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Stabilization of continuous-time switched nonlinear systems
Systems & Control Letters 57 (2008) 95 – 103 www.elsevier.com/locate/sysconle
Stabilization of continuous-time switched nonlinear systems夡 Patrizio Colaneri a , José C. Geromel b , Alessandro Astolfi c,d,∗ a Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza leonardo da Vinci 32, Milano, Italy b DSCE / School of Electrical and Computer Engineering, UNICAMP, CP 6101, 13083 - 970 Campinas, SP, Brazil c Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK d DISP, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy
Received 23 January 2007; received in revised form 20 July 2007; accepted 22 July 2007 Available online 6 September 2007
Abstract The paper considers three problems for continuous-time nonlinear switched systems. The first result of this paper is a open-loop stabilization strategy based on dwell time computation. The second considers a state switching strategy for global stabilization. The strategy is of closed loop nature (trajectory dependent) and is designed from the solution of what we call nonlinear Lyapunov–Metzler inequalities from which the stability condition is expressed. Finally, results on the stabilization of nonlinear time varying polytopic systems are provided. © 2007 Elsevier B.V. All rights reserved. Keywords: Switched systems; Stability; Stabilization
while the second one is such that, for each t 0,
1. Introduction This paper aims at providing new results on stabilizing control synthesis for a continuous-time switched nonlinear system of the following general form: x(t) ˙ = f(t) (x(t)),
x(0) = x0 ,
(1)
where x(t) ∈ Rn is the state, (t) is the switching rule and x0 is the initial condition. Given a set of vector fields {f1 (x), . . . , fN (x)}, such that fi (0) = 0 for all i = 1, . . . , N, two different classes of switched systems are studied. The first is characterized by the fact that the switching rule, for each t 0, is such that f(t) ∈ {f1 , . . . , fN },
(2)
夡 This research was supported by grants from “Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq” - Brazil, and supported by the Italian National Research Council (CNR) and by MIUR project “Advanced control methodologies for hybrid dynamical systems”. ∗ Corresponding author. Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK. Tel.: +44 171 594 6289; fax: +44 171 594 6282. E-mail addresses: [email protected] (P. Colaneri), [email protected] (J.C. Geromel), a.astolfi@ic.ac.uk (A. Astolfi).
0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.07.001
f(t) ∈ co{f1 , . . . , fN },
(3)
where co{·} denotes the convex hull. It is important to make clear the basic difference between these two classes of switched systems. The model (2) naturally imposes a discontinuity on f(t) since this vector must jump instantaneously from fi to fj for some i = j once switching occurs. In other words, f(t) is constrained to jump among the N vertices of the vector polytope co{f1 , . . . , fN }. The model defined by (3) is more general in the sense that the interior of the same polytope is now feasible for f(t) and so it supports switching rules with no discontinuity with respect to time. As it will become clear in the sequel there are some important relationship between the stability conditions of both models. The paper contributes in three directions, namely dwell time computation, state-feedback stabilization and polytopic stability analysis. The idea underlying the first issue has been spurred by reading the recent paper [12], where a method for stability analysis of switched and hybrid systems is provided by using polynomial and piecewise polynomial Lyapunov functions. A result is provided on the computation of an upper bound of the dwell time, by means of a suitable Lyapunov inequality. This result provides a characterization of some time switching stabilizing strategies in terms of dwell time computation. It finds
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its counterpart in the linear case in [6], and it is shown to be less conservative than that provided in [10], although in general less computationally attractive, see also [3] for a simple time switching stabilizing strategy for nonholonomic systems. The second issue concerns the state-feedback stabilization of autonomous switched nonlinear systems. It is rooted in the setting in which multiple Lyapunov functions are used to find a stabilizing switching strategy, see [2]. The new idea, inherited from the linear case, see [6], is to use the so-called Lyapunov–Metzler inequalities, thanks to which the final Lyapunov function is built as the minimum over a finite number of functions. It is worth mentioning that this idea extends a result (for the linear case) in [11], valid in the case N = 2. The extension of La-Salle’s principle for linear switched systems can be found in [9]. The third issue concerns the stability of nonlinear polytopic systems. Among the extensive literature in the linear case, the results provided here is based on [7]. A sufficient condition for the stability of polytopic nonlinear systems is provided in terms of a Lyapunov inequality. The paper is organized as follows. In Section 2 a time switching stabilizing control strategy is provided based on the computation of an upper bound on the minimum dwell time. In Section 3 a state switching globally stabilizing strategy is devised based on the solution of Lyapunov–Metzler inequalities. Finally the stability of polytopic systems is considered and relations with the previous material are pointed out.
where x(T ) is the solution of the differential equation x˙ =fj (x) with initial state x at t = 0. Then the time switching control (4) with tk+1 − tk T makes the equilibrium x = 0 of (1) globally exponentially stable.
2. Time switching control
where the second inequality holds from the fact that for every = Tk − T 0 it is true that (d/dt)Vi (x()) < 0. The consequence is that there exists ∈ (0, 1) such that Vi (x) Vj (x(T )), for each i and j, so that
This section is entirely dedicated to design a time switching control law for the switched linear system defined by the models (1) and (2) where it is assumed that each vector field in the set {f1 , . . . , fN } is stable. The problem under consideration can be stated as follows. Determine a minimum dwell time T∗ > 0 such that the equilibrium point x = 0 of the system (1) is globally asymptotically stable with the time switching control (t) = i ∈ {1, . . . , N},
t ∈ [tk , tk+1 ),
(4)
where tk and tk+1 are successive switching time instants satisfying tk+1 − tk T∗ for all k ∈ N and the index i ∈ {1, . . . , N}, selected at each instant of t 0, is arbitrary. Hence, asymptotic stability is preserved whenever (t) remains unchanged for a period of time greater or equal to the minimum dwell time T∗ . The next theorem provides the theoretical basis toward a possible solution of this problem by characterizing an upper bound for T∗ . It uses the concept of multiple Lyapunov functions.
Proof. To begin with note that if there are no jumps the result trivially follows by Eq. (6). Consider now, in accordance with (4), that (t) = i ∈ {1, . . . , N} for all t ∈ [tk , tk+1 ) where tk+1 = tk + Tk with Tk T > 0 and that at t = tk+1 the time switching control jumps to (t) = j ∈ {1, . . . , N}. From (6), it is seen that, for all t ∈ [tk , tk+1 ), the time derivative of the Lyapunov function v(x(t)) = V(t) , along an arbitrary solution of (1) satisfies, for x = 0, v(x(t)) ˙ =
jVi fi < 0, jx
(8)
which enables to conclude that x(t)2 ai−1 e−i (t−tk ) v(x(tk )),
∀t ∈ [tk , tk+1 ).
(9)
On the other hand, using the inequalities (7) we have v(x(tk+1 )) = Vj (x(tk+1 )) < Vi (x(tk + Tk − T )) tk +Tk −T d < Vi (x(tk )) + Vi (x()) d dt tk < v(x(tk )),
v(x(tk )) k v(x0 ),
∀k ∈ N,
(10)
(11)
which together with (9) implies that the equilibrium x = 0 of (1) is globally exponentially stable. This result deserves some comments. First, since all systems in the set {f1 , . . . , fN } are supposed to be asymptotically stable, the constraints (6) are always feasible and the constraints (7) are satisfied when T > 0 is taken large enough. Second, an upper bound for the dwell time can be computed directly from (5)–(6). Indeed, a simple computation shows that Vj (x(T )) bj x(T )2 bj ai−1 Vi (x(T )) bj ai−1 e−i T Vi (x) so that (7) holds selecting bj 1 T max log . (12) i,j i ai
Theorem 1. Assume that, for some T > 0, there exist a collection of positive definite radially unbounded functions {V1 , . . . , VN } and positive numbers ai , bi , i such that, for each x,
Of course the upper bound (12) is a consequence of condition (7), hence it is more conservative. Third, assume that the vector fields fi share the same Lyapunov function V (x), i.e.
ai x2 Vi (x)bi x2 , ∀ i = 1, . . . , N, jVi fi < − i Vi , ∀ i = 1, . . . , N, x = 0, jx Vi (x) > Vj (x(T )), ∀ i = j ,
jV fi < − i V , jx
(5) (6) (7)
∀i = 1, . . . , N,
(13)
then the inequality (7) is satisfied for any T > 0 meaning that the switching policy (4) may jump from i to j arbitrarily fast
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
preserving asymptotic stability. In the linear setting this corresponds to the quadratic stability condition. Finally, with T > 0 fixed, it is always possible to define a time switching control strategy (4) such that f(t) (x) is periodic as a function of t. As a consequence, a necessary condition for the feasibility of constraints (6) and (7) is that the nonlinear system with vector field f(t) (x) is globally exponentially stable, for each periodic switching time-policy with period greater than T. However, since (4) may produce nonperiodic policies as well, this condition for the existence of a feasible solution to inequalities (6)–(7) is, in general, not sufficient. Finally, it is possible to generalize the result of Theorem 1 in order to define a guaranteed cost to go from an arbitrary initial point to the origin, associated to the stabilizing time switching rule (4) with tk+1 − tk T for any fixed T > 0. To this end we make the assumption that T > 0 is known such that tk+1 − tk T for all k ∈ N. Clearly, these quantities are related through T T T∗ where the second inequality assures global stability. Now consider the switched system (1) and the output variable y(t) = h(t) (x(t)),
(14)
where hi (x), i = 1, 2, . . . , N, are given p-valued maps. The next result provides an upper bound to the L2 norm of y(t) for any switching signals (t) with residence time between any two consecutive switching instants not less than T.
