- Email: [email protected]
RELAXED CHARACTERIZATIONS OF SMOOTH PATCHY CONTROL LYAPUNOV FUNCTIONS
RELAXED CHARACTERIZATIONS OF SMOOTH PATCHY CONTROL LYAPUNOV FUNCTIONS Rafal Goebel ∗ Christophe Prieur ∗∗,1 Andrew R. Teel ∗∗∗,2
∗
Department of Mathematics, University of Washington, Seattle, WA 98195-4350 [email protected] ∗∗ LAAS–CNRS, 7, Avenue du Colonel Roche 31077 Toulouse, Cedex 4, France [email protected] ∗∗∗ Center for Control, Dynamical Systems & Computation, Department of Electrical and Computer Engineering, University of California Santa Barbara, CA 93106-9560 [email protected]
Abstract: A smooth patchy control Lyapunov function for a nonlinear control system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a certain arrangement property. In contrast to (Goebel et al., IEEE CDC, 2006), on each intersection of the domains, we do not impose a decrease condition of the local control Lyapunov functions when the index increases. We show a construction, based on such a patchy control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise. Copyright © 2007 IFAC Keywords: hybrid systems, control Lyapunov function, stabilization
1. INTRODUCTION While every asymptotically controllable nonlinear control system admits a locally Lipschitz, semiconcave clf ((Sontag, 1983), (Rifford, 2002)), not every such system (even if it is affine in the control variable) admits a continuously differentiable clf. Systems that do not admit a contin1
Research partly done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368. 2 Research partially supported by ARO under Grant no. DAAD19-03-1-0144, NSF under Grant no. CCR-0311084, ECS-0324679, and ECS-0622253 and by AFOSR under Grant no. AFOSR F9550-06-1-0134.
uously differentiable clf include the systems that fail Brockett’s condition, see (Ryan, 1994). However it is shown in (Goebel et al., 2006) that each asymptotically controllable system admits a proper smooth patchy control Lyapunov function. Roughly speaking, a smooth patchy control Lyapunov function consists of a family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space, and which satisfy a certain arrangement property. It is said to be proper if, moreover, on each intersection of the domains, the local control Lyapunov functions decrease when the index increases. Given such a proper smooth patchy control Lyapunov
function, it is shown in (Goebel et al., 2006) that we can build a hybrid stabilizer. Moreover the stability is stated to be robust with respect to (small) measurement noise and (small) external disturbances.
the notion of pclf, and the main result. Section 6 contains concluding remarks. Due to space limitation, the proofs are omitted.
Other kinds of feedback stabilizers can be considered in the absence of a continuously differentiable clf. Consider e.g. (Clarke et al., 1997) and (Ancona and Bressan, 1999) where asymptotic feedback stabilization is achieved with some robustness to additive disturbances (but no robustness to measurement noise). The feedback of (Ancona and Bressan, 1999) was later shown, in (Ancona and Bressan, 2003), to have some robustness to measurement noise. Sample and hold implementation of the feedback of (Clarke et al., 1997) also induces some robustness to measurement noise; see (Clarke et al., 1997; Sontag, 1999). Such implementation adds a dynamic discrete variable to the closed loop system. Together with a hysteresis-based implementation of the feedback of (Ancona and Bressan, 1999), as used in (Prieur, 2005; Prieur et al., 2007), this motivates the study of hybrid systems where two kinds of dynamic equations are considered: a discrete one and a continuous one. The closed-loop system resulting from the feedback constructed in (Prieur et al., 2007) essentially fits the form of the general class of hybrid systems studied in (Goebel and Teel, 2006). In particular, under appropriate regularity assumptions, it is shown in (Prieur et al., 2007) that all asymptotically stable hybrid systems are robust to measurement noise and external disturbances.
2. PRELIMINARIES
The previous work (Goebel et al., 2006) and this historical perspective motivate us to consider systems having a smooth patchy control Lyapunov function. In the present work, we state that, with a smooth patchy control Lyapunov in hand, it is possible to build a hybrid stabilizer so that the closed-loop system fits the class of hybrid systems consider in (Goebel and Teel, 2006; Prieur et al., 2007). We generalize the result of (Goebel et al., 2006) since, on the intersection of the domains, we do not need a decreasing condition of the control Lyapunov functions as the index increases. This condition is replaced by a weaker assumption, more precisely by an arrangement property (see Theorem 4.3 below). The paper is organized as follows. First we introduce some background material of hybrid systems in Section 2. Then we introduce the notion of a smooth patchy control Lyapunov function (pclf for short) for a finite number of patches, and we state a first sufficient condition for the feedback stabilization by means of a hybrid controller in Section 3. The case of an infinite case of patches is considered in Section 4. In Section 5 we illustrate
e ⊂ Rn is an open set and Throughout the paper, O e A ⊂ O is compact. We will be interested in hybrid feedback stabilization of the nonlinear system x(t) ˙ = f (x(t), u(t)), u(t) ∈ U, for all t ≥ 0,(1) e × U → Rn where U ⊂ Rk is a set and f : O is a (nonlinear in general) continuous mapping. The state space for the continuous variable of the e but hybrid feedback will not necessarily cover O, e will be an open set O such that O ⊂ O ⊂ O, e \ A. where O = O Definition 2.1. A hybrid feedback consists of • a totally ordered countable set Q • for each q ∈ Q, · sets Cq ⊂ O and Dq ⊂ O, · a function kq : Cq → U , · a set-valued mapping Gq : Dq → → Q. In closed loop with the nonlinear system (1), a hybrid feedback as in Definition 2.1 leads to a hybrid system x˙ = f (x, kq (x)) if x ∈ Cq , q + ∈ Gq (x) if x ∈ Dq .
(2)
The state space for (2) is O × Q. To make the definition of the hybrid system (2) and the notion of solutions to such a system precise, we recall some concepts from (Goebel and Teel, 2006). A subset S ⊂ R≥0 × N is a hybrid time domain if S is a union of a finite or infinite sequence of intervals [tj , tj+1 ] × {j}, with the last interval, if it exists, possibly of the form [tj , T ) with T finite or T = +∞. A solution to the hybrid system (2) consists of: a nonempty hybrid time domain S, a function x : S → O with x(t, j) locally absolutely continuous in t for a fixed j and constant in j for a fixed t, and a function q : S → Q with q(t, j) constant in t for a fixed j, meeting the following conditions: x(0, 0) ∈ Cq(0,0) ∪ Dq(0,0) and (S1) For all j ∈ N and almost all t such that (t, j) ∈ S, x(t, ˙ j) = f (x(t, j), kq(t,j) (x(t, j)), x(t, j) ∈ Cq(t,j) . (S2) For all (t, j) ∈ S such that (t, j + 1) ∈ S, q(t, j+1) ∈ Gq(t,j) (x(t, j)), x(t, j) ∈ Dq(t,j) .
