all k ∈ Z≥0 , |φ(k, x, u(k))|A ≤ β(|x|A , k), |u(k)| ≤ σ(|φ(k, x, u)|A ).
(6)
When the function σ ∈ K, we say that (1) is UGAC to A with vanishing controls. Messina, Tuna, and Teel [18, Corollary 4] use the notion of UGAC to A with vanishing controls, coupled with uniform complete observability, to show that, for a controlled difference equation, the set A can be robustly stabilized via dynamic output feedback Definition 7 A function V : Rn → R≥0 is said to be a control Lyapunov function with U ∩ σ controls for the system (1) if the function V (·) is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (7) min
V (f (x, u)) ≤ V (x)e−1 .
(8)
u∈U∩σ(|x|A )BU
The following theorem is the discrete-time equivalent of the result in [14] (see also [12]). Theorem 2 Suppose (1) is UGAC to A with U ∩ σ controls and that the set-valued map F (x) := f (x, U ∩ σ(|x|A )B U ) satisfies the basic conditions and is continuous on Rn \A. Then there exists a control Lyapunov function with U ∩ σ controls for (1).
5962
Proof: By assumption, the set-valued map satisfies all the conditions of Theorem 1. Let β ∈ KL, σ(·) nondecreasing, and φ, a solution of x+ = f (x, u) satisfying (6), come from the assumption of UGAC to A with U ∩ σ controls. By construction, for all k ∈ Z≥0 , φ(k + 1, x, u) ∈ F (φ(k, x, u(k))). Therefore, A is weakly UGAS for x+ ∈ F (x). Appealing to Theorem 1, there exist functions α1 , α2 ∈ K∞ , and V : Rn → R≥0 such that, for all x ∈ Rn α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and min V (f ) ≤ V (x)e−1 , f ∈F (x)
where we may replace the infimum with a minimum since F (x) is compact for every x ∈ Rn . Therefore, from the definition of F (·), we have min
V (f ) ≤ V (x)e−1 ,
f ∈f (x,U ∩σ(|x|A )BU )
and so V (·) is a control Lyapunov function with U ∩ σ controls for (1). 4 Robust Stabilization We now turn to the design of a feedback stabilizer that yields robustness to measurement noise and additive perturbations. For this purpose, we restrict our attention to compact sets A. It is possible to discuss robustness for noncompact sets, but it is more involved (see [10]). We require the notion of strong global asymptotic stability (strong GAS) that governs how all solutions to a difference inclusion behave (as opposed to the previous concept of weak GAS that constrains at least one solution but that says nothing about how all solutions behave). Definition 8 We say that the compact set A ⊂ Rn is strongly GAS for the difference inclusion x+ ∈ F (x) if there exists a function β ∈ KL such that for every initial condition x ∈ Rn all solutions φ ∈ S(x) satisfy |φ(k, x)|A ≤ β(|x|A , k), for all k ∈ Z≥0 . For a continuous function δ : Rn → R≥0 we define the δ-perturbation of F (·) as Fδ (x) := F {x} + δ(x)B + δ(x)B. (9) We may clearly interpret the “inner perturbation” above as measurement noise and the “outer perturbation” as an additive disturbance. Definition 9 We say that the compact set A is robustly strongly GAS for the difference inclusion x+ ∈ F (x) if there exists a continuous function δ : Rn → R≥0 such that δ(x) > 0 for x ∈ Rn \A and A is strongly GAS for the difference inclusion x+ ∈ Fδ (x). Definition 10 A function V : Rn → R≥0 is said to be a strong discrete-time Lyapunov function for the difference inclusion x+ ∈ F (x) if it is smooth and there exist α1 , α2 ∈ K∞ such that for all x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and (10)
