Dissipativity Based Stability of Switched Systems with ... - CiteSeerX

Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007

ThC04.3

Dissipativity Based Stability of Switched Systems with State-dependent Switchings Jun Zhao and David J.Hill Abstract— Stability problem of switched systems with statedependent switchings is addressed. Sufficient conditions for stability are presented using dissipativity property of subsystems on their active regions. In these conditions, each storage function of a subsystem is allowed to grow on the “switched on” time sequence but the total growth is bounded in certain ways. Asymptotic stability is achieved under further assumptions of a detectability property of a local form and boundedness of the total change of some storage function on its inactive intervals. A necessary and sufficient condition for all subsystems to be dissipative on their active regions is given and a statedependent switching law is designed. As a particular case, localized Kalman-Yakubovich-Popov conditions are derived for passivity. A condition for piecewise dissipativity property and a design method of switching laws are also proposed.

I. INTRODUCTION The dissipativity concept was introduced by Willems [20] and developed further by Hill and Moylan [8], [9]. Dissipativity of nonlinear systems has attracted great interest in the control area mainly because of the link between Lyapunov stability and dissipativity. Stability and stabilization problems can be solved once the dissipativity property is assured. Storage functions that represent abstract energy and characterize dissipativity usually come from simple observation of physical variables, or can be constructed intuitively, and such storage functions often qualify as Lyapunov functions. As a particular form of dissipativity, passivity is even important because it is preserved under interconnection [20]. Thus, passivity is considered to be an efficient tool for the analysis and design of composite (large-scale) systems. There are many works concerning dissipativity and passivity-based control (see, for example, the results summarized in the books [12], [19]). Switched systems have drawn considerable attention in recent years [4], [10], [11], [17]. Stability issues have been a major focus in studying switched systems. The multiple Lypunov function technique, proposed by Peleties and DeCarlo [13], and further extended by Branicky [2] and Ye, Michel and Hou [21] has become accepted as a more powerful This work was supported by the Australian Research Council Federation Fellowship and Centres of Excellence Schemes and the NNSF of China under Grant 60574013. J. Zhao is with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education of China, Northeastern University, Shenyang, 110004, P. R. China, and also with the Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia.

[email protected] D. J. Hill is with the Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia.

[email protected]

1-4244-1498-9/07/$25.00 ©2007 IEEE.

tool with less conservativeness. However, when applying this method, a non-increasing condition of the switched-on sequence is required and this is usually hard to check. How to weaken or remove this assumption is a challenging issue. The dissipativity property which is a powerful concept for the study of stability for non-switched systems is of course expected to be useful for switched systems. This has not received much attention until recently with few results appearing on the topic focusing on passivity. Passivity of nonlinear systems via controller switching was discussed in [15]. A passivity based design method for switched control systems was proposed in [3]. Dissipativity and stability analysis for interconnections of hybrid systems were addressed in [6]. Passivity analysis of discrete-time hybrid systems was carried out in [1]. Besides, there are applications of passivitybased control to electrical systems with a hybrid nature ([5] and [14]). All of the results mentioned above adopt a common storage function to characterize passivity. However, this classic version of the passivity property is much too restrictive for a switched system. A single storage function for all subsystems is usually difficult to find or may not exist at all. Each subsystem will have an individual storage function when this subsystem is activated. The adoption of all storage functions of the subsystems to describe the passivity property for the switched system is thus natural. However, a simple adoption of multiple storage functions may cause the loss of desirable properties that should be inferred by passivity. To overcome this restriction, [22] proposed a notion of passivity by using multiple storage functions. But this passivity concept requires each storage function to be non-increasing on the switching sequence of consecutive “switched on” times for zero input as a prerequisite to meet Branicky’s non-increasing condition of multiple Lyapunov functions which in turn guarantees stability. For general dissipativity of switched systems, the research results are very few. [24] proposed a dissipativity concept for switched systems with time-dependent switchings without the nonincreasing condition. Switching times are needed to be known to characterize this dissipativity. This paper studies stability of switched systems with state-dependent switchings based on dissipativity property of subsystems. Under some other conditions, dissipativity of subsystems on their active regions is shown to induce stability. Each storage function is allowed to grow on the “switched on” time sequence but the total growth is bounded in certain ways. The design issue of state-dependent switching laws is also studied.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 II. PRELIMINARIES A. Switched systems We consider a switched nonlinear system described by: x˙ = fσ (x, uσ ), y = hσ (x),