97
that Vi (x) > ri (x) and, for x = 0, j(Vi − ri ) jri fi < − hi hi − fi jx jx < − hi (x(T)) hi (x(T)) <0
(19)
for all i =1, · · · , N . The important consequence of this calculation is that for each i =1, · · · , N the inequality d(Vi −ri )/dt 0 holds. Using this property, taking into account the switching strategy (4) with tk+1 − tk = Tk T and the inequalities (17), one gets v(x(tk+1 )) = V(tk+1 ) (x(tk+1 )) < V(tk ) (x(tk + Tk − T )) − r(tk ) (x(tk + Tk − T )) < V(tk ) (x(tk )) − r(tk ) (x(tk )) < v(x(tk )) − r(tk ) (x(tk )),
which summing up for all k ∈ N and taking into account that Ttk+1 − tk , yields ∞ ∞ Tk h(t) (x(t)) h(t) (x(t)) dt= h(tk ) (x(t+tk )) h(tk ) k=0 0
0
× (x(t + tk )) dt ∞ T h(tk ) (x(t+tk )) h(tk )
Theorem 2. Let T T > 0 be given. Define the set of nonnegative maps ri (x) :=
T 0
hi (x(t)) hi (x(t)) dt,
i = 1, . . . , N,
∀ i = 1, . . . , N, x = 0,
(16) (17)
The time switching control (4) with Ttk+1 − tk T makes the equilibrium x = 0 of (1) globally asymptotically stable and
0
∞
× (x(t+tk )) dt
∞
r(tk ) (x(tk ))
k=0
jVi fi + hi hi < − i Vi , ∀i = 1, . . . , N, jx Vi (x) > Vj (x(T )) + ri (x), ∀ i = j .
k=0 0
(15)
where x =x(0) and x(t) is the solution of x˙ =fi (x). Assume that there exist a collection of positive definite radially unbounded functions {V1 , . . . , VN } and positive numbers ai , bi , i such that, for each x, ai x2 Vi (x)bi x2 ,
(20)
h(t) (x(t)) h(t) (x(t)) dt < V(0) (x0 ).
(18)
Proof. Since each function ri (x) defined in (15) is nonnegative and inequalities (16)–(17) are satisfied, then inequalities (6)–(7) are also satisfied. As a consequence, global asymptotical stability follows from Theorem 1. On the other hand, using (15) together with the inequalities (16) and (17) we have
< v(x0 ), which proves the claim.
(21)
It is interesting to observe that the conditions of Theorem 2 are feasible if and only if TT T∗ and from (18) it is seen that a more accurate guaranteed cost is obtained whenever the value of T is chosen as small as possible. In addition, the choice T = +∞ enables to conclude that the proposed time switching rule (4), with tk+1 − tk T∗ , makes the signal y(t) = h(x(t)), t 0 square integrable. Theorem 2 admits the extreme situation T = T = +∞ for which no jump occurs and inequalities (16)–(17) are verified for ∞ hi (x(t)) hi (x(t)) + εx(t)2 dt > ri 0, Vi (x) = 0
i = 1, . . . , N
(22)
with ε > 0. When ε > 0 goes to zero, Vi goes to ri and (18) becomes a well-known result. On the other hand, for T > 0 arbitrarily small and any T T , feasibility holds whenever the set of vectors {f1 , . . . , fN } admits a common Lyapunov function.
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globally asymptotically stabilizes the equilibrium point x = 0 of the nonlinear system (1).
3. State switching control In this section we consider the system (1) where the switching rule satisfies (2). It is assumed that the state vector x(t) is available for feedback for all t 0, and the goal is to determine a function u(·) : Rn → {1, . . . , N}, such that the switching rule (t) = u(x(t))
(23)
assures that the equilibrium x = 0 of (1) is globally asymptotically stable. Note that we do not assume that any of the vector fields in the set {f1 , . . . , fN } be either locally or globally asymptotically stable. Define the simplex N N i = 1, i 0 (24) := ∈ R : i=1
and the function v(x) :=
min Vi (x),
(25)
i=1,...,N
where {V1 , . . . , VN } is a set of differentiable, positive definite and radially unbounded functions, which are zero at x = 0. As it will be clear in the sequel, the function v(x) is a candidate Lyapunov function, crucial for our purposes. However, even if the functions Vi (x) are differentiable, the function v(x) is not (in general) differentiable everywhere. To address this issue the set I (x) = {i : v(x) = Vi (x)} plays a central role since v(x) fails to be differentiable at all x ∈ Rn such that card I (x) is discontinuous [16]. Before proceeding, recall the class of Metzler matrices denoted by M and constituted by all matrices ∈ RN×N with elements ij , such that ij 0 ∀i = j,
N
ij = 0, ∀j .
(26)
i=1
It is clear that any ∈ M presents an eigenvalue at the origin of the complex plane since c = 0 where c = [1 · · · 1]. In addition it is well known from the Frobenius–Perron’s theorem that the eigenvector associated to the null eigenvalue of is nonnegative, yielding the conclusion that there always exists ∞ ∈ such that ∞ = 0. The next theorem summarizes the main result of this section. Theorem 3. Assume that there exist a set of functions {V1 , . . . , VN }, which are all differentiable, positive definite, radially unbounded and zero at zero, and a matrix ∈ M satisfying the Lyapunov–Metzler inequalities jVi fi + jx
N
j i Vj < 0,
i = 1, . . . , N
(27)
j =1
for all x = 0. Then the state switching control (23) with u(x(t)) = arg min Vi (x(t)) i=1,...,N
(28)
Proof. To begin with note that the Lyapunov function (25) is piecewise differentiable, positive definite, radially unbounded and zero at x = 0. Moreover, the Lyapunov function (25) is not differentiable for all x ∈ Rn . For this reason we need to deal with the Dini derivative (see [5]) v(x(t + h)) − v(x(t)) . h
D + v(x(t)) = lim sup h→0+
(29)
Assume, in accordance with (28), that at an arbitrary t 0, the state switching control is given by (t) = u(x(t)) = i for some i ∈ I (x(t)). Hence, from Eqs. (29) and (1), we have D + v(x(t)) = min
l∈I (x(t))
jVl jVi fi fi , jx jx
(30)
where the inequality holds from the fact that i ∈ I (x(t)). Finally, remembering that (27) is valid for ∈ M and that Vj (x) Vi (x) for all j = i = 1, . . . , N once again due to the fact that i ∈ I (x(t)), using the Lyapunov–Metzler inequalities (27) one gets ⎛ ⎞ N D + v(x(t)) < − ⎝ j i Vj (x)⎠ ⎛ −⎝
j =1 N
⎞ j i ⎠ Vi (x) = 0
(31)
j =1
for all x = 0, which proves the claim.
Remark 1. Theorem 3 is particularly important when the set {f1 , . . . , fN } is composed of unstable vector fields. Indeed, if a function Vi (x) is locally quadratic, a necessary condition for the Lyapunov–Metzler inequalities to be feasible with respect to {V1 , . . . , VN } is that the vector fields fi + (ii /2)x be locally asymptotically stable. Since ii 0 this condition does not imply local asymptotic stability of any of the fi ’s. On the other hand, in general, the Lyapunov–Metzler inequalities imply that D + Vi (x(t)) < |ii |Vi (x(t))
∀i
and, since the functions Vi are radially unbounded, this implies that the vector fields {f1 , . . . , fN } are complete and that, along the trajectories of fi the functions Vi are such that Vi (x(t)) Vi (x(0))e|ii |t
∀i.
Remark 2. An interesting case occurs when all vector fields {f1 , . . . , fN } are globally asymptotically stable for which the choice = 0 is possible and the state switching strategy proposed preserves stability. Furthermore, if the set {f1 , . . . , fN } admits a unique Lyapunov function V, then the Lyapunov–Metzler inequalities admit a solution V1 = · · · = VN = V and I (x(t)) = {1, . . . , N} for all t 0. In this classical but particular case, at any t 0, the control
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
99
law u(x(t)) being an arbitrary logic state i ∈ {1, . . . , N}, asymptotic stability is once again guaranteed.
