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ℋ2 Performance Optimization of Singularly Perturbed Switched Linear Systems
Proceedings of the 4th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 12) June 6-8, 2012. Eindhoven, The Netherlands
H2 Performance Optimization of Singularly Perturbed Switched Linear Systems Grace S. Deaecto ∗ Jamal Daafouz ∗∗ Jose C. Geromel ∗∗∗ ∗
FEEC - UNICAMP, Campinas, SP, Brazil, [email protected] ∗∗ CRAN - CNRS, Institut Universitaire de France, Vandoeuvre-l`es-Nancy, France, [email protected] ∗∗∗ FEEC - UNICAMP, Campinas, SP, Brazil, [email protected] Abstract: This paper addresses the H2 performance problem for singularly perturbed switched linear systems in continuous-time. We propose a new and unified approach for stability analysis and norm computation for singularly perturbed LTI systems and generalize it to the case of singularly perturbed switched linear systems to obtain numerically well conditioned conditions for the design of a state-switching control. Such a switching strategy guarantees that the closed-loop switched system is globally asymptotically stable for sufficiently small values of the perturbation parameter and provides an upper bound to an H2 performance index. We provide a comparison with existing methods. 1. INTRODUCTION Singular perturbation theory is a well known topic for mathematicians, physicists, and other researchers. For control scientists, this setting has been used to model systems with dynamics operating on multiple time scales and also for model reduction purposes (see Kokotovic et al. [1986], Naidu [2002] and references therein). In this context, the simultaneous occurrence of slow and fast phenomena gives rise to time scales and poses difficult computational complexities. As far as time invariant models are considered, a solution is obtained using time scale separation which simplifies the analysis and avoids ill-conditioning by decomposing the problem into two independent well conditioned problems, one for the slow subsystem and another one for the fast subsystem. When the dynamical system is characterized by a hybrid nature, that is, its behavior is the result of the interaction of two different dynamics: discrete and continuous, both phenomena, switching and multiple time scale, may arise simultaneously and classical results may not necessarily hold. For switched systems with arbitrary switching in general we can cite Liberzon [2003] and Shorten et al. [2007]. It has been shown in Malloci et al. [2010] that, stability of the slow and the fast switched subsystems separately does not imply stability of the original switched system for small values of the singular perturbation parameter. In other words, the Tikhonov theorem, cannot be used in its classical form. ⋆ This work was supported in part by the by grants from ”Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico - CNPq”, “Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Capes”, Brazil and by the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n257462 HYCON2 Network of excellence.
978-3-902823-00-7/12/$20.00 © 2012 IFAC
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To emphasize the interest in studying mathematical problems related to singularly perturbed hybrid systems, we mention three classes of problems. The first class has been introduced in Heck and Haddad [1989] where singularly perturbed piecewise systems have been considered and a technique that allows decoupling of such systems into fast and slow subsystems is proposed. The second class has been introduced in Sanfelice and Teel [2011] where a semiglobal practical asymptotic stability result has been proposed for a class of singularly perturbed hybrid systems. The third class concerns singularly perturbed switched systems which can be motivated by several practical applications from power electronics or biological networks. It is also motivated by the steering control for the last phase of the rolling process, called the tail end phase. This application is described in Malloci et al. [2009] where it is shown that a steering control for the tail end phase reduces to a control design for a singularly perturbed switched system. Despite this practical interest, a complete characterization of the stability and performance properties of singularly perturbed hybrid systems and a general singular perturbation theory are missing. Indeed, taking into account the effects of vector field discontinuities and multiple time scales simultaneously for stability, performance analysis and design is a very difficult and challenging task. In this paper, we propose a new approach for stability analysis and norms computations for singularly perturbed LTI systems which can be generalized to the case of singularly perturbed switched linear systems. It is shown that using the proposed approach, we obtain well conditioned conditions for switching control design with performance constraints for a class of hybrid problems where both switching and multiple time scale phenomena arise. The notation throughout is standard. For square matrices Tr(·) denotes the trace function and for a symmetric ma10.3182/20120606-3-NL-3011.00005
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
trix λmax (·) is its maximum eigenvalue. For real matrices or vectors (′ ) indicates transpose. For symmetric matrices, the symbol (•) denotes each of its symmetric blocks. The set M is composed by all Metzler matrices with nonnegative off diagonal elements P πji ≥ 0, ∀i 6= j satisfying the normalization constraint j6=i πji = −πii ≥ 0, ∀i. The set of real numbers is denoted as R whereas the set of the N first natural numbers is denoted as K = {1, 2, · · · , N }. The squared norm of a trajectory ξ(t)R defined for all t ≥ 0, ∞ denoted by kξk22 , is equal to kξk22 = 0 ξ(t)′ ξ(t)dt. 2. PROBLEM MOTIVATION AND PRELIMINARIES
Consider a singularly perturbed linear system evolving from the rest with the following state space realization x˙ 1 A11 A12 x1 B1 = + w (1) x˙ 2 A21 /ε A22 /ε x2 B2 x (2) z = [ C1 C2 ] 1 x2
where ε > 0 is a small scalar, x1 ∈ Rn1 and x2 ∈ Rn2 are the slow and fast states, w ∈ Rr is an external disturbing input and z ∈ Rp is the system output. The system matrices have compatible dimensions and A22 is supposed to be nonsingular. It is well known that to avoid numerical difficulties that arise when ε > 0 is small, which is usually the case for singularly perturbed systems, two smaller−order subsystems can be defined using a time-scale decomposition Chow and Kokotovic [1976], Kokotovic et al. [1986]. Namely, the system (1)(2) is decomposed into slow and fast subsystems, where G = A−1 22 A21 is the coupling static gain between them: x˙ s = As xs + Bs w (3) evolving from xs (0) = x1 (0) = 0 with As = A11 − A12 G and Bs = B1 , representing the slow dynamics and x˙ f = (Af /ε) xf + Bf w (4) evolving from xf (0) = Gx1 (0) + x2 (0) = 0 with Af = A22 and Bf = B2 +GB1 , representing the fast one. The system output is given by z = Cs xs + Cf xf (5) with Cs = C1 −C2 G and Cf = C2 . This decomposition can be generalized to switched linear systems evolving from the rest with the state space realization σ σ x1 B1 x˙ 1 A11 Aσ12 + w (6) = x2 B2σ x˙ 2 Aσ21 /ε Aσ22 /ε x z = [ C1σ C2σ ] 1 (7) x2
where w ∈ Rr is an external input and z ∈ Rp is the corresponding controlled output. The switching function is defined as σ(t) : t ≥ 0 −→ K, making clear that at each instant of time t ≥ 0, one and only one among N known linear subsystems corresponding to some i ∈ K is switched on. The switched system (6)-(7) is decoupled in two smaller order subsystems where the coupling static −1 gain is defined as Gi = Ai22 Ai21 , for all i ∈ K. x˙ s = Asσ xs + Bsσ w, xs (0) = 0 (8) i i i with Asi = A11 − A12 Gi and Bsi = B1 , ∀i ∈ K, and x˙ f = (Af σ /ε) xf + Bf σ w, xf (0) = 0 (9)
229