Given a solution to (2) we will usually not mention the hybrid time domain explicitly, but will identify the solution by (x, q), and when needed, refer to the associated domain by dom(x, q). In what follows, we will write supt (S) for the supremum of all t such that (t, j) ∈ S for some j.
and the following conditions are met: there exist a continuous function α : (0, ∞) → (0, ∞), classK∞ functions γ, γ, and a function ω which is a e such that: proper indicator of A with respect to O S (iv) for all q ∈ Q, all x ∈ Ωq \ r>q Ω0r ,
We will say that the set A is stable for the hybrid system (2) if for any ε > 0 there exists δ > 0 such that any solution (x, q) with distA (x(0, 0)) ≤ δ satisfies distA (x(t, j)) ≤ ε for all (t, j) ∈ dom(x, q). The set A is globally attractive for the hybrid system (2) if
γ(ω(x)) ≤ Vq (x) ≤ γ(ω(x)) ; S (v) for all q ∈ Q, all x ∈ Ωq \ r>q Ω0r , there exists uq,x ∈ U such that ∇Vq (x) · f (x, uq,x ) ≤ −α(ω(x)); S (vi) for all q ∈ Q, all x ∈ ∂Ωq \ r>q Ω0r ∩ O, the uq,x of (v) can be chosen such that
• for any (x0 , q0 ) ∈ O × Q there exists a solution to (2) with x(0, 0) = x0 , q(0, 0) = q0 ; • for any maximal solution (x, q) to (2) we have distA (x(t, j)) → 0 as t → supt (dom(x, q)). Finally, we will say that A is (globally) asymptotically stable for (2) if it is both stable and globally attractive. In particular, note that our concept of asymptotic stability of A for (2) refers only to the behavior of the “continuous” part of solutions and also that we are allowing for solutions reaching A in finite time.
nq (x) · f (x, uq,x ) ≤ −α(ω(x)), where nq (x) is the unit (outward) normal vector to Ωq at x. We say that a smooth patchy control Lyapunov function is proper if additionally the following condition holds: (vii) for all q, r ∈ Q, r > q, all x ∈ Ωq ∩ ∂Ω0r , Vr (x) ≤ Vq (x). If this inequality is strict, we say that a proper smooth patchy control Lyapunov function is strict.
3. FINITE NUMBER OF PATCHES AND A SUFFICIENT CONDITION FOR FEEDBACK STABILIZATION 3.2 Stabilizing feedback 3.1 PCFL with finite number of patches Below, given a set Ω, its boundary is denoted by ∂Ω. By a proper indicator of A with respect to e we will understand a function ω : O e → R≥0 O that is continuous, positive definite with respect to A, and that approaches ∞ if the argument e or its norm apapproaches the boundary of O proaches ∞. Definition 3.1. A smooth patchy control Lyapunov function (with finite number of patches) for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that (i) Q ⊂ Z is a finite set; (ii) {Ωq }q∈Q and {Ω0q }q∈Q are families of nonempty e such that open subsets of O [ [ e where O := O ⊂ O ⊂ O, Ωq = Ω0q , q∈Q
With a patchy control Lyapunov function, and under a convexity assumption on the nonlinear system, we can design a hybrid feedback on O for (1) that renders A asymptotically stable with the basin of attraction equal to O. We note that the convexity assumption on the map f below is automatically satisfied when the system (1) is affine with respect to the control variable. Theorem 3.2. Suppose that • there exists a smooth patchy control Lyapunov function (with finitely many patches) for (1) with the attractor A; • for any v ∈ Rn , c ∈ R, the set {u ∈ U | v · f (x, u) ≤ c} is convex. Then, there exists a hybrid feedback on O for (1) that renders A asymptotically stable with the basin of attraction equal to O.
q∈Q
and for all q ∈ Q, the unit (outward) normal S vector to ∂Ωq is continuous on ∂Ωq \ r>q Ω0r ∩ O, and Ω0q ∩ O ⊂ Ωq ; (iii) for each q, Vq is a smooth function defined Son a (relative to O) neighborhood of Ωq \ r>q Ω0r ;
Remark 3.3. Some observations are in order. • Combining (Prieur et al., 2007, Theorem 4.3) and this result, and using the regularity of the data of our hybrid feedback, we may state the stronger result that A is asymptotically stable robustly to measurement noise, actuator errors and external disturbances. For more details see (Prieur et al., 2007).
• This result generalizes (Goebel et al., 2006, Theorem 4.1) since in this paper it is considered only proper smooth patchy control Lyapunov functions. The proof of this result has three main steps. First, for each q ∈ Q, we construct a “local” continuous feedback on each patch. Lemma 3.4. Under the assumptions of Theorem 3.2, for each q ∈S Q there exists a continuous mapping kq : Ωq \ r>q Ω0r ∩ O → U such that S (a) for all x ∈ Ωq \ r>q Ω0r ∩ O, ∇Vq (x) · f (x, kq (x)) ≤ −α(ω(x))/2; S (b) for all x ∈ ∂Ωq \ r>q Ω0r ∩ O, nq (x) · f (x, kq (x)) ≤ −α(ω(x))/2. The second step defines the remaining data that is needed to turn the collection of (continuous time) feedbacks kq into a hybrid feedback. We do this explicitly, by setting: Cq = Ωq \
[
Ω0r ∩ O
r>q
Dq =
[
Ω0r ∩ O ∪ (O \ Ωq )
r>q
{r ∈ Q | x ∈ Ω0r ∩ O, r!> q} (3) [ if x ∈ Ω0r ∩ Ωq r>q Gq (x) = {r ∈ Q | x ∈ Ω0 ∩ O} r if x ∈ O \ Ωq
It is shown in (Goebel et al., 2006) that for all asymptotically controllable systems, there exists a strict proper patchy control Lyapunov function 3 . Let us investigate the sufficiency part. To do that we need the following: Definition 4.2. A patchy control Lyapunov function for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that Q ⊂ Z, conditions (ii)(vi) of Definition 3.1 hold, and the hybrid feedback given by (3) and any functions kq satisfying the conditions of Lemma 3.4 renders A asymptotically stable. We have shown, in Theorem 3.2, that a patchy control Lyapunov function with finitely many patches is a patchy control Lyapunov function (in the sense of the definition above) subject to a convexity assumption on the dynamics. Below, we will show that a proper smooth PCLF is indeed a PCLF in the sense of the definition above. First, we propose a different sufficient condition for a collection of local control Lyapunov functions to be a PCLF. This condition does not rely on a decrease condition of Vq on the regions where the patches overlap. Rather, it places certain conditions on how the patches need to be arranged. Theorem 4.3. Suppose that a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q is such that Q ⊂ Z, conditions (ii)-(vi) of Definition 3.1 hold and, for each N ∈ N ∪ {∞}, (pa1 ) for each ε > 0 there exists δ such that, for all increasing sequence (qn )0≤n≤N in Q, and for all sequence (xn )0≤n≤N in O satisfying x0 ∈ Ωq0 , Vq0 (x0 ) ≤ δ , and, for all 0 ≤ n ≤ N ,
In the third step, we use the collection of local control Lyapunov functions Vq to show that the constructed feedback is stabilizing.