Theorem 3 Let F (·) mapping Rn to subsets of Rn be compact and nonempty. If there exists a strong discretetime Lyapunov function for the difference inclusion x+ ∈ F (x) then A is robustly strongly GAS for the inclusion. Theorem 4 Suppose the set-valued map F (x) := f (x, U ∩ σ(|x|A )B U ) satisfies the basic conditions and is continuous on Rn . If the system x+ = f (x, u) is UGAC to A with U ∩ σ controls then there exists a feedback function κ : Rn → U such that A is robustly strongly GAS. If the system is asymptotically controllable to A with vanishing controls then, moreover, κ(·) can be taken to satisfy lim supx→0 |κ(x)| = 0. Proof: Since x+ = f (x, u) is UGAC to A and F (·) satisfies the basic conditions and is continuous on Rn , we appeal to Theorem 2 to obtain functions α1 , α2 ∈ K∞ and a smooth function V : Rn → R≥0 such that, for each x ∈ Rn , α1 (|x|A ) ≤ V (x) ≤ α2 (|x|A ), and min V (f (x, u)) ≤ V (x)e−1 . u∈U∩σ(|x|A )BU
Let λ ∈ (0, e−1 ] and then, for each x ∈ Rn , we let κ(x) ∈ U ∩ σ(|x|A )B U satisfy V (f (x, κ(x))) ≤ λV (x). We see that V (·) is then a strong discrete-time Lyapunov function for the closed-loop system x+ = f (x, κ(x)). Therefore, appealing to Theorem 3 yields robust strong UGAS of the set A for the closed loop system. Furthermore, when σ ∈ K we obtain lim supx→0 |κ(x)| = 0. 5 Proof of Theorem 1 We now demonstrate that weak-UGAS of a closed set A implies the existence of a weak discrete-time Lyapunov function. We require several preliminary results. The proofs of Lemmas 1-4 can be found in [10, Chapter 2]. 5.1 Preliminary Lemmas In what follows, we will make use of the perturbed difference inclusion x+ ∈ Fδ (x) where Fδ (·) is defined by (9). We denote the solution set of this inclusion from the point x ∈ Rn by Sδ (x). This lemma resembles continuous-time results [26, Lemmas 4 & 5] which derived from [7, §7, Theorem 3]. Lemma 1 Let F (·) mapping Rn to subsets of Rn satisfy the basic conditions. Let x ∈ Rn be given and suppose that all solutions φ ∈ S(x) are defined for all k ≥ 0. Then each sequence {φn }∞ n=1 of solutions in S(x) has a subsequence converging to a function φ ∈ S(x) and the convergence is uniform on each finite time interval. The following result is similar to a result on closeness of solutions for differential inclusions (see, for example, [7, §8 Cor. 2]) and is a simplification of [10, Lemma 2.8].
(11)
Lemma 2 Let F : Rn → subsets of Rn satisfy the basic conditions. Let the triple (K, ε, x) be such that K ∈ Z>0 , ε > 0, x ∈ Rn . Then there exists δ > 0 such that for every xδ ∈ {x} + δB and every φδ ∈ Sδ (xδ ) there exists φ ∈ S(x) such that for all k ∈ {0, . . . , K} we have ||φδ (k, xδ )|A − |φ(k, x)|A | ≤ ε . (12)
The following theorem is a consequence of [10, Theorem 7.1] and [10, Remark 7.3] (see also [15, Theorem 1]):
Lemma 3 Suppose F (·) is a set-valued map from Rn to subsets of Rn continuous on an open set O ⊆ Rn and that, for each x ∈ Rn , F (x) is compact and nonempty.