ThC04.3 B. Dissipativity Recall that the dissipativity property in state space form for a continuous control system x˙ = f (x, u), y = h(x)

(1)

(4)

where σ : R+ = [0, ∞) → M = {1, 2, · · · , m} is a switching signal which is a piecewise constant function of time t, x ∈ Rn is the state, ui and hi (x) are the input vector and output vector of the i-th subsystem respectively. Further, fi (x, u) ∈ Rn , fi (0, 0) = 0 and hi (0) = 0, i = 1, 2, · · · , m. Here, we adopt the standard notations from [2], [11]. A switching law σ = σ(t) can be characterized by the switching sequence:

is characterized by a storage function S(x) and a supply rate ω(u, h) satisfying Z t S(x(t)) − S(x(t0 )) ≤ ω(u(t), h(x(t)))dt (5)

Σs : σ(x) = i when x ∈ Ωi .

Theorem 3.1. Suppose that all subsystems of the system (1) are dissipative on their active regions with continuous positive definite storage functions Si (x) satisfying Si (0) = 0 and the supply rates ωi (ui , hi ) satisfying ωi (0, hi ) ≤ 0, i.e. Z s2 ωi (ui (τ ), hi (x(τ )))dτ, Si (x(s2 )) ≤ Si (x(s1 )) +

t0

for ∀t ≥ t0 (see [20]). This classical form of dissipativity is obviously applicable to the switched system (1) as Z t Σt = {x0 ; (i0 , t0 ), (i1 , t1 ), · · · , (in , tn ), · · · , |in ∈ M, n ∈ N }, S(x(t)) − S(x(t )) ≤ ω(uσ(t) (t), hσ(t) (x(t)))dt. (6) (2) 0 t 0 in which t0 is the initial time, x0 is the initial state and N is the set of nonnegative integers. When t ∈ [tk , tk+1 ), However, this classic dissipativity property is much too σ(t) = ik , that is, the ik -th subsystem is active. Therefore, restrictive for switched systems because each subsystem the trajectory x(t) of the switched system (1) is the trajectory usually has its individual storage function Si (x) and supply xik (t) of the ik -th subsystem when t ∈ [tk , tk+1 ). rate ωi (ui , hi ) when this subsystem is active, and a common For any j ∈ M , let storage function S(x) or a common supply rate ω(ui , hi ) for all subsystems is usually difficult to find or may not exist at Σt | j = {tj1 , tj2 , · · · , tjn , · · · , ijq = j, q ∈ N } all. Therefore, it is reasonable and necessary to adopt all be the sequence of switching times when the j-th subsystem storage functions and supply rates of subsystems to describe is switched on, and thus the dissipativity property for switched systems. However, a simple adoption of the dissipativity property of individual {tj1 +1 , tj2 +1 , · · · , tjn +1 , · · · , ijq = j, q ∈ N } subsystems is inadequate to achieve desirable properties such is the sequence of switching times when the j-th subsystem as stability that are expected to be induced by dissipativity. is switched off. This may happen mainly because of the negative impact of Assumption 2.1. For any finite T > t0 , there exists a inactive subsystems on the behavior of the whole switched positive integer K = KT , dependent of T , such that during system. In particular, unlike Branicky’s multiple Lyapunov the time interval [t0 , T ] the system (1) switches no more functions method, where a non-increasing condition on a than K times for any initial state x0 . “switched on” time sequence is a basic assumption, a storage This assumption is not restrictive. It merely excludes function may increase when the corresponding subsystem systems where the switching can be arbitrarily fast. is inactive, and thus the non-increasing condition may be In this paper, we consider a state-dependent switching of destroyed. Therefore, the change of storage functions of the form σ(t) = σ(x(t)), which is based on a partition of inactive subsystems must be properly described. The purthe state space. pose here is to find conditions under which dissipativity of Let {Ωi , i = 1, 2, · · ·S, m} be a family of closed regions of subsystems implies stability. m Rn with the property i=1 Ωi = Rn . Denote the boundary of ΩT we have closed sets Hij ⊂ III. STABILITY ANALYSIS i by ∂Ωi . Suppose that T ∂Ωi ∂Ωj , i 6= j and Hij Hji ⊂ {0}, i 6= j. We In this section we will show how dissipativity of individual will consider state-dependent switching laws which can be subsystems on their active regions induces stability under described by: some further conditions. (3)