Theorem 4. Consider inequalities (34) and assume that, pointwise, the limits lim →∞ Vj exist. Then, the system
Remark 3. (Chattering). Another important feature of Theorem 3 is that chattering in the switching, when it occurs, is always stable in the Filippov sense. Indeed, assume that x ∈ Rn belongs to a certain region C of the state space where the cardinality of I (x) is greater than one. From the Lyapunov function (25), a switching from i ∈ I (x) to j ∈ I (x) is possible only if (jVj /jx)fi (jVi /jx)fi < 0 where the last inequality follows directly from (27). Hence, we conclude that whenever x ∈ C the time derivative of the positive definite function Vj is strictly negative along all trajectories such that x˙ ∈ co{fi : i ∈ I (x)} which implies that they are asymptotically stable. This idea finds its roots in the result in [11], for N = 2 and in the linear case.
x(t) ˙ = f∞ (x(t))
Remark 4. Theorem 3 also holds if the matrix is a function of x, i.e. = (x), provided that, for each fixed x ∈ Rn , it satisfies (x) ∈ M.
is globally asymptotically stable, where f ∞ =
N
Proof. Pointwise it follows that lim
→∞
N
N
jVi fi + j i j (Vj ) < 0, jx
i = 1, . . . , N
(32)
j =1
for all x = 0, and for some positive definite functions j ’s. Remark 6. In the literature of linear systems the Lyapunov–Metzler inequalities, with ∈ M fixed, have been introduced in order to study the mean-square (MS) stability of Markov Jump Linear Systems (MJLS), see e.g. [4]. In that context, the Metzler matrix ∈ M is given and represents the infinitesimal transition matrix of a Markov chain (t) governing the dynamical system. In this respect, each component of the vector (t) ∈ is the probability of the Markov chain to be on the ith logical state and obeys the differential equation ˙ = (t), (t)
j i Vj 0,
∀i.
j =1
Moreover, recalling the role of the vector ∞ in the Metzler matrix , we have N
∞i
i=1
Remark 5. Condition (27) in Theorem 3 can be replaced by the condition
∞i fi .
i=1
N
j i Vj = 0.
j =1
It then follows that ⎡ ⎤ V1 ⎢ V2 ⎥ ⎥ lim ⎢ ⎣ ... ⎦ = 0,
→∞ VN
which implies that lim →∞ Vi =V , for all i =1, . . . , N. Finally, (34) yields N i=1
∞i
jVi fi < 0 jx
and hence jV f < 0, jx ∞
f∞ =
N
∞i fi .
i=1
The proof is then concluded. (0) = 0 ∈ ,
where the eigenvector ∞ ∈ associated to the null eigenvalue of represents the stationary probability vector. Consider now the modified Lyapunov–Metzler inequalities defined as N
jVi fi +
j i Vj < 0, jx
i = 1, . . . , N,
(33)
(34)
j =1
for all x = 0, where is a positive parameter. This parameter multiplies all elements of the matrix , therefore the matrix
is still a Metzler matrix, i.e. ∈ M whenever ∈ M. Notice that these new inequalities are those relative to vector fields {f1 / , . . . , fN / }, obtained by the time scaling t → t/ .
The above theorem relates the Lyapunov–Metzler inequalities with the stability of the “average” system characterized by the vector field f∞ . Indeed, if the inequalities (34) hold for a sufficiently large , then f∞ is globally asymptotically stable. This is a relevant point, since it recovers the classical stability condition (in the linear setting) provided in [11,19]. To see this fact in the present context assume that there exists ∞ ∈ such that f∞ is globally asymptotically stable, making possible the determination of V > 0 satisfying the Lyapunov inequality (jV /jx)f∞ < 0. Hence, the switching rule (23), with u(x(t)) = arg min
i=1,...,N
jV fi , jx
(35)
makes the equilibrium x = 0 of the switched system (1) globally asymptotically stable. Indeed, considering the Lyapunov
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P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
function V (x) we have jV jV jV jV f(t) = min fi = min f f < 0 V˙ (x(t))= i=1,...,N jx jx jx ∞ ∈ jx for all x = 0. It is important to keep in mind that the numerical determination (if any) of ∈ and V > 0 such that N jV i f i < 0 jx i=1
is not a simple task even in the simplest case of linear time invariant systems. We conclude this section introducing a guaranteed cost associated to the proposed state switching control law (28). To this end, consider again the output variable y(t) in (14). The next result provides an upper bound for the L2 norm of y(t) under the switching rule (23). Corollary 1. Assume that there exist a set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at zero, and a matrix ∈ M satisfying the Lyapunov–Metzler inequalities N
jVi fi + j i Vj + hi hi < 0, jx
i = 1, . . . , N,
(36)
j =1
system does not admit, in general, a closed form solution, see e.g. [18] for some insight in the problem. We finally discuss a possible stabilizing switching strategy that includes the previous one as a particular case and hence may provide less conservative results. To this end, define a set of positive definite functions {W1 (x), . . . , WN (x)} and the functions Hi (x) =
N
j i Wj (x),
i = 1, 2, . . . , N.
j =1
Due to the structure of the Metzler matrices, these functions cannot be strictly negative for all i = 1, . . . , N, since N n i=1 ∞i Hi (x) = 0. As a result, for each x ∈ R the set I˜(x) = {i : Hi (x) 0} is not empty, and it is possible to define the candidate Lyapunov function v(x) := min Vi (x).
(40)
i∈I˜(x)
Now, assume that there exist a set of functions {V1 , . . . , VN }, which are all differentiable, positive definite, radially unbounded and zero at zero, a set of positive definite functions {W1 , . . . , WN } and a matrix ∈ M satisfying the extended Lyapunov–Metzler inequalities N
jVi fi + j i Wj < 0, jx
for all x = 0. Then the state switching control (23) with u(x(t)) given by the Eq. (28) globally asymptotically stabilizes the equilibrium point x = 0 of (1) and it is such that the inequality ∞ J= h(t) (x) h(t) (x) dt < min Vi (x0 ) (37)
for all x = 0. Finally, consider the switching control rule (23) with
holds.
u(x(t)) = arg min Vi (x).
i=1,...,N
0
i = 1, . . . , N
(42)
i∈I˜(x)
Proof. The proof has the same structure as the proof of Theorem 3. The Lyapunov function (25) and the Lyapunov–Metzler inequalities (36) yield D + v(x(t)) < − h(t) (x) h(t) (x),
x = 0,
(38)
and, by integration it is readily verified that t D + v(x()) d v(x(t)) − v(x(0)) = 0
t
< − 0
h() (x) h() (x) d
(39)
for all t 0, proving thus the claim since, by asymptotic stability, v(x(t)) goes to zero as t goes to infinity. Notice that Corollary 1 provides a pre-computable upper bound to the control strategy minimizing the cost J. This upper bound depends on the choice of the parameters in the inequality (36) so that, in principle, it is possible to minimize it with respect to ∈ M. The optimal control problem for switched
(41)
j =1
Notice that the Lyapunov–Metzler inequalities (27) are recovered by imposing Wi (x) = Vi (x) in the extended Lyapunov–Metzler inequalities (41). These inequalities imply the partition of the state-space into subsets where the Lyapunov function (25) is decreasing. However, this Lyapunov function is not continuous with respect to x ∈ Rn and the stabilizing property of the switching rule (42) depends on the jumps of v(x) in the switching instances and by the possible presence of sliding modes. Despite this fact, inequalities (41) are easy to be handled and verified, and this is the main advantage for their use (see the illustrative example in Section 5). Further research is necessary to incorporate to (42) additional constraints that imply global stability of the equilibrium point x = 0. Notice however that the switching rule (23), if stabilizable, furnishes a computable upper bound of an integral cost. To be precise, let hi (x), i = 1, 2, . . . , N be assigned maps and consider the extended Lyapunov–Metzler inequalities N
jVi j i Wj + hi hi < 0, fi + jx j =1
i = 1, . . . , N.
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
Then, if (23) is stabilizable, it follows that (the proof is similar to the one of Corollary 1) ∞ h(t) (x) h(t) (x) dt < min Vi (x0 ). i∈I˜(x0 )
0
Remark 7. The role of the strictly positive parameters j i is immaterial in the case N = 2. Indeed, in this case, the inequalities reduce to jV¯1 f1 + < 0, jx
jV¯2 f2 + (− ) < 0, jx
x = 0,
101
for all x, y = 0. Then, provided (t) is given by the rule (43), the equilibrium point x = 0 is a globally asymptotically stable equilibrium of (1). Proof. Assume that (45) holds and (t) ∈ for all t 0. Multiplying each inequality by i (t), adding up for all i = 1, . . . , N, and letting y = N i=1 i fi (x) one gets 0>
jVm (x, t) jVm (x, t) fm (x, t) + , jx jt
where
where = W2 − W1 , V¯1 = V1 /21 and V¯2 = V2 /12 .
