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Switching rule design for affine switched systems using a max-type composition rule
Systems & Control Letters 68 (2014) 1–8
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Switching rule design for affine switched systems using a max-type composition rule✩ César C. Scharlau a,∗ , Mauricio C. de Oliveira b , Alexandre Trofino a , Tiago J.M. Dezuo a a
Department of Automation and Systems Engineering, Federal University of Santa Catarina, DAS/CTC/UFSC, Florianópolis, SC, Brazil
b
Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA, United States
article
info
Article history: Received 14 June 2012 Received in revised form 30 November 2013 Accepted 7 February 2014
Keywords: Affine switched systems Switching rule design Sliding mode Lyapunov function
abstract This paper presents conditions for designing a switching rule that drives the state of the switched dynamic system to a desired equilibrium point. The proposed method deals with the class of switched systems where each subsystem has an affine vector field and considers a switching rule using ‘max’ composition. The results guarantee global asymptotic stability of the tracking error dynamics even if sliding mode occurs at any switching surface of the system. In addition, the method does not require a Hurwitz convex combination of the dynamic matrices of the subsystems. Two numerical examples are used to illustrate the results. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A switched system can be defined as a dynamical system composed by a set of subsystems with continuous time dynamic and a rule that organizes the switching among them [1]. Each of these subsystems corresponds to a particular operation mode of the switched system. Switched systems can be seen as a particular class of hybrid systems or also as a variable structure system [2,3]. The problem of designing switching rules for switched systems has been largely studied and several results are available in the literature [4,5]. One may classify switching-based control strategies as time-dependent, state-dependent or time–state-dependent [2]. This paper will focus on the problem of designing state-dependent switching strategies. In this problem the type of Lyapunov function is an important issue. One type of approach is based on a common quadratic Lyapunov function, i.e. a function that is the same for all subsystems [6–8]. Another type of approach is based on multiple
✩ This work was supported in part by National Council for Scientific and Technological Development (CNPq) and Coordination for the Improvement of Higher Education Personnel (CAPES), Brazil, under grants 304834/2009-2, 550136/2009-6, 201638/2010-0, 473724/2009-0, 558642/2010-1 and BEX0421/103. ∗ Correspondence to: PO Box 476, 88040-900, Florianópolis, SC, Brazil. Tel.: +55 48 3721 7793; fax: +55 48 3721 7793. E-mail addresses: [email protected] (C.C. Scharlau), [email protected] (M.C. de Oliveira), [email protected] (A. Trofino), [email protected] (T.J.M. Dezuo).
http://dx.doi.org/10.1016/j.sysconle.2014.02.007 0167-6911/© 2014 Elsevier B.V. All rights reserved.
Lyapunov functions, i.e. the Lyapunov function is a composition of auxiliary functions that are different for each subsystem, as for instance in [9,10]. The motivation for using a multiple instead of a single Lyapunov function approach is that the first is more general and encompass the second as a particular case. Thus, this paper will focus on the problem of designing state-dependent switching rules using the multiple Lyapunov function approach. Another important aspect of the study of switched systems is the system behavior in sliding motions. Sliding motions play an important role in switched systems as they can ideally represent some complicated dynamics found in the real world [11]. However, control strategies based on sliding motions cannot be implemented in the real world because real actuators cannot operate under the unlimited switching frequency regime of a sliding mode. To avoid chattering problems, it is possible to introduce dwell time restrictions or suitable structural state dependent constraints to the switching rule design [12,5,13,14]. For this reason, many results found in the literature assume that some kind of switching rule constraint exists so that only a finite number of switches occur in any finite time. This paper presents conditions to the design of a switching rule that asymptotically drives the state of an affine switched system to a given constant reference. The switching rule is based on a ‘max’ composition of auxiliary functions, i.e. the maximum over a set of auxiliary functions. This particular type of composition was considered for instance in [9,10,15,16]. The ‘max’ composition rule has interesting properties as, for instance, it does not require all the auxiliary functions to be positive functions, which is
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necessary when using the ‘min’ composition for switched linear systems. However, some technical difficulties may appear when dealing with sliding motions. See [2,15] for details. As in [16], the conditions presented in this paper guarantee global asymptotic stability of the closed-loop switched system even if sliding motions occur at any sliding surface of the system. The main contribution of this paper is that the proposed conditions for switching rule design do not require the existence of a Hurwitz convex combination of the affine subsystems. The results in [16] can only be applied if it is possible to find a Hurwitz convex combination of the subsystems. This paper is organized as follows. This section ends with the notation used in the paper. The next section presents some preliminary definitions and two illustrative examples. Section 3 presents some aspects regarding the use of ‘max’ composition for the switching rule. The main results are presented in Section 4 and are illustrated through the numerical examples previously presented. The paper ends with some concluding remarks.
e(t ) := x(t ) − x¯
Notation. R denotes the n-dimensional Euclidean space, R is the set of n × m real matrices. For a real matrix S , S T denotes its transpose and S > 0 (S < 0) means that S is symmetric and positive-definite (negative-definite). For a set of real numbers {v1 , . . . , vm } we use arg max{v1 , . . . , vm } to denote a set of indexes that is the subset of {1, . . . , m} associated with the maximum element of {v1 , . . . , vm }. For a differentiable scalar function V (e) the column vector ∇ V (e) denotes the gradient of V (e). Co{g1 , . . . , gk } denotes the convex hull obtained from the set of k vectors {g1 , . . . , gk }. 2. Affine switched systems Consider a switched dynamical system composed of m affine subsystems indicated below i ∈ M := {1, . . . , m}
(1)
(4)
and rewrite the dynamics of the switched subsystems in terms of e(t ) as follows ki := bi + Ai x¯ .
e˙ (t ) = Ai e(t ) + ki ,
(5)
Since x¯ is a constant reference, we can therefore reformulate our switching rule design problem in terms of e(t ). In order to take into account the sliding motions, if they occur, we assume that the dynamics of the switched error system can be represented as a convex combination of the subsystem’s vector fields (5) [11], i.e. by the differential inclusion
e˙ (t ) =
θi (e(t )) (Ai e(t ) + ki ),
θ (e(t )) ∈ Θ
(6)
i∈σ (e(t ))
where
n×m
n
x˙ (t ) = Ai x(t ) + bi ,
Given x¯ , it is convenient to define the state error vector
θ ∈R :
Θ :=
m
m
θi = 1, θi ≥ 0
(7)
i =1
and θ (e(t )) is the vector with entries θi (e(t )) defined according to Filippov [11, p. 50]. Observe that θi (e(t )) = 0 if i ̸∈ σ (e(t )). If σ (e(t )) = {i} is singleton, we have θi (e(t )) = 1 and in this case (6) can be represented as in (5). If σ (e(t )) is not singleton the system (6) can be alternatively represented as e˙ (t ) ∈ Co{Ai e(t ) + ki , i ∈ σ (e(t ))}. The vector θ (e(t )) in (6) represents a parametrization of the elements of the above convex hull. When σ (e(t )) is not singleton, a sliding motion may occur at a point e(t ) if it is possible to find a vector θ (e(t )) such that e˙ (t ) is an element of the convex hull belonging to the tangent hyperplane of the switching surface at the point e(t ). In order to achieve globally the tracking objective in (3), the origin must be a globally asymptotically stable equilibrium point of the differential inclusion (6).
where x ∈ Rn is the system state supposed to be available from measurements and Ai ∈ Rn×n , bi ∈ Rn are given matrices of structure. Let us suppose the changes among the m subsystems of (1) occur according to a switching rule represented by the set valued switching signal
Lemma 1. The origin is an equilibrium point of the differential inclusion (6) only if there exists θ¯ ∈ Θ such that
σ (x(t )) : Rn → M
Proof. Since θi (e(t )) for i ∈ σ (e(t )) is defined according to Filippov [11, p. 50] and θi (e(t )) = 0 if i ̸∈ σ (e(t )), (6) can be rewritten as
(2)
that is assumed to be piecewise constant and may be viewed as a mapping from the state vector, taken at each time instant t, to the index set σ (x(t )) ⊆ M of the current (active) operation mode. If, at a given time, σ (x(t )) is a singleton, the element of σ (x(t )) defines the active subsystem and the switched system dynamics is given by (1). On the other hand, if σ (x(t )) is not a singleton, a sliding mode may be occurring at that time and the switched system dynamics can be represented by the differential inclusion x˙ (t ) ∈ Co{Ai x(t ) + bi , i ∈ σ (x(t ))} where Co denotes the convex hull. Recall that the vector field characterizing a switched system is discontinuous and therefore does not satisfy the usual Lipschitz conditions for the existence and uniqueness of the solutions to the differential equations. For this reason, additional considerations must be done in order to characterize solutions to a differential inclusion representing a switched system. In this paper, the solutions to the above differential inclusion is taken in the sense of Filippov [11, p. 50]. We seek to design a switching rule, σ (x(t )), that drives the switched system state asymptotically to a given constant reference x¯ , that is lim x(t ) = x¯ .
t →∞
(3)
m
θ¯i ki = 0.
(8)
i =1
e˙ (t ) =
m
θi (e(t )) (Ai e(t ) + ki ).
i=1
At equilibrium e˙ = e = 0, θ (0) = θ¯ and thus (8) is obtained.