The j-th subsystem is switched on when x ∈ Hij for some i. The switching from the i-th subsystem to the j-th subsystem only occurs on Hij . Note that the switching law thus defined may not be well-defined on a subset of measure zero. This problem can be easily fixed by the method in [11] and by the following assumption. Sm Assumption 2.2. No sliding modes occur in i=1 ∂Ωi . Alternatively, for convenience, we will adopt the the time-dependent description Σt for the analysis of the statedependent case.

s1

x(t) ∈ Ωi for s1 ≤ t ≤ s2 .

(7) If, in addition, either one of the following conditions is satisfied, then, the origin of the system (1) with ui = 0 is stable.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Condition (i). There exist locally integral functions φji (·, ·, ·, ·) : Rn × Rl × Rl × R → R satisfying φji (x, u, y, ω) ≤ 0 for any x ∈ Rn , u, y ∈ Rl and ω ≤ 0, and φii (x, u, y, ω) ≡ ω, continuous functions αji (·, ·) : R+ × R+ → R+ , which are nondecreasing with respect to the second variable and αjj (t, s) ≡ s, and class K functions γj (·) such that Sj (x(s2 )) ≤ αji(s2 −s1 ) (Sj (x(s1 ))) Z +

s2

s1

(8)

and when ui = 0, i = 1, · · · , m, it holds that αjik τk ◦ αjik−1 τk−1 ◦ · · · αjip τp (s) ≤ γj (s), for k ≥ p, ip = j and λ − µ ≤ K(τµ +···+τλ ) , 1 ≤ µ ≤ λ ≤ k

(9)

where KT is the integer given in Assumption 2.1, and for simplicity of notations, we define αjit (s) = αji (t, s). Condition (ii). There exist a continuous function µ(·) : R+ → R+ with µ(0) = 0, such that p X ¡

k=1

¢ Sj (x(tjk+1 )) − Sj (x(tjk +1 )) ≤ µ(k x0 k), ∀p (10)

Proof. If Condition (i) holds, since αjit (s) is nondecreasing with respect to the variable s, for any k satisfying tk ≥ tj1 we have Sj (x(tk )) αjik−1 (tk −tk−1 ) (Sj (x(tk−1 ))) αjik−1 (tk −tk−1 ) ◦ αjik−2 (tk−1 −tk−2 ) (Sj (x(tk−2 ))) ··· αjik−1 (tk −tk−1 ) ◦ αjik−2 (tk−1 −tk−2 ) ◦ · · · ◦ αjj(tj1 +1 −tj1 ) (Sj (x(tj1 ))) . (11) Applying (9) gives rise to ≤ ≤ ≤ ≤

Sj (x(tk )) ≤ γj (Sj (x(tj1 ))) .