Vm (x, t) =
4. Stability of time varying polytopic systems
N
i (t)Vi (x),
fm (x, t) =
i=1
In this section we discuss the stability of systems defined by (1) which are classified in the literature as polytopic systems [8]. In this case, the very basic requirement on each trajectory of (t) is that (t) ∈ for all t 0. Since this property alone does not suffice to define the way (t) evolves with time, we consider further that (t) ˙ = (t),
(0) = 0 ,
(43)
where ∈ RN×N is a Metzler matrix a priori known or to be determined by the designer. The rationale behind this choice follows from a well-known property of this class of matrices. Whenever the initial condition 0 ∈ then (t) ∈ for all t 0. From now on it is assumed that 0 ∈ . Due to the fact that ∈ M is necessarily marginally stable, and there exists ∞ ∈ such that ∞ = 0 then (t) evolves inside and goes to ∞ as t goes to infinity. The time evolution of (t) toward ∞ depends, of course, on each particular choice of ∈ M. The results given in the sequel are based on the parameter dependent Lyapunov function N v(x(t)) := i (t)Vi (x(t)) (44) i=1
defined by an adequately determined set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at x = 0. The next theorem provides the way to determine the Lyapunov function (44) and a sufficient condition for asymptotic stability of the zero equilibrium of the considered system. Theorem 5. Assume that there exist a set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at zero, a matrix ∈ M, and a function G(x, y) satisfying, for each x and y and for each i = 1, . . . , N the following inequalities N
jVi (x) y+ j i Vj (x) jx j =1 jG(x, y) jG(x, y) + + (fi (x) − y) jx jy
(46)
0>
(45)
N
i (t)fi (x).
i=1
This states that the time derivative of the parameter dependent Lyapunov function (44), which is differentiable, positive definite, radially unbounded and zero at zero, is negative along all trajectories of x(t) ˙ = f(t) (x(t)), proving thus global asymptotical stability of (1). The role of the function G(x, y) is to extend the classical condition of stability under arbitrary signal, i.e. the existence of a positive definite function V (x) such as (jV (x)/jx)fi (x) < 0 for each x = 0. Indeed, such a condition is recovered from (45) by letting Vi (x) = V (x) and G(x, y) = V (x). Theorem 5 does not require the set {f1 , . . . , fN } to be composed only by asymptotically stable vector fields. Of course, this is a consequence of our previous assumption which implies that the variables (t) and (t) ˙ are not independent but are coupled together by the linear model (43). The inequalities (45) must be satisfied for all x, y = 0 and in particular for all x, y = 0 satisfying the additional constraint y = fi (x) for each i = 1, . . . , N implying that the inequalities N
jVi fi + j i Vj < 0, jx
i = 1, . . . , N
(47)
j =1
must hold for all x = 0, which is nothing else than the stability condition provided by Theorem 3 for state switching control. The conclusion is that the stability condition of Theorem 5 is more exigent than the one of Theorem 3. This fact is expected, since the set of vector fields defined in (2) is a subset of that defined in (3). 5. An illustrative example In this section we propose an illustrative example of application of some of the theoretical tools discussed. The purpose of this example is twofold. First, to illustrate the theory, and then, to underscore that the proposed switching law may be nonrobust (this is actually true also if the underlying system is linear), hence further research is needed to derive a robustly stabilizing switching mechanism.
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1.6 1.4 1.2
v
1 0.8 0.6 0.4 0.2 0 1 1
0.5 x2
0.5
0
0
−0.5 −1
−0.5
x1
−1
Fig. 1. The function v(x) for the Artstein circle.
Consider the so-called Artstein circle [1,13,17], namely the system described by the equation x˙1 = (−x12 + x22 )u, x˙2 = − 2x1 x2 u.
(48)
This system is asymptotically controllable and it is (robustly) asymptotically stabilizable exploiting the results in [13]. We exploit the observation in Remark 5 to design a stabilizing control law. For, consider the set of vector fields F = {f1 , f2 }
f1 =
−x12 + x22
⎧ x1 0, ⎨ V+ (x1 , x2 ), 2 V1 (x1 , x2 ) = 1 + 2x1 ⎩ V− (x1 , x2 ) x1 < 0 1 + x12 and ⎧ 2 ⎨ 1 + 2x1 V+ (x1 , x2 ), x1 0, 1 + x12 V2 (x1 , x2 ) = ⎩ V− (x1 , x2 ), x1 < 0. Note that V1 and V2 are continuous, and differentiable for any x1 = 0. Moreover, for i = 1, 2,
with
Let
(x)x Vi (x1 , x2 ) K(x)x,
−2x1 x2
and f2 = −f1 . Note that f1 (resp. f2 ) is obtained from system (48) setting u = 1 (resp. u = −1). Consider now the functions V+ (x1 , x2 ) = (x12 + x22 )
sign(x2 ) − 2 arctan(x1 /x2 ) 2x2
for x1 0 and V− (x1 , x2 ) = (x12 + x22 )
sign(x2 ) + 2 arctan(x1 /x2 ) 2x2
for x1 0. Notice that V+ (0, x2 ) = V− (0, x2 ) =
|x2 | 2
and that V1 is differentiable for all x1 > 0 and V2 is differentiable for all x1 < 0.
for some functions 0 < (x) < K(x). The function v(x) = min{V1 (x), V2 (x)}, depicted in Fig. 1, is continuous, positive definite, radially unbounded, zero at zero and it is differentiable for any x1 = 0. For this example, and after simple computations, the conditions (32) yield − (x12 + x22 ) + 1 ( 2 (V2 ) − 1 (V1 )) 0, − (x12 + x22 ) + 2 ( 1 (V1 ) − 2 (V2 )) 0, for all nonzero x, and for some 1 and 2 positive. Selecting 1 (v) = 2 (v) = v 2 , and 1 = 2 = yields the unique condition x2 (−1 + (K 2 (x) − 2 (x))) < 0
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
for all nonzero x, and this holds selecting, for example, =
1 2(K 2 (x) − 2 (x))
.
In conclusion, taking the minimum argument between V1 (x) and V2 (x) yields the control law 1 if x1 0, u(x) = −1 if x1 < 0 or, alternatively 1 if x1 > 0, u(x) = −1 if x1 0 that, by Theorem 3, are both globally asymptotically stabilizing the zero equilibrium of the system (48). This is the same control law proposed in [13]. Therein (see also [17]) it is, however, argued that this control law is not robust against measurement noises, and a (simple) robust modification of this controller (in the spirit of the result in [14]) is proposed. Hence, the results presented in this paper, and in its linear counterparts, see [6], have to be understood as first step toward a general (robust) stabilization theory for switched systems. We believe that this theory could be developed exploiting the results in this paper and the results in [13–15]. 6. Conclusions In this paper we have introduced stability conditions for switched systems. They have been used for control synthesis of state dependent (closed-loop) switching rules using nonlinear Lyapunov–Metzler inequalities. The determination of a guaranteed cost associated to each control strategy has been addressed. A relationship between stability of switched systems and stability of time varying polytopic systems has been investigated, yielding useful mathematical properties for both classes of dynamical systems. Various issues deserve more attention. The first is related to the development of numerical algorithms for the solution of the introduced nonlinear Lyapunov–Metzler inequalities. The second one is the possible generalization of the stability conditions to cope with an optimal guaranteed cost. Taking into account the nonlinear nature of the involved stability conditions, this point constitutes a real theoretical challenge. Finally, the
103
crucial and difficult issue of robust stability requires an in-depth investigation. These issues are being currently studied. References [1] Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal. TMA 7 (1983) 1163–1173. [2] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hydrid systems, IEEE Trans. Automat. Control 43 (1998) 475–482. [3] D. Casagrande, A. Astolfi, T. Parisini, Control of nonholonomic systems: a simple stabilizing time-switching strategy, in: Proceedings of the IFAC World Congress, Prague, Czech Republic, 2005. [4] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control 37 (1992) 38–53. [5] K.M. Garg, Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives, WileyInterscience, New York, 1998. [6] J.C. Geromel, P. Colaneri, Stability and stabilization of continuous-time switched systems, SIAM J. Control Optim. 45 (2006) 1915–1930. [7] J.C. Geromel, P. Colaneri, Robust stability of time varying polytopic systems, Systems Control Lett. 55 (2006) 81–85. [8] J.C. Geromel, P.L.D. Peres, J. Bernussou, On a convex parameter space method for linear control design of uncertain systems, SIAM J. Control Optim. 29 (1991) 381–402. [9] J.P. Hespanha, Uniform stability of switched linear systems: extensions of LaSalle’s principle, IEEE Trans. Automat. Control 49 (2004) 470–482. [10] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, New York, 1999, pp. 2655–2660. [11] D. Liberzon, Switching in Systems and Control, Birkhauser, Basel, 2003. [12] S. Prajna, A. Papachristodoulou, Analysis of switched and hybrid systems, in: Proceedings of the American Control Conference, Denver, 2003. [13] C. Prieur, A robust globally asymptotically stabilizing feedback: the example of the Artstein’s circles, in: A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek (Eds.), Nonlinear Control in the Year 2000, vol. 2 (NCN), Lecture Notes in Control and Information Sciences, vol. 258, Springer, London, 2000, pp. 279–300. [14] C. Prieur, Uniting local and global controllers with robustness to vanishing noise, Math. Control Signals Systems 14 (2001) 143–172. [15] C. Prieur, A. Astolfi, Robust stabilization of chained systems via hybrid control, IEEE Trans. Automat. Control 48 (2003) 1768–1772. [16] R. Rockafellar, Convex Analysis, Princeton Press, Princeton, NJ, 1970. [17] E.D. Sontag, Clocks and insensitivity to small measurement errors, ESIAM: COCV, www.emath.fr/cocv/4 (1999), pp. 537–557. [18] X. Xu, P.J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. Control 49 (2004) 2–16. [19] J. Zhao, G.M. Dimirovski, Quadratic stability of a class of switched nonlinear systems, IEEE Trans. Automat. Control 49 (2004) 574–578.