4. INFINITE NUMBER OF PATCHES AND A NECESSARY AND SUFFICIENT CONDITION
xn+1 ∈ Ωqn ∩∂Ω0qn+1 , Vqn (xn+1 ) ≤ Vqn (xn ) , we have Vqn (xn ) ≤ ε, for all 0 ≤ n ≤ N ; (pa2 ) for all increasing sequence (qn )0≤n≤N in Q, and for all sequence (xn )0≤n≤N in O satisfying x0 ∈ Ωq0 , and, for all 0 ≤ n ≤ N ,
We will say that a family {Ωq }q∈Q is a locally finite family if for any compact K subset of O, there is finitely many q’s such that K ∩ Ωq 6= ∅.
xn+1 ∈ Ωqn ∩∂Ω0qn+1 , Vqn (xn+1 ) ≤ Vqn (xn ) ,
Definition 4.1. A proper patchy control Lyapunov function for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that Q ⊂ Z and conditions (ii)-(vii) of Definition 3.1 hold. A proper patchy control Lyapunov function is strict if the inequality in condition (vii) is strict.
Then this collection is a patchy control Lyapunov function for (1).
there exists M > 0 such that Vqn (xn ) ≤ M , for all 0 ≤ n ≤ N .
3
In fact, it is stated in (Goebel et al., 2006, Theorem 5.3) that there exists a proper patchy control Lyapunov function. However a small modification of the proof of this result gives the existence of a strict proper pclf.
Proposition 4.4. A proper smooth patchy control Lyapunov function for (1) is a smooth patchy control Lyapunov function for (1). Remark 4.5. Let us note that we recover (Goebel et al., 2006, Theorem 4.1) since we get that, under the convexity assumption of Theorem 3.2, if there exists a proper smooth patchy control Lyapunov function, then the hybrid feedback given by (3) renders A asymptotically stable.
5. ILLUSTRATION To illustrate the definitions and the results of the present work, we consider the Brockett integrator: x˙ 1 = u1 , x˙ 2 = u2 , (4) x˙ 3 = x1 u2 − x2 u1 . It is known that the necessary condition (Brockett, 1983) for the stabilization by means of a continuous feedback or robust stabilization by locally bounded feedback (Ryan, 1994) does not hold for this system. Let us check that we can prove the existence of a hybrid stabilizing feedback by applying Theorem 3.2. Let us denote f (x, u) the right-hand side of the system (4), where x = (x1 , x2 , p x3 ) and u = (u1 , u2 ). We use the notation: r = x21 + x22 . We consider a simplified version of the hybrid controller of (Hespanha and Morse, 1999) (see also (Goebel et al., 2004)) and we define a patchy control Lyapunov function as a collection of two patches. These patches Ω0q ⊂ Ωq , for q = 1, 2 are sketched in Figure 1 in the (r, x3 ) plane (they are surfaces of revolution around the x3 axis and symmetric with respect to (0, r) axis). x3
Ω2 Ω2’
r
Fig. 1. Sketch of the sets Ω02 ⊂ Ω2 in the (r, x3 ) plane, for x3 > 0 To define the first one, let us consider the following sets Ω01 = Ω1 = R3 , and the function V1 : R3 → R, p defined by V1 (x) = (ρ + ε) |x3 | − x1 , for all x ∈ R3 , where 1 < ρ and 0 < ε < 1 will be prescribed below.
Now let us define the second patch, and let us consider the following sets Ω02 = {x 6= 0, r2 > ρ|x3 |} , Ω2 = {x 6= 0, r2 > |x3 |} , and the smooth function V2 : R3 → R, defined by V2 (x) = 21 (r2 + x23 ). Observe that V1 is a smooth function on a (relative to R3 \ {0}) neighborhood of Ω1 \ Ω02 . We have (iii) of Definition 3.1. The index set Q = {1, 2} verifies (i) of Definition 3.1. The families {Ω1 , Ω2 }, {Ω01 , Ω02 } consist of nonempty and open sets and cover O = R3 \ {0}, and we have (ii) of Definition 3.1. For all x ∈ Ω1 \ Ω02 , we have ε
p p |x3 | εr + √ ≤ ε |x3 | ≤ V1 (x) , 2 2 ρ p V1 (x) ≤ (ρ + ε) |x3 | + r .