(12)

Using a method similar to [24] and in view of γj being class K functions, we can have stability. If Condition (ii) holds, we only give the sketch of the proof. For any given ǫ > 0, if the initial state x0 is sufficiently close to the origin, it is not difficult to deduce from (7) and (10) that Sj (x(tjk )) − Sj (x(tj1 )) ≤ ǫ ∀k ≥ 2.

(13)

A simple argument of nested neighborhoods gives stability. Example 3.2. Consider the switched system x˙ = fσ (x) + gσ (x)uσ , y = hσ (x)

f1 (x) =

µ

x2 −2x1 − 2x2 T

¶

, f2 (x) =

µ

x2 −x1 − x2

¶

,

g1 (x) = g2 (x) = (0, 1) and h1 (x) = h2 (x) = x2 . This switched system can be viewed as a mass-springdamper system with jump stiffness and damping [16]. Now we consider a state-dependent switching law ½ 1, if x ∈ Ω1 = {x|x1 x2 ≥ 0, x2 6= 0}, σ = σ(x) = 2, if x ∈ Ω2 = {x|x1 x2 ≤ 0, x1 6= 0}

It is easy to see that the two subsystems are dissipative with ωi (ui , hi ) = ui hi , i.e., passive on their active regions. A direct computation shows that Condition (i) is satisfied with α11 (t, s) = α22 (t, s) = s,

³ ´ φji x(τ ), ui (τ ), yi (τ ), ωi (ui (τ ), hi (x(τ ))) dτ,

1 ≤ i, j ≤ m, x(t) ∈ Ωi for s1 ≤ t ≤ s2 ,

ThC04.3

(14)

√    exp(t)s, 0 ≤ t ≤ 3π , √6 √ α12 (t, s) =  3π  exp( 3π − t)s, ≤ t < ∞, 3 6  3   exp(−t)s, 0 ≤ t ≤ arctan{ },   4  3 π 3 α21 (t, s) = exp(t − 2 arctan{ })s, arctan{ } ≤ t ≤ ,  4 4 4     exp( π − t − 2 arctan{ 3 })s, π ≤ t < ∞, 2 4 4 φ21 = exp(t)u1 h1 (x), φ12 = exp(t)u2 h2 (x) and γ1 (s) = γ2 (s) = exp(π)s. Remark 3.3. It is worthwhile noticing that when x(t) evolves in Ωi and j 6= i, that is, the j-th subsystem is inactive, the “energy” Sj (x) of the j-th subsystem still changes or even increases because all subsystems share the same state variable. The change of Sj (x) can be viewed as the result of “imported energy” from the active i-th subsystem into the inactive j-th subsystem. When all ui = 0, namely, no external energy is supplied to System (1), on any time interval the total “energy” coming from the active subsystems to an inactive subsystem is finite. This feature is described by either Condition (i) or Condition (ii). Condition (i) adopts functions αjit to measure the energy growth of inactive subsystems and the total energy growth of an inactive subsystem is bounded by a class K function γj . It can be seen that the growth tends to zero as the initial state x0 goes to the origin. Condition (ii) directly measures the energy change of an inactive subsystem. When all subsystems share a common storage function Sj (x) = S(x), then, Condition (i) and Condition (ii) are both automatically satisfied. Besides, when Branicky’s non-increasing condition holds, Condition (ii) is automatically satisfied. Remark 3.4. In some cases, a piecewise storage function and a piecewise supply rate for the switched system (1) with a state-dependent switching law (3) may exist. That is, there exists a unified storage function S(x), such that Z s2 S(x(s2 )) ≤ S(x(s1 )) + ωi (ui (τ ), hi (x(τ )))dτ, s1

x(t) ∈ Ωi for s1 ≤ t ≤ s2 .

with σ taking values in {1, 2},

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(15)