Stabilization of continuous-time switched nonlinear systems夡 Patrizio Colaneri a , José C. Geromel b , Alessandro Astolfi c,d,∗ a Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza leonardo da Vinci 32, Milano, Italy b DSCE / School of Electrical and Computer Engineering, UNICAMP, CP 6101, 13083 - 970 Campinas, SP, Brazil c Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK d DISP, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy
Received 23 January 2007; received in revised form 20 July 2007; accepted 22 July 2007 Available online 6 September 2007
Abstract The paper considers three problems for continuous-time nonlinear switched systems. The first result of this paper is a open-loop stabilization strategy based on dwell time computation. The second considers a state switching strategy for global stabilization. The strategy is of closed loop nature (trajectory dependent) and is designed from the solution of what we call nonlinear Lyapunov–Metzler inequalities from which the stability condition is expressed. Finally, results on the stabilization of nonlinear time varying polytopic systems are provided. © 2007 Elsevier B.V. All rights reserved. Keywords: Switched systems; Stability; Stabilization
while the second one is such that, for each t 0,
1. Introduction This paper aims at providing new results on stabilizing control synthesis for a continuous-time switched nonlinear system of the following general form: x(t) ˙ = f(t) (x(t)),
x(0) = x0 ,
(1)
where x(t) ∈ Rn is the state, (t) is the switching rule and x0 is the initial condition. Given a set of vector fields {f1 (x), . . . , fN (x)}, such that fi (0) = 0 for all i = 1, . . . , N, two different classes of switched systems are studied. The first is characterized by the fact that the switching rule, for each t 0, is such that f(t) ∈ {f1 , . . . , fN },
(2)
夡 This research was supported by grants from “Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq” - Brazil, and supported by the Italian National Research Council (CNR) and by MIUR project “Advanced control methodologies for hybrid dynamical systems”. ∗ Corresponding author. Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK. Tel.: +44 171 594 6289; fax: +44 171 594 6282. E-mail addresses: [email protected] (P. Colaneri), [email protected] (J.C. Geromel), a.astolfi@ic.ac.uk (A. Astolfi).
0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.07.001
f(t) ∈ co{f1 , . . . , fN },
(3)
where co{·} denotes the convex hull. It is important to make clear the basic difference between these two classes of switched systems. The model (2) naturally imposes a discontinuity on f(t) since this vector must jump instantaneously from fi to fj for some i = j once switching occurs. In other words, f(t) is constrained to jump among the N vertices of the vector polytope co{f1 , . . . , fN }. The model defined by (3) is more general in the sense that the interior of the same polytope is now feasible for f(t) and so it supports switching rules with no discontinuity with respect to time. As it will become clear in the sequel there are some important relationship between the stability conditions of both models. The paper contributes in three directions, namely dwell time computation, state-feedback stabilization and polytopic stability analysis. The idea underlying the first issue has been spurred by reading the recent paper [12], where a method for stability analysis of switched and hybrid systems is provided by using polynomial and piecewise polynomial Lyapunov functions. A result is provided on the computation of an upper bound of the dwell time, by means of a suitable Lyapunov inequality. This result provides a characterization of some time switching stabilizing strategies in terms of dwell time computation. It finds
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its counterpart in the linear case in [6], and it is shown to be less conservative than that provided in [10], although in general less computationally attractive, see also [3] for a simple time switching stabilizing strategy for nonholonomic systems. The second issue concerns the state-feedback stabilization of autonomous switched nonlinear systems. It is rooted in the setting in which multiple Lyapunov functions are used to find a stabilizing switching strategy, see [2]. The new idea, inherited from the linear case, see [6], is to use the so-called Lyapunov–Metzler inequalities, thanks to which the final Lyapunov function is built as the minimum over a finite number of functions. It is worth mentioning that this idea extends a result (for the linear case) in [11], valid in the case N = 2. The extension of La-Salle’s principle for linear switched systems can be found in [9]. The third issue concerns the stability of nonlinear polytopic systems. Among the extensive literature in the linear case, the results provided here is based on [7]. A sufficient condition for the stability of polytopic nonlinear systems is provided in terms of a Lyapunov inequality. The paper is organized as follows. In Section 2 a time switching stabilizing control strategy is provided based on the computation of an upper bound on the minimum dwell time. In Section 3 a state switching globally stabilizing strategy is devised based on the solution of Lyapunov–Metzler inequalities. Finally the stability of polytopic systems is considered and relations with the previous material are pointed out.
where x(T ) is the solution of the differential equation x˙ =fj (x) with initial state x at t = 0. Then the time switching control (4) with tk+1 − tk T makes the equilibrium x = 0 of (1) globally exponentially stable.
2. Time switching control
where the second inequality holds from the fact that for every = Tk − T 0 it is true that (d/dt)Vi (x()) < 0. The consequence is that there exists ∈ (0, 1) such that Vi (x) Vj (x(T )), for each i and j, so that
This section is entirely dedicated to design a time switching control law for the switched linear system defined by the models (1) and (2) where it is assumed that each vector field in the set {f1 , . . . , fN } is stable. The problem under consideration can be stated as follows. Determine a minimum dwell time T∗ > 0 such that the equilibrium point x = 0 of the system (1) is globally asymptotically stable with the time switching control (t) = i ∈ {1, . . . , N},
t ∈ [tk , tk+1 ),
(4)
where tk and tk+1 are successive switching time instants satisfying tk+1 − tk T∗ for all k ∈ N and the index i ∈ {1, . . . , N}, selected at each instant of t 0, is arbitrary. Hence, asymptotic stability is preserved whenever (t) remains unchanged for a period of time greater or equal to the minimum dwell time T∗ . The next theorem provides the theoretical basis toward a possible solution of this problem by characterizing an upper bound for T∗ . It uses the concept of multiple Lyapunov functions.
Proof. To begin with note that if there are no jumps the result trivially follows by Eq. (6). Consider now, in accordance with (4), that (t) = i ∈ {1, . . . , N} for all t ∈ [tk , tk+1 ) where tk+1 = tk + Tk with Tk T > 0 and that at t = tk+1 the time switching control jumps to (t) = j ∈ {1, . . . , N}. From (6), it is seen that, for all t ∈ [tk , tk+1 ), the time derivative of the Lyapunov function v(x(t)) = V(t) , along an arbitrary solution of (1) satisfies, for x = 0, v(x(t)) ˙ =
jVi fi < 0, jx
(8)
which enables to conclude that x(t)2 ai−1 e−i (t−tk ) v(x(tk )),
∀t ∈ [tk , tk+1 ).
(9)
On the other hand, using the inequalities (7) we have v(x(tk+1 )) = Vj (x(tk+1 )) < Vi (x(tk + Tk − T )) tk +Tk −T d < Vi (x(tk )) + Vi (x()) d dt tk < v(x(tk )),
v(x(tk )) k v(x0 ),
∀k ∈ N,
(10)
(11)
which together with (9) implies that the equilibrium x = 0 of (1) is globally exponentially stable. This result deserves some comments. First, since all systems in the set {f1 , . . . , fN } are supposed to be asymptotically stable, the constraints (6) are always feasible and the constraints (7) are satisfied when T > 0 is taken large enough. Second, an upper bound for the dwell time can be computed directly from (5)–(6). Indeed, a simple computation shows that Vj (x(T )) bj x(T )2 bj ai−1 Vi (x(T )) bj ai−1 e−i T Vi (x) so that (7) holds selecting bj 1 T max log . (12) i,j i ai
Theorem 1. Assume that, for some T > 0, there exist a collection of positive definite radially unbounded functions {V1 , . . . , VN } and positive numbers ai , bi , i such that, for each x,
Of course the upper bound (12) is a consequence of condition (7), hence it is more conservative. Third, assume that the vector fields fi share the same Lyapunov function V (x), i.e.
ai x2 Vi (x)bi x2 , ∀ i = 1, . . . , N, jVi fi < − i Vi , ∀ i = 1, . . . , N, x = 0, jx Vi (x) > Vj (x(T )), ∀ i = j ,
jV fi < − i V , jx
(5) (6) (7)
∀i = 1, . . . , N,
(13)
then the inequality (7) is satisfied for any T > 0 meaning that the switching policy (4) may jump from i to j arbitrarily fast
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
preserving asymptotic stability. In the linear setting this corresponds to the quadratic stability condition. Finally, with T > 0 fixed, it is always possible to define a time switching control strategy (4) such that f(t) (x) is periodic as a function of t. As a consequence, a necessary condition for the feasibility of constraints (6) and (7) is that the nonlinear system with vector field f(t) (x) is globally exponentially stable, for each periodic switching time-policy with period greater than T. However, since (4) may produce nonperiodic policies as well, this condition for the existence of a feasible solution to inequalities (6)–(7) is, in general, not sufficient. Finally, it is possible to generalize the result of Theorem 1 in order to define a guaranteed cost to go from an arbitrary initial point to the origin, associated to the stabilizing time switching rule (4) with tk+1 − tk T for any fixed T > 0. To this end we make the assumption that T > 0 is known such that tk+1 − tk T for all k ∈ N. Clearly, these quantities are related through T T T∗ where the second inequality assures global stability. Now consider the switched system (1) and the output variable y(t) = h(t) (x(t)),
(14)
where hi (x), i = 1, 2, . . . , N, are given p-valued maps. The next result provides an upper bound to the L2 norm of y(t) for any switching signals (t) with residence time between any two consecutive switching instants not less than T.