(5) (6)
Thus V is positive definite and proper, and therefore it verifies (iv) of Definition 3.1. Now set u1,x = (1, 0), for all x ∈ R3 . Observe that 4 , for all x ∈ Ω1 \ Ω02 , 1 sgn(x3 ) ∇V1 (x) · f (x, u1,x ) = − (ρ + ε) p x2 − 1 , 2 |x3 | √ (ρ + ε) ρ −1 . (7) ≤ 2 √ Thus by picking ρ and ε such that (ρ + ε) ρ < 2, this verifies (v) for q = 1 of Definition 3.1 with √ (ρ+ε) ρ α(|x|) ≤ 1 − being prescribed below. 2 Now set u2,x = (−x1 + 4 xr2 x2 3 , −x2 − 4 xr1 x2 3 ), for all x ∈ Ω2 . We compute, for all x ∈ Ω2 , ∇V2 (x) · f (x, u2,x ) = −r2 − 4x23 . This verifies (v) for q = 2 of Definition 3.1, with α(|x|) ≤ |x|2 being prescribed below. Since ∂Ω1 \ Ω02 is empty, (vi) of Definition 3.1 verifies for q = 1. Let us consider now x ∈ ∂Ω2 ∩O. The unit normal vector n2 (x) to Ω2 at x is n2 (x) = √ 1 (−2x1 , −2x2 , sgn(x3 ))0 . Using r2 = |x3 |, 4r 2 +1 and |x|2 = |x3 | + |x3 |2 , for all x ∈ ∂Ω2 ∩ O, we compute 2r2 − 4|x3 | , n2 (x) · f (x, u2,x ) = √ 4r2 + 1 p ≤ −2|x3 | = 1 − 1 + 4|x|2 . This verifies (vi) of Definition 3.1 for q = 2 with √ p (ρ+ε) ρ 2 α(|x|) = min(1 − , |x| , 1 + 4|x|2 − 1). 2 As a conclusion of these computations, we can claim that the set Q, the families of open sets {Ωq }q∈Q and {Ω0q }q∈Q , and the family of functions {Vq }q∈Q consist a patchy control Lyapunov function (with finite number of patches) for (4). 4
Here and in what follows, sgn(x3 ) is the sign of x3 6= 0, and |.| denotes the Euclidian norm.
Control, Optimisation and Calculus of Variations 4, 445–471. Ancona, F. and A. Bressan (2003). Flow stability of patchy vector fields and robust feedback stabilization. SIAM Journal on Control and Optimization 41(5), 1455–1476. It is also possible to scale V1 to get a proper, Brockett, R.W. (1983). Asymptotic stability in fact strict proper, patchy control Lyapunov and feedback stabilization. In: Differenfunction. First, we may prove that there exists tial geometric control theory. pp. 181–191. a continuously differentiable class-K∞ function γ Birkhauser. such that, for all x ∈ Ω1 \ ∂Ω02 , Clarke, F.H., Y.S. Ledyaev, E.D. Sontag and A.I. Subbotin (1997). Asymptotic controllability |x|2 ). (8) V1 (x) ≥ γ( implies feedback stabilization. IEEE Trans. 2 Automat. Control 42(10), 1394–1407. Goebel, R. and A.R. Teel (2006). Solutions to hyNow, let us define W1 : R3 → R by W1 (x) = brid inclusions via set and graphical converγ −1 (V1 (x)). Since V1 verifies (iv) of Definition 3.1, gence with stability theory applications. AuW1 also verifies this condition. tomatica 42(4), 573–587. With (7) and since the γ −1 has a positive derivaGoebel, R., C. Prieur and A.R. Teel (2006). tive, we compute, for all x ∈ Ω1 \ Ω02 , Smooth patchy control Lyapunov functions. In: Proc. 45th IEEE Conf. on Decision and dγ −1 Control. San Diego, CA. pp. 3271–3276. (V1 (x)) ∇V1 (x) · f (x, u1,x ), ∇W1 (x) · f (x, u1,x ) = ds Goebel, R., J. Hespanha, A.R. Teel, C. Cai and √ (ρ + ε) ρ dγ −1 R. Sanfelice (2004). Hybrid systems: gen≤( − 1) (V1 (x)), 2 ds eralized solutions and robust stability. In: ≤ −e α(|x|) , IFAC Symp. on Nonlinear Control Systems. Stuttgart, Germany. pp. 1–12. where, for all s ∈ (0, ∞), Hespanha, J.P. and A.S. Morse (1999). Stabiliza√ (ρ + ε) ρ dγ −1 tion of nonholonomic integrators via logicα e(s) = max (1− ) (V1 (x)) , 2 ds x∈Ω1 \Ω02 , |x|≤s based switching. Automatica 35(3), 385–393. Prieur, C. (2005). Asymptotic controllability and Due to (5)-(6), α e : (0, ∞) → (0, ∞) is well-defined robust asymptotic stabilizability. SIAM J. and the function α e is continuous. Thus W1 verifies Control Opt. 43, 1888–1912. (v) of Definition 3.1, and due to (8), it verifies (vii) Prieur, C. and E. Tr´elat (2005). Robust optimal of Definition 3.1. stabilization of the Brockett integrator via a As a conclusion of these computations, we can hybrid feedback. Math. Control Signals Sysclaim that the set Q, the families of open sets tems 17, 201–216. {Ωq }q∈Q and {Ω0q }q∈Q , and the family of functions Prieur, C. and E. Tr´elat (2006). Quasi-optimal {W1 , V2 } consist a strict proper pclf for (4). robust stabilization of control systems. SIAM J. Control Opt. 45(5), 1875–1897. We recover partially the result of (Prieur and Prieur, C., R. Goebel and A.R. Teel (2007). HyTr´elat, 2005) (see also (Prieur and Tr´elat, 2006)) brid feedback control and robust stabilization where an optimization criterion is also considered. of nonlinear systems. IEEE Trans. Autom. Control. To appear. Rifford, L. (2002). Semiconcave control-Lyapunov 6. CONCLUSION functions and stabilizing feedbacks. SIAM J. This work considered the concept of a smooth Control Optim. 41(3), 659–681. patchy control Lyapunov function (pclf). The Ryan, E.P. (1994). On Brockett’s condition for main result states that, under a mild convexity smooth stabilizability and its necessity in assumption, the existence of such pclf implies the a context of nonsmooth feedback. SIAM J. construction of a stabilizing hybrid feedback. It Control Optim. 32(6), 1597–1604. generalizes (Goebel et al., 2006). Moreover, using Sontag, E.D. (1983). A Lyapunov-like characteri(Prieur et al., 2007), we can show that this hybrid zation of asymptotic controllability. SIAM J. controller is robust with respect to measurement Control Optim. 21(3), 462–471. noise and external disturbances. Sontag, E.D. (1999). Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var. 4, 537–557. REFERENCES Moreover, since the system (4) is affine with respect to u, the convexity assumption in Theorem 3.2 holds. Thus by defining uq : Cq → R2 by uq (x) = uq,x , for all x ∈ Cq , the hybrid controller (3) renders 0 asymptotically stable on R3 \ {0}.