with Af i = Ai22 and Bf i = B2i +Gi B1i , ∀i ∈ K. The system output is given by z = Csσ xs + Cf σ xf (10) with Csi = C1i − C2i Gi and Cf i = C2i , ∀i ∈ K. Hence, for switched linear systems the same time scale decomposition in slow and fast subsystems can be adopted as well. The main interest in performing the previous decomposition is that a separate analysis of the slow and fast subsystems can be done avoiding numerical difficulties and allowing the derivation of an approximated solution which coincides with the direct solution to a full-order problem as ε → 0 (that is for 0 < ε < εmax with εmax a bound to be characterized). For linear systems such as (1)-(2) the validity of this decomposition has been demonstrated for stability and performance properties including H2 problems Kokotovic et al. [1986]. For the first time in the literature, a solution based on convex optimization has been proposed in Garcia et al. [2002] but it is specific to the LQ problem. The characterization and the computation of the bound εmax has attracted a number of researchers since the end of the seventies (see Cao and Schwartz [2004] and references therein), but most of the proposed approaches are restricted to stability analysis. For switched systems, the problem is totally open and there is no complete and satisfactory solution. The case of arbitrary switching has been investigated in Malloci et al. [2009], but as illustrated in El Hachemi et al. [2011] for planar systems, the proposed solution can be conservative. We provide a new convex optimization based approach for H2 norm computation with the determination of εmax . Lemma 1. Consider Φ and Γ two symmetric matrices of compatible dimensions with the four-blocks form Φ1 Φ2 0 Γ2 Φ= , Γ= (11) • Φ3 • Γ3 where Γ3 is nonsingular. There exists εmax > 0 such that the inequality Φ + ε−1 Γ < 0 (12) holds for all ε ∈ (0, εmax ) if and only if the conditions Φ1 < 0, Γ2 = 0 and Γ3 < 0 simultaneously hold. Proof: For the inequality (12) to hold for ε > 0 sufficiently small it is required that Γ ≤ 0. Hence, it is seen that Γ ≤ 0 holds if and only if Γ2 = 0 and Γ3 < 0 (because Γ3 is supposed to be nonsingular). On the other hand, the inequality (12) written as Φ1 Φ2 <0 (13) Φ + ε−1 Γ = • Φ3 + ε−1 Γ3 enables us to conclude that it holds for ε > 0 sufficiently small if and only if Φ1 < 0. Indeed, in this case, applying the Schur complement in the first row and column we get ε−1 (−Γ3 ) > Φ3 − Φ′2 Φ−1 (14) 1 Φ2 which implies that (13) is satisfied whenever ε > 0 is taken sufficiently small and the claim follows. 2 The determination of εmax > 0 such that (14) holds for all ε ∈ (0, εmax ) is simple. From −1/2 βmax = λmax (−Γ3 )−1/2 (Φ3 − Φ′2 Φ−1 1 Φ2 )(−Γ3 ) we get εmax = 1/βmax whenever βmax > 0 and εmax = +∞ otherwise. On the other hand, if we suppose (as usual in the literature of singularly perturbed linear systems) that matrices Φ and Γ depend on ε > 0 and
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
limε→0+ Φ(ε) = Φ0 and limε→0+ Γ(ε) = Γ0 exist and are finite then, in many instances, the existence of εmax > 0 such that Φ(ε) + ε−1 Γ(ε) < 0 holds for all ε ∈ (0, εmax ) is assured whenever Φ0 and Γ0 satisfy Lemma 1. In that precise context, the importance of Lemma 1 is apparent since the dependence of matrices Φ and Γ with respect to ε > 0 does not need to be explicit. 3. H2 ANALYSIS The aim is to characterize stability and calculate the H2 norm of system (1)-(2) denoted by S(ε) = (A(ε), B, C) and where A(ε) is defined as A11 A12 A(ε) = (15) A21 /ε A22 /ε through the data of the slow and fast subsystems. In the classical literature, Boyd et al. [1994], it is well known that S(ε) is asymptotically stable and the constraint kS(ε)k22 < γ 2 holds if and only if there exists P > 0 such that Tr(B ′ P B) < γ 2 , A(ε)′ P + P A(ε) + C ′ C < 0 (16) and B ′ = [B1′ B2′ ], C = [C1 C2 ]. The next theorem provides conditions to calculate the H2 norm of system (1)-(2) avoiding numerical problems due to ill conditioning. Theorem 2. Assume there exist X > 0, Z > 0 satisfying As X + XA′s XCs′ < 0 (17) Tr Bs′ X −1 Bs < γs2 , • −I (18) Tr Bf′ Z −1 Bf < γf2 , Af Z + ZA′f < 0
In the affirmative case, there exists εmax > 0 such that the system S(ε) is asymptotically stable and satisfies the H2 norm constraint kS(ε)k22 < γs2 + γf2 for all ε ∈ (0, εmax ). Proof: We start by writing A(ε) as 0 0 A11 A12 −1 A(ε) = +ε A21 A22 0 0 and splitting the inverse P −1 as X V >0 P −1 = • Y
(19)
(20)
Multiplying both sides of inequality (16) by P −1 , it can be written as Φ + ε−1 Γ < 0 with Φ and Γ presenting the structure (11). From Lemma 1, there exists εmax such that the inequality Φ + ε−1 Γ < 0 is satisfied for ε ∈ (0, εmax ) if and only if Φ1 < 0, Γ2 = 0 and Γ3 < 0. Hence, from Γ2 = 0 we get V ′ = −GX which together with (17) verify the condition Φ1 = As X + XA′s + XCs′ Cs X < 0. This is clear after performing the Schur Complement with respect to the last row and column of (17). Finally Γ3 = =
A22 Y + Y A′22 − A21 XG′ Af Z + ZA′f < 0
−
GXA′21 (21)
taking into account the definition Y = Z + GXG′ and (18). This choice is made with no loss of generality since P −1 > 0 whenever X > 0 and Z > 0. Now, it is a matter of simple calculation to verify that −1 X + G′ Z −1 G G′ Z −1 P = (22) • Z −1 enables the conclusion that the system S(ε) is asymptotically stable and yields the equality B ′ P B = Bs′ X −1 Bs + 230
Bf′ Z −1 Bf from which we immediately see that the condition kS(ε)k22 < γs2 + γf2 holds for all ε ∈ (0, εmax ). 2 Consider the maximum value εmax = supε>0,X>0,Z>0 ε subject to A(X, Z) + ε−1 B(Z) < 0 where the matrix functions A(X, Z) and B(Z) are obtained directly from the multiplication of both sides of the Lyapunov inequality (16) by P −1 given in (20), that is As X + XA′s A12 Z − As XG′ A(X, Z) = + • 0 ′ XCs′ XCs′ + (23) ′ ′ ′ ′ ZCf − GXCs ZCf − GXCs 0 0 B(Z) = (24) ′ • Af Z + ZAf
A remarkable property of (23) is that it can be converted into an LMI with respect to the matrix variables X > 0 and Z > 0. It assures that for any γs > 0 and γf > 0 such that Tr(Bs′ X −1 Bs ) < γs2 and Tr(Bf′ Z −1 Bf ) < γf2 , where X > 0 and Z > 0 are the optimal solution that define εmax , then kS(ε)k22 < γs2 + γf2 for all ε ∈ (0, εmax ) and this is the maximum interval of validity of this inequality. Second, it is also possible to determine εmax > 0 (generally smaller than the previous one) by taking scalars γs2 , γf2 and matrices X > 0, Z > 0 satisfying the conditions of Theorem 2 and maximizing ε > 0 exclusively. Finally, from the fact that Af is Hurwitz, we notice that there exists Z −1 → 0 satisfying Af Z + ZA′f < 0. On the other hand, any X > 0 satisfying the LMI constraint (17) is such that Z ∞ ′ −1 X > eAs t Cs′ Cs eAs t dt (25) 0
which enables us to conclude that γs2 > kCs (sI − As )−1 Bs k22 and consequently kCs (sI − As )−1 Bs k22 is the minimum value of γs2 + γf2 preserving feasibility. Based on this discussion a relevant function to be determined is ρ(ω) =
inf
{ρ : Tr(Bs′ X −1 Bs ) +
ρ>0,X>0,Z>0
+Tr(Bf′ Z −1 Bf ) < ρ}
(26)
−1
subject to the LMI A(X, Z)+ω B(Z) < 0 for ω > 0 fixed. The fact that this inequality being feasible for (X, Z, ω) is also feasible for (X, Z, ε) for all 0 < ε ≤ ω allows us to say that ρ(ω) remains an upper bound to the true cost for all ε in this interval. In other words, for each εmax > 0 the associated cost ρ(εmax ) is a valid upper bound to the true singularly perturbed system for all ε ∈ (0, εmax ). 4. SWITCHED LINEAR SYSTEMS The H2 control problems for switched linear systems have been solved in Geromel et al. [2008], using the min-type piecewise positive definite quadratic Lyapunov function v(x) = mini∈K x′ Pi x where Pi > 0, ∀i ∈ K and the state dependent switching function σ(x) = arg min x′ Pi x (27) i∈K
introduced in Geromel and Colaneri [2006]. Accordingly to (6)-(7) we denote i A11 Ai12 (28) Ai (ε) = Ai21 /ε Ai21 /ε
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
′
′