97
that Vi (x) > ri (x) and, for x = 0, j(Vi − ri ) jri fi < − hi hi − fi jx jx < − hi (x(T)) hi (x(T)) <0
(19)
for all i =1, · · · , N . The important consequence of this calculation is that for each i =1, · · · , N the inequality d(Vi −ri )/dt 0 holds. Using this property, taking into account the switching strategy (4) with tk+1 − tk = Tk T and the inequalities (17), one gets v(x(tk+1 )) = V(tk+1 ) (x(tk+1 )) < V(tk ) (x(tk + Tk − T )) − r(tk ) (x(tk + Tk − T )) < V(tk ) (x(tk )) − r(tk ) (x(tk )) < v(x(tk )) − r(tk ) (x(tk )),
which summing up for all k ∈ N and taking into account that Ttk+1 − tk , yields ∞ ∞ Tk h(t) (x(t)) h(t) (x(t)) dt= h(tk ) (x(t+tk )) h(tk ) k=0 0
0
× (x(t + tk )) dt ∞ T h(tk ) (x(t+tk )) h(tk )
Theorem 2. Let T T > 0 be given. Define the set of nonnegative maps ri (x) :=
T 0
hi (x(t)) hi (x(t)) dt,
i = 1, . . . , N,
∀ i = 1, . . . , N, x = 0,
(16) (17)
The time switching control (4) with Ttk+1 − tk T makes the equilibrium x = 0 of (1) globally asymptotically stable and
0
∞
× (x(t+tk )) dt
∞
r(tk ) (x(tk ))
k=0
jVi fi + hi hi < − i Vi , ∀i = 1, . . . , N, jx Vi (x) > Vj (x(T )) + ri (x), ∀ i = j .
k=0 0
(15)
where x =x(0) and x(t) is the solution of x˙ =fi (x). Assume that there exist a collection of positive definite radially unbounded functions {V1 , . . . , VN } and positive numbers ai , bi , i such that, for each x, ai x2 Vi (x)bi x2 ,
(20)
h(t) (x(t)) h(t) (x(t)) dt < V(0) (x0 ).
(18)
Proof. Since each function ri (x) defined in (15) is nonnegative and inequalities (16)–(17) are satisfied, then inequalities (6)–(7) are also satisfied. As a consequence, global asymptotical stability follows from Theorem 1. On the other hand, using (15) together with the inequalities (16) and (17) we have
< v(x0 ), which proves the claim.
(21)
It is interesting to observe that the conditions of Theorem 2 are feasible if and only if TT T∗ and from (18) it is seen that a more accurate guaranteed cost is obtained whenever the value of T is chosen as small as possible. In addition, the choice T = +∞ enables to conclude that the proposed time switching rule (4), with tk+1 − tk T∗ , makes the signal y(t) = h(x(t)), t 0 square integrable. Theorem 2 admits the extreme situation T = T = +∞ for which no jump occurs and inequalities (16)–(17) are verified for ∞ hi (x(t)) hi (x(t)) + εx(t)2 dt > ri 0, Vi (x) = 0
i = 1, . . . , N
(22)
with ε > 0. When ε > 0 goes to zero, Vi goes to ri and (18) becomes a well-known result. On the other hand, for T > 0 arbitrarily small and any T T , feasibility holds whenever the set of vectors {f1 , . . . , fN } admits a common Lyapunov function.
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globally asymptotically stabilizes the equilibrium point x = 0 of the nonlinear system (1).
3. State switching control In this section we consider the system (1) where the switching rule satisfies (2). It is assumed that the state vector x(t) is available for feedback for all t 0, and the goal is to determine a function u(·) : Rn → {1, . . . , N}, such that the switching rule (t) = u(x(t))
(23)
assures that the equilibrium x = 0 of (1) is globally asymptotically stable. Note that we do not assume that any of the vector fields in the set {f1 , . . . , fN } be either locally or globally asymptotically stable. Define the simplex N N i = 1, i 0 (24) := ∈ R : i=1
and the function v(x) :=
min Vi (x),
(25)
i=1,...,N
where {V1 , . . . , VN } is a set of differentiable, positive definite and radially unbounded functions, which are zero at x = 0. As it will be clear in the sequel, the function v(x) is a candidate Lyapunov function, crucial for our purposes. However, even if the functions Vi (x) are differentiable, the function v(x) is not (in general) differentiable everywhere. To address this issue the set I (x) = {i : v(x) = Vi (x)} plays a central role since v(x) fails to be differentiable at all x ∈ Rn such that card I (x) is discontinuous [16]. Before proceeding, recall the class of Metzler matrices denoted by M and constituted by all matrices ∈ RN×N with elements ij , such that ij 0 ∀i = j,
N
ij = 0, ∀j .
(26)
i=1
It is clear that any ∈ M presents an eigenvalue at the origin of the complex plane since c = 0 where c = [1 · · · 1]. In addition it is well known from the Frobenius–Perron’s theorem that the eigenvector associated to the null eigenvalue of is nonnegative, yielding the conclusion that there always exists ∞ ∈ such that ∞ = 0. The next theorem summarizes the main result of this section. Theorem 3. Assume that there exist a set of functions {V1 , . . . , VN }, which are all differentiable, positive definite, radially unbounded and zero at zero, and a matrix ∈ M satisfying the Lyapunov–Metzler inequalities jVi fi + jx
N
j i Vj < 0,
i = 1, . . . , N
(27)
j =1
for all x = 0. Then the state switching control (23) with u(x(t)) = arg min Vi (x(t)) i=1,...,N
(28)
Proof. To begin with note that the Lyapunov function (25) is piecewise differentiable, positive definite, radially unbounded and zero at x = 0. Moreover, the Lyapunov function (25) is not differentiable for all x ∈ Rn . For this reason we need to deal with the Dini derivative (see [5]) v(x(t + h)) − v(x(t)) . h
D + v(x(t)) = lim sup h→0+
(29)
Assume, in accordance with (28), that at an arbitrary t 0, the state switching control is given by (t) = u(x(t)) = i for some i ∈ I (x(t)). Hence, from Eqs. (29) and (1), we have D + v(x(t)) = min
l∈I (x(t))
jVl jVi fi fi , jx jx
(30)
where the inequality holds from the fact that i ∈ I (x(t)). Finally, remembering that (27) is valid for ∈ M and that Vj (x) Vi (x) for all j = i = 1, . . . , N once again due to the fact that i ∈ I (x(t)), using the Lyapunov–Metzler inequalities (27) one gets ⎛ ⎞ N D + v(x(t)) < − ⎝ j i Vj (x)⎠ ⎛ −⎝
j =1 N
⎞ j i ⎠ Vi (x) = 0
(31)
j =1
for all x = 0, which proves the claim.
Remark 1. Theorem 3 is particularly important when the set {f1 , . . . , fN } is composed of unstable vector fields. Indeed, if a function Vi (x) is locally quadratic, a necessary condition for the Lyapunov–Metzler inequalities to be feasible with respect to {V1 , . . . , VN } is that the vector fields fi + (ii /2)x be locally asymptotically stable. Since ii 0 this condition does not imply local asymptotic stability of any of the fi ’s. On the other hand, in general, the Lyapunov–Metzler inequalities imply that D + Vi (x(t)) < |ii |Vi (x(t))
∀i
and, since the functions Vi are radially unbounded, this implies that the vector fields {f1 , . . . , fN } are complete and that, along the trajectories of fi the functions Vi are such that Vi (x(t)) Vi (x(0))e|ii |t
∀i.
Remark 2. An interesting case occurs when all vector fields {f1 , . . . , fN } are globally asymptotically stable for which the choice = 0 is possible and the state switching strategy proposed preserves stability. Furthermore, if the set {f1 , . . . , fN } admits a unique Lyapunov function V, then the Lyapunov–Metzler inequalities admit a solution V1 = · · · = VN = V and I (x(t)) = {1, . . . , N} for all t 0. In this classical but particular case, at any t 0, the control
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
99
law u(x(t)) being an arbitrary logic state i ∈ {1, . . . , N}, asymptotic stability is once again guaranteed.