Ancona, F. and A. Bressan (1999). Patchy vector fields and asymptotic stabilization. ESAIM:
∗
Department of Mathematics, University of Washington, Seattle, WA 98195-4350 [email protected] ∗∗ LAAS–CNRS, 7, Avenue du Colonel Roche 31077 Toulouse, Cedex 4, France [email protected] ∗∗∗ Center for Control, Dynamical Systems & Computation, Department of Electrical and Computer Engineering, University of California Santa Barbara, CA 93106-9560 [email protected]
Abstract: A smooth patchy control Lyapunov function for a nonlinear control system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a certain arrangement property. In contrast to (Goebel et al., IEEE CDC, 2006), on each intersection of the domains, we do not impose a decrease condition of the local control Lyapunov functions when the index increases. We show a construction, based on such a patchy control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise. Copyright © 2007 IFAC Keywords: hybrid systems, control Lyapunov function, stabilization
1. INTRODUCTION While every asymptotically controllable nonlinear control system admits a locally Lipschitz, semiconcave clf ((Sontag, 1983), (Rifford, 2002)), not every such system (even if it is affine in the control variable) admits a continuously differentiable clf. Systems that do not admit a contin1
Research partly done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368. 2 Research partially supported by ARO under Grant no. DAAD19-03-1-0144, NSF under Grant no. CCR-0311084, ECS-0324679, and ECS-0622253 and by AFOSR under Grant no. AFOSR F9550-06-1-0134.
uously differentiable clf include the systems that fail Brockett’s condition, see (Ryan, 1994). However it is shown in (Goebel et al., 2006) that each asymptotically controllable system admits a proper smooth patchy control Lyapunov function. Roughly speaking, a smooth patchy control Lyapunov function consists of a family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space, and which satisfy a certain arrangement property. It is said to be proper if, moreover, on each intersection of the domains, the local control Lyapunov functions decrease when the index increases. Given such a proper smooth patchy control Lyapunov
function, it is shown in (Goebel et al., 2006) that we can build a hybrid stabilizer. Moreover the stability is stated to be robust with respect to (small) measurement noise and (small) external disturbances.
the notion of pclf, and the main result. Section 6 contains concluding remarks. Due to space limitation, the proofs are omitted.
Other kinds of feedback stabilizers can be considered in the absence of a continuously differentiable clf. Consider e.g. (Clarke et al., 1997) and (Ancona and Bressan, 1999) where asymptotic feedback stabilization is achieved with some robustness to additive disturbances (but no robustness to measurement noise). The feedback of (Ancona and Bressan, 1999) was later shown, in (Ancona and Bressan, 2003), to have some robustness to measurement noise. Sample and hold implementation of the feedback of (Clarke et al., 1997) also induces some robustness to measurement noise; see (Clarke et al., 1997; Sontag, 1999). Such implementation adds a dynamic discrete variable to the closed loop system. Together with a hysteresis-based implementation of the feedback of (Ancona and Bressan, 1999), as used in (Prieur, 2005; Prieur et al., 2007), this motivates the study of hybrid systems where two kinds of dynamic equations are considered: a discrete one and a continuous one. The closed-loop system resulting from the feedback constructed in (Prieur et al., 2007) essentially fits the form of the general class of hybrid systems studied in (Goebel and Teel, 2006). In particular, under appropriate regularity assumptions, it is shown in (Prieur et al., 2007) that all asymptotically stable hybrid systems are robust to measurement noise and external disturbances.
2. PRELIMINARIES
The previous work (Goebel et al., 2006) and this historical perspective motivate us to consider systems having a smooth patchy control Lyapunov function. In the present work, we state that, with a smooth patchy control Lyapunov in hand, it is possible to build a hybrid stabilizer so that the closed-loop system fits the class of hybrid systems consider in (Goebel and Teel, 2006; Prieur et al., 2007). We generalize the result of (Goebel et al., 2006) since, on the intersection of the domains, we do not need a decreasing condition of the control Lyapunov functions as the index increases. This condition is replaced by a weaker assumption, more precisely by an arrangement property (see Theorem 4.3 below). The paper is organized as follows. First we introduce some background material of hybrid systems in Section 2. Then we introduce the notion of a smooth patchy control Lyapunov function (pclf for short) for a finite number of patches, and we state a first sufficient condition for the feedback stabilization by means of a hybrid controller in Section 3. The case of an infinite case of patches is considered in Section 4. In Section 5 we illustrate
e ⊂ Rn is an open set and Throughout the paper, O e A ⊂ O is compact. We will be interested in hybrid feedback stabilization of the nonlinear system x(t) ˙ = f (x(t), u(t)), u(t) ∈ U, for all t ≥ 0,(1) e × U → Rn where U ⊂ Rk is a set and f : O is a (nonlinear in general) continuous mapping. The state space for the continuous variable of the e but hybrid feedback will not necessarily cover O, e will be an open set O such that O ⊂ O ⊂ O, e \ A. where O = O Definition 2.1. A hybrid feedback consists of • a totally ordered countable set Q • for each q ∈ Q, · sets Cq ⊂ O and Dq ⊂ O, · a function kq : Cq → U , · a set-valued mapping Gq : Dq → → Q. In closed loop with the nonlinear system (1), a hybrid feedback as in Definition 2.1 leads to a hybrid system x˙ = f (x, kq (x)) if x ∈ Cq , q + ∈ Gq (x) if x ∈ Dq .
(2)
The state space for (2) is O × Q. To make the definition of the hybrid system (2) and the notion of solutions to such a system precise, we recall some concepts from (Goebel and Teel, 2006). A subset S ⊂ R≥0 × N is a hybrid time domain if S is a union of a finite or infinite sequence of intervals [tj , tj+1 ] × {j}, with the last interval, if it exists, possibly of the form [tj , T ) with T finite or T = +∞. A solution to the hybrid system (2) consists of: a nonempty hybrid time domain S, a function x : S → O with x(t, j) locally absolutely continuous in t for a fixed j and constant in j for a fixed t, and a function q : S → Q with q(t, j) constant in t for a fixed j, meeting the following conditions: x(0, 0) ∈ Cq(0,0) ∪ Dq(0,0) and (S1) For all j ∈ N and almost all t such that (t, j) ∈ S, x(t, ˙ j) = f (x(t, j), kq(t,j) (x(t, j)), x(t, j) ∈ Cq(t,j) . (S2) For all (t, j) ∈ S such that (t, j + 1) ∈ S, q(t, j+1) ∈ Gq(t,j) (x(t, j)), x(t, j) ∈ Dq(t,j) .