46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Note that this piecewise dissipativity property is a weaker property than the classic one because no unified supply rate exists over the whole state space. This piecewise dissipativity property is still very useful because almost all desirable properties of the classic dissipativity are preserved. Thus, a natural question arises: when does the dissipativity of individual subsystems on their active regions turn to be a piecewise dissipativity of the overall switched system? Obviously, this question T is possible if the condition Si (x) = Sj (x) holds on ∂Ωi ∂Ωj . In this case, a piecewise storage function can be defined as S(x) = Si (x), x ∈ Ωi . We now turn to the asymptotic stability problem. Similarly to non-switched systems, in order to derive asymptotic stability from dissipativity, we need certain detectability. Here, because of switchings, we have to consider detectability of a finite time form. Definition 3.5.([24]) A system x˙ = f (x), y = h(x)

(16)

is called asymptotically zero state detectable if for any ǫ > 0, there exists δ > 0, such that when k y(t + s) k< δ holds for some t ≥ 0, ∆ > 0 and 0 ≤ s ≤ ∆, we have k x(t) k< ǫ. Remark 3.6. This asymptotic zero state detectability is a weaker version of small-time norm observability [7]. Theorem 3.7. Suppose that all subsystems of the system (1) are dissipative on their active regions with ωi (ui , hi ) satisfying ωi (0, hi ) ≤ 0, all Si are globally defined radially unbounded with Si (0) = 0, and either Condition (i) or (ii) holds. If for each i ∈ {1, 2, · · · m} the system x˙ = fi (x, 0), y = ωi (0, hi (x))

ThC04.3 In view of x ¯ ∈ Hi0 l , the solution x(t) of the system (12) exists on [0, t¯] with some constant t¯ > 0 and x(t) ∈ Ωl . Thus, when k is large enough, the solution of the l-th subsystem x˙ = fl (x, 0), x(0) = x(tlk )

exists for t ∈ [0, 12 t¯] and x(t) ∈ Ωl . This means that tlk +1 − tlk ≥ 21 t¯, which contradicts lim (tlk +1 − tlk ) = 0. k→∞

Now that limk→∞ x(tjk ) = 0, we have limk→∞ x(t) = 0 because of the stability and the continuity of the solution. Case 2. lim (tjk +1 − tjk ) 6= 0, ∀j. k→∞ Dissipativity means Z t − ωik (0, hik (x(t))dt ≤ Sik (x(s)) − Sik (x(t)), (21) s tk ≤ s ≤ t < tk+1 .

Choose an integer j such that (18) holds. We can select δ > 0 such that the set Λ = {k|tjk +1 − tjk ≥ δ} is infinite. Define  [  ωj (0, hj (x(t))), t ∈ [tjk , tjk +1 ), (22) ω ˜ j (t) = k∈Λ  0, otherwise.

For any t > 0, if tjk ≤ t < tjk +1 for some k ∈ Λ, (21) gives Z t ˜ j (t)dt − ω t0 X Z tjp +1 ωj (0, hj (x(t)))dt = − tjp

p∈Λ,p k. In this case, we have ω ˜ j (s) ≡ 0, s ∈ [tjZ , t], and (23) still holds. It k +1 ∞

ω ˜ j (t)dt is finite. We can

follows from (18) and (23) that

t0

easily show ω ˜ j (t) → 0 as t → ∞. Therefore, x(tjk ) → 0 as k → ∞ and k ∈ Λ follows from the asymptotic zero state detectability of the systems (17). This in turn implies x(t) → 0 as t → ∞ due to stability of the system and continuity of x(t). Remark 3.8. If Condition (ii) holds, (18) is automatically satisfied.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Remark 3.9. In the strict output dissipativivty case, that is, ωi (ui , hi ) = ωi∗ (ui , hi ) − εi hTi hi