sup V (f ) ≤ V (x)e−1 . f ∈F (x)
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For each triple (K, ε, x0 ) such that K ∈ Z>0 , ε > 0, and x0 ∈ Rn , and for each solution φ ∈ S(x0 ) such that φ(k, x0 ) ∈ O for all k ∈ {0, . . . , K} there exists a δ > 0 such that, for every xδ ∈ {x0 }+δB there exists a solution ψ ∈ S(xδ ) such that, for all k ∈ {0, . . . , K + 1}, ||φ(k, x0 )|A − |ψ(k, xδ )|A | ≤ ε. The following lemma appeared as [10, Lemma 2.4]. Lemma 4 If γ > 1 and µ ∈ K∞ satisfies (µ − Id) ∈ K∞ then there exists ρ ∈ K∞ such that ρ ◦ µ(s) = γ · ρ(s) for all s ≥ 0. The following lemma is [24, Proposition 7]. Lemma 5 For each β ∈ KL and λ > 0, there exist α1 , α2 ∈ K∞ such that, for all (s, t) ∈ R≥0 × R≥0 , α1 (β(s, t)) ≤ α2 (s)e−λt . The following appeared as [19, Theorem 8]. Lemma 6 Let V : Rn → R≥0 and suppose that, for α ∈ K∞ , V (φ(k +1, x))−V (φ(k, x)) ≤ −α(V (φ(k, x))). Then there exists β ∈ KL such that V (φ(k, x)) ≤ β(V (x), k), ∀k ∈ Z≥0 . (13) The following is a standard approximation result. Lemma 7 Let O ⊂ Rn be open and let µ : O → (0, ∞) be continuous. Suppose V : O → R is continuous. Then there exists a smooth function Vs : O → R such that, for all x ∈ O, |V (x) − Vs (x)| ≤ µ(x) . The following result appeared as [26, Lemma 17] which derives from [17, Lemma 4.3] and [16, Theorem 6]. Lemma 8 Let A ⊂ Rn be a closed set and assume that Vs : Rn → R≥0 is continuous, the restriction of Vs to Rn \A is smooth, Vs (x) = 0 for all x ∈ A, and Vs (x) > 0 for all x ∈ Rn \A. Then there exists a function ρ ∈ K∞ , strictly convex, and smooth on (0, ∞) such that V := ρ ◦ Vs is smooth on Rn . 5.2 Core of Proof of Theorem 1 Let β ∈ KL and φˆ ∈ S(x) come from the assumption of weak UGAS; i.e., ˆ x)|A ≤ β(|x|A , k), ∀k ∈ Z≥0 . |φ(k, (14) Let α ¯1, α ˆ 2 ∈ K∞ come from Lemma 5 such that α ¯ 1 (β(s, k)) ≤ α ˆ 2 (s)e−k . (15) n We define a function V2 : R → R≥0 by ∞ X V2 (x) := inf α ¯ 1 (|φ(k, x)|A ) , ∀x ∈ Rn . (16) φ∈S(x)
k=0
Upper and Lower bounds: We see that V2 (x) ≥ inf α ¯ 1 (|φ(0, x)|A ) = α ¯ 1 (|x|A ).
(17)
The upper bound follows from (14) and (15): ∞ ∞ X X V2 (x) ≤ α ¯ 1 (β(|x|A , k)) ≤ α ˆ 2 (|x|A ) e−k e α ˆ 2 (|x|A ) =: α ¯ 2 (|x|A ). = e−1 Preliminary Claims:
k=0
Proof: From the definition P∞ of V2∗ we have that for any φ∗ ∈ S(x), V2 (x) ≤ ¯ 1 (|φ (k, x)|A ). So we just k=0 α need to construct a solution φ∗ ∈ S(x) such that ∞ X V2 (x) ≥ α ¯ 1 (|φ∗ (k, x)|A ) . (20) k=0
∞
We let {φi }i=1 be such that ∞ X V2 (x) ≥ α ¯ 1 (|φi (k, x)|A ) − 1/i . k=0
(21)
∞
From Lemma 1, the sequence {φi }i=1 has a converging subsequence (which we won’t relabel) converging to a solution φ∗ ∈ S(x) and the convergence is uniform on finite intervals. We first claim that ∞ X V ∗ := α ¯ 1 (|φ∗ (k, x)|A ) < ∞ . (22) k=0
Suppose not. Then there exists κ ∈ Z>0 such that κ X α ¯ 1 (|φ∗ (k, x)|A ) ≥ α ¯ 2 (|x|A ) + 1 . (23) k=0
Then, using the continuity of α ¯ 1 , the uniform convergence of φi to φ∗ on {0, . . . , κ}, (21), and (23), for each ε > 0 there exists i∗ such that for all i ≥ i∗ , κ X α ¯ 1 (|φi (k, x)|A ) − 1/i V2 (x) ≥ ≥
k=0 κ X
α ¯ 1 (|φ∗ (k, x)|A ) − ε − 1/i
k=0
≥ α ¯ 2 (|x|A ) + 1 − ε − 1/i . (24) Taking the limit as ε → 0+ and i → ∞ establishes a contradiction with the upper bound (18). Using each ε > 0 there exists κε ∈ Z>0 such that Pκε (22), for ¯ 1 (|φ∗ (k, x)|A ) ≥ V ∗ −ε . Then, as above, for each k=0 α ε > 0 there exists i∗ ∈ Z>0 such that for all i ≥ i∗ , we have κε X α ¯ 1 (|φi (k, x)|A ) − 1/i V2 (x) ≥ ≥
k=0 κε X
α ¯ 1 (|φ∗ (k, x)|A ) − ε − 1/i
k=0
≥ V ∗ − 2ε − 1/i . (25) Taking the limit as ε → 0+ and i → ∞ gives V2 (x) ≥ V ∗ ; i.e., gives (20), and thus establishes the result. Claim 2 For an optimal solution φ∗ ∈ S(x), k−1 X V2 (φ∗ (k, x)) = V2 (x) − α ¯ 1 (|φ∗ (j, x)|A ).