Given a solution to (2) we will usually not mention the hybrid time domain explicitly, but will identify the solution by (x, q), and when needed, refer to the associated domain by dom(x, q). In what follows, we will write supt (S) for the supremum of all t such that (t, j) ∈ S for some j.
and the following conditions are met: there exist a continuous function α : (0, ∞) → (0, ∞), classK∞ functions γ, γ, and a function ω which is a e such that: proper indicator of A with respect to O S (iv) for all q ∈ Q, all x ∈ Ωq \ r>q Ω0r ,
We will say that the set A is stable for the hybrid system (2) if for any ε > 0 there exists δ > 0 such that any solution (x, q) with distA (x(0, 0)) ≤ δ satisfies distA (x(t, j)) ≤ ε for all (t, j) ∈ dom(x, q). The set A is globally attractive for the hybrid system (2) if
γ(ω(x)) ≤ Vq (x) ≤ γ(ω(x)) ; S (v) for all q ∈ Q, all x ∈ Ωq \ r>q Ω0r , there exists uq,x ∈ U such that ∇Vq (x) · f (x, uq,x ) ≤ −α(ω(x)); S (vi) for all q ∈ Q, all x ∈ ∂Ωq \ r>q Ω0r ∩ O, the uq,x of (v) can be chosen such that
• for any (x0 , q0 ) ∈ O × Q there exists a solution to (2) with x(0, 0) = x0 , q(0, 0) = q0 ; • for any maximal solution (x, q) to (2) we have distA (x(t, j)) → 0 as t → supt (dom(x, q)). Finally, we will say that A is (globally) asymptotically stable for (2) if it is both stable and globally attractive. In particular, note that our concept of asymptotic stability of A for (2) refers only to the behavior of the “continuous” part of solutions and also that we are allowing for solutions reaching A in finite time.
nq (x) · f (x, uq,x ) ≤ −α(ω(x)), where nq (x) is the unit (outward) normal vector to Ωq at x. We say that a smooth patchy control Lyapunov function is proper if additionally the following condition holds: (vii) for all q, r ∈ Q, r > q, all x ∈ Ωq ∩ ∂Ω0r , Vr (x) ≤ Vq (x). If this inequality is strict, we say that a proper smooth patchy control Lyapunov function is strict.
3. FINITE NUMBER OF PATCHES AND A SUFFICIENT CONDITION FOR FEEDBACK STABILIZATION 3.2 Stabilizing feedback 3.1 PCFL with finite number of patches Below, given a set Ω, its boundary is denoted by ∂Ω. By a proper indicator of A with respect to e we will understand a function ω : O e → R≥0 O that is continuous, positive definite with respect to A, and that approaches ∞ if the argument e or its norm apapproaches the boundary of O proaches ∞. Definition 3.1. A smooth patchy control Lyapunov function (with finite number of patches) for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that (i) Q ⊂ Z is a finite set; (ii) {Ωq }q∈Q and {Ω0q }q∈Q are families of nonempty e such that open subsets of O [ [ e where O := O ⊂ O ⊂ O, Ωq = Ω0q , q∈Q
With a patchy control Lyapunov function, and under a convexity assumption on the nonlinear system, we can design a hybrid feedback on O for (1) that renders A asymptotically stable with the basin of attraction equal to O. We note that the convexity assumption on the map f below is automatically satisfied when the system (1) is affine with respect to the control variable. Theorem 3.2. Suppose that • there exists a smooth patchy control Lyapunov function (with finitely many patches) for (1) with the attractor A; • for any v ∈ Rn , c ∈ R, the set {u ∈ U | v · f (x, u) ≤ c} is convex. Then, there exists a hybrid feedback on O for (1) that renders A asymptotically stable with the basin of attraction equal to O.
q∈Q
and for all q ∈ Q, the unit (outward) normal S vector to ∂Ωq is continuous on ∂Ωq \ r>q Ω0r ∩ O, and Ω0q ∩ O ⊂ Ωq ; (iii) for each q, Vq is a smooth function defined Son a (relative to O) neighborhood of Ωq \ r>q Ω0r ;
Remark 3.3. Some observations are in order. • Combining (Prieur et al., 2007, Theorem 4.3) and this result, and using the regularity of the data of our hybrid feedback, we may state the stronger result that A is asymptotically stable robustly to measurement noise, actuator errors and external disturbances. For more details see (Prieur et al., 2007).
• This result generalizes (Goebel et al., 2006, Theorem 4.1) since in this paper it is considered only proper smooth patchy control Lyapunov functions. The proof of this result has three main steps. First, for each q ∈ Q, we construct a “local” continuous feedback on each patch. Lemma 3.4. Under the assumptions of Theorem 3.2, for each q ∈S Q there exists a continuous mapping kq : Ωq \ r>q Ω0r ∩ O → U such that S (a) for all x ∈ Ωq \ r>q Ω0r ∩ O, ∇Vq (x) · f (x, kq (x)) ≤ −α(ω(x))/2; S (b) for all x ∈ ∂Ωq \ r>q Ω0r ∩ O, nq (x) · f (x, kq (x)) ≤ −α(ω(x))/2. The second step defines the remaining data that is needed to turn the collection of (continuous time) feedbacks kq into a hybrid feedback. We do this explicitly, by setting: Cq = Ωq \
[
Ω0r ∩ O
r>q
Dq =
[
Ω0r ∩ O ∪ (O \ Ωq )
r>q
{r ∈ Q | x ∈ Ω0r ∩ O, r!> q} (3) [ if x ∈ Ω0r ∩ Ωq r>q Gq (x) = {r ∈ Q | x ∈ Ω0 ∩ O} r if x ∈ O \ Ωq
It is shown in (Goebel et al., 2006) that for all asymptotically controllable systems, there exists a strict proper patchy control Lyapunov function 3 . Let us investigate the sufficiency part. To do that we need the following: Definition 4.2. A patchy control Lyapunov function for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that Q ⊂ Z, conditions (ii)(vi) of Definition 3.1 hold, and the hybrid feedback given by (3) and any functions kq satisfying the conditions of Lemma 3.4 renders A asymptotically stable. We have shown, in Theorem 3.2, that a patchy control Lyapunov function with finitely many patches is a patchy control Lyapunov function (in the sense of the definition above) subject to a convexity assumption on the dynamics. Below, we will show that a proper smooth PCLF is indeed a PCLF in the sense of the definition above. First, we propose a different sufficient condition for a collection of local control Lyapunov functions to be a PCLF. This condition does not rely on a decrease condition of Vq on the regions where the patches overlap. Rather, it places certain conditions on how the patches need to be arranged. Theorem 4.3. Suppose that a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q is such that Q ⊂ Z, conditions (ii)-(vi) of Definition 3.1 hold and, for each N ∈ N ∪ {∞}, (pa1 ) for each ε > 0 there exists δ such that, for all increasing sequence (qn )0≤n≤N in Q, and for all sequence (xn )0≤n≤N in O satisfying x0 ∈ Ωq0 , Vq0 (x0 ) ≤ δ , and, for all 0 ≤ n ≤ N ,
In the third step, we use the collection of local control Lyapunov functions Vq to show that the constructed feedback is stabilizing.