with ωi∗ (0, hi ) ≤ 0 and some constants εi > 0, the asymptotic zero state detectability of the i-th subsystem of the system (1) implies the asymptotic zero state detectability of the system (17). IV. DESIGN OF SWITCHING LAWS The previous section shows how dissipativity of individual subsystems, together with some other conditions, induces stability under a given state-dependent switching law. In this section, we study the problem of design of state-dependent laws to achieve dissipativity of subsystems on their active regions and to meet stability conditions. We only consider smooth storage functions here. Definition 4.1. Let Λ1 and Λ2 be two index sets. A collection of functions {ζi (x, θ), i ∈ Λ1 } is said to be uniformly complete with respect to the parameter θ if for any x ∈ Rn , there exists an i ∈ Λ1 , which is independent of θ, such that ζi (x, θ) ≤ 0 for all θ. A group of collections of functions {ζik (x, θ)}, i ∈ Λ1 }, k ∈ Λ2 are said to be simultaneously uniformly complete with respect to the parameter θ if for each k ∈ Λ2 , {ζik (x, θ)}, i ∈ Λ} is uniformly complete with respect to the parameter θ, and any x satisfying {ζik (x, θ) ≤ 0} for some k ∈ Λ2 also satisfies {ζik (x, θ) ≤ 0} for all k ∈ Λ2 . Theorem 4.2. A necessary and sufficient condition for the existence of a state-dependent switching law (3) under which all subsystems of the system (1) are dissipative on their active regions is that there exist positive definite smooth functions Si (x) with Si (0) = 0, functions ωi : Rl × Rl → R, a collection of functions {ζi (x, ui , hi (x)), i = 1, · · · m} which is uniformly complete with respect to ui and hi , such that ∂Si f (x, u ) − ω (u , h (x)) − ζ (x, u , h (x)) ≤ 0, i i i i i i i ∂x i i = 1, 2, · · · , m. (24) Proof. Sufficiency. Let Ωi = {x|ζi (x, ui , hi (x)) ≤ 0}, which is well-defined by the uniform completeness. Define a switching laws as: σ = σ(x) = arg min{ζi (x, ui , hi (x))}, i

which can be equivalently described by σ = σ(x) = i, x ∈ Ωi . Here, we neglect some possible overlaps of Ωi ’s, which can be easily fixed using the existing methods ( for example, see [11] ). m [ Ωi = Rn . On each Obviously, completeness implies Ωi (24) gives

i=1

∂Si fi (x, ui ) − ωi (ui , hi (x)) ≤ 0, i = 1, 2, · · · , m. (25) ∂x

ThC04.3 which means the dissipativity of the i-th subsystem on Ωi . Necessity. Suppose that there exists a switching law (3) under which all subsystems have dissipativity on their active regions. The set of functions {ζi (x, ui , hi (x))} defined by ζi (x, ui , hi (x)) =

(

¯0, x ∈ Ωi , ¯ ¯ ∂Si ¯ ¯ ∂x fi (x, ui ) − ωi (ui , hi (x))¯ , x 6= Ωi

(26)

obviously has uniform completeness and satisfies (24). Corollary 4.3. Suppose that there exist positive definite smooth functions Si (x) with Si (0) = 0, functions ωi : Rl × Rl → R, functions βij (·, ·, ·) : Rn × Rl × Rl → [0, ∞), such that for any x, and ui ∂Si f (x, u ) − ω (u , h (x)) i i i i ∂x i m X βij (x, ui , hi (x))(Si (x) − Sj (x)) ≤ 0. +

(27)

j=1

Then, under the state-dependent switching law σ = σ(x) = arg max{Si (x)}, i

the system (1) possesses piecewise dissipativity with the piecewise storage function △

S(x) = max{Si (x)} i

and the piecewise supply rate △

ω = ωσ(x) (uσ(x) , hσ(x) (x)). Proof. The uniform completeness of the collection of functions {ζi (x, ui , hi (x)), i = 1, 2, · · · , m} defined by m X βij (x, ui , hi (x))(Si (x) − Sj (x)) ζi (x, ui , hi (x)) = − j=1