Theorem 4. Consider inequalities (34) and assume that, pointwise, the limits lim →∞ Vj exist. Then, the system
Remark 3. (Chattering). Another important feature of Theorem 3 is that chattering in the switching, when it occurs, is always stable in the Filippov sense. Indeed, assume that x ∈ Rn belongs to a certain region C of the state space where the cardinality of I (x) is greater than one. From the Lyapunov function (25), a switching from i ∈ I (x) to j ∈ I (x) is possible only if (jVj /jx)fi (jVi /jx)fi < 0 where the last inequality follows directly from (27). Hence, we conclude that whenever x ∈ C the time derivative of the positive definite function Vj is strictly negative along all trajectories such that x˙ ∈ co{fi : i ∈ I (x)} which implies that they are asymptotically stable. This idea finds its roots in the result in [11], for N = 2 and in the linear case.
x(t) ˙ = f∞ (x(t))
Remark 4. Theorem 3 also holds if the matrix is a function of x, i.e. = (x), provided that, for each fixed x ∈ Rn , it satisfies (x) ∈ M.
is globally asymptotically stable, where f ∞ =
N
Proof. Pointwise it follows that lim
→∞
N
N
jVi fi + j i j (Vj ) < 0, jx
i = 1, . . . , N
(32)
j =1
for all x = 0, and for some positive definite functions j ’s. Remark 6. In the literature of linear systems the Lyapunov–Metzler inequalities, with ∈ M fixed, have been introduced in order to study the mean-square (MS) stability of Markov Jump Linear Systems (MJLS), see e.g. [4]. In that context, the Metzler matrix ∈ M is given and represents the infinitesimal transition matrix of a Markov chain (t) governing the dynamical system. In this respect, each component of the vector (t) ∈ is the probability of the Markov chain to be on the ith logical state and obeys the differential equation ˙ = (t), (t)
j i Vj 0,
∀i.
j =1
Moreover, recalling the role of the vector ∞ in the Metzler matrix , we have N
∞i
i=1
Remark 5. Condition (27) in Theorem 3 can be replaced by the condition
∞i fi .
i=1
N
j i Vj = 0.
j =1
It then follows that ⎡ ⎤ V1 ⎢ V2 ⎥ ⎥ lim ⎢ ⎣ ... ⎦ = 0,
→∞ VN
which implies that lim →∞ Vi =V , for all i =1, . . . , N. Finally, (34) yields N i=1
∞i
jVi fi < 0 jx
and hence jV f < 0, jx ∞
f∞ =
N
∞i fi .
i=1
The proof is then concluded. (0) = 0 ∈ ,
where the eigenvector ∞ ∈ associated to the null eigenvalue of represents the stationary probability vector. Consider now the modified Lyapunov–Metzler inequalities defined as N
jVi fi +
j i Vj < 0, jx
i = 1, . . . , N,
(33)
(34)
j =1
for all x = 0, where is a positive parameter. This parameter multiplies all elements of the matrix , therefore the matrix
is still a Metzler matrix, i.e. ∈ M whenever ∈ M. Notice that these new inequalities are those relative to vector fields {f1 / , . . . , fN / }, obtained by the time scaling t → t/ .
The above theorem relates the Lyapunov–Metzler inequalities with the stability of the “average” system characterized by the vector field f∞ . Indeed, if the inequalities (34) hold for a sufficiently large , then f∞ is globally asymptotically stable. This is a relevant point, since it recovers the classical stability condition (in the linear setting) provided in [11,19]. To see this fact in the present context assume that there exists ∞ ∈ such that f∞ is globally asymptotically stable, making possible the determination of V > 0 satisfying the Lyapunov inequality (jV /jx)f∞ < 0. Hence, the switching rule (23), with u(x(t)) = arg min
i=1,...,N
jV fi , jx
(35)
makes the equilibrium x = 0 of the switched system (1) globally asymptotically stable. Indeed, considering the Lyapunov
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P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
function V (x) we have jV jV jV jV f(t) = min fi = min f f < 0 V˙ (x(t))= i=1,...,N jx jx jx ∞ ∈ jx for all x = 0. It is important to keep in mind that the numerical determination (if any) of ∈ and V > 0 such that N jV i f i < 0 jx i=1
is not a simple task even in the simplest case of linear time invariant systems. We conclude this section introducing a guaranteed cost associated to the proposed state switching control law (28). To this end, consider again the output variable y(t) in (14). The next result provides an upper bound for the L2 norm of y(t) under the switching rule (23). Corollary 1. Assume that there exist a set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at zero, and a matrix ∈ M satisfying the Lyapunov–Metzler inequalities N
jVi fi + j i Vj + hi hi < 0, jx
i = 1, . . . , N,
(36)
j =1
system does not admit, in general, a closed form solution, see e.g. [18] for some insight in the problem. We finally discuss a possible stabilizing switching strategy that includes the previous one as a particular case and hence may provide less conservative results. To this end, define a set of positive definite functions {W1 (x), . . . , WN (x)} and the functions Hi (x) =
N
j i Wj (x),
i = 1, 2, . . . , N.
j =1
Due to the structure of the Metzler matrices, these functions cannot be strictly negative for all i = 1, . . . , N, since N n i=1 ∞i Hi (x) = 0. As a result, for each x ∈ R the set I˜(x) = {i : Hi (x) 0} is not empty, and it is possible to define the candidate Lyapunov function v(x) := min Vi (x).
(40)
i∈I˜(x)
Now, assume that there exist a set of functions {V1 , . . . , VN }, which are all differentiable, positive definite, radially unbounded and zero at zero, a set of positive definite functions {W1 , . . . , WN } and a matrix ∈ M satisfying the extended Lyapunov–Metzler inequalities N
jVi fi + j i Wj < 0, jx
for all x = 0. Then the state switching control (23) with u(x(t)) given by the Eq. (28) globally asymptotically stabilizes the equilibrium point x = 0 of (1) and it is such that the inequality ∞ J= h(t) (x) h(t) (x) dt < min Vi (x0 ) (37)
for all x = 0. Finally, consider the switching control rule (23) with
holds.
u(x(t)) = arg min Vi (x).
i=1,...,N
0
i = 1, . . . , N
(42)
i∈I˜(x)
Proof. The proof has the same structure as the proof of Theorem 3. The Lyapunov function (25) and the Lyapunov–Metzler inequalities (36) yield D + v(x(t)) < − h(t) (x) h(t) (x),
x = 0,
(38)
and, by integration it is readily verified that t D + v(x()) d v(x(t)) − v(x(0)) = 0
t
< − 0
h() (x) h() (x) d
(39)
for all t 0, proving thus the claim since, by asymptotic stability, v(x(t)) goes to zero as t goes to infinity. Notice that Corollary 1 provides a pre-computable upper bound to the control strategy minimizing the cost J. This upper bound depends on the choice of the parameters in the inequality (36) so that, in principle, it is possible to minimize it with respect to ∈ M. The optimal control problem for switched
(41)
j =1
Notice that the Lyapunov–Metzler inequalities (27) are recovered by imposing Wi (x) = Vi (x) in the extended Lyapunov–Metzler inequalities (41). These inequalities imply the partition of the state-space into subsets where the Lyapunov function (25) is decreasing. However, this Lyapunov function is not continuous with respect to x ∈ Rn and the stabilizing property of the switching rule (42) depends on the jumps of v(x) in the switching instances and by the possible presence of sliding modes. Despite this fact, inequalities (41) are easy to be handled and verified, and this is the main advantage for their use (see the illustrative example in Section 5). Further research is necessary to incorporate to (42) additional constraints that imply global stability of the equilibrium point x = 0. Notice however that the switching rule (23), if stabilizable, furnishes a computable upper bound of an integral cost. To be precise, let hi (x), i = 1, 2, . . . , N be assigned maps and consider the extended Lyapunov–Metzler inequalities N
jVi j i Wj + hi hi < 0, fi + jx j =1
i = 1, . . . , N.
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
Then, if (23) is stabilizable, it follows that (the proof is similar to the one of Corollary 1) ∞ h(t) (x) h(t) (x) dt < min Vi (x0 ). i∈I˜(x0 )
0
Remark 7. The role of the strictly positive parameters j i is immaterial in the case N = 2. Indeed, in this case, the inequalities reduce to jV¯1 f1 + < 0, jx
jV¯2 f2 + (− ) < 0, jx
x = 0,
101
for all x, y = 0. Then, provided (t) is given by the rule (43), the equilibrium point x = 0 is a globally asymptotically stable equilibrium of (1). Proof. Assume that (45) holds and (t) ∈ for all t 0. Multiplying each inequality by i (t), adding up for all i = 1, . . . , N, and letting y = N i=1 i fi (x) one gets 0>
jVm (x, t) jVm (x, t) fm (x, t) + , jx jt
where
where = W2 − W1 , V¯1 = V1 /21 and V¯2 = V2 /12 .