(26)
j=0
φ∈S(x)
k=0
Claim 1 There exists a solution φ∗ ∈ S(x) (which we will call optimal) such that ∞ X V2 (x) = α ¯ 1 (|φ∗ (k, x)|A ) . (19)
Proof: Let ψ ∗ ∈ S(φ∗ (k, x)) be optimal from the point φ∗ (k, x). Using the definition of V2 (φ∗ (k, x)), the optimality of ψ ∗ , and the definition of V2 (x) we may write ∞ X V2 (φ∗ (k, x)) = α ¯ 1 (|ψ ∗ (j, φ∗ (k, x))|A )
k=0
j=0
(18) ≤ V2 (x) −
k−1 X i=0
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α ¯ 1 (|φ∗ (i, x)|A ) .
(27)
Reversing directions, and this time using the optimality of φ∗ , it is easy to see that k−1 X V2 (x) ≤ α ¯ 1 (|φ∗ (i, x)|A ) + V2 (φ∗ (k, x)). i=0
Continuity: We define 1 α ˜ (s) := min s, α ¯1 ◦ α ¯ 2−1 (s) , ∀s ∈ R≥0 , (28) 2 and note that Id − α ˜ ∈ K∞ . We first address lower semicontinuity of V2 (·). Appealing to Claim 2 twice and using (18) we see that, along an optimal trajectory φ∗ ∈ S(x), V2 (φ∗ (k + 1, x)) − V2 (φ∗ (k, x)) = −¯ α1 (|φ∗ (k, x)|A ) ≤ −¯ α1 ◦ α ¯ 2−1 (V2 (φ∗ (k, x))) ≤ −˜ α (V2 (φ∗ (k, x))) . Using Lemma 6 we have the existence of a function βs ∈ KL such that, along an optimal trajectory φ∗ ∈ S(x), V2 (φ∗ (k, x)) ≤ βs (V2 (x), k), ∀k ∈ Z≥0 . (29) Therefore, using (17) and (18), ˜ A , k). (30) |φ∗ (k, x)|A ≤ α ¯ 1−1 (βs (¯ α2 (|x|A ), k)) =: β(|x| ∗ Let ε > 0. Let xn → x and φn ∈ S(xn ) be optimal for each n. Let ε˜ := min α ¯ 2−1 2ε , 1 and choose κ ∈ Z>0 such that ˜ A + 1, κ) ≤ ε˜. 2β(|x| (31) ε Since α ¯ 1 (·) is continuous, for 2κ > 0 on the compact ˜ A + 1, 0) + 1] there exists δ ∈ (0, ε˜ ] such interval [0, β(|x| 2 ˜ A + 1, 0) + 1] then that, if s1 , s2 ∈ [0, β(|x| ε |s1 − s2 | ≤ δ =⇒ |¯ α1 (s1 ) − α ¯ 1 (s2 )| ≤ . (32) 2κ With the triple (κ, δ, x) Lemma 2 yields a δ1 > 0 and a solution φˆn ∈ S(x) defined on {0, . . . , κ} such that, for each n such that |xn − x| < δ1 , ||φˆn (k, x)|A − |φ∗n (k, xn )|A | ≤ δ, ∀k ∈ {0, . . . , κ}. (33) Since | · |A is globally Lipschitz with Lipschitz constant 1, using (32) we may write that, for all k ∈ {0, . . . , κ}, ε . (34) |¯ α1 (|φˆn (k, x)|A ) − α ¯ 1 (|φ∗n (k, xn )|A )| ≤ 2κ Let φ¯n ∈ S(φˆn (κ, x)) be optimal. We define φˆn (k, x), k ∈ {0, . . . , κ} ˜ φn (k, x) := φ¯n (k − κ, φˆn (κ, x)), k ∈ Z>κ . (35) Then, by definition, ∞ X V2 (φ˜n (κ, x)) = α ¯ 1 (|φ˜n (k, x)|A ). (36)