is evident. It follows from the well-known Max-switching strategy [11] that S(x) = max{Si (x)} is a well defined i positive definite function which qualifies as a piecewise storage function. Next, we consider a special case that all subsystems are dissipative on their active regions and Condition (i) holds. Theorem 4.4. Suppose that we have positive definite smooth functions Si (x) with Si (0) = 0, functions ωi : Rl ×Rl → R, functions φki (·, ·, ·, ·) : Rn ×Rl ×Rl ×R → R satisfying φki (x, u, y, ω) ≤ 0 for any x ∈ Rn , u, y ∈ Rl and ω ≤ 0, and φii (x, u, y, ω) ≡ ω, m groups of functions Γk = {ζki (x, ui , hi (x)), i = 1, · · · m}, k = 1, 2, · · · , m which are simultaneously uniformly complete with respect to ui and hi , such that ³ ´ ∂Sk f (x, u ) − φ x, u , h , ω (u , h (x)) i i ki i i i i i ∂x (28) −ζki (x, ui , hi (x)) ≤ 0, i, k = 1, 2, · · · , m. Let

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Ωi = {x | ζki (x, ui , hi (x)) ≤ 0, ∀ui , hi , k}.

46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Sm Then, i=1 Ωi = Rn and the i-th subsystem of the system (1) is dissipative on Ωi and Condition (i) holds. Proof. Straightforward. Next, we consider an affine switched system x˙ = fσ (x) + gσ (x)uσ , y = hσ (x).

(29)

Then, (27) becomes: ∂Si f (x) + ∂Si g (x)u − ω (u , h (x)) i i i i ∂xm i ∂x i X + βij (x, ui , hi (x))(Si (x) − Sj (x))

(30)

j=1

≤ 0, i = 1, 2, · · · , m.

Let Ωi = {x | Si (x) − Sj (x) ≥ 0, ∀j}. For the passivity case, namely, ωi (ui , hi ) = uTi hi , (30) implies the following conditions ∂Si fi (x) ≤ 0, x ∈ Ωi , (31) ∂x ∂Si gi (x) = hTi (x), x ∈ Ωi , (32) ∂x which can be regarded as a localized form of K-Y-P condition. In particular, for a switched linear system x˙ = Aσ x + Bσ uσ y = Cσ x,

(33)

and quadratic storage functions Si (x) = xT Pi x with positive definite matrices Pi , we have the following conditions: ¡ ¢ xT Pi Ai + ATi Pi x ≤ 0, x ∈ Ωi , (34) ¢ ¡ xT Pi Bi − CiT = 0, x ∈ Ωi . Sometimes, one may apply more checkable condition: Pi Ai + ATi Pi +

m X j=1

βij (Pi − Pj ) ≤ 0,

Pi Bi =

(35)

CiT .

This condition, though much stronger than (34), is easier to implement. V. CONCLUDING REMARKS We have established dissipativity-based stability conditions for switched systems with state-dependent switchings. Multiple storage functions and multiple supply rates are adopted in these conditions. When a subsystem is active, the dissipativity inequality is consistent with the classical condition for continuous systems. For inactive subsystems, the associated storage functions may grow but are bounded by certain ways. We have also designed a state-dependent switching law that renders all subsystems dissipative. The main feature of the results is the adoption of multiple storage functions and supply rates without the sequence nonincreasing condition. The idea can be applied to the study of gains for switched systems.

ThC04.3 Further study on dissipativity needs to be conducted. An important problem is to meet the proposed conditions via feedback. How to search for more checkable dissipativitybased conditions is another challenging issue. R EFERENCES [1] A.Bemporad, G.Bianchini, F.Brogi and F.Barbagli, “Passivity analysis and passification of discrete-time hybrid systems”, in Proc. 16th IFAC World Congress, Prague, July, 2005. [2] M.S.Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Automat. Contr., vol.43, pp.475-482, 1998. [3] W.Chen and M.Saif, “Passivity and passivity based controller design of a class of switched control systems”, in Proc. 16th IFAC World Congress, Prague, July, 2005. [4] R.A.Decarlo, M.S.Branicky, S.Pettersson and B.Lennartson, “Perspectives and results on the stability and stabilization of hybrid systems,” Proceedings of the IEEE, vol.88, no.7, 1069-1082, 2000. [5] G.Escobar, R.Ortega, H.Sira-Ramirez and H.Ludvigsen, “A hybrid passivity based controller design for a three phase voltage sourced reversible boost type rectifier,” in Proc. 37th IEEE Conf.on Decision and Control, Tampa, December, 1998, 2035-2040. [6] W.M.Haddad and V.Chellaboina, “Dissipativity theory and stability of feedback interconnections for hybrid dynamical systems,” Mathematical Problems in Engineering, vol.7, 299-335, 2001. [7] J.P.Hespanha, D.Liberzon D.Angeli and E.D.Sontag, “Nonlinear Norm Observability Notions and Stability of Switched Systems,” IEEE Trans.Automat.Contr., vol.50, 2005, pp.154-168. [8] D.J.Hill and P.J.Moylan, “Dissipative dynamical systems: Basic inputoutput and state properties,” J. Franklin Inst., vol.309, pp.327-357, 1980. [9] D.J.Hill and P.J.Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Contr., Vol.21, pp.708-711, 1976. [10] D.Liberzon and A.S.Morse, “Basic problems in stability and design of switched systems,” IEEE Contr. Syst. Magazine, vol.19, pp.59-70, 1999. [11] D.Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003. [12] R.Lozano, B.Brogliato, O.Egeland and B.Maschke, Dissipative Systems Analysis and Control, Springer, 2000. [13] P.Peleties and R.DeCarlo, “Asymptotic stability of m-switched systems using Lyapunov-like functions,” in Proc. American Control Conf., Boston, MA, 1991, pp.1679-1684. [14] G.E.Perez, P.M.Ortiz, M.V.Villa and H.S.Ramirez, “Passivity-based control of switched reluctance motors with nonlinear magnetic circuits,” IEEE Trans. Control Systems Technology, vol.12, pp.439-448, 2004. [15] A.Y.Pogromsky, M. Jirstrand and P.Spangeus, “On stability and passivity of a class of hybrid systems,” in Proc. 37th IEEE Conf.on Decision and Control, Tempa, Florida, 1998, pp.3705-3710. [16] C.A.Schwartz, E.Maben, “A Minimum energy approach to switching control for mechanical systems,” in Control Using Logic-Based Switching, Edit A.S.Morse, 142-150, Springer, 1997. [17] Z.Sun and S.S.Ge, Switched Linear Systems – Control and Design, Springer, London, 2005. [18] X.Sun, J.Zhao, D.J.Hill, “Stability and L2-gain analysis for switched delay systems: Adelay-dependent method,” Automatica, vol.42 pp.1769 1774, 2006. [19] A.J.van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, Springer, London, 2000. [20] J.C.Willems, “Dissipative dynamical systems Part I: General theory,” Arch. Rational Mech.Anal., vol.45, pp.321-351, 1972. [21] H.Ye, A.N.Michel and L.Hou, “Stability theory for hybrid dynamical systems,” IEEE Trans. Automat. Contr., Vol.43, pp.461-474, 1998. [22] M.Zefran, F.Bullo and M.Stein, “A notion of passivity for hybrid systems,” in Proc. 40th IEEE Conf.on Decision and Control, Las Vegas, Nevada, 2001, 768-773. [23] J.Zhao and G.M.Dimirovski, “Quadratic stability of a class of switched nonlinear systems,” IEEE Trans.Automat.Contr., vol.49, pp.574-578, 2004. [24] J.Zhao and D.J.Hill, “Dissipativity theory for switched systems,” Proc. of the 44th IEEE Conf.on Decision and Control, pp.7003-7008, 2005. [25] J.Zhao and M.W.Spong, “Hybrid control for global stabilization of the cart-pendulum system,” Automatica, vol.37, pp.1941-1951, 2001.

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