Vm (x, t) =
4. Stability of time varying polytopic systems
N
i (t)Vi (x),
fm (x, t) =
i=1
In this section we discuss the stability of systems defined by (1) which are classified in the literature as polytopic systems [8]. In this case, the very basic requirement on each trajectory of (t) is that (t) ∈ for all t 0. Since this property alone does not suffice to define the way (t) evolves with time, we consider further that (t) ˙ = (t),
(0) = 0 ,
(43)
where ∈ RN×N is a Metzler matrix a priori known or to be determined by the designer. The rationale behind this choice follows from a well-known property of this class of matrices. Whenever the initial condition 0 ∈ then (t) ∈ for all t 0. From now on it is assumed that 0 ∈ . Due to the fact that ∈ M is necessarily marginally stable, and there exists ∞ ∈ such that ∞ = 0 then (t) evolves inside and goes to ∞ as t goes to infinity. The time evolution of (t) toward ∞ depends, of course, on each particular choice of ∈ M. The results given in the sequel are based on the parameter dependent Lyapunov function N v(x(t)) := i (t)Vi (x(t)) (44) i=1
defined by an adequately determined set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at x = 0. The next theorem provides the way to determine the Lyapunov function (44) and a sufficient condition for asymptotic stability of the zero equilibrium of the considered system. Theorem 5. Assume that there exist a set of functions {V1 , . . . , VN }, which are differentiable, positive definite, radially unbounded and zero at zero, a matrix ∈ M, and a function G(x, y) satisfying, for each x and y and for each i = 1, . . . , N the following inequalities N
jVi (x) y+ j i Vj (x) jx j =1 jG(x, y) jG(x, y) + + (fi (x) − y) jx jy
(46)
0>
(45)
N
i (t)fi (x).
i=1
This states that the time derivative of the parameter dependent Lyapunov function (44), which is differentiable, positive definite, radially unbounded and zero at zero, is negative along all trajectories of x(t) ˙ = f(t) (x(t)), proving thus global asymptotical stability of (1). The role of the function G(x, y) is to extend the classical condition of stability under arbitrary signal, i.e. the existence of a positive definite function V (x) such as (jV (x)/jx)fi (x) < 0 for each x = 0. Indeed, such a condition is recovered from (45) by letting Vi (x) = V (x) and G(x, y) = V (x). Theorem 5 does not require the set {f1 , . . . , fN } to be composed only by asymptotically stable vector fields. Of course, this is a consequence of our previous assumption which implies that the variables (t) and (t) ˙ are not independent but are coupled together by the linear model (43). The inequalities (45) must be satisfied for all x, y = 0 and in particular for all x, y = 0 satisfying the additional constraint y = fi (x) for each i = 1, . . . , N implying that the inequalities N
jVi fi + j i Vj < 0, jx
i = 1, . . . , N
(47)
j =1
must hold for all x = 0, which is nothing else than the stability condition provided by Theorem 3 for state switching control. The conclusion is that the stability condition of Theorem 5 is more exigent than the one of Theorem 3. This fact is expected, since the set of vector fields defined in (2) is a subset of that defined in (3). 5. An illustrative example In this section we propose an illustrative example of application of some of the theoretical tools discussed. The purpose of this example is twofold. First, to illustrate the theory, and then, to underscore that the proposed switching law may be nonrobust (this is actually true also if the underlying system is linear), hence further research is needed to derive a robustly stabilizing switching mechanism.
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P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
1.6 1.4 1.2
v
1 0.8 0.6 0.4 0.2 0 1 1
0.5 x2
0.5
0
0
−0.5 −1
−0.5
x1
−1
Fig. 1. The function v(x) for the Artstein circle.
Consider the so-called Artstein circle [1,13,17], namely the system described by the equation x˙1 = (−x12 + x22 )u, x˙2 = − 2x1 x2 u.
(48)
This system is asymptotically controllable and it is (robustly) asymptotically stabilizable exploiting the results in [13]. We exploit the observation in Remark 5 to design a stabilizing control law. For, consider the set of vector fields F = {f1 , f2 }
f1 =
−x12 + x22
⎧ x1 0, ⎨ V+ (x1 , x2 ), 2 V1 (x1 , x2 ) = 1 + 2x1 ⎩ V− (x1 , x2 ) x1 < 0 1 + x12 and ⎧ 2 ⎨ 1 + 2x1 V+ (x1 , x2 ), x1 0, 1 + x12 V2 (x1 , x2 ) = ⎩ V− (x1 , x2 ), x1 < 0. Note that V1 and V2 are continuous, and differentiable for any x1 = 0. Moreover, for i = 1, 2,
with
Let
(x)x Vi (x1 , x2 ) K(x)x,
−2x1 x2
and f2 = −f1 . Note that f1 (resp. f2 ) is obtained from system (48) setting u = 1 (resp. u = −1). Consider now the functions V+ (x1 , x2 ) = (x12 + x22 )
sign(x2 ) − 2 arctan(x1 /x2 ) 2x2
for x1 0 and V− (x1 , x2 ) = (x12 + x22 )
sign(x2 ) + 2 arctan(x1 /x2 ) 2x2
for x1 0. Notice that V+ (0, x2 ) = V− (0, x2 ) =
|x2 | 2
and that V1 is differentiable for all x1 > 0 and V2 is differentiable for all x1 < 0.
for some functions 0 < (x) < K(x). The function v(x) = min{V1 (x), V2 (x)}, depicted in Fig. 1, is continuous, positive definite, radially unbounded, zero at zero and it is differentiable for any x1 = 0. For this example, and after simple computations, the conditions (32) yield − (x12 + x22 ) + 1 ( 2 (V2 ) − 1 (V1 )) 0, − (x12 + x22 ) + 2 ( 1 (V1 ) − 2 (V2 )) 0, for all nonzero x, and for some 1 and 2 positive. Selecting 1 (v) = 2 (v) = v 2 , and 1 = 2 = yields the unique condition x2 (−1 + (K 2 (x) − 2 (x))) < 0
P. Colaneri et al. / Systems & Control Letters 57 (2008) 95 – 103
for all nonzero x, and this holds selecting, for example, =
1 2(K 2 (x) − 2 (x))
.
In conclusion, taking the minimum argument between V1 (x) and V2 (x) yields the control law 1 if x1 0, u(x) = −1 if x1 < 0 or, alternatively 1 if x1 > 0, u(x) = −1 if x1 0 that, by Theorem 3, are both globally asymptotically stabilizing the zero equilibrium of the system (48). This is the same control law proposed in [13]. Therein (see also [17]) it is, however, argued that this control law is not robust against measurement noises, and a (simple) robust modification of this controller (in the spirit of the result in [14]) is proposed. Hence, the results presented in this paper, and in its linear counterparts, see [6], have to be understood as first step toward a general (robust) stabilization theory for switched systems. We believe that this theory could be developed exploiting the results in this paper and the results in [13–15]. 6. Conclusions In this paper we have introduced stability conditions for switched systems. They have been used for control synthesis of state dependent (closed-loop) switching rules using nonlinear Lyapunov–Metzler inequalities. The determination of a guaranteed cost associated to each control strategy has been addressed. A relationship between stability of switched systems and stability of time varying polytopic systems has been investigated, yielding useful mathematical properties for both classes of dynamical systems. Various issues deserve more attention. The first is related to the development of numerical algorithms for the solution of the introduced nonlinear Lyapunov–Metzler inequalities. The second one is the possible generalization of the stability conditions to cope with an optimal guaranteed cost. Taking into account the nonlinear nature of the involved stability conditions, this point constitutes a real theoretical challenge. Finally, the
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crucial and difficult issue of robust stability requires an in-depth investigation. These issues are being currently studied. References [1] Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal. TMA 7 (1983) 1163–1173. [2] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hydrid systems, IEEE Trans. Automat. Control 43 (1998) 475–482. [3] D. Casagrande, A. Astolfi, T. Parisini, Control of nonholonomic systems: a simple stabilizing time-switching strategy, in: Proceedings of the IFAC World Congress, Prague, Czech Republic, 2005. [4] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control 37 (1992) 38–53. [5] K.M. Garg, Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives, WileyInterscience, New York, 1998. [6] J.C. Geromel, P. Colaneri, Stability and stabilization of continuous-time switched systems, SIAM J. Control Optim. 45 (2006) 1915–1930. [7] J.C. Geromel, P. Colaneri, Robust stability of time varying polytopic systems, Systems Control Lett. 55 (2006) 81–85. [8] J.C. Geromel, P.L.D. Peres, J. Bernussou, On a convex parameter space method for linear control design of uncertain systems, SIAM J. Control Optim. 29 (1991) 381–402. [9] J.P. Hespanha, Uniform stability of switched linear systems: extensions of LaSalle’s principle, IEEE Trans. Automat. Control 49 (2004) 470–482. [10] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, New York, 1999, pp. 2655–2660. [11] D. Liberzon, Switching in Systems and Control, Birkhauser, Basel, 2003. [12] S. Prajna, A. Papachristodoulou, Analysis of switched and hybrid systems, in: Proceedings of the American Control Conference, Denver, 2003. [13] C. Prieur, A robust globally asymptotically stabilizing feedback: the example of the Artstein’s circles, in: A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek (Eds.), Nonlinear Control in the Year 2000, vol. 2 (NCN), Lecture Notes in Control and Information Sciences, vol. 258, Springer, London, 2000, pp. 279–300. [14] C. Prieur, Uniting local and global controllers with robustness to vanishing noise, Math. Control Signals Systems 14 (2001) 143–172. [15] C. Prieur, A. Astolfi, Robust stabilization of chained systems via hybrid control, IEEE Trans. Automat. Control 48 (2003) 1768–1772. [16] R. Rockafellar, Convex Analysis, Princeton Press, Princeton, NJ, 1970. [17] E.D. Sontag, Clocks and insensitivity to small measurement errors, ESIAM: COCV, www.emath.fr/cocv/4 (1999), pp. 537–557. [18] X. Xu, P.J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. Control 49 (2004) 2–16. [19] J. Zhao, G.M. Dimirovski, Quadratic stability of a class of switched nonlinear systems, IEEE Trans. Automat. Control 49 (2004) 574–578.