Therefore, since ε > 0 was arbitrary, V2 (·) is lower semicontinuous on Rn . We now demonstrate upper semicontinuity of V2 (·) on A. Let x ∈ A and recall that this implies V2 (x) = 0. Using the upper bound (18), since α ¯ 2 (·) is continuous, for any ε > 0 there exists δ > 0 such that if |x − z| < δ we may write V2 (z) ≤ α ¯ 2 (|z|A ) = α ¯ 2 (|z|A ) + V2 (x) ≤ ε + V2 (x). Therefore, V2 (·) is upper semicontinuous on A. All that remains is to prove upper semicontinuity of V2 (·) on Rn \A. For x ∈ Rn \A, letφ∗ ∈ S(x) be optimal. Pick ε > 0 and define ε˜ := α ¯ 2−1 2ε . Let κ ∈ Z>0 be such that κ − 1 is the smallestP value such that φ∗ (κ − 1, x) ∈ Rn \A ∞ ∗ and V2 (φ (κ, x)) = k=κ α ¯ 1 (|φ∗ (k, x)|A ) < α ¯ 1 2ε˜ . Note that, with (17), this implies that ε˜ (40) |φ∗ (κ, x)|A ≤ . 2 Since α ¯ 1 (·) is continuous, with the modulus of continuity ε interval [0, α ¯ 2 (|x|A + 1) + 1] there 2κ , on the compact ε˜ exists δ ∈ 0, 2 such that for s1 , s2 ∈ [0, α ¯ 2 (|x|A +1)+1], ε |s1 − s2 | ≤ δ =⇒ |¯ α1 (s1 ) − α ¯ 1 (s2 )| ≤ . (41) 2κ Since F (·) is continuous on Rn \A, with the triple (κ − 1, δ, x) and the solution φ∗ ∈ S(x), Lemma 3 yields δ1 > 0 such that for all z ∈ {x} + δ1 B there exists ψ ∈ S(z) such that, for all k ∈ {0, . . . , κ}, ||φ∗ (k, x)|A − |ψ(k, z)|A | ≤ δ. With (41) this implies that κ−1 X
k=0
≥ V2 (x) − ε.
κ−1 ε X ≤ α ¯ 1 (|φ∗ (k, x)|A ) . 2
(42)
k=0
k=0
Let ψ ∗ ∈ S(ψ(κ, z)) be optimal; i.e., ∞ X V2 (ψ(κ, z)) = α ¯ 1 (|ψ ∗ (k − κ, ψ(κ, z))|A ) .
(43)
k=κ
Since | · |A is globally Lipschitz with Lipschitz constant 1 and using (40) with the fact that δ ≤ 2ε˜ we have that |φ∗ (κ, x)|A + |φ∗ (κ, x) − ψ(κ, z)| ε˜ + δ ≤ ε˜. (44) ≤ 2 Therefore, using the upper bound (18) and the definition of ε˜ we see that ε V2 (ψ(κ, z)) ≤ α ¯ 2 (|ψ(κ, z)|A ) ≤ α ¯ 2 (˜ ε) = . (45) 2 Therefore, using Claim 2, (42), (43), the nonoptimality of ψ ∈ S(z), and (45) one may show |ψ(κ, z)|A
k=κ
With (30) and (31), we see that, for n sufficiently large ˜ A + 1, κ) ≤ ε˜ . |φ∗n (κ, xn )|A ≤ β(|x| (37) 2 Therefore, from (33) (with δ ≤ 2ε˜ ) and (35) we have |φ˜n (κ, x)|A ≤ ε˜ and, with the upper bound (18) ε V2 (φ˜n (κ, x)) ≤ α ¯ 2 (|φ˜n (κ, x)|A ) ≤ α ¯ 2 (˜ ε) = . (38) 2 Using Claim 2, (34), (35), (36), and (38) we have κ−1 X ε V2 (xn ) ≥ α ¯ 1 (|φ˜n (k, x)|A ) − + V2 (φ∗n (κ, xn )) 2
α ¯ 1 (|ψ(k, z)|A ) −
V2 (x) =
≤
κ−1 X
α ¯ 1 (|φ∗ (k, x)|A ) + V2 (φ∗ (κ, x))
k=0
ε − V2 (ψ(κ, z)) ≥ V2 (z) − ε, 2 which establishes upper semicontinuity on Rn \A. Decrease Condition: Since φ∗ (1, x) ∈ F (x) we may use Claim 2 and (28) to write min V2 (f ) ≤ V2 (x) − α ¯ 1 (|x|A ) f ∈F (x) ≤ V2 (x) − α ¯1 α ¯ 2−1 (V2 (x)) ≤ V2 (x) − α ˜ (V2 (x)). (46) ≥ V2 (z) −
Recall that (Id − α ˜ ) ∈ K∞ . Let µ ∈ K∞ be defined as
(39)
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µ(s) := (Id − α ˜ )−1 (s),
∀s ≥ 0.
(47)
We see then that µ−1 (s) = s − α ˜ (s) (48) −1 or s − µ (s) = α ˜ (s) or µ(s) − s = α ˜ ◦ µ(s) from which it follows that (µ − Id) ∈ K∞ . Let γ = e1 > 1 and let ρ ∈ K∞ come from Lemma 4 and define V1 := ρ(V2 ). Combining (46) and (48) we see that min V2 (f ) ≤ µ−1 (V2 (x)), (49) f ∈F (x)
which allows us to write min V1 (f ) = min ρ(V2 (f )) ≤ ρ ◦ µ−1 (V2 (x)) f ∈F (x)
f ∈F (x)
= e−1 ρ(V2 (x)) = e−1 V1 (x).
(50)
We note that V1 (·) is continuous by virtue of the continuity of V2 (·) and of ρ(·). Finally, we define K∞ functions α := ρ ◦ α ¯ 1 , and α := ρ◦α ¯ 2 so that α(|x|A ) ≤ V1 (x) ≤ α(|x|A ). 1/2
Smoothing: Let ε > 0 be such that ε ≤ ee1/2 −1 . We +1 appeal to Lemma 7 with µ(x) = εV1 (x) for all x ∈ Rn \A. Then we have that, for all x ∈ Rn \A, (1 − ε)V1 (x) ≤ Vs (x) ≤ (1 + ε)V1 (x). (51) We complete the definition of Vs by defining Vs (x) = 0 for x ∈ A. Using (50), (51), and the bound on ε we may write min Vs (f ) ≤ (1 + ε) min V1 (f ) ≤ (1 + ε)e−1 V1 (x)
f ∈F (x)
f ∈F (x)
≤
1 + ε −1 e Vs (x) ≤ e−1/2 Vs (x). 1−ε
Let V¯s = (Vs )2 . Then min V¯s (f ) = min Vs2 (f ) ≤ e−1 Vs2 (x) = e−1 V¯s (x). f ∈F (x)
f ∈F (x)
We note that V¯s satisfies all the conditions of Lemma 8. Let ρ ∈ K∞ come from Lemma 8. Then V := ρ ◦ V¯s is smooth on Rn and, since ρ(·) is convex, we may write ¯ min V (f ) = ρ min Vs (f ) f ∈F (x) f ∈F (x) ≤ ρ e−1 V¯s (x) ≤ e−1 ρ(V¯s (x)) = e−1 V (x). The upper and lower bounds on V (·) are then α1 (s) := ρ 21 α(s) , and α2 (s) := ρ(2α(s)), for all s ≥ 0. References [1] F. Albertini and E. D. Sontag. Continuous controlLyapunov functions for asymptotically controllable timevarying systems. Int. J. Control, 72:1630–1641, 1999. [2] J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkh¨ auser, 1990. [3] F. H. Clarke, Y. S. Ledyaev, L. Rifford, and R. J. Stern. Feedback stabilization and Lyapunov functions. SIAM J. Control Optim., 39(1):25–48, 2000. [4] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag, and A. I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control, 42:1394– 1407, 1997. [5] F. H. Clarke, Y. S. Ledyaev, and R. J. Stern. Asymptotic stability and smooth Lyapunov functions. J. Differential Equations, 149:69–114, 1998. [6] K. Deimling. Multivalued Differential Equations. Walter de Gruyter, 1992. [7] A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Pub., 1988.
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