4. INFINITE NUMBER OF PATCHES AND A NECESSARY AND SUFFICIENT CONDITION
xn+1 ∈ Ωqn ∩∂Ω0qn+1 , Vqn (xn+1 ) ≤ Vqn (xn ) , we have Vqn (xn ) ≤ ε, for all 0 ≤ n ≤ N ; (pa2 ) for all increasing sequence (qn )0≤n≤N in Q, and for all sequence (xn )0≤n≤N in O satisfying x0 ∈ Ωq0 , and, for all 0 ≤ n ≤ N ,
We will say that a family {Ωq }q∈Q is a locally finite family if for any compact K subset of O, there is finitely many q’s such that K ∩ Ωq 6= ∅.
xn+1 ∈ Ωqn ∩∂Ω0qn+1 , Vqn (xn+1 ) ≤ Vqn (xn ) ,
Definition 4.1. A proper patchy control Lyapunov function for (1) with the attractor A consists of a set Q and a collection of functions Vq and sets Ωq , Ω0q for each q ∈ Q, such that Q ⊂ Z and conditions (ii)-(vii) of Definition 3.1 hold. A proper patchy control Lyapunov function is strict if the inequality in condition (vii) is strict.
Then this collection is a patchy control Lyapunov function for (1).
there exists M > 0 such that Vqn (xn ) ≤ M , for all 0 ≤ n ≤ N .
3
In fact, it is stated in (Goebel et al., 2006, Theorem 5.3) that there exists a proper patchy control Lyapunov function. However a small modification of the proof of this result gives the existence of a strict proper pclf.
Proposition 4.4. A proper smooth patchy control Lyapunov function for (1) is a smooth patchy control Lyapunov function for (1). Remark 4.5. Let us note that we recover (Goebel et al., 2006, Theorem 4.1) since we get that, under the convexity assumption of Theorem 3.2, if there exists a proper smooth patchy control Lyapunov function, then the hybrid feedback given by (3) renders A asymptotically stable.
5. ILLUSTRATION To illustrate the definitions and the results of the present work, we consider the Brockett integrator: x˙ 1 = u1 , x˙ 2 = u2 , (4) x˙ 3 = x1 u2 − x2 u1 . It is known that the necessary condition (Brockett, 1983) for the stabilization by means of a continuous feedback or robust stabilization by locally bounded feedback (Ryan, 1994) does not hold for this system. Let us check that we can prove the existence of a hybrid stabilizing feedback by applying Theorem 3.2. Let us denote f (x, u) the right-hand side of the system (4), where x = (x1 , x2 , p x3 ) and u = (u1 , u2 ). We use the notation: r = x21 + x22 . We consider a simplified version of the hybrid controller of (Hespanha and Morse, 1999) (see also (Goebel et al., 2004)) and we define a patchy control Lyapunov function as a collection of two patches. These patches Ω0q ⊂ Ωq , for q = 1, 2 are sketched in Figure 1 in the (r, x3 ) plane (they are surfaces of revolution around the x3 axis and symmetric with respect to (0, r) axis). x3
Ω2 Ω2’
r
Fig. 1. Sketch of the sets Ω02 ⊂ Ω2 in the (r, x3 ) plane, for x3 > 0 To define the first one, let us consider the following sets Ω01 = Ω1 = R3 , and the function V1 : R3 → R, p defined by V1 (x) = (ρ + ε) |x3 | − x1 , for all x ∈ R3 , where 1 < ρ and 0 < ε < 1 will be prescribed below.
Now let us define the second patch, and let us consider the following sets Ω02 = {x 6= 0, r2 > ρ|x3 |} , Ω2 = {x 6= 0, r2 > |x3 |} , and the smooth function V2 : R3 → R, defined by V2 (x) = 21 (r2 + x23 ). Observe that V1 is a smooth function on a (relative to R3 \ {0}) neighborhood of Ω1 \ Ω02 . We have (iii) of Definition 3.1. The index set Q = {1, 2} verifies (i) of Definition 3.1. The families {Ω1 , Ω2 }, {Ω01 , Ω02 } consist of nonempty and open sets and cover O = R3 \ {0}, and we have (ii) of Definition 3.1. For all x ∈ Ω1 \ Ω02 , we have ε
p p |x3 | εr + √ ≤ ε |x3 | ≤ V1 (x) , 2 2 ρ p V1 (x) ≤ (ρ + ε) |x3 | + r .