Before proceeding with the technical results we introduce two illustrative examples. 2.1. Example #1 Consider a buck–boost converter with a linear (resistor) load [17–19]. This is an affine linear switched system with two modes of operation, M = {1, 2}, with state space representation (1) where
A1 =
0 0
−
Ein
b1 =
L 0
0 1
,
−
RC
,
0 A2 = 1
b2 =
0 . 0
C
1
−
L 1 , RC
C.C. Scharlau et al. / Systems & Control Letters 68 (2014) 1–8
(a) Subsystem 1.
(b) Subsystem 2.
3
(c) Subsystem 3.
Fig. 1. Example #2: trajectories of each individual subsystem.
The states are the inductor current (x1 ) and the output capacitor voltage (x2 ). The constants Ein = 15 V, L = 1 mH, C = 1 µF, R = 30 are, respectively, the external source voltage, the inductance of the input circuit, the capacitance of the output filter, and the nominal load resistance. The matrix A1 is not Hurwitz, because it has an eigenvalue at 0. For the values of L, C , and R given above, the matrix A2 is Hurwitz. We would like to design a switching rule so that the output capacitor voltage, x2 , should be equal to a given Eout , i.e. x2 = Eout ̸= 0. Under this consideration, we look for an equilibrium point such
T
that x¯ = x¯ 1 Eout , where x¯ 1 is a constant to be determined. Computing k1 and k2 from (5) we obtain
E in L k1 = , Eout −
k2 =
−
RC
Eout x¯ 1 C
L
−
RLC
RC (θ¯1 Ein + θ¯2 Eout ) −L[θ¯1 Eout + θ¯2 (R x¯ 1 + Eout )]
θ¯1 + θ¯2 = 1,
0 , 0
θ¯1 ≥ 0, θ¯2 ≥ 0
from which
θ¯1 =
Eout Eout − Ein
,
θ¯2 = 1 − θ¯1 ,
x¯ 1 =
2 Eout − Eout Ein
Ein R
.
(9)
Note that in this example, given Ein , Eout , the choice of θ¯ is unique. However, this might not always be the case (for instance, see Example #2). The system operates as a ‘‘voltage buck’’ (Ein > Eout ) if θ¯1 < 0.5 and as a ‘‘voltage boost’’ (Ein < Eout ) if θ¯1 > 0.5. 2.2. Example #2 Consider an affine switched system with three subsystems,
M = {1, 2, 3}, and state space representation (1), where
0 A1 = A2 = 0
b1 =
−1 0
,
0 , 0
0 A3 = 0
0 , −1
b2 =
1 , 0
b3 =
0 . 0
(10)
In this example, we seek to drive the trajectories of the switched system to the origin, that is, x¯ = 0. Note that none of the subsystem matrices Ai , i = 1, . . . , 3 is Hurwitz. The trajectories of each individual subsystem in the phase plane is shown in Fig. 1. Using Lemma 1 we compute
θ¯2 − θ¯1 0
=
0 , 0
θ¯1 + θ¯2 + θ¯3 = 1,
θ¯1 ≥ 0, θ¯2 ≥ 0, θ¯3 ≥ 0
(12)
Any choice of β in the range [0, 1/2] makes the origin an equilibrium point of the differential inclusion. 3. Switching rule using a max composition As discussed earlier, we are interested in designing a switching rule that drives the switched system to the origin of the differential inclusion (6). In this paper we focus on rules obtained from the application of a ‘max’ composition of the form (13)
where Vi : Rn → R, i ∈ M are auxiliary functions associated with the subsystems (5). The switching rule σ (e(t )) must render the origin of the error system globally asymptotically stable. The following lemma sheds light on a subclass of affine switched systems for which the problem of global stabilization is trivial.
=
β ∈ [0, 1/2].
i∈M
RC
θ¯ = (β, β, 1 − 2β),
σ (e(t )) := arg max{Vi (e(t ))}
Eout .
Using Lemma 1, there should exist θ¯ ∈ Θ such that 1
from which
(11)
Lemma 2. Consider the affine switched error system (5). Assume m ¯ there exists θ¯ ∈ Θ such that i=1 θi ki = 0. If there exists i such that Ai is Hurwitz and ki = 0 then the closed-loop system under the switching law σ (e(t )) = {i} is globally asymptotically stable. The proof of this lemma is trivial and is omitted. As a consequence of Lemma 2, the challenging problems will be the ones for which the subsystems (5) satisfy the following condition: Assumption 1. If Ai is Hurwitz, then ki ̸= 0. Both examples introduced earlier fulfill Assumption 1. The methods which we will present shortly will apply when Assumption 1 is satisfied and, of course, also when it is not satisfied. At this point, it is important to identify a significant difference between linear and affine switched systems: even if all Ai ’s are Hurwitz, one does not automatically obtain convergence to the origin due to the presence of a constant term in the vector field. Another consequence of the presence of the constant term and Lemma 1 is that a stable equilibrium at the origin of the error system will often be enforced by a sliding mode, since under Assumption 1 the only possibility of a single subsystem to be active at the origin, i.e. θi = 1 for some i, is when the corresponding Ai is not Hurwitz. These observations help to illustrate how the stabilization problem of switched affine system can be significantly more challenging than the stabilization of switched linear systems. In particular, the need to consider the stability on sliding modes is a significant complication, which is often tactfully excluded from the analysis of many existing results for linear switched systems, e.g. [12]. Moreover, the literature is rich in switching laws based
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on ‘min’ composition rules as opposed to ‘max’ composition rules, and one reason for this fact is the simplified treatment of sliding modes in the case of linear switched systems [2, Section 3.4] when the ‘min’ composition is used. As we will see in the next section, the use of a ‘max’ composition rule will however add flexibility to the stability analysis because the functions Vi ’s are not required to be positive functions. Recall the positiveness requirement is present when using the ‘min’ composition combined with a Lyapunov stability argument [20,21,19].
∇ Vi (e)T (Ai e + ki ) < Q (e) ≤ 0 whenever Vi (e) ≥ θ¯j Vj (e).
In this section we present our main stability result. When the conditions of Theorem 1 are satisfied we get a switching rule σ (x(t )) that asymptotically drives the switched system state to a given constant reference x¯ . We pay special attention to stability on sliding modes. Theorem 1. Consider the affine switched error system (5). Assume m ¯ there exists θ¯ ∈ Θ such that i=1 θi ki = 0, and functions Vi : n R → R, with Vi (0) = 0 for all i ∈ M , Q : Rn → R, all in C 1 , and α : Θ → R such that V (e) = max{Vi (e)} > 0
(14)
i∈M
V (e) is radially unbounded
(15)
Q (e) ≤ 0,
(16)
α(θ ) ≥ 0
θi θj ∇ Vi (e)T (Aj e + kj ) + 2α(θ )(θi − θ¯i )Vi (e) < Q (e) (17)
Therefore, for e(t ) such that σ (e(t )) = {i} we have V (e(t )) = Vi (e(t )) > 0, hence V (e(t )) is differentiable, so that inequality (20) implies
Now consider a point e(t ) in which σ (e(t )) is not a singleton. In this case the trajectories of the closed-loop switched system either go to a region where σ (e(t )) is a singleton, in which case V (e(t )) decreases because of the argument above, or enter a sliding mode. In a sliding mode, the system trajectory is described by the differential inclusion (6). In particular, the switched system will remain in a sliding mode for as long as θ (e(t )) ∈ Θ is such that e˙ (t ) ∈ T (e(t )), where T (e) := h ∈ Rn : φi (e) = 0, ∇φi (e)T h = 0, ∀i ∈ σ (e)
for all θ ∈ Θ , and e ̸= 0. Then the origin of closed-loop differential inclusion (6) under the switching rule (13) is globally asymptotically stable. Proof. The assumptions on Vi ∈ C 1 ensure that V is continuous with V (0) = 0 while (15) ensures that V is radially unbounded. Hence V qualifies as a Lyapunov function candidate and can be used to prove global asymptotic stability [22,23]. The fact that V does not have a derivative at all points will not be important. As we will show soon, it is possible to use a standard Lyapunov stability argument in this proof. Condition (14) requires that V should be positive definite while the condition (17) ensures that V decreases along any trajectory of the closed-loop switched system, a fact we prove next. Firstly observe that θ∈Θ
i∈M
θi Vi (e) ≥
i∈M
θ¯i Vi (e)
(18)
i∈M
¯ which in turn implies i∈M (θi − θi )Vi (e) ≥ 0. As α(θ ) ≥ 0 it follows that (17) can be obtained by applying the S-procedure [24], [25, p. 23] to the conditions
θi θj ∇ Vi (e)T (Aj e + kj ) < Q (e) ≤ 0 (θi − θ¯i )Vi (e) ≥ 0
(19)
∀(θ, e) ̸= (θ¯ , 0). While the first inequality is related to the time derivative of the Lyapunov function, as shown in the sequel, the second inequality is related to the ‘max’ composition in (18). Firstly, let us consider the particular case when σ (e) is a singleton, namely σ (e) = {i}. In this case we have
j∈M
θ¯j Vj (e) ≥ 0.
θi (e(t ))Vi (e(t ))
= max θ∈Θ
θi Vi (e(t )) ≥
θ¯i Vi (e(t ))
i∈M
i∈M
which implies
(θi (e(t )) − θ¯i )Vi (e(t )) ≥ 0.
(22)
i∈M
Therefore, with the same S-procedure arguments previously used we conclude from (17) that for e(t ) and all θ (e(t )) ∈ Θ we get
θi (e(t ))θj (e(t ))∇ Vi (e(t ))T (Aj e(t ) + kj )
i∈M j∈M
=
θi (e(t ))θj (e(t ))∇ Vi (e(t ))T (Aj e(t ) + kj )
i∈σ (e(t )) j∈σ (e(t ))
< Q (e(t )) ≤ 0 whenever (22) is satisfied. In particular, for all θ (e(t )) ∈ Θ such that e˙ (t ) =
θj (e(t ))(Aj e(t ) + kj )
j∈M
θj (e(t ))(Aj e(t ) + kj )
(23)
j∈σ (e(t ))
i∈M
Vi (e) −
(21)
i∈M
=
i∈M j∈M
whenever
where φi (e) := V (e) − Vi (e). The condition e˙ (t ) ∈ T (e(t )), implies that the system’s vector field in (6) is orthogonal to the set of gradients {∇φi (e), i ∈ σ (e)}. For each i ∈ σ (e) the condition ∇φi (e)T h = 0 can be viewed as a requirement that h belongs to the tangent hyperplane of the sliding surface φi (e) = 0. Back to stability, for such an e(t ) and all θ (e(t )) ∈ Θ , inequality (18) implies that V (e(t )) =
i∈M j∈M
V (e) = max{Vi (e)} = max
(20)
j∈M
V˙ (e(t )) = V˙ i (e(t )) = ∇ Vi (e(t ))T (Ai e(t ) + ki ) < Q (e(t )) ≤ 0.
4. Global asymptotic stabilization
According to Filippov’s results [11, p. 50] in this situation we have θi = 1 and θj = 0 for all j ̸= i and inequality (19) reduces to
with e˙ (t ) ∈ T (e(t )), we have DV (e(t ))[˙e(t )] = max ∇ Vi (e(t ))T e˙ (t ) i∈σ (e(t ))
=
θi (e(t ))∇ Vi (e(t ))T e˙ (t )
i∈σ (e(t ))
< Q (e(t )) ≤ 0
(24)
where DV (e(t ))[h] is Dini’s directional derivative of V (e(t )) [26, p. 420]. Observe that DV (e(t ))[h] is therefore guaranteed to decrease along all trajectories in the sliding manifold.
C.C. Scharlau et al. / Systems & Control Letters 68 (2014) 1–8
5
(b) Composite V (red) and lower bound V¯ (brown).
(a) Components V1 (blue) and V2 (green).
Fig. 2. Buck–boost converter: Lyapunov function V (e) = max{V1 (e), V2 (e)} and lower bound V¯ (e).
The last situation that needs to be considered is when σ (e(t )) is not a singleton and the sliding manifold is not differentiable at e(t ). Due to the continuity and differentiability of the functions Vi ’s, such points can only occur at the intersection of two sliding manifolds, therefore creating a lower dimensional sub-manifold. The argument to be used in this case is as follows: the system will either exit the sliding mode and becomes singleton or will remain on a sliding motion on one of sub-manifolds that intersect at e(t ). In both cases V (e(t )) decreases because of the previous arguments. Moreover, continuity of V (e(t )) implies that the value of V (e(t )) cannot increase at these points where V (e(t )) is not differentiable. Furthermore, condition (24) is satisfied also at points e(t ) in which the sliding manifold is not differentiable. At those points θ (e(t )) might be discontinuous but DV (e(t ))[˙e(t )] < 0 because (19) holds for all θ ∈ Θ − {θ¯ }. See [11, p. 155] for details regarding this point. Remark 1. Theorem 1 and its proof can be generalized to general nonlinear switched systems where the functions fi (x(t )), i ∈ M , defining the subsystems x˙ (t ) = fi (x(t )) do not necessarily need to be homogeneous. We will address this case in future works. Remark 2. Observe from (22) that θ is in fact a function of e(t ). However in condition (17) this dependence is not taken into account which may introduce some extra conservatism to (17). The role of α(θ ), that appears in (17) thanks to the S-Procedure, is to reduce this conservatism to some extent. Remark 3. It should be mentioned that the inequalities appearing in Theorem 1 are not convex with respect to θ and α(θ ). However, for some particular choices of Vi (e(t )) and α(θ ) and some additional conditions, it is possible to obtain sufficient LMI conditions to check expressions which are similar to the ones in Theorem 1. For instance, LMI conditions are proposed in [16] under the assumption of the existence of a coefficient θ˜ ∈ Θ such that ˜ i∈M θi Ai is Hurwitz stable. Observe this requirement does not appear in the conditions of Theorem 1. Remark 4. In some cases it is possible to use the following sufficient condition for checking the positivity of V . Suppose that there exists θ˜ ∈ Θ such that
θ˜i Vi (e) > 0,
∀e ̸= 0.
(25)
Then we get V (e) > 0 for all e ̸= 0. This follows from the fact that i∈M
θ∈Θ
Consider the buck–boost converter with a linear (resistor) load previously introduced in Section 2.1. As in this case there exists a coefficient θ¯ ∈ Θ such that i∈M θ¯i Ai is Hurwitz stable, according to Remark 3 the conditions of Theorem 1 can be checked by solving an LMI problem given in [16]. After tweaking numerical results obtained from [16] we arrived at the following choices of auxiliary functions in C 1 V1 (e) = −6000e1 + 180e2 − 300 e21 + 3e22 + 20 e1 e2 V2 (e) = 3600e1 − 108 e2 + 4000e21 + 7e22 + 40 e1 e2 . In the sequel we show that the conditions of Theorem 1 are satisfied with the above auxiliary functions leading to a switching rule σ (e) that globally drives the output voltage to the reference value Eout = −9 V. It is worth noticing that the arguments below are based on the conditions of Theorem 1 and not on the methodology of [16]. Note that V1 (0) = V2 (0) = 0 and V1 and V2 are not positive definite functions because they have a linear term. A threedimensional plot of V1 and V2 is shown in Fig. 2(a). Our first challenge is to prove that the ‘max’ composition of V1 and V2 is positive definite and radially unbounded, as shown in the threedimensional plot in Fig. 2(b). We will also prove that, for this example, there exists a positive definite quadratic function that bounds V from below globally by using the arguments in Remark 4. This lower bound function is represented by V¯ (e(t )) in Fig. 2(b). When Eout = −9 V, Ein = 15 V, and L, C , R are as in Section 2.1, the converter operates as a buck. Furthermore, θ¯1 = 3/8 and θ¯2 = 5/8, from which we compute V¯ (e) := θ¯1 V1 (e) + θ¯2 V2 (e) = 2387.5 e21 + 5.5 e22 + 32.5 e1 e2 . One can easily verify that V¯ (e) > 0 for all e ̸= 0 hence, from Remark 4 we conclude that V (e) ≥ V¯ (e) > 0 for all e ̸= 0. As V¯ (e) is radially unbounded, so is V (e). Having proved positivity of V , we now show that condition (17) is satisfied with the particular choices Q (e) = Q = 0, α(θ ) = α = 200. We start by rewriting (17) in the form 3 100
i∈M
V (e) = max{Vi (e)} = max
4.1. Example #1 (continued)
i∈M
θi Vi (e) ≥
θ˜i Vi (e) > 0.
i∈M
In particular θ˜ = θ¯ from Lemma 1 seems to be a natural choice. By ˜ the same token, if i∈M θi Vi (e) is radially unbounded then V (e) will also be radially unbounded.
T e 1
M (θ ) r (θ )
r T (θ )
γ (θ )
e 1
> 0,
for all θ ∈ Θ , e ̸= 0 (26)
where the matrices M (θ ), r (θ ) and γ (θ ) can be computed from the choices of V1 (e), V2 (e) given above. In [16], a sufficient LMI condition is derived based on (26). Here we take a different path based on a condition that depends exclusively on θ . A sufficient condition for (26) to hold for all e ̸= 0 is that M (θ ) r (θ )
r T (θ )
γ (θ )
≥ 0,
M (θ ) > 0,
for all θ ∈ Θ .
(27)
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Fig. 3. Buck–boost converter in buck type operation with Eout = −9 V.
(a) Components V1 (blue), V2 (green), and V3 (yellow).
(b) Composite V (light green).
Fig. 4. Example #2: Lyapunov function V (e) = max{V1 (e), V2 (e), V3 (e)}; solid black curve shown in (b) is the switching surface.
Using Schur-Complement [25, p. 7], the above conditions are equivalent to
γ (θ) ≥ r (θ ) M −1 (θ ) r T (θ ),
M (θ ) > 0,
for all θ ∈ Θ .
Since θ2 = 1 − θ1 it follows that M (θ ) is a polynomial matrix function of θ1 ∈ [0, 1]. In this case we have M (θ ) > 0 as the principal minors of M (θ ) are positive scalar functions of θ1 in the interval 0 ≤ θ1 ≤ 1. The condition γ (θ )− r (θ ) M −1 (θ ) r T (θ ) ≥ 0 is a bit more subtle to prove. First, define p(θ) := γ (θ ) − r (θ ) M −1 (θ ) r T (θ ). r (θ) = (3 − 8 θ1 ) 5400 (25 θ1 − 28)
4.2. Example #2 (continued) Consider the system introduced in Section 2.2 and the auxiliary functions in C 1 V1 (e) = e1 (t ) + e2 (t )2 ,
Now verify that
x¯ = (480 m A, −9 V) is shown in Fig. 3. The state response is shown at the left hand side and the corresponding trajectory on the error phase-plane is on the right. When the trajectory touches the switching surface for the second time a sliding motion drives the error towards the origin, e = 0.
396 (25 θ1 − 54) ,
γ (θ) = 43200 (3 − 8 θ1 )2 . Therefore, p(θ ) can be factored into p(θ ) = 864 (3 − 8 θ1 )2 p1 (θ1 ), where the rational function p1 (θ1 ) > 0 in the interval 0 ≤ θ1 ≤ 1. Therefore p(θ ) > 0 for all θ1 ̸= θ¯1 = 3/8 and p(θ¯ ) = 0. The conclusion is that (26) holds true for all e ̸= 0 and θ ∈ Θ . The above factorization of p is not a coincidence, and it comes from a careful choice of the linear coefficients in the functions V1 and V2 . The response of the system starting from a zero initial state, x = (0, 0), in the original coordinates (1) towards the target
V2 (e) = −e1 (t ) + e2 (t )2 ,
V3 (e) = 2e2 (t )2 .
(28)
A three-dimensional plot of these functions is shown in Fig. 4(a). We will show that these auxiliary functions produce a switching rule σ (e) that can drive the system asymptotically to zero, i.e. towards e = x¯ = 0. First consider the question of whether the ‘max’ composition of the above functions is positive definite. Since V (e) = max{Vi (e)} = max{|e1 | + e22 , 2e22 }
(29)
i∈M
we can readily conclude that V (e) > 0 for all e ̸= 0. V (e) is also radially unbounded. Interestingly, since V (e1 , 0) = |e1 | it
C.C. Scharlau et al. / Systems & Control Letters 68 (2014) 1–8
7
equilibrium. Moreover for any initial condition where e2 (0) = 0 or e1 (0) = 0, marked with • in the figure, the system trajectory goes to the origin without switchings during the whole trajectory. If e2 (0) = 0 and e1 (0) > 0 we have σ (e(t )) = {1}, ∀t ≥ 0. If e2 (0) = 0 and e1 (0) < 0 we have σ (e(t )) = {2}, ∀t ≥ 0. If e1 (0) = 0 we have σ (e(t )) = {3}, ∀t ≥ 0. At the equilibrium we have e(t ) = 0 and θ = θ¯ for any θ¯ in (12). 5. Concluding remarks
Fig. 5. Phase plane of the switched system. Solid (black) lines are error trajectories; Dashed (color) lines are switching surfaces.
follows that there exists no positive definite quadratic function that bounds V from below globally, as any positive definite quadratic form would grow larger than |e1 | for large enough e1 . This is therefore an example in which Remark 4 does not apply for the choice of Vi (e) in (28). Indeed, θ¯1 V1 (e) + θ¯2 V2 (e) + θ¯3 V3 (e) = 2(1 −β)e22 , which is positive semidefinite for β ∈ [0, 1/2]. A threedimensional plot of V (e) is shown in Fig. 4(b). In the sequel we check condition (17) with the particular choice of Q (e) = Q = 0, α(θ ) = α = 0. Using elementary algebra manipulation, it can be shown that
θi θj ∇ Vi (e)T (Aj e + kj ) = −f (θ ) − g (θ , e2 )
i∈M j∈M
where f (θ ) = (θ1 − θ2 )2 ≥ 0 and g (θ , e) = 2θ3 (θ1 + θ2 + 2θ3 )e22 . Note that f (θ ) + g (θ , e) ≥ 0 for all e and θ ∈ Θ , hence (17) is only negative semidefinite. One might be tempted to use La Salle’s extension to prove stability under this weaker constraint. The discussion is however complicated (see for instance [27]), and here we use an ad hoc argument which we will elaborate in future publications. As V is positive definite and (17) is only negative semi-definite, the Lyapunov machinery can only prove convergence to a subset1 of the invariant set
Ω := {(θ , e) : f (θ ) + g (θ , e) = 0, θ ∈ Θ } = Ω1 ∪ Ω2 where
Ω1 := {(θ , e) : θ3 = 1, θ1 = θ2 = e2 = 0, e ∈ R2 } Ω2 := {(θ , e) : θ1 = θ2 = 1/2, θ3 = 0, e ∈ R2 }. Note that, for this example, the switched system with the switching rule σ (e(t )) leads to
(θ(e(t )), e(t )) ∈ Ω H⇒ e(t ) = 0. Indeed, if θ3 (e(t )) = 1 and e2 (t ) = 0, it follows from (28) and (29) that e1 (t ) = 0. On the other hand, if θ1 (e(t )) = θ2 (e(t )) = 1/2 it follows that V1 (e(t )) = V2 (e(t )) < V (e(t )) except when e2 (t ) = 0. In summary, V˙ = 0 only at the equilibrium point (e, θ ) = (0, θ¯ ) and V˙ < 0 outside the equilibrium. The system trajectory for several initial conditions are indicated in Fig. 5. Observe that all the trajectories starting at a point where e1 (0) ̸= 0, e2 (0) ̸= 0 have sliding motion dynamics outside the
1 Indeed Ω (e) := {(θ(e), e) ∈ Ω : θ(e) parametrizes a feasible trajectory } ⊆ Ω .
This paper presents conditions for switching rules design that asymptotically drives the state of an affine switched system to a given constant reference. The results are summarized by Theorem 1 and they guarantee global asymptotic stability of the closed-loop switched system even if sliding motions occur at any sliding surface of the system. It is important to emphasize that the results in this paper do not require the existence of a Hurwitz convex combination of the dynamic matrices of the subsystems. This point is discussed in Example #2. This requirement is present in several results found in the literature, as for instance in [16,8]. See [4] for more references on this point. One may observe that the conditions appearing in Theorem 1 are not convex with respect to θ and α(θ ). However, for some particular choices of Vi (e(t )) and α(θ ), it is possible to get sufficient LMI conditions to check the expressions in Theorem 1. For instance, LMI conditions are proposed in [16]. However, the results in this reference require the existence of a Hurwitz convex combination as previously mentioned. Currently, the authors are working on how to express the conditions of Theorem 1 as an LMI problem without this requirement. We are also investigating extensions to include performance requirements such as guaranteed cost and disturbance attenuation. References [1] D. Liberzon, A. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag. 19 (5) (1999) 59–70. [2] D. Liberzon, Switching in Systems and Control, Birkhauser, 2003. [3] R.A. DeCarlo, S.H. Zak, G.P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proc. IEEE 76 (3) (1988) 212–232. [4] R.A. DeCarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proc. IEEE 88 (7) (2000) 1069–1082. [5] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a short survey of recent results, in: Proc. of the 2005 IEEE International Symposium on Intelligent Control, Limassol, Cyprus, 2005, pp. 24–29. [6] M.A. Wicks, P. Peleties, R.A. DeCarlo, Switched controller design for the quadratic stabilization of a pair of unstable linear systems, Eur. J. Control 4 (2) (1998) 140–147. [7] G. Zhai, H. Lin, P. Antsaklis, Quadratic stabilizability of switched linear systems with polytopic uncertainties, Internat. J. Control 76 (7) (2003) 747–753. [8] P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, in: Proceedings of the 2004 American Control Conference, Boston, MA, 2004, pp. 3890–3895. [9] M. Wicks, R.A. DeCarlo, Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, in: Proc. of the 15th American Control Conference, Albuquerque, NM, 1997, pp. 1709–1713. [10] S. Pettersson, Synthesis of switched linear systems, in: Proc. of the 42nd IEEE Conference on Decision and Control, Maui, HI, 2003, pp. 5283–5288. [11] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Norwell, MA, 1988. [12] Z. Sun, Combined stabilizing strategies for switched linear systems, IEEE Trans. Automat. Control 51 (4) (2006) 666–674. [13] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Automat. Control 57 (7) (2012) 1809–1815. [14] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell time switching, Automatica 48 (6) (2012) 1132–1137. [15] T. Hu, L. Ma, Z. Lin, On several composite quadratic Lyapunov functions for switched systems, in: Proc. of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006, pp. 113–118. [16] A. Trofino, C.C. Scharlau, T. Dezuo, M.C. de Oliveira, Stabilizing switching rule design for affine switched systems, in: Proc. of the 50th IEEE Conference on Decision and Control, Orlando, FL, 2011, pp. 1183–1188. [17] M. Rashid, Power Electronics Handbook, third ed., Butterworth-Heinemann, 2010.
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[18] M. Di Bernardo, F. Vasca, Discrete-time maps for the analysis of bifurcations and chaos in DC/DC converters, IEEE Trans. Circuits Syst. I 47 (2) (2000) 130–143. [19] G. Deaecto, J. Geromel, F. Garcia, J. Pomilio, Switched affine systems control design with application to dc–dc converters, IET Control Theory Appl. 4 (7) (2010) 1201–1210. [20] T. Hu, L. Ma, Z. Lin, Stabilization of switched systems via composite quadratic functions, IEEE Trans. Automat. Control 53 (11) (2008) 2571–2585. [21] J. Geromel, P. Colaneri, P. Bolzern, Dynamic output feedback control of switched linear systems, IEEE Trans. Automat. Control 53 (3) (2008) 720–733.
[22] F. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [23] H.K. Khalil, Nonlinear Systems, third ed., Prentice Hall, 2002. [24] V. Yakubovich, S-procedure in nonlinear control theory, Vestn. Leningr. Univ. 1 (1971) 62–77. [25] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [26] L.S. Lasdon, Optimization Theory for Large Systems, Macmillan, 1970. [27] J. Hespanha, Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle, IEEE Trans. Automat. Control 49 (4) (2004) 470–482.
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Switching rule design for affine switched systems using a max-type composition rule✩ César C. Scharlau a,∗ , Mauricio C. de Oliveira b , Alexandre Trofino a , Tiago J.M. Dezuo a a
Department of Automation and Systems Engineering, Federal University of Santa Catarina, DAS/CTC/UFSC, Florianópolis, SC, Brazil
b
Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA, United States
article
info
Article history: Received 14 June 2012 Received in revised form 30 November 2013 Accepted 7 February 2014
Keywords: Affine switched systems Switching rule design Sliding mode Lyapunov function
abstract This paper presents conditions for designing a switching rule that drives the state of the switched dynamic system to a desired equilibrium point. The proposed method deals with the class of switched systems where each subsystem has an affine vector field and considers a switching rule using ‘max’ composition. The results guarantee global asymptotic stability of the tracking error dynamics even if sliding mode occurs at any switching surface of the system. In addition, the method does not require a Hurwitz convex combination of the dynamic matrices of the subsystems. Two numerical examples are used to illustrate the results. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A switched system can be defined as a dynamical system composed by a set of subsystems with continuous time dynamic and a rule that organizes the switching among them [1]. Each of these subsystems corresponds to a particular operation mode of the switched system. Switched systems can be seen as a particular class of hybrid systems or also as a variable structure system [2,3]. The problem of designing switching rules for switched systems has been largely studied and several results are available in the literature [4,5]. One may classify switching-based control strategies as time-dependent, state-dependent or time–state-dependent [2]. This paper will focus on the problem of designing state-dependent switching strategies. In this problem the type of Lyapunov function is an important issue. One type of approach is based on a common quadratic Lyapunov function, i.e. a function that is the same for all subsystems [6–8]. Another type of approach is based on multiple
✩ This work was supported in part by National Council for Scientific and Technological Development (CNPq) and Coordination for the Improvement of Higher Education Personnel (CAPES), Brazil, under grants 304834/2009-2, 550136/2009-6, 201638/2010-0, 473724/2009-0, 558642/2010-1 and BEX0421/103. ∗ Correspondence to: PO Box 476, 88040-900, Florianópolis, SC, Brazil. Tel.: +55 48 3721 7793; fax: +55 48 3721 7793. E-mail addresses: [email protected] (C.C. Scharlau), [email protected] (M.C. de Oliveira), [email protected] (A. Trofino), [email protected] (T.J.M. Dezuo).
http://dx.doi.org/10.1016/j.sysconle.2014.02.007 0167-6911/© 2014 Elsevier B.V. All rights reserved.
Lyapunov functions, i.e. the Lyapunov function is a composition of auxiliary functions that are different for each subsystem, as for instance in [9,10]. The motivation for using a multiple instead of a single Lyapunov function approach is that the first is more general and encompass the second as a particular case. Thus, this paper will focus on the problem of designing state-dependent switching rules using the multiple Lyapunov function approach. Another important aspect of the study of switched systems is the system behavior in sliding motions. Sliding motions play an important role in switched systems as they can ideally represent some complicated dynamics found in the real world [11]. However, control strategies based on sliding motions cannot be implemented in the real world because real actuators cannot operate under the unlimited switching frequency regime of a sliding mode. To avoid chattering problems, it is possible to introduce dwell time restrictions or suitable structural state dependent constraints to the switching rule design [12,5,13,14]. For this reason, many results found in the literature assume that some kind of switching rule constraint exists so that only a finite number of switches occur in any finite time. This paper presents conditions to the design of a switching rule that asymptotically drives the state of an affine switched system to a given constant reference. The switching rule is based on a ‘max’ composition of auxiliary functions, i.e. the maximum over a set of auxiliary functions. This particular type of composition was considered for instance in [9,10,15,16]. The ‘max’ composition rule has interesting properties as, for instance, it does not require all the auxiliary functions to be positive functions, which is
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necessary when using the ‘min’ composition for switched linear systems. However, some technical difficulties may appear when dealing with sliding motions. See [2,15] for details. As in [16], the conditions presented in this paper guarantee global asymptotic stability of the closed-loop switched system even if sliding motions occur at any sliding surface of the system. The main contribution of this paper is that the proposed conditions for switching rule design do not require the existence of a Hurwitz convex combination of the affine subsystems. The results in [16] can only be applied if it is possible to find a Hurwitz convex combination of the subsystems. This paper is organized as follows. This section ends with the notation used in the paper. The next section presents some preliminary definitions and two illustrative examples. Section 3 presents some aspects regarding the use of ‘max’ composition for the switching rule. The main results are presented in Section 4 and are illustrated through the numerical examples previously presented. The paper ends with some concluding remarks.
e(t ) := x(t ) − x¯
Notation. R denotes the n-dimensional Euclidean space, R is the set of n × m real matrices. For a real matrix S , S T denotes its transpose and S > 0 (S < 0) means that S is symmetric and positive-definite (negative-definite). For a set of real numbers {v1 , . . . , vm } we use arg max{v1 , . . . , vm } to denote a set of indexes that is the subset of {1, . . . , m} associated with the maximum element of {v1 , . . . , vm }. For a differentiable scalar function V (e) the column vector ∇ V (e) denotes the gradient of V (e). Co{g1 , . . . , gk } denotes the convex hull obtained from the set of k vectors {g1 , . . . , gk }. 2. Affine switched systems Consider a switched dynamical system composed of m affine subsystems indicated below i ∈ M := {1, . . . , m}
(1)
(4)
and rewrite the dynamics of the switched subsystems in terms of e(t ) as follows ki := bi + Ai x¯ .
e˙ (t ) = Ai e(t ) + ki ,
(5)
Since x¯ is a constant reference, we can therefore reformulate our switching rule design problem in terms of e(t ). In order to take into account the sliding motions, if they occur, we assume that the dynamics of the switched error system can be represented as a convex combination of the subsystem’s vector fields (5) [11], i.e. by the differential inclusion
e˙ (t ) =
θi (e(t )) (Ai e(t ) + ki ),
θ (e(t )) ∈ Θ
(6)
i∈σ (e(t ))
where
n×m
n
x˙ (t ) = Ai x(t ) + bi ,
Given x¯ , it is convenient to define the state error vector
θ ∈R :
Θ :=
m
m
θi = 1, θi ≥ 0
(7)
i =1
and θ (e(t )) is the vector with entries θi (e(t )) defined according to Filippov [11, p. 50]. Observe that θi (e(t )) = 0 if i ̸∈ σ (e(t )). If σ (e(t )) = {i} is singleton, we have θi (e(t )) = 1 and in this case (6) can be represented as in (5). If σ (e(t )) is not singleton the system (6) can be alternatively represented as e˙ (t ) ∈ Co{Ai e(t ) + ki , i ∈ σ (e(t ))}. The vector θ (e(t )) in (6) represents a parametrization of the elements of the above convex hull. When σ (e(t )) is not singleton, a sliding motion may occur at a point e(t ) if it is possible to find a vector θ (e(t )) such that e˙ (t ) is an element of the convex hull belonging to the tangent hyperplane of the switching surface at the point e(t ). In order to achieve globally the tracking objective in (3), the origin must be a globally asymptotically stable equilibrium point of the differential inclusion (6).
where x ∈ Rn is the system state supposed to be available from measurements and Ai ∈ Rn×n , bi ∈ Rn are given matrices of structure. Let us suppose the changes among the m subsystems of (1) occur according to a switching rule represented by the set valued switching signal
Lemma 1. The origin is an equilibrium point of the differential inclusion (6) only if there exists θ¯ ∈ Θ such that
σ (x(t )) : Rn → M
Proof. Since θi (e(t )) for i ∈ σ (e(t )) is defined according to Filippov [11, p. 50] and θi (e(t )) = 0 if i ̸∈ σ (e(t )), (6) can be rewritten as
(2)
that is assumed to be piecewise constant and may be viewed as a mapping from the state vector, taken at each time instant t, to the index set σ (x(t )) ⊆ M of the current (active) operation mode. If, at a given time, σ (x(t )) is a singleton, the element of σ (x(t )) defines the active subsystem and the switched system dynamics is given by (1). On the other hand, if σ (x(t )) is not a singleton, a sliding mode may be occurring at that time and the switched system dynamics can be represented by the differential inclusion x˙ (t ) ∈ Co{Ai x(t ) + bi , i ∈ σ (x(t ))} where Co denotes the convex hull. Recall that the vector field characterizing a switched system is discontinuous and therefore does not satisfy the usual Lipschitz conditions for the existence and uniqueness of the solutions to the differential equations. For this reason, additional considerations must be done in order to characterize solutions to a differential inclusion representing a switched system. In this paper, the solutions to the above differential inclusion is taken in the sense of Filippov [11, p. 50]. We seek to design a switching rule, σ (x(t )), that drives the switched system state asymptotically to a given constant reference x¯ , that is lim x(t ) = x¯ .
t →∞
(3)
m
θ¯i ki = 0.
(8)
i =1
e˙ (t ) =
m
θi (e(t )) (Ai e(t ) + ki ).
i=1
At equilibrium e˙ = e = 0, θ (0) = θ¯ and thus (8) is obtained.
Before proceeding with the technical results we introduce two illustrative examples. 2.1. Example #1 Consider a buck–boost converter with a linear (resistor) load [17–19]. This is an affine linear switched system with two modes of operation, M = {1, 2}, with state space representation (1) where
A1 =
0 0
−
Ein
b1 =
L 0
0 1
,
−
RC
,
0 A2 = 1
b2 =
0 . 0
C
1
−
L 1 , RC
C.C. Scharlau et al. / Systems & Control Letters 68 (2014) 1–8
(a) Subsystem 1.
(b) Subsystem 2.
3
(c) Subsystem 3.
Fig. 1. Example #2: trajectories of each individual subsystem.
The states are the inductor current (x1 ) and the output capacitor voltage (x2 ). The constants Ein = 15 V, L = 1 mH, C = 1 µF, R = 30 are, respectively, the external source voltage, the inductance of the input circuit, the capacitance of the output filter, and the nominal load resistance. The matrix A1 is not Hurwitz, because it has an eigenvalue at 0. For the values of L, C , and R given above, the matrix A2 is Hurwitz. We would like to design a switching rule so that the output capacitor voltage, x2 , should be equal to a given Eout , i.e. x2 = Eout ̸= 0. Under this consideration, we look for an equilibrium point such
T
that x¯ = x¯ 1 Eout , where x¯ 1 is a constant to be determined. Computing k1 and k2 from (5) we obtain
E in L k1 = , Eout −
k2 =
−
RC
Eout x¯ 1 C
L
−
RLC
RC (θ¯1 Ein + θ¯2 Eout ) −L[θ¯1 Eout + θ¯2 (R x¯ 1 + Eout )]
θ¯1 + θ¯2 = 1,
0 , 0
θ¯1 ≥ 0, θ¯2 ≥ 0
from which
θ¯1 =
Eout Eout − Ein
,
θ¯2 = 1 − θ¯1 ,
x¯ 1 =
2 Eout − Eout Ein
Ein R
.
(9)
Note that in this example, given Ein , Eout , the choice of θ¯ is unique. However, this might not always be the case (for instance, see Example #2). The system operates as a ‘‘voltage buck’’ (Ein > Eout ) if θ¯1 < 0.5 and as a ‘‘voltage boost’’ (Ein < Eout ) if θ¯1 > 0.5. 2.2. Example #2 Consider an affine switched system with three subsystems,
M = {1, 2, 3}, and state space representation (1), where
0 A1 = A2 = 0
b1 =
−1 0
,
0 , 0
0 A3 = 0
0 , −1
b2 =
1 , 0
b3 =
0 . 0
(10)
In this example, we seek to drive the trajectories of the switched system to the origin, that is, x¯ = 0. Note that none of the subsystem matrices Ai , i = 1, . . . , 3 is Hurwitz. The trajectories of each individual subsystem in the phase plane is shown in Fig. 1. Using Lemma 1 we compute
θ¯2 − θ¯1 0
=
0 , 0
θ¯1 + θ¯2 + θ¯3 = 1,
θ¯1 ≥ 0, θ¯2 ≥ 0, θ¯3 ≥ 0
(12)
Any choice of β in the range [0, 1/2] makes the origin an equilibrium point of the differential inclusion. 3. Switching rule using a max composition As discussed earlier, we are interested in designing a switching rule that drives the switched system to the origin of the differential inclusion (6). In this paper we focus on rules obtained from the application of a ‘max’ composition of the form (13)
where Vi : Rn → R, i ∈ M are auxiliary functions associated with the subsystems (5). The switching rule σ (e(t )) must render the origin of the error system globally asymptotically stable. The following lemma sheds light on a subclass of affine switched systems for which the problem of global stabilization is trivial.
=
β ∈ [0, 1/2].
i∈M
RC
θ¯ = (β, β, 1 − 2β),
σ (e(t )) := arg max{Vi (e(t ))}
Eout .
Using Lemma 1, there should exist θ¯ ∈ Θ such that 1
from which
(11)
Lemma 2. Consider the affine switched error system (5). Assume m ¯ there exists θ¯ ∈ Θ such that i=1 θi ki = 0. If there exists i such that Ai is Hurwitz and ki = 0 then the closed-loop system under the switching law σ (e(t )) = {i} is globally asymptotically stable. The proof of this lemma is trivial and is omitted. As a consequence of Lemma 2, the challenging problems will be the ones for which the subsystems (5) satisfy the following condition: Assumption 1. If Ai is Hurwitz, then ki ̸= 0. Both examples introduced earlier fulfill Assumption 1. The methods which we will present shortly will apply when Assumption 1 is satisfied and, of course, also when it is not satisfied. At this point, it is important to identify a significant difference between linear and affine switched systems: even if all Ai ’s are Hurwitz, one does not automatically obtain convergence to the origin due to the presence of a constant term in the vector field. Another consequence of the presence of the constant term and Lemma 1 is that a stable equilibrium at the origin of the error system will often be enforced by a sliding mode, since under Assumption 1 the only possibility of a single subsystem to be active at the origin, i.e. θi = 1 for some i, is when the corresponding Ai is not Hurwitz. These observations help to illustrate how the stabilization problem of switched affine system can be significantly more challenging than the stabilization of switched linear systems. In particular, the need to consider the stability on sliding modes is a significant complication, which is often tactfully excluded from the analysis of many existing results for linear switched systems, e.g. [12]. Moreover, the literature is rich in switching laws based
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on ‘min’ composition rules as opposed to ‘max’ composition rules, and one reason for this fact is the simplified treatment of sliding modes in the case of linear switched systems [2, Section 3.4] when the ‘min’ composition is used. As we will see in the next section, the use of a ‘max’ composition rule will however add flexibility to the stability analysis because the functions Vi ’s are not required to be positive functions. Recall the positiveness requirement is present when using the ‘min’ composition combined with a Lyapunov stability argument [20,21,19].
∇ Vi (e)T (Ai e + ki ) < Q (e) ≤ 0 whenever Vi (e) ≥ θ¯j Vj (e).
In this section we present our main stability result. When the conditions of Theorem 1 are satisfied we get a switching rule σ (x(t )) that asymptotically drives the switched system state to a given constant reference x¯ . We pay special attention to stability on sliding modes. Theorem 1. Consider the affine switched error system (5). Assume m ¯ there exists θ¯ ∈ Θ such that i=1 θi ki = 0, and functions Vi : n R → R, with Vi (0) = 0 for all i ∈ M , Q : Rn → R, all in C 1 , and α : Θ → R such that V (e) = max{Vi (e)} > 0
(14)
i∈M
V (e) is radially unbounded
(15)
Q (e) ≤ 0,
(16)
α(θ ) ≥ 0
θi θj ∇ Vi (e)T (Aj e + kj ) + 2α(θ )(θi − θ¯i )Vi (e) < Q (e) (17)
Therefore, for e(t ) such that σ (e(t )) = {i} we have V (e(t )) = Vi (e(t )) > 0, hence V (e(t )) is differentiable, so that inequality (20) implies
Now consider a point e(t ) in which σ (e(t )) is not a singleton. In this case the trajectories of the closed-loop switched system either go to a region where σ (e(t )) is a singleton, in which case V (e(t )) decreases because of the argument above, or enter a sliding mode. In a sliding mode, the system trajectory is described by the differential inclusion (6). In particular, the switched system will remain in a sliding mode for as long as θ (e(t )) ∈ Θ is such that e˙ (t ) ∈ T (e(t )), where T (e) := h ∈ Rn : φi (e) = 0, ∇φi (e)T h = 0, ∀i ∈ σ (e)
for all θ ∈ Θ , and e ̸= 0. Then the origin of closed-loop differential inclusion (6) under the switching rule (13) is globally asymptotically stable. Proof. The assumptions on Vi ∈ C 1 ensure that V is continuous with V (0) = 0 while (15) ensures that V is radially unbounded. Hence V qualifies as a Lyapunov function candidate and can be used to prove global asymptotic stability [22,23]. The fact that V does not have a derivative at all points will not be important. As we will show soon, it is possible to use a standard Lyapunov stability argument in this proof. Condition (14) requires that V should be positive definite while the condition (17) ensures that V decreases along any trajectory of the closed-loop switched system, a fact we prove next. Firstly observe that θ∈Θ
i∈M
θi Vi (e) ≥
i∈M
θ¯i Vi (e)
(18)
i∈M
¯ which in turn implies i∈M (θi − θi )Vi (e) ≥ 0. As α(θ ) ≥ 0 it follows that (17) can be obtained by applying the S-procedure [24], [25, p. 23] to the conditions
θi θj ∇ Vi (e)T (Aj e + kj ) < Q (e) ≤ 0 (θi − θ¯i )Vi (e) ≥ 0
(19)
∀(θ, e) ̸= (θ¯ , 0). While the first inequality is related to the time derivative of the Lyapunov function, as shown in the sequel, the second inequality is related to the ‘max’ composition in (18). Firstly, let us consider the particular case when σ (e) is a singleton, namely σ (e) = {i}. In this case we have
j∈M
θ¯j Vj (e) ≥ 0.
θi (e(t ))Vi (e(t ))
= max θ∈Θ
θi Vi (e(t )) ≥
θ¯i Vi (e(t ))
i∈M
i∈M
which implies
(θi (e(t )) − θ¯i )Vi (e(t )) ≥ 0.
(22)
i∈M
Therefore, with the same S-procedure arguments previously used we conclude from (17) that for e(t ) and all θ (e(t )) ∈ Θ we get
θi (e(t ))θj (e(t ))∇ Vi (e(t ))T (Aj e(t ) + kj )
i∈M j∈M
=
θi (e(t ))θj (e(t ))∇ Vi (e(t ))T (Aj e(t ) + kj )
i∈σ (e(t )) j∈σ (e(t ))
< Q (e(t )) ≤ 0 whenever (22) is satisfied. In particular, for all θ (e(t )) ∈ Θ such that e˙ (t ) =
θj (e(t ))(Aj e(t ) + kj )
j∈M
θj (e(t ))(Aj e(t ) + kj )
(23)
j∈σ (e(t ))
i∈M
Vi (e) −
(21)
i∈M
=
i∈M j∈M
whenever
where φi (e) := V (e) − Vi (e). The condition e˙ (t ) ∈ T (e(t )), implies that the system’s vector field in (6) is orthogonal to the set of gradients {∇φi (e), i ∈ σ (e)}. For each i ∈ σ (e) the condition ∇φi (e)T h = 0 can be viewed as a requirement that h belongs to the tangent hyperplane of the sliding surface φi (e) = 0. Back to stability, for such an e(t ) and all θ (e(t )) ∈ Θ , inequality (18) implies that V (e(t )) =
i∈M j∈M
V (e) = max{Vi (e)} = max
(20)
j∈M
V˙ (e(t )) = V˙ i (e(t )) = ∇ Vi (e(t ))T (Ai e(t ) + ki ) < Q (e(t )) ≤ 0.
4. Global asymptotic stabilization
According to Filippov’s results [11, p. 50] in this situation we have θi = 1 and θj = 0 for all j ̸= i and inequality (19) reduces to
with e˙ (t ) ∈ T (e(t )), we have DV (e(t ))[˙e(t )] = max ∇ Vi (e(t ))T e˙ (t ) i∈σ (e(t ))
=
θi (e(t ))∇ Vi (e(t ))T e˙ (t )
i∈σ (e(t ))
< Q (e(t )) ≤ 0
(24)
where DV (e(t ))[h] is Dini’s directional derivative of V (e(t )) [26, p. 420]. Observe that DV (e(t ))[h] is therefore guaranteed to decrease along all trajectories in the sliding manifold.
C.C. Scharlau et al. / Systems & Control Letters 68 (2014) 1–8
5
(b) Composite V (red) and lower bound V¯ (brown).
(a) Components V1 (blue) and V2 (green).
Fig. 2. Buck–boost converter: Lyapunov function V (e) = max{V1 (e), V2 (e)} and lower bound V¯ (e).
The last situation that needs to be considered is when σ (e(t )) is not a singleton and the sliding manifold is not differentiable at e(t ). Due to the continuity and differentiability of the functions Vi ’s, such points can only occur at the intersection of two sliding manifolds, therefore creating a lower dimensional sub-manifold. The argument to be used in this case is as follows: the system will either exit the sliding mode and becomes singleton or will remain on a sliding motion on one of sub-manifolds that intersect at e(t ). In both cases V (e(t )) decreases because of the previous arguments. Moreover, continuity of V (e(t )) implies that the value of V (e(t )) cannot increase at these points where V (e(t )) is not differentiable. Furthermore, condition (24) is satisfied also at points e(t ) in which the sliding manifold is not differentiable. At those points θ (e(t )) might be discontinuous but DV (e(t ))[˙e(t )] < 0 because (19) holds for all θ ∈ Θ − {θ¯ }. See [11, p. 155] for details regarding this point. Remark 1. Theorem 1 and its proof can be generalized to general nonlinear switched systems where the functions fi (x(t )), i ∈ M , defining the subsystems x˙ (t ) = fi (x(t )) do not necessarily need to be homogeneous. We will address this case in future works. Remark 2. Observe from (22) that θ is in fact a function of e(t ). However in condition (17) this dependence is not taken into account which may introduce some extra conservatism to (17). The role of α(θ ), that appears in (17) thanks to the S-Procedure, is to reduce this conservatism to some extent. Remark 3. It should be mentioned that the inequalities appearing in Theorem 1 are not convex with respect to θ and α(θ ). However, for some particular choices of Vi (e(t )) and α(θ ) and some additional conditions, it is possible to obtain sufficient LMI conditions to check expressions which are similar to the ones in Theorem 1. For instance, LMI conditions are proposed in [16] under the assumption of the existence of a coefficient θ˜ ∈ Θ such that ˜ i∈M θi Ai is Hurwitz stable. Observe this requirement does not appear in the conditions of Theorem 1. Remark 4. In some cases it is possible to use the following sufficient condition for checking the positivity of V . Suppose that there exists θ˜ ∈ Θ such that
θ˜i Vi (e) > 0,
∀e ̸= 0.
(25)
Then we get V (e) > 0 for all e ̸= 0. This follows from the fact that i∈M
θ∈Θ
Consider the buck–boost converter with a linear (resistor) load previously introduced in Section 2.1. As in this case there exists a coefficient θ¯ ∈ Θ such that i∈M θ¯i Ai is Hurwitz stable, according to Remark 3 the conditions of Theorem 1 can be checked by solving an LMI problem given in [16]. After tweaking numerical results obtained from [16] we arrived at the following choices of auxiliary functions in C 1 V1 (e) = −6000e1 + 180e2 − 300 e21 + 3e22 + 20 e1 e2 V2 (e) = 3600e1 − 108 e2 + 4000e21 + 7e22 + 40 e1 e2 . In the sequel we show that the conditions of Theorem 1 are satisfied with the above auxiliary functions leading to a switching rule σ (e) that globally drives the output voltage to the reference value Eout = −9 V. It is worth noticing that the arguments below are based on the conditions of Theorem 1 and not on the methodology of [16]. Note that V1 (0) = V2 (0) = 0 and V1 and V2 are not positive definite functions because they have a linear term. A threedimensional plot of V1 and V2 is shown in Fig. 2(a). Our first challenge is to prove that the ‘max’ composition of V1 and V2 is positive definite and radially unbounded, as shown in the threedimensional plot in Fig. 2(b). We will also prove that, for this example, there exists a positive definite quadratic function that bounds V from below globally by using the arguments in Remark 4. This lower bound function is represented by V¯ (e(t )) in Fig. 2(b). When Eout = −9 V, Ein = 15 V, and L, C , R are as in Section 2.1, the converter operates as a buck. Furthermore, θ¯1 = 3/8 and θ¯2 = 5/8, from which we compute V¯ (e) := θ¯1 V1 (e) + θ¯2 V2 (e) = 2387.5 e21 + 5.5 e22 + 32.5 e1 e2 . One can easily verify that V¯ (e) > 0 for all e ̸= 0 hence, from Remark 4 we conclude that V (e) ≥ V¯ (e) > 0 for all e ̸= 0. As V¯ (e) is radially unbounded, so is V (e). Having proved positivity of V , we now show that condition (17) is satisfied with the particular choices Q (e) = Q = 0, α(θ ) = α = 200. We start by rewriting (17) in the form 3 100
i∈M
V (e) = max{Vi (e)} = max
4.1. Example #1 (continued)
i∈M
θi Vi (e) ≥
θ˜i Vi (e) > 0.
i∈M
In particular θ˜ = θ¯ from Lemma 1 seems to be a natural choice. By ˜ the same token, if i∈M θi Vi (e) is radially unbounded then V (e) will also be radially unbounded.
T e 1
M (θ ) r (θ )
r T (θ )
γ (θ )
e 1
> 0,
for all θ ∈ Θ , e ̸= 0 (26)
where the matrices M (θ ), r (θ ) and γ (θ ) can be computed from the choices of V1 (e), V2 (e) given above. In [16], a sufficient LMI condition is derived based on (26). Here we take a different path based on a condition that depends exclusively on θ . A sufficient condition for (26) to hold for all e ̸= 0 is that M (θ ) r (θ )
r T (θ )
γ (θ )
≥ 0,
M (θ ) > 0,
for all θ ∈ Θ .
(27)
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Fig. 3. Buck–boost converter in buck type operation with Eout = −9 V.
(a) Components V1 (blue), V2 (green), and V3 (yellow).
(b) Composite V (light green).
Fig. 4. Example #2: Lyapunov function V (e) = max{V1 (e), V2 (e), V3 (e)}; solid black curve shown in (b) is the switching surface.
Using Schur-Complement [25, p. 7], the above conditions are equivalent to
γ (θ) ≥ r (θ ) M −1 (θ ) r T (θ ),
M (θ ) > 0,
for all θ ∈ Θ .
Since θ2 = 1 − θ1 it follows that M (θ ) is a polynomial matrix function of θ1 ∈ [0, 1]. In this case we have M (θ ) > 0 as the principal minors of M (θ ) are positive scalar functions of θ1 in the interval 0 ≤ θ1 ≤ 1. The condition γ (θ )− r (θ ) M −1 (θ ) r T (θ ) ≥ 0 is a bit more subtle to prove. First, define p(θ) := γ (θ ) − r (θ ) M −1 (θ ) r T (θ ). r (θ) = (3 − 8 θ1 ) 5400 (25 θ1 − 28)
4.2. Example #2 (continued) Consider the system introduced in Section 2.2 and the auxiliary functions in C 1 V1 (e) = e1 (t ) + e2 (t )2 ,
Now verify that
x¯ = (480 m A, −9 V) is shown in Fig. 3. The state response is shown at the left hand side and the corresponding trajectory on the error phase-plane is on the right. When the trajectory touches the switching surface for the second time a sliding motion drives the error towards the origin, e = 0.
396 (25 θ1 − 54) ,
γ (θ) = 43200 (3 − 8 θ1 )2 . Therefore, p(θ ) can be factored into p(θ ) = 864 (3 − 8 θ1 )2 p1 (θ1 ), where the rational function p1 (θ1 ) > 0 in the interval 0 ≤ θ1 ≤ 1. Therefore p(θ ) > 0 for all θ1 ̸= θ¯1 = 3/8 and p(θ¯ ) = 0. The conclusion is that (26) holds true for all e ̸= 0 and θ ∈ Θ . The above factorization of p is not a coincidence, and it comes from a careful choice of the linear coefficients in the functions V1 and V2 . The response of the system starting from a zero initial state, x = (0, 0), in the original coordinates (1) towards the target
V2 (e) = −e1 (t ) + e2 (t )2 ,
V3 (e) = 2e2 (t )2 .
(28)
A three-dimensional plot of these functions is shown in Fig. 4(a). We will show that these auxiliary functions produce a switching rule σ (e) that can drive the system asymptotically to zero, i.e. towards e = x¯ = 0. First consider the question of whether the ‘max’ composition of the above functions is positive definite. Since V (e) = max{Vi (e)} = max{|e1 | + e22 , 2e22 }
(29)
i∈M
we can readily conclude that V (e) > 0 for all e ̸= 0. V (e) is also radially unbounded. Interestingly, since V (e1 , 0) = |e1 | it
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7
equilibrium. Moreover for any initial condition where e2 (0) = 0 or e1 (0) = 0, marked with • in the figure, the system trajectory goes to the origin without switchings during the whole trajectory. If e2 (0) = 0 and e1 (0) > 0 we have σ (e(t )) = {1}, ∀t ≥ 0. If e2 (0) = 0 and e1 (0) < 0 we have σ (e(t )) = {2}, ∀t ≥ 0. If e1 (0) = 0 we have σ (e(t )) = {3}, ∀t ≥ 0. At the equilibrium we have e(t ) = 0 and θ = θ¯ for any θ¯ in (12). 5. Concluding remarks
Fig. 5. Phase plane of the switched system. Solid (black) lines are error trajectories; Dashed (color) lines are switching surfaces.
follows that there exists no positive definite quadratic function that bounds V from below globally, as any positive definite quadratic form would grow larger than |e1 | for large enough e1 . This is therefore an example in which Remark 4 does not apply for the choice of Vi (e) in (28). Indeed, θ¯1 V1 (e) + θ¯2 V2 (e) + θ¯3 V3 (e) = 2(1 −β)e22 , which is positive semidefinite for β ∈ [0, 1/2]. A threedimensional plot of V (e) is shown in Fig. 4(b). In the sequel we check condition (17) with the particular choice of Q (e) = Q = 0, α(θ ) = α = 0. Using elementary algebra manipulation, it can be shown that
θi θj ∇ Vi (e)T (Aj e + kj ) = −f (θ ) − g (θ , e2 )
i∈M j∈M
where f (θ ) = (θ1 − θ2 )2 ≥ 0 and g (θ , e) = 2θ3 (θ1 + θ2 + 2θ3 )e22 . Note that f (θ ) + g (θ , e) ≥ 0 for all e and θ ∈ Θ , hence (17) is only negative semidefinite. One might be tempted to use La Salle’s extension to prove stability under this weaker constraint. The discussion is however complicated (see for instance [27]), and here we use an ad hoc argument which we will elaborate in future publications. As V is positive definite and (17) is only negative semi-definite, the Lyapunov machinery can only prove convergence to a subset1 of the invariant set
Ω := {(θ , e) : f (θ ) + g (θ , e) = 0, θ ∈ Θ } = Ω1 ∪ Ω2 where
Ω1 := {(θ , e) : θ3 = 1, θ1 = θ2 = e2 = 0, e ∈ R2 } Ω2 := {(θ , e) : θ1 = θ2 = 1/2, θ3 = 0, e ∈ R2 }. Note that, for this example, the switched system with the switching rule σ (e(t )) leads to
(θ(e(t )), e(t )) ∈ Ω H⇒ e(t ) = 0. Indeed, if θ3 (e(t )) = 1 and e2 (t ) = 0, it follows from (28) and (29) that e1 (t ) = 0. On the other hand, if θ1 (e(t )) = θ2 (e(t )) = 1/2 it follows that V1 (e(t )) = V2 (e(t )) < V (e(t )) except when e2 (t ) = 0. In summary, V˙ = 0 only at the equilibrium point (e, θ ) = (0, θ¯ ) and V˙ < 0 outside the equilibrium. The system trajectory for several initial conditions are indicated in Fig. 5. Observe that all the trajectories starting at a point where e1 (0) ̸= 0, e2 (0) ̸= 0 have sliding motion dynamics outside the
1 Indeed Ω (e) := {(θ(e), e) ∈ Ω : θ(e) parametrizes a feasible trajectory } ⊆ Ω .
This paper presents conditions for switching rules design that asymptotically drives the state of an affine switched system to a given constant reference. The results are summarized by Theorem 1 and they guarantee global asymptotic stability of the closed-loop switched system even if sliding motions occur at any sliding surface of the system. It is important to emphasize that the results in this paper do not require the existence of a Hurwitz convex combination of the dynamic matrices of the subsystems. This point is discussed in Example #2. This requirement is present in several results found in the literature, as for instance in [16,8]. See [4] for more references on this point. One may observe that the conditions appearing in Theorem 1 are not convex with respect to θ and α(θ ). However, for some particular choices of Vi (e(t )) and α(θ ), it is possible to get sufficient LMI conditions to check the expressions in Theorem 1. For instance, LMI conditions are proposed in [16]. However, the results in this reference require the existence of a Hurwitz convex combination as previously mentioned. Currently, the authors are working on how to express the conditions of Theorem 1 as an LMI problem without this requirement. We are also investigating extensions to include performance requirements such as guaranteed cost and disturbance attenuation. References [1] D. Liberzon, A. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag. 19 (5) (1999) 59–70. [2] D. Liberzon, Switching in Systems and Control, Birkhauser, 2003. [3] R.A. DeCarlo, S.H. Zak, G.P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proc. IEEE 76 (3) (1988) 212–232. [4] R.A. DeCarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proc. IEEE 88 (7) (2000) 1069–1082. [5] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a short survey of recent results, in: Proc. of the 2005 IEEE International Symposium on Intelligent Control, Limassol, Cyprus, 2005, pp. 24–29. [6] M.A. Wicks, P. Peleties, R.A. DeCarlo, Switched controller design for the quadratic stabilization of a pair of unstable linear systems, Eur. J. Control 4 (2) (1998) 140–147. [7] G. Zhai, H. Lin, P. Antsaklis, Quadratic stabilizability of switched linear systems with polytopic uncertainties, Internat. J. Control 76 (7) (2003) 747–753. [8] P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, in: Proceedings of the 2004 American Control Conference, Boston, MA, 2004, pp. 3890–3895. [9] M. Wicks, R.A. DeCarlo, Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, in: Proc. of the 15th American Control Conference, Albuquerque, NM, 1997, pp. 1709–1713. [10] S. Pettersson, Synthesis of switched linear systems, in: Proc. of the 42nd IEEE Conference on Decision and Control, Maui, HI, 2003, pp. 5283–5288. [11] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Norwell, MA, 1988. [12] Z. Sun, Combined stabilizing strategies for switched linear systems, IEEE Trans. Automat. Control 51 (4) (2006) 666–674. [13] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Automat. Control 57 (7) (2012) 1809–1815. [14] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell time switching, Automatica 48 (6) (2012) 1132–1137. [15] T. Hu, L. Ma, Z. Lin, On several composite quadratic Lyapunov functions for switched systems, in: Proc. of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006, pp. 113–118. [16] A. Trofino, C.C. Scharlau, T. Dezuo, M.C. de Oliveira, Stabilizing switching rule design for affine switched systems, in: Proc. of the 50th IEEE Conference on Decision and Control, Orlando, FL, 2011, pp. 1183–1188. [17] M. Rashid, Power Electronics Handbook, third ed., Butterworth-Heinemann, 2010.
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[18] M. Di Bernardo, F. Vasca, Discrete-time maps for the analysis of bifurcations and chaos in DC/DC converters, IEEE Trans. Circuits Syst. I 47 (2) (2000) 130–143. [19] G. Deaecto, J. Geromel, F. Garcia, J. Pomilio, Switched affine systems control design with application to dc–dc converters, IET Control Theory Appl. 4 (7) (2010) 1201–1210. [20] T. Hu, L. Ma, Z. Lin, Stabilization of switched systems via composite quadratic functions, IEEE Trans. Automat. Control 53 (11) (2008) 2571–2585. [21] J. Geromel, P. Colaneri, P. Bolzern, Dynamic output feedback control of switched linear systems, IEEE Trans. Automat. Control 53 (3) (2008) 720–733.
[22] F. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [23] H.K. Khalil, Nonlinear Systems, third ed., Prentice Hall, 2002. [24] V. Yakubovich, S-procedure in nonlinear control theory, Vestn. Leningr. Univ. 1 (1971) 62–77. [25] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [26] L.S. Lasdon, Optimization Theory for Large Systems, Macmillan, 1970. [27] J. Hespanha, Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle, IEEE Trans. Automat. Control 49 (4) (2004) 470–482.