Bi′ = [B1i B2i ] and Ci = [C1i C2i ] for all i ∈ K. Our main goal is to obtain conditions such that a state feedback switching rule of type (27) renders the associated closedloop switched linear system (6)-(7) globally asymptotically stable and assures an upper bound for a given H2 performance criterion. This goal is accomplished by adopting the previous min-type Lyapunov function and, consequently, the state dependent switching function (27) where −1 Xi + G′i Zi−1 Gi G′i Zi−1 >0 (29) Pi = • Zi−1 with appropriate matrices Xi > 0 and Zi > 0 for all i ∈ K. Lemma 3. Consider the relative static gains Gij = Gi −Gj and Pi > 0 of the form (29). Any feasible solution of Tij + Xi Xi G′ji Xi (30) • Zj 0 >0 • • Xj is such that ′ I I (31) Tij > (Pi−1 Pj Pi−1 − Pi−1 ) 0 0 for all i 6= j ∈ K × K and vice-versa. Proof: Assuming (30) is feasible, the Schur Complement applied to the last two rows and columns provides Tij > Xi Xj−1 + (Gj − Gi )′ Zj−1 (Gj − Gi ) Xi − Xi ′ Xi Xi > − Xi Pj −Gi Xi −Gi Xi ′ I I > (32) (Pi−1 Pj Pi−1 − Pi−1 ) 0 0 where the second and third inequalities follow from the inverse of Pi given in (29). 2
This result induces a linearization without introducing any kind of conservatism. Consider that for each k = 1, · · · , r an external input of the form w(t) = ek δ(t) ∈ Rr , where ek is the kth column of the identity matrix, is applied to the system (6)-(7) and the corresponding controlled output is denoted as zk ∈ Rp . Then, it is possible to define an H2 functional cost as r X kzk k22 (33) J2 (σ) = k=1
The adopted denomination is done with a little abuse but it is based on the fact that imposing the switching rule σ(·) = i ∈ K fixed for all t ≥ 0, the cost (33) equals the squared H2 norm of the i-th subsystem transfer function from the input w to the output z. It was proven in Geromel et al. [2008] that if there exist Pi > 0, ∀i ∈ K, and a Metzler matrix Π ∈ M satisfying the following LyapunovMetzler inequalities N X ′ (34) πji Pj + Ci ′ Ci < 0 Ai (ε) Pi + Pi Ai (ε) + j=1
for all i ∈ K, the switching rule (27) is globally stabilizing and the following guaranteed cost (35) J2 (σ) < min Tr(Bi ′ Pi Bi ) i∈K
is satisfied, provided that σ(0) = i∗ where i∗ ∈ K is the optimal index solution of problem appearing in the right hand side of (35). 231
Theorem 4. Assume there exist Xi > 0, Zi > 0, matrices Tij and a Metzler matrix Π ∈ M satisfying (30), ′ 2 Tr(Bsi Xi−1 Bsi ) < γsi , Tr(Bf′ i Zi−1 Bf i ) < γf2i and Asi Xi + Xi A′si +
N X
′ πji Tij + Xi Csi Csi Xi < 0 (36)
j6=i=1
Af i Zi + Zi A′f i < 0 (37)
for all i 6= j ∈ K × K. In the affirmative case, there exists εmax > 0 such that the min switching strategy n o ′ σ(x) = min x′1 Xi−1 x1 + x2 + Gi x1 Zi−1 x2 + Gi x1 i∈K
2 is stabilizing and J2 (σ) < mini∈K γsi + γf2i , ∀ε ∈ (0, εmax ).
Proof: Multiplying both sides of (34) by Pi−1 with the fourblocks structure Xi Vi −1 Pi = >0 (38) • Yi and writing Ai (ε) as in (19), we obtain matrix inequalities expressed as Φi + ε−1 Γi < 0. According to Lemma 1 it is satisfied if and only if Φ1i < 0, Γ2i = 0 and Γ3i < 0 for each i ∈ K. Hence, from Γ2i = 0 we have Vi′ = −Gi Xi , which together with (37) assures Γ3i < 0. In addition, Φ1i < Asi Xi + Xi A′si +
N X
′ πji Tij + Xi Csi Csi Xi (39)
j6=i=1
as a direct consequence of Lemma 3. Hence, from (36), the conclusion is that Φ1i < 0 for all i ∈ K. Finally, the switching rule follows directly from (27) and the particular structure (38) of Pi , for all i ∈ K. Moreover, it is simple to ′ verify from (29) that Bi′ Pi Bi = Bsi Xi−1 Bsi + Bf′ i Zi−1 Bf i 2 from which the inequality J2 (σ) < mini∈K γsi + γf2i follows for all ε ∈ (0, εmax ). 2 To find a solution to the matrix inequalities stated in Theorem 4 is not a simple task due to the nonconvexity inherited by the product of variables. There are several possibilities to circumvent this difficulty but always at expense of introducing some conservatism, see Geromel and Colaneri [2006]. One of the simplest is to adopt Π = µΠ0 where Π0 ∈ M and µ ≥ 0. In this case, the matrix inequalities stated in Theorem 4 becomes LMIs whenever µ ≥ 0 is determined by a line search procedure. As a consequence of matrices Af i be Hurwitz for all i ∈ K, matrices Zi−1 → 0 also satisfy Af i Zi + Zi A′f i < 0 implying that, in this particular situation, the H2 guaranteed cost depends only on the slow subsystems. It is clear that, if 2 + γf2i is fixed the right hand side of J2 (σ) < mini∈K γsi close enough to the H2 guaranteed cost, the value of εmax approaches zero. As a consequence, a possible way to increase εmax is by solving the optimization problem ρ(ω) = min
inf
{ρi : Tr(Bsi ′ Xi−1 Bsi ) +
i∈K ρi ,µ>0,Xi >0,Zi >0
+Tr(Bf i ′ Zi−1 Bf i ) < ρi }
(40) −1
subject to Aij (Xi , Xj , Zi , Zj , µ) + ω Bi (Zi ) < 0 for all i 6= j ∈ K × K. These constraints are calculated as before and are expressed through LMIs for ω > 0 and µ > 0 possibly fixed. Moreover, the fact that these inequalities are feasible for ω > 0 means that they also are feasible for
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
all ε ∈ (0, ω]. Hence, the cost ρ(ω) remains an upper bound to the true H2 cost for all ε in this interval. That is, for εmax calculated, the associated cost ρ(εmax ) is an upper bound for the true cost J2 (σ) of the singularly perturbed switched linear system (6)-(7) for all ε ∈ (0, εmax ). 5. ARBITRARY SWITCHING LINEAR SYSTEMS The goal of this section is to show how an extension of the proposed approach to the case of arbitrary switching signals outperforms the available results in the literature, in particular Malloci et al. [2009]. Consider a subclass of switching linear systems described by σ x˙ 1 A11 Aσ12 x1 = (41) x˙ 2 A21 /ε A22 /ε x2
evolving from initial condition x(0) = [x′1 (0) x′2 (0)]′ and with matrix A22 nonsingular. The switching function σ(·) : t ≥ 0 → K is arbitrary. We consider two different Lyapunov functions to study the stability of (41), the quadratic function v(x) = x′ P x with P > 0, also used in Malloci et al. [2009], and the min-type piecewise quadratic function. For this special subclass of switching linear systems characterized by presenting the switching function acting only on the slow dynamics, the asymptotic stability of the slow and fast subsystems suffices to assure the stability of the two time-scale switching linear system (41). Using the same procedure adopted for the LTI case, the slow and fast subsystems are x˙ s = Asσ xs with xs (0) = x1 (0), Asi = Ai11 − Ai12 G, ∀i ∈ K and x˙ f = (Af /ε) xf evolving from xf (0) = Gx1 (0) + x2 (0), with Af = A22 . Theorem 5. Assume there exist X > 0, Z > 0 satisfying Asi X + XA′si < 0
(42)
ZA′f
Af Z + <0 (43) for all i ∈ K. There exists εmax > 0 such that ∀ε ∈ (0, εmax ) the system (41) is asymptotically stable for σ(t) ∈ K arbitrary for all t ≥ 0. Proof: The proof follows the same procedure of the one of Theorem 2. Details are omitted. 2 An important point of the proof to be stressed is the use of Lemma 1 in this case. Since P > 0 does not depend on i ∈ K, the condition Γ2i = 0 can be solved if and only if the static gain G is also independent of the index i ∈ K. This condition is fulfilled whenever the matrices related to the fast dynamics are independent of i ∈ K. Now, adopting the min type piecewise quadratic Lyapunov function. The stability conditions for system (41) stems from the existence of symmetric matrices Pi > 0, i ∈ K and a scalar µ > 0 such that the following inequalities Ai (ε)′ Pj + Pj Ai (ε) + µ(Pi − Pj ) < 0 (44) for all i, j ∈ K×K hold. Indeed, suppose that at some time instant t ≥ 0 the switching rule is given by σ(t) = i ∈ K and define the set I(x) = {j ∈ K : v(x) = x′ Pj x}. The Dini derivative of a generic trajectory of system (41) is D+ v(x(t)) = min x(t)′ Ai (ε)′ Pℓ + Pℓ Ai (ε) x(t) ℓ∈I(x(t))
= x(t)′ (A′i (ε)Pj + Pj Ai (ε))x(t) < µx(t)′ (Pj − Pi )x(t) < 0
(45) 232
where the first inequality follows from (44) and the last one is a consequence of x(t)′ Pi x(t) ≥ x(t)′ Pj x(t) since i ∈ K and j ∈ I(x(t)). These conditions contain the quadratic ones, which can be easily verified by setting Pi = Pj = P, i, j ∈ K × K. Theorem 6. Assume there exist matrices Xj > 0, Zj > 0 for all j ∈ K and a scalar µ > 0 satisfying Asi Xj + Xj A′si − µXj µXj <0 (46) • −µXi Af Zj + Zj A′f < 0 (47) for all i, j ∈ K × K. There exists εmax > 0 such that ∀ε ∈ (0, εmax ) the system (41) is asymptotically stable for σ(t) ∈ K arbitrary for all t ≥ 0.
Proof: The proof follows that of Theorem 2.
2
The min-type Lyapunov function associated to the system (41) is constructed with Pi > 0 given by −1 Xi + G′ Zi−1 G G′ Zi−1 Pi = (48) • Zi−1 Moreover, it is important to notice that the conditions of Theorem 6 contain those of quadratic stability presented in Theorem 5. This can be verified by simply imposing Xi = Xj = X and Zi = Zj = Z for all i, j ∈ K × K which indicates that conservativeness is reduced. The value of εmax for which the result of Theorem 6 is valid can be calculated through an optimization problem similar to the previous one where µ > 0 is determined by a line search. The conditions of Theorems 5 and 6 are valid for a subclass of switching linear systems where the fast dynamics are not dependent on the switching function σ(t). Although the result of Malloci et al. [2009] can be applied for more general switching linear systems, including the case where the fast dynamics are arbitrarily dependent on the switching function σ(t), it is more conservative than our results when we restrict the analysis to (41). Indeed, for this subclass, the conditions of Malloci et al. [2009] stem from the existence of matrices Ps > 0, Qsi ≥ 0, Pf > 0 and Qf i ≥ 0 such that the LMIs Asi Ps + Ps A′si + Qsi < 0 Pf A′f
Af Pf + + Qf i < 0 # " ′ Qf i −A22 Y − Pf Ai12 ′ ≥ 0 • Qsi − Ai12 Y − Y ′ Ai12
(49) (50) (51)
are verified for all i ∈ K, with Y = (1−N )GPs . In addition to the conditions of Theorem 5, from inequalities (49) and the second diagonal block of (51), we see that the result of Malloci et al. [2009] includes an additional necessary condition: Ai11 − N Ai12 G should be Hurwitz for all i ∈ K. 6. ILLUSTRATIVE EXAMPLES The first example, inspired from Malloci et al. [2009], is dedicated to the analysis of the behavior of the H2 norm as a function of ω. We consider two systems having the state space realization (1)-(2), that is 0 1 20 10 A11 A12 0 6 5 2 (52) = A21 A22 0 0 0 1 −1.3 −0.46 −3 −5
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
3
assumes that the fast dynamics are not dependent on the switching function. When we restrict the analysis to this subclass of switching linear systems, it is shown that existing approaches are more conservative than our results.
45
2
γdB
γdB
40 1
35
0
30 −1
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Fig. 1. H2 norm 0 1 0.7 0 2 −0.3 −0.2 A11 A12 0 = (53) A21 A22 1 5 0 1 151 46.9 −3.8 −6 10 0 0 (54) , C1 = C2 = , B2 = B1 = 01 1 −1 We have solved the problem (26) for ω > 0 belonging to the interval (10−5 , 1). Figure 1 presents the H2 norm of each subsystem in dB as a function of ω. Continuous line represents the solution of problem (26) and dashed line the actual H2 norm obtained from classical theory. Notice that the maximum value of εmax for which the result of Theorem 2 is valid, is εmax = 0.0919 for the first subsystem and εmax = 0.1841 for the second one. As already expected, for ω ≈ 0, the value of the H2 norm obtained from the classical theory and that obtained from Theorem 2 are arbitrarily close and the difference becomes larger when ω increases.
For the second example, let the switching linear system (41) be defined by matrices 1 −6 −1 1 −2 1 (55) A11 A12 = 2 −5 −0.5 −4 2 −10 −1 3 −6 (56) A11 A212 = −4 −5 −1 2 2 1 −5 1 [ A21 A22 ] = (57) 3 5 −3 −2.5 As it can be verified the conditions of Malloci et al. [2009] are not satisfied since matrix A211 − 2A212 G is not Hurwitz. On the other hand, the conditions of Theorem 6 are feasible and yield εmax = 0.5943. Using the approach of Theorem 5, we obtain εmax = 0.5378. This puts in evidence that the conditions of Theorem 6 are valid for a bigger range of ε than that for Theorem 5 and hence the method obtained using a min-type Lyapunov function is strictly less conservative. 7. CONCLUSION It has been shown in the literature that for singularly perturbed switched linear systems, stability of the fast and slow subsystems independently does not necessarily imply stability of the original switched system for small values of the singular perturbation parameter. In this paper, a method that allows decoupling of singularly perturbed switched linear systems into fast and slow subsystems has been proposed. It also provides an upper bound to an H2 performance index, as well as, a theoretical characterization of the maximum singular perturbation for both, linear time invariant and switched linear systems. In the case of arbitrary switching rules, the proposed solution 233
S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. LMI in System and Control Theory. SIAM, Philadelphia, 1994. L. Cao and H. M. Schwartz. Complementary results on the stability bounds of singularly perturbed systems. IEEE Trans. Automat. Control, 49:2017–2021, 2004. J. Chow and P. Kokotovic. A decomposition of nearoptimum regulators for systems with slow and fast modes. IEEE Trans. on Automat. Control, 21(5):701 – 705, 1976. F. El Hachemi, M. Sigalotti, and J. Daafouz. Stability of singularly perturbed switched systems. In Proceedings of IEEE American Control Conference, pages 4061–4066, San Francisco, USA, 2011. G. Garcia, J. Daafouz, and J. Bernussou. The infinite time near optimal decentralized regulator problem for singularly perturbed systems: a convex optimization approach. Automatica, 38(8):1397–1406, 2002. J. C. Geromel and P. Colaneri. Stability and stabilization of continuous-time switched linear systems. SIAM J. Contr. Optim., 45(5):1915–1930, 2006. J. C. Geromel, P. Colaneri, and P. Bolzern. Dynamic output feedback feedback control of switched linear systems. IEEE Trans. Automat. Control, 53:720–733, 2008. B. S. Heck and A. H. Haddad. Singular perturbation in piecewise-linear systems. IEEE Trans. Automat. Control, 34(1):87–90, 1989. Petar V. Kokotovic, Hassan K. Khalil, and John O’Reilly. Singular perturbation methods in control: analysis and design. Academic Press Inc., London, 1986. Daniel Liberzon. Switching in systems and control. Birkh¨auser Boston Inc., Boston, MA, 2003. I. Malloci, J. Daafouz, C. Iung, R. Bonidal, and P. Szczepanski. Switched system modeling and robust steering control of the tail end phase in a hot strip mill. Nonlinear Analysis: Hybrid Systems, 3(3):239–250, August 2009. Ivan Malloci, Jamal Daafouz, and Claude Iung. Stabilization of continuous-time singularly perturbed switched systems. In 48th IEEE Conference on Decision and Control, pages 6371–6376, Shanghai, China, 2009. Ivan Malloci, Jamal Daafouz, and Claude Iung. Stability and stabilization of two time scale switched systems in discrete time. IEEE Trans. Automat. Control, 55(6): 1434–1438, 2010. D. Subbaram Naidu. Singular perturbations and time scales in control theory and applications: An overview. Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applciations and algorithms, 9:223–278, 2002. Ricardo G. Sanfelice and Andrew R. Teel. On singular perturbations due to fast actuators in hybrid control systems. Automatica, 47(4):692–701, 2011. R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King. Stability criteria for switched and hybrid systems. SIAM Review, 49(4):545–592, 2007.
H2 Performance Optimization of Singularly Perturbed Switched Linear Systems Grace S. Deaecto ∗ Jamal Daafouz ∗∗ Jose C. Geromel ∗∗∗ ∗
FEEC - UNICAMP, Campinas, SP, Brazil, [email protected] ∗∗ CRAN - CNRS, Institut Universitaire de France, Vandoeuvre-l`es-Nancy, France, [email protected] ∗∗∗ FEEC - UNICAMP, Campinas, SP, Brazil, [email protected] Abstract: This paper addresses the H2 performance problem for singularly perturbed switched linear systems in continuous-time. We propose a new and unified approach for stability analysis and norm computation for singularly perturbed LTI systems and generalize it to the case of singularly perturbed switched linear systems to obtain numerically well conditioned conditions for the design of a state-switching control. Such a switching strategy guarantees that the closed-loop switched system is globally asymptotically stable for sufficiently small values of the perturbation parameter and provides an upper bound to an H2 performance index. We provide a comparison with existing methods. 1. INTRODUCTION Singular perturbation theory is a well known topic for mathematicians, physicists, and other researchers. For control scientists, this setting has been used to model systems with dynamics operating on multiple time scales and also for model reduction purposes (see Kokotovic et al. [1986], Naidu [2002] and references therein). In this context, the simultaneous occurrence of slow and fast phenomena gives rise to time scales and poses difficult computational complexities. As far as time invariant models are considered, a solution is obtained using time scale separation which simplifies the analysis and avoids ill-conditioning by decomposing the problem into two independent well conditioned problems, one for the slow subsystem and another one for the fast subsystem. When the dynamical system is characterized by a hybrid nature, that is, its behavior is the result of the interaction of two different dynamics: discrete and continuous, both phenomena, switching and multiple time scale, may arise simultaneously and classical results may not necessarily hold. For switched systems with arbitrary switching in general we can cite Liberzon [2003] and Shorten et al. [2007]. It has been shown in Malloci et al. [2010] that, stability of the slow and the fast switched subsystems separately does not imply stability of the original switched system for small values of the singular perturbation parameter. In other words, the Tikhonov theorem, cannot be used in its classical form. ⋆ This work was supported in part by the by grants from ”Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico - CNPq”, “Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Capes”, Brazil and by the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n257462 HYCON2 Network of excellence.
978-3-902823-00-7/12/$20.00 © 2012 IFAC
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To emphasize the interest in studying mathematical problems related to singularly perturbed hybrid systems, we mention three classes of problems. The first class has been introduced in Heck and Haddad [1989] where singularly perturbed piecewise systems have been considered and a technique that allows decoupling of such systems into fast and slow subsystems is proposed. The second class has been introduced in Sanfelice and Teel [2011] where a semiglobal practical asymptotic stability result has been proposed for a class of singularly perturbed hybrid systems. The third class concerns singularly perturbed switched systems which can be motivated by several practical applications from power electronics or biological networks. It is also motivated by the steering control for the last phase of the rolling process, called the tail end phase. This application is described in Malloci et al. [2009] where it is shown that a steering control for the tail end phase reduces to a control design for a singularly perturbed switched system. Despite this practical interest, a complete characterization of the stability and performance properties of singularly perturbed hybrid systems and a general singular perturbation theory are missing. Indeed, taking into account the effects of vector field discontinuities and multiple time scales simultaneously for stability, performance analysis and design is a very difficult and challenging task. In this paper, we propose a new approach for stability analysis and norms computations for singularly perturbed LTI systems which can be generalized to the case of singularly perturbed switched linear systems. It is shown that using the proposed approach, we obtain well conditioned conditions for switching control design with performance constraints for a class of hybrid problems where both switching and multiple time scale phenomena arise. The notation throughout is standard. For square matrices Tr(·) denotes the trace function and for a symmetric ma10.3182/20120606-3-NL-3011.00005
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trix λmax (·) is its maximum eigenvalue. For real matrices or vectors (′ ) indicates transpose. For symmetric matrices, the symbol (•) denotes each of its symmetric blocks. The set M is composed by all Metzler matrices with nonnegative off diagonal elements P πji ≥ 0, ∀i 6= j satisfying the normalization constraint j6=i πji = −πii ≥ 0, ∀i. The set of real numbers is denoted as R whereas the set of the N first natural numbers is denoted as K = {1, 2, · · · , N }. The squared norm of a trajectory ξ(t)R defined for all t ≥ 0, ∞ denoted by kξk22 , is equal to kξk22 = 0 ξ(t)′ ξ(t)dt. 2. PROBLEM MOTIVATION AND PRELIMINARIES
Consider a singularly perturbed linear system evolving from the rest with the following state space realization x˙ 1 A11 A12 x1 B1 = + w (1) x˙ 2 A21 /ε A22 /ε x2 B2 x (2) z = [ C1 C2 ] 1 x2
where ε > 0 is a small scalar, x1 ∈ Rn1 and x2 ∈ Rn2 are the slow and fast states, w ∈ Rr is an external disturbing input and z ∈ Rp is the system output. The system matrices have compatible dimensions and A22 is supposed to be nonsingular. It is well known that to avoid numerical difficulties that arise when ε > 0 is small, which is usually the case for singularly perturbed systems, two smaller−order subsystems can be defined using a time-scale decomposition Chow and Kokotovic [1976], Kokotovic et al. [1986]. Namely, the system (1)(2) is decomposed into slow and fast subsystems, where G = A−1 22 A21 is the coupling static gain between them: x˙ s = As xs + Bs w (3) evolving from xs (0) = x1 (0) = 0 with As = A11 − A12 G and Bs = B1 , representing the slow dynamics and x˙ f = (Af /ε) xf + Bf w (4) evolving from xf (0) = Gx1 (0) + x2 (0) = 0 with Af = A22 and Bf = B2 +GB1 , representing the fast one. The system output is given by z = Cs xs + Cf xf (5) with Cs = C1 −C2 G and Cf = C2 . This decomposition can be generalized to switched linear systems evolving from the rest with the state space realization σ σ x1 B1 x˙ 1 A11 Aσ12 + w (6) = x2 B2σ x˙ 2 Aσ21 /ε Aσ22 /ε x z = [ C1σ C2σ ] 1 (7) x2
where w ∈ Rr is an external input and z ∈ Rp is the corresponding controlled output. The switching function is defined as σ(t) : t ≥ 0 −→ K, making clear that at each instant of time t ≥ 0, one and only one among N known linear subsystems corresponding to some i ∈ K is switched on. The switched system (6)-(7) is decoupled in two smaller order subsystems where the coupling static −1 gain is defined as Gi = Ai22 Ai21 , for all i ∈ K. x˙ s = Asσ xs + Bsσ w, xs (0) = 0 (8) i i i with Asi = A11 − A12 Gi and Bsi = B1 , ∀i ∈ K, and x˙ f = (Af σ /ε) xf + Bf σ w, xf (0) = 0 (9)
229
with Af i = Ai22 and Bf i = B2i +Gi B1i , ∀i ∈ K. The system output is given by z = Csσ xs + Cf σ xf (10) with Csi = C1i − C2i Gi and Cf i = C2i , ∀i ∈ K. Hence, for switched linear systems the same time scale decomposition in slow and fast subsystems can be adopted as well. The main interest in performing the previous decomposition is that a separate analysis of the slow and fast subsystems can be done avoiding numerical difficulties and allowing the derivation of an approximated solution which coincides with the direct solution to a full-order problem as ε → 0 (that is for 0 < ε < εmax with εmax a bound to be characterized). For linear systems such as (1)-(2) the validity of this decomposition has been demonstrated for stability and performance properties including H2 problems Kokotovic et al. [1986]. For the first time in the literature, a solution based on convex optimization has been proposed in Garcia et al. [2002] but it is specific to the LQ problem. The characterization and the computation of the bound εmax has attracted a number of researchers since the end of the seventies (see Cao and Schwartz [2004] and references therein), but most of the proposed approaches are restricted to stability analysis. For switched systems, the problem is totally open and there is no complete and satisfactory solution. The case of arbitrary switching has been investigated in Malloci et al. [2009], but as illustrated in El Hachemi et al. [2011] for planar systems, the proposed solution can be conservative. We provide a new convex optimization based approach for H2 norm computation with the determination of εmax . Lemma 1. Consider Φ and Γ two symmetric matrices of compatible dimensions with the four-blocks form Φ1 Φ2 0 Γ2 Φ= , Γ= (11) • Φ3 • Γ3 where Γ3 is nonsingular. There exists εmax > 0 such that the inequality Φ + ε−1 Γ < 0 (12) holds for all ε ∈ (0, εmax ) if and only if the conditions Φ1 < 0, Γ2 = 0 and Γ3 < 0 simultaneously hold. Proof: For the inequality (12) to hold for ε > 0 sufficiently small it is required that Γ ≤ 0. Hence, it is seen that Γ ≤ 0 holds if and only if Γ2 = 0 and Γ3 < 0 (because Γ3 is supposed to be nonsingular). On the other hand, the inequality (12) written as Φ1 Φ2 <0 (13) Φ + ε−1 Γ = • Φ3 + ε−1 Γ3 enables us to conclude that it holds for ε > 0 sufficiently small if and only if Φ1 < 0. Indeed, in this case, applying the Schur complement in the first row and column we get ε−1 (−Γ3 ) > Φ3 − Φ′2 Φ−1 (14) 1 Φ2 which implies that (13) is satisfied whenever ε > 0 is taken sufficiently small and the claim follows. 2 The determination of εmax > 0 such that (14) holds for all ε ∈ (0, εmax ) is simple. From −1/2 βmax = λmax (−Γ3 )−1/2 (Φ3 − Φ′2 Φ−1 1 Φ2 )(−Γ3 ) we get εmax = 1/βmax whenever βmax > 0 and εmax = +∞ otherwise. On the other hand, if we suppose (as usual in the literature of singularly perturbed linear systems) that matrices Φ and Γ depend on ε > 0 and
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limε→0+ Φ(ε) = Φ0 and limε→0+ Γ(ε) = Γ0 exist and are finite then, in many instances, the existence of εmax > 0 such that Φ(ε) + ε−1 Γ(ε) < 0 holds for all ε ∈ (0, εmax ) is assured whenever Φ0 and Γ0 satisfy Lemma 1. In that precise context, the importance of Lemma 1 is apparent since the dependence of matrices Φ and Γ with respect to ε > 0 does not need to be explicit. 3. H2 ANALYSIS The aim is to characterize stability and calculate the H2 norm of system (1)-(2) denoted by S(ε) = (A(ε), B, C) and where A(ε) is defined as A11 A12 A(ε) = (15) A21 /ε A22 /ε through the data of the slow and fast subsystems. In the classical literature, Boyd et al. [1994], it is well known that S(ε) is asymptotically stable and the constraint kS(ε)k22 < γ 2 holds if and only if there exists P > 0 such that Tr(B ′ P B) < γ 2 , A(ε)′ P + P A(ε) + C ′ C < 0 (16) and B ′ = [B1′ B2′ ], C = [C1 C2 ]. The next theorem provides conditions to calculate the H2 norm of system (1)-(2) avoiding numerical problems due to ill conditioning. Theorem 2. Assume there exist X > 0, Z > 0 satisfying As X + XA′s XCs′ < 0 (17) Tr Bs′ X −1 Bs < γs2 , • −I (18) Tr Bf′ Z −1 Bf < γf2 , Af Z + ZA′f < 0
In the affirmative case, there exists εmax > 0 such that the system S(ε) is asymptotically stable and satisfies the H2 norm constraint kS(ε)k22 < γs2 + γf2 for all ε ∈ (0, εmax ). Proof: We start by writing A(ε) as 0 0 A11 A12 −1 A(ε) = +ε A21 A22 0 0 and splitting the inverse P −1 as X V >0 P −1 = • Y
(19)
(20)
Multiplying both sides of inequality (16) by P −1 , it can be written as Φ + ε−1 Γ < 0 with Φ and Γ presenting the structure (11). From Lemma 1, there exists εmax such that the inequality Φ + ε−1 Γ < 0 is satisfied for ε ∈ (0, εmax ) if and only if Φ1 < 0, Γ2 = 0 and Γ3 < 0. Hence, from Γ2 = 0 we get V ′ = −GX which together with (17) verify the condition Φ1 = As X + XA′s + XCs′ Cs X < 0. This is clear after performing the Schur Complement with respect to the last row and column of (17). Finally Γ3 = =
A22 Y + Y A′22 − A21 XG′ Af Z + ZA′f < 0
−
GXA′21 (21)
taking into account the definition Y = Z + GXG′ and (18). This choice is made with no loss of generality since P −1 > 0 whenever X > 0 and Z > 0. Now, it is a matter of simple calculation to verify that −1 X + G′ Z −1 G G′ Z −1 P = (22) • Z −1 enables the conclusion that the system S(ε) is asymptotically stable and yields the equality B ′ P B = Bs′ X −1 Bs + 230
Bf′ Z −1 Bf from which we immediately see that the condition kS(ε)k22 < γs2 + γf2 holds for all ε ∈ (0, εmax ). 2 Consider the maximum value εmax = supε>0,X>0,Z>0 ε subject to A(X, Z) + ε−1 B(Z) < 0 where the matrix functions A(X, Z) and B(Z) are obtained directly from the multiplication of both sides of the Lyapunov inequality (16) by P −1 given in (20), that is As X + XA′s A12 Z − As XG′ A(X, Z) = + • 0 ′ XCs′ XCs′ + (23) ′ ′ ′ ′ ZCf − GXCs ZCf − GXCs 0 0 B(Z) = (24) ′ • Af Z + ZAf
A remarkable property of (23) is that it can be converted into an LMI with respect to the matrix variables X > 0 and Z > 0. It assures that for any γs > 0 and γf > 0 such that Tr(Bs′ X −1 Bs ) < γs2 and Tr(Bf′ Z −1 Bf ) < γf2 , where X > 0 and Z > 0 are the optimal solution that define εmax , then kS(ε)k22 < γs2 + γf2 for all ε ∈ (0, εmax ) and this is the maximum interval of validity of this inequality. Second, it is also possible to determine εmax > 0 (generally smaller than the previous one) by taking scalars γs2 , γf2 and matrices X > 0, Z > 0 satisfying the conditions of Theorem 2 and maximizing ε > 0 exclusively. Finally, from the fact that Af is Hurwitz, we notice that there exists Z −1 → 0 satisfying Af Z + ZA′f < 0. On the other hand, any X > 0 satisfying the LMI constraint (17) is such that Z ∞ ′ −1 X > eAs t Cs′ Cs eAs t dt (25) 0
which enables us to conclude that γs2 > kCs (sI − As )−1 Bs k22 and consequently kCs (sI − As )−1 Bs k22 is the minimum value of γs2 + γf2 preserving feasibility. Based on this discussion a relevant function to be determined is ρ(ω) =
inf
{ρ : Tr(Bs′ X −1 Bs ) +
ρ>0,X>0,Z>0
+Tr(Bf′ Z −1 Bf ) < ρ}
(26)
−1
subject to the LMI A(X, Z)+ω B(Z) < 0 for ω > 0 fixed. The fact that this inequality being feasible for (X, Z, ω) is also feasible for (X, Z, ε) for all 0 < ε ≤ ω allows us to say that ρ(ω) remains an upper bound to the true cost for all ε in this interval. In other words, for each εmax > 0 the associated cost ρ(εmax ) is a valid upper bound to the true singularly perturbed system for all ε ∈ (0, εmax ). 4. SWITCHED LINEAR SYSTEMS The H2 control problems for switched linear systems have been solved in Geromel et al. [2008], using the min-type piecewise positive definite quadratic Lyapunov function v(x) = mini∈K x′ Pi x where Pi > 0, ∀i ∈ K and the state dependent switching function σ(x) = arg min x′ Pi x (27) i∈K
introduced in Geromel and Colaneri [2006]. Accordingly to (6)-(7) we denote i A11 Ai12 (28) Ai (ε) = Ai21 /ε Ai21 /ε
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′
′
Bi′ = [B1i B2i ] and Ci = [C1i C2i ] for all i ∈ K. Our main goal is to obtain conditions such that a state feedback switching rule of type (27) renders the associated closedloop switched linear system (6)-(7) globally asymptotically stable and assures an upper bound for a given H2 performance criterion. This goal is accomplished by adopting the previous min-type Lyapunov function and, consequently, the state dependent switching function (27) where −1 Xi + G′i Zi−1 Gi G′i Zi−1 >0 (29) Pi = • Zi−1 with appropriate matrices Xi > 0 and Zi > 0 for all i ∈ K. Lemma 3. Consider the relative static gains Gij = Gi −Gj and Pi > 0 of the form (29). Any feasible solution of Tij + Xi Xi G′ji Xi (30) • Zj 0 >0 • • Xj is such that ′ I I (31) Tij > (Pi−1 Pj Pi−1 − Pi−1 ) 0 0 for all i 6= j ∈ K × K and vice-versa. Proof: Assuming (30) is feasible, the Schur Complement applied to the last two rows and columns provides Tij > Xi Xj−1 + (Gj − Gi )′ Zj−1 (Gj − Gi ) Xi − Xi ′ Xi Xi > − Xi Pj −Gi Xi −Gi Xi ′ I I > (32) (Pi−1 Pj Pi−1 − Pi−1 ) 0 0 where the second and third inequalities follow from the inverse of Pi given in (29). 2
This result induces a linearization without introducing any kind of conservatism. Consider that for each k = 1, · · · , r an external input of the form w(t) = ek δ(t) ∈ Rr , where ek is the kth column of the identity matrix, is applied to the system (6)-(7) and the corresponding controlled output is denoted as zk ∈ Rp . Then, it is possible to define an H2 functional cost as r X kzk k22 (33) J2 (σ) = k=1
The adopted denomination is done with a little abuse but it is based on the fact that imposing the switching rule σ(·) = i ∈ K fixed for all t ≥ 0, the cost (33) equals the squared H2 norm of the i-th subsystem transfer function from the input w to the output z. It was proven in Geromel et al. [2008] that if there exist Pi > 0, ∀i ∈ K, and a Metzler matrix Π ∈ M satisfying the following LyapunovMetzler inequalities N X ′ (34) πji Pj + Ci ′ Ci < 0 Ai (ε) Pi + Pi Ai (ε) + j=1
for all i ∈ K, the switching rule (27) is globally stabilizing and the following guaranteed cost (35) J2 (σ) < min Tr(Bi ′ Pi Bi ) i∈K
is satisfied, provided that σ(0) = i∗ where i∗ ∈ K is the optimal index solution of problem appearing in the right hand side of (35). 231
Theorem 4. Assume there exist Xi > 0, Zi > 0, matrices Tij and a Metzler matrix Π ∈ M satisfying (30), ′ 2 Tr(Bsi Xi−1 Bsi ) < γsi , Tr(Bf′ i Zi−1 Bf i ) < γf2i and Asi Xi + Xi A′si +
N X
′ πji Tij + Xi Csi Csi Xi < 0 (36)
j6=i=1
Af i Zi + Zi A′f i < 0 (37)
for all i 6= j ∈ K × K. In the affirmative case, there exists εmax > 0 such that the min switching strategy n o ′ σ(x) = min x′1 Xi−1 x1 + x2 + Gi x1 Zi−1 x2 + Gi x1 i∈K
2 is stabilizing and J2 (σ) < mini∈K γsi + γf2i , ∀ε ∈ (0, εmax ).
Proof: Multiplying both sides of (34) by Pi−1 with the fourblocks structure Xi Vi −1 Pi = >0 (38) • Yi and writing Ai (ε) as in (19), we obtain matrix inequalities expressed as Φi + ε−1 Γi < 0. According to Lemma 1 it is satisfied if and only if Φ1i < 0, Γ2i = 0 and Γ3i < 0 for each i ∈ K. Hence, from Γ2i = 0 we have Vi′ = −Gi Xi , which together with (37) assures Γ3i < 0. In addition, Φ1i < Asi Xi + Xi A′si +
N X
′ πji Tij + Xi Csi Csi Xi (39)
j6=i=1
as a direct consequence of Lemma 3. Hence, from (36), the conclusion is that Φ1i < 0 for all i ∈ K. Finally, the switching rule follows directly from (27) and the particular structure (38) of Pi , for all i ∈ K. Moreover, it is simple to ′ verify from (29) that Bi′ Pi Bi = Bsi Xi−1 Bsi + Bf′ i Zi−1 Bf i 2 from which the inequality J2 (σ) < mini∈K γsi + γf2i follows for all ε ∈ (0, εmax ). 2 To find a solution to the matrix inequalities stated in Theorem 4 is not a simple task due to the nonconvexity inherited by the product of variables. There are several possibilities to circumvent this difficulty but always at expense of introducing some conservatism, see Geromel and Colaneri [2006]. One of the simplest is to adopt Π = µΠ0 where Π0 ∈ M and µ ≥ 0. In this case, the matrix inequalities stated in Theorem 4 becomes LMIs whenever µ ≥ 0 is determined by a line search procedure. As a consequence of matrices Af i be Hurwitz for all i ∈ K, matrices Zi−1 → 0 also satisfy Af i Zi + Zi A′f i < 0 implying that, in this particular situation, the H2 guaranteed cost depends only on the slow subsystems. It is clear that, if 2 + γf2i is fixed the right hand side of J2 (σ) < mini∈K γsi close enough to the H2 guaranteed cost, the value of εmax approaches zero. As a consequence, a possible way to increase εmax is by solving the optimization problem ρ(ω) = min
inf
{ρi : Tr(Bsi ′ Xi−1 Bsi ) +
i∈K ρi ,µ>0,Xi >0,Zi >0
+Tr(Bf i ′ Zi−1 Bf i ) < ρi }
(40) −1
subject to Aij (Xi , Xj , Zi , Zj , µ) + ω Bi (Zi ) < 0 for all i 6= j ∈ K × K. These constraints are calculated as before and are expressed through LMIs for ω > 0 and µ > 0 possibly fixed. Moreover, the fact that these inequalities are feasible for ω > 0 means that they also are feasible for
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all ε ∈ (0, ω]. Hence, the cost ρ(ω) remains an upper bound to the true H2 cost for all ε in this interval. That is, for εmax calculated, the associated cost ρ(εmax ) is an upper bound for the true cost J2 (σ) of the singularly perturbed switched linear system (6)-(7) for all ε ∈ (0, εmax ). 5. ARBITRARY SWITCHING LINEAR SYSTEMS The goal of this section is to show how an extension of the proposed approach to the case of arbitrary switching signals outperforms the available results in the literature, in particular Malloci et al. [2009]. Consider a subclass of switching linear systems described by σ x˙ 1 A11 Aσ12 x1 = (41) x˙ 2 A21 /ε A22 /ε x2
evolving from initial condition x(0) = [x′1 (0) x′2 (0)]′ and with matrix A22 nonsingular. The switching function σ(·) : t ≥ 0 → K is arbitrary. We consider two different Lyapunov functions to study the stability of (41), the quadratic function v(x) = x′ P x with P > 0, also used in Malloci et al. [2009], and the min-type piecewise quadratic function. For this special subclass of switching linear systems characterized by presenting the switching function acting only on the slow dynamics, the asymptotic stability of the slow and fast subsystems suffices to assure the stability of the two time-scale switching linear system (41). Using the same procedure adopted for the LTI case, the slow and fast subsystems are x˙ s = Asσ xs with xs (0) = x1 (0), Asi = Ai11 − Ai12 G, ∀i ∈ K and x˙ f = (Af /ε) xf evolving from xf (0) = Gx1 (0) + x2 (0), with Af = A22 . Theorem 5. Assume there exist X > 0, Z > 0 satisfying Asi X + XA′si < 0
(42)
ZA′f
Af Z + <0 (43) for all i ∈ K. There exists εmax > 0 such that ∀ε ∈ (0, εmax ) the system (41) is asymptotically stable for σ(t) ∈ K arbitrary for all t ≥ 0. Proof: The proof follows the same procedure of the one of Theorem 2. Details are omitted. 2 An important point of the proof to be stressed is the use of Lemma 1 in this case. Since P > 0 does not depend on i ∈ K, the condition Γ2i = 0 can be solved if and only if the static gain G is also independent of the index i ∈ K. This condition is fulfilled whenever the matrices related to the fast dynamics are independent of i ∈ K. Now, adopting the min type piecewise quadratic Lyapunov function. The stability conditions for system (41) stems from the existence of symmetric matrices Pi > 0, i ∈ K and a scalar µ > 0 such that the following inequalities Ai (ε)′ Pj + Pj Ai (ε) + µ(Pi − Pj ) < 0 (44) for all i, j ∈ K×K hold. Indeed, suppose that at some time instant t ≥ 0 the switching rule is given by σ(t) = i ∈ K and define the set I(x) = {j ∈ K : v(x) = x′ Pj x}. The Dini derivative of a generic trajectory of system (41) is D+ v(x(t)) = min x(t)′ Ai (ε)′ Pℓ + Pℓ Ai (ε) x(t) ℓ∈I(x(t))
= x(t)′ (A′i (ε)Pj + Pj Ai (ε))x(t) < µx(t)′ (Pj − Pi )x(t) < 0
(45) 232
where the first inequality follows from (44) and the last one is a consequence of x(t)′ Pi x(t) ≥ x(t)′ Pj x(t) since i ∈ K and j ∈ I(x(t)). These conditions contain the quadratic ones, which can be easily verified by setting Pi = Pj = P, i, j ∈ K × K. Theorem 6. Assume there exist matrices Xj > 0, Zj > 0 for all j ∈ K and a scalar µ > 0 satisfying Asi Xj + Xj A′si − µXj µXj <0 (46) • −µXi Af Zj + Zj A′f < 0 (47) for all i, j ∈ K × K. There exists εmax > 0 such that ∀ε ∈ (0, εmax ) the system (41) is asymptotically stable for σ(t) ∈ K arbitrary for all t ≥ 0.
Proof: The proof follows that of Theorem 2.
2
The min-type Lyapunov function associated to the system (41) is constructed with Pi > 0 given by −1 Xi + G′ Zi−1 G G′ Zi−1 Pi = (48) • Zi−1 Moreover, it is important to notice that the conditions of Theorem 6 contain those of quadratic stability presented in Theorem 5. This can be verified by simply imposing Xi = Xj = X and Zi = Zj = Z for all i, j ∈ K × K which indicates that conservativeness is reduced. The value of εmax for which the result of Theorem 6 is valid can be calculated through an optimization problem similar to the previous one where µ > 0 is determined by a line search. The conditions of Theorems 5 and 6 are valid for a subclass of switching linear systems where the fast dynamics are not dependent on the switching function σ(t). Although the result of Malloci et al. [2009] can be applied for more general switching linear systems, including the case where the fast dynamics are arbitrarily dependent on the switching function σ(t), it is more conservative than our results when we restrict the analysis to (41). Indeed, for this subclass, the conditions of Malloci et al. [2009] stem from the existence of matrices Ps > 0, Qsi ≥ 0, Pf > 0 and Qf i ≥ 0 such that the LMIs Asi Ps + Ps A′si + Qsi < 0 Pf A′f
Af Pf + + Qf i < 0 # " ′ Qf i −A22 Y − Pf Ai12 ′ ≥ 0 • Qsi − Ai12 Y − Y ′ Ai12
(49) (50) (51)
are verified for all i ∈ K, with Y = (1−N )GPs . In addition to the conditions of Theorem 5, from inequalities (49) and the second diagonal block of (51), we see that the result of Malloci et al. [2009] includes an additional necessary condition: Ai11 − N Ai12 G should be Hurwitz for all i ∈ K. 6. ILLUSTRATIVE EXAMPLES The first example, inspired from Malloci et al. [2009], is dedicated to the analysis of the behavior of the H2 norm as a function of ω. We consider two systems having the state space realization (1)-(2), that is 0 1 20 10 A11 A12 0 6 5 2 (52) = A21 A22 0 0 0 1 −1.3 −0.46 −3 −5
ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands
3
assumes that the fast dynamics are not dependent on the switching function. When we restrict the analysis to this subclass of switching linear systems, it is shown that existing approaches are more conservative than our results.
45
2
γdB
γdB
40 1
35
0
30 −1
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25 −2
−3 −4
10
−3
10
−2
−1
10
0
10
ω
10
20 −5 10
−4
10
−3
10
−2
10
ω
−1
10
0
10
Fig. 1. H2 norm 0 1 0.7 0 2 −0.3 −0.2 A11 A12 0 = (53) A21 A22 1 5 0 1 151 46.9 −3.8 −6 10 0 0 (54) , C1 = C2 = , B2 = B1 = 01 1 −1 We have solved the problem (26) for ω > 0 belonging to the interval (10−5 , 1). Figure 1 presents the H2 norm of each subsystem in dB as a function of ω. Continuous line represents the solution of problem (26) and dashed line the actual H2 norm obtained from classical theory. Notice that the maximum value of εmax for which the result of Theorem 2 is valid, is εmax = 0.0919 for the first subsystem and εmax = 0.1841 for the second one. As already expected, for ω ≈ 0, the value of the H2 norm obtained from the classical theory and that obtained from Theorem 2 are arbitrarily close and the difference becomes larger when ω increases.
For the second example, let the switching linear system (41) be defined by matrices 1 −6 −1 1 −2 1 (55) A11 A12 = 2 −5 −0.5 −4 2 −10 −1 3 −6 (56) A11 A212 = −4 −5 −1 2 2 1 −5 1 [ A21 A22 ] = (57) 3 5 −3 −2.5 As it can be verified the conditions of Malloci et al. [2009] are not satisfied since matrix A211 − 2A212 G is not Hurwitz. On the other hand, the conditions of Theorem 6 are feasible and yield εmax = 0.5943. Using the approach of Theorem 5, we obtain εmax = 0.5378. This puts in evidence that the conditions of Theorem 6 are valid for a bigger range of ε than that for Theorem 5 and hence the method obtained using a min-type Lyapunov function is strictly less conservative. 7. CONCLUSION It has been shown in the literature that for singularly perturbed switched linear systems, stability of the fast and slow subsystems independently does not necessarily imply stability of the original switched system for small values of the singular perturbation parameter. In this paper, a method that allows decoupling of singularly perturbed switched linear systems into fast and slow subsystems has been proposed. It also provides an upper bound to an H2 performance index, as well as, a theoretical characterization of the maximum singular perturbation for both, linear time invariant and switched linear systems. In the case of arbitrary switching rules, the proposed solution 233
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