(5) (6)
Thus V is positive definite and proper, and therefore it verifies (iv) of Definition 3.1. Now set u1,x = (1, 0), for all x ∈ R3 . Observe that 4 , for all x ∈ Ω1 \ Ω02 , 1 sgn(x3 ) ∇V1 (x) · f (x, u1,x ) = − (ρ + ε) p x2 − 1 , 2 |x3 | √ (ρ + ε) ρ −1 . (7) ≤ 2 √ Thus by picking ρ and ε such that (ρ + ε) ρ < 2, this verifies (v) for q = 1 of Definition 3.1 with √ (ρ+ε) ρ α(|x|) ≤ 1 − being prescribed below. 2 Now set u2,x = (−x1 + 4 xr2 x2 3 , −x2 − 4 xr1 x2 3 ), for all x ∈ Ω2 . We compute, for all x ∈ Ω2 , ∇V2 (x) · f (x, u2,x ) = −r2 − 4x23 . This verifies (v) for q = 2 of Definition 3.1, with α(|x|) ≤ |x|2 being prescribed below. Since ∂Ω1 \ Ω02 is empty, (vi) of Definition 3.1 verifies for q = 1. Let us consider now x ∈ ∂Ω2 ∩O. The unit normal vector n2 (x) to Ω2 at x is n2 (x) = √ 1 (−2x1 , −2x2 , sgn(x3 ))0 . Using r2 = |x3 |, 4r 2 +1 and |x|2 = |x3 | + |x3 |2 , for all x ∈ ∂Ω2 ∩ O, we compute 2r2 − 4|x3 | , n2 (x) · f (x, u2,x ) = √ 4r2 + 1 p ≤ −2|x3 | = 1 − 1 + 4|x|2 . This verifies (vi) of Definition 3.1 for q = 2 with √ p (ρ+ε) ρ 2 α(|x|) = min(1 − , |x| , 1 + 4|x|2 − 1). 2 As a conclusion of these computations, we can claim that the set Q, the families of open sets {Ωq }q∈Q and {Ω0q }q∈Q , and the family of functions {Vq }q∈Q consist a patchy control Lyapunov function (with finite number of patches) for (4). 4
Here and in what follows, sgn(x3 ) is the sign of x3 6= 0, and |.| denotes the Euclidian norm.
Control, Optimisation and Calculus of Variations 4, 445–471. Ancona, F. and A. Bressan (2003). Flow stability of patchy vector fields and robust feedback stabilization. SIAM Journal on Control and Optimization 41(5), 1455–1476. It is also possible to scale V1 to get a proper, Brockett, R.W. (1983). Asymptotic stability in fact strict proper, patchy control Lyapunov and feedback stabilization. In: Differenfunction. First, we may prove that there exists tial geometric control theory. pp. 181–191. a continuously differentiable class-K∞ function γ Birkhauser. such that, for all x ∈ Ω1 \ ∂Ω02 , Clarke, F.H., Y.S. Ledyaev, E.D. Sontag and A.I. Subbotin (1997). Asymptotic controllability |x|2 ). (8) V1 (x) ≥ γ( implies feedback stabilization. IEEE Trans. 2 Automat. Control 42(10), 1394–1407. Goebel, R. and A.R. Teel (2006). Solutions to hyNow, let us define W1 : R3 → R by W1 (x) = brid inclusions via set and graphical converγ −1 (V1 (x)). Since V1 verifies (iv) of Definition 3.1, gence with stability theory applications. AuW1 also verifies this condition. tomatica 42(4), 573–587. With (7) and since the γ −1 has a positive derivaGoebel, R., C. Prieur and A.R. Teel (2006). tive, we compute, for all x ∈ Ω1 \ Ω02 , Smooth patchy control Lyapunov functions. In: Proc. 45th IEEE Conf. on Decision and dγ −1 Control. San Diego, CA. pp. 3271–3276. (V1 (x)) ∇V1 (x) · f (x, u1,x ), ∇W1 (x) · f (x, u1,x ) = ds Goebel, R., J. Hespanha, A.R. Teel, C. Cai and √ (ρ + ε) ρ dγ −1 R. Sanfelice (2004). Hybrid systems: gen≤( − 1) (V1 (x)), 2 ds eralized solutions and robust stability. In: ≤ −e α(|x|) , IFAC Symp. on Nonlinear Control Systems. Stuttgart, Germany. pp. 1–12. where, for all s ∈ (0, ∞), Hespanha, J.P. and A.S. Morse (1999). Stabiliza√ (ρ + ε) ρ dγ −1 tion of nonholonomic integrators via logicα e(s) = max (1− ) (V1 (x)) , 2 ds x∈Ω1 \Ω02 , |x|≤s based switching. Automatica 35(3), 385–393. Prieur, C. (2005). Asymptotic controllability and Due to (5)-(6), α e : (0, ∞) → (0, ∞) is well-defined robust asymptotic stabilizability. SIAM J. and the function α e is continuous. Thus W1 verifies Control Opt. 43, 1888–1912. (v) of Definition 3.1, and due to (8), it verifies (vii) Prieur, C. and E. Tr´elat (2005). Robust optimal of Definition 3.1. stabilization of the Brockett integrator via a As a conclusion of these computations, we can hybrid feedback. Math. Control Signals Sysclaim that the set Q, the families of open sets tems 17, 201–216. {Ωq }q∈Q and {Ω0q }q∈Q , and the family of functions Prieur, C. and E. Tr´elat (2006). Quasi-optimal {W1 , V2 } consist a strict proper pclf for (4). robust stabilization of control systems. SIAM J. Control Opt. 45(5), 1875–1897. We recover partially the result of (Prieur and Prieur, C., R. Goebel and A.R. Teel (2007). HyTr´elat, 2005) (see also (Prieur and Tr´elat, 2006)) brid feedback control and robust stabilization where an optimization criterion is also considered. of nonlinear systems. IEEE Trans. Autom. Control. To appear. Rifford, L. (2002). Semiconcave control-Lyapunov 6. CONCLUSION functions and stabilizing feedbacks. SIAM J. This work considered the concept of a smooth Control Optim. 41(3), 659–681. patchy control Lyapunov function (pclf). The Ryan, E.P. (1994). On Brockett’s condition for main result states that, under a mild convexity smooth stabilizability and its necessity in assumption, the existence of such pclf implies the a context of nonsmooth feedback. SIAM J. construction of a stabilizing hybrid feedback. It Control Optim. 32(6), 1597–1604. generalizes (Goebel et al., 2006). Moreover, using Sontag, E.D. (1983). A Lyapunov-like characteri(Prieur et al., 2007), we can show that this hybrid zation of asymptotic controllability. SIAM J. controller is robust with respect to measurement Control Optim. 21(3), 462–471. noise and external disturbances. Sontag, E.D. (1999). Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var. 4, 537–557. REFERENCES Moreover, since the system (4) is affine with respect to u, the convexity assumption in Theorem 3.2 holds. Thus by defining uq : Cq → R2 by uq (x) = uq,x , for all x ∈ Cq , the hybrid controller (3) renders 0 asymptotically stable on R3 \ {0}.
Ancona, F. and A. Bressan (1999). Patchy vector fields and asymptotic stabilization. ESAIM: