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Suboptimal Switching Control Consistency Analysis for Switched Linear Systems José C. Geromel, Grace S. Deaecto and Jamal Daafouz
Abstract This paper introduces the concept of consistency for continuous-time switched linear systems having the switching function as a primary control signal to be designed. A switching control strategy is strictly consistent whenever it improves performance compared to the ones of all isolated subsystems. Conditions under which a min-type switching strategy is strictly consistent for the classes of H2 and H∞ performance indexes are determined. This property makes clear the importance of switching systems control design in both theoretical and practical application frameworks. Moreover, with this property it is not necessary to assume that all the subsystems are not stable in order to make a switching strategy design problem well posed. The theory is illustrated by means of several academic examples.
Index Terms Switched systems, Optimality, Stability, Time varying systems
I. I NTRODUCTION Switched dynamical systems and, in particular, switched linear systems analysis and control design have been the concern of many researchers in the last decades since J. C. Geromel is with FEEC-UNICAMP, Campinas, SP, Brazil. G. S. Deaecto is with Institute of Science and Technology, UNIFESP, São José dos Campos, SP, Brazil. J. Daafouz is with CRAN - CNRS, Institut Universitaire de France, Vandoeuvre-lès-Nancy, France. November 27, 2012
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their introduction by Morse and collaborators in the early 70’s, see the survey papers [3], [12], [13], [15], the interesting and useful books [11], [16] and the references therein. For continuous-time switched linear systems stability analysis many results are available. They put in evidence the important fact that it is possible to orchestrate the subsystems through an adequate switching strategy in order to impose global stability. This property is valid even though all isolated subsystems are unstable, see [5], [7], [12] and [18]. For control design, several results are also available in two main different contexts. In the first one, controllers are designed in order to maintain global stability in the presence of switching viewed as unknown and arbitrary trajectories, see [9] for details. In the other framework the switching function is a control strategy that is used to improve performance, see for instance [2], [6], [10], [14] and [17] among others. In this paper, the concept of consistency is introduced. Roughly speaking, a switching strategy is said to be strictly consistent if it improves performance when compared to all performances produced by the isolated subsystems. Clearly, an optimal switching strategy is consistent, but since it may be difficult to determine, we think that consistency is a valid certificate for suboptimal switching strategies quality. The consistency property clarifies the formulation of a switching strategy design problem in general in the sense that we do not need to assume that all the subsystems are not stable to make the problem well posed. We show in this paper how to design a min-type switching strategy that is strictly consistent as far as H2 and H∞ performance indexes are adopted. For matrices or vectors (′ ) indicates transpose and Tr(·) denotes the trace function. For symmetric matrices, the symbol (•) denotes each of its symmetric blocks. The symbol ≪ (≫) denotes “much less (greater) than”. The set of real numbers is R and K = {1, 2, · · · , N} where N is the number of subsystems. The convex combination of P matrices {Ji : ∀i ∈ K} is denoted by Jλ = j∈K λj Jj where λ ∈ Λ is the unitary P simplex composed by all nonnegative vectors λ ∈ RN such that j∈K λj = 1. The R∞ squared norm of ξ(t), ∀t ≥ 0, denoted by kξk22 , is equal to kξk22 = 0 ξ(t)′ ξ(t)dt. The November 27, 2012
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set of all trajectories such that kξk22 < ∞ is denoted by L2 . II. P ROBLEM
STATEMENT AND PRELIMINARIES
Consider the following switched linear system x(t) ˙ = Aσ x(t) + Hσ w(t)
(1)
z(t) = Eσ x(t) + Gσ w(t)
(2)
evolving from x(0) = 0. The vectors x ∈ Rn , w ∈ Rm , and z ∈ Rr are the state, the exogenous disturbance and the controlled output, respectively. The switching function denoted by σ(t) selects at each time instant t ≥ 0 a subsystem among those belonging to the set K. The state space realization of each subsystem is defined by matrices (Ai , Hi , Ei , Gi ) of appropriate dimensions and its transfer function is denoted by Si (s) = Ei (sI − Ai )−1 Hi + Gi . It is assumed that the differential equation (1) admits a solution in the sense of Filippov, [4]. For a given switching trajectory σ(t), ∀t ≥ 0, two different performance indexes can be associated with system (1)-(2) depending on the class of external perturbation w considered. With a little abuse of notation they are denominated H2 and H∞ performance indexes, defined as: •
H2 performance: For strictly proper subsystems (Gi = 0, ∀i ∈ K), the controlled output z(t) associated with impulsive disturbances of the form w(t) = ek δ(t), where ek ∈ Rm is the k th column of the identity matrix, provides the index J2 (σ) =
m X
kzk k22
(3)
k=1
•
H∞ performance: The controlled output z(t) associated with arbitrary square integrable disturbances w ∈ L2 provides the index J∞ (σ) = sup 06=w∈L2
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kzk22 kwk22
(4)
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Clearly, whenever the switching rule is kept constant, that is, σ(t) = i ∈ K for all t ≥ 0, and matrix Ai , i ∈ K is Hurwitz then the indexes (3) and (4) equal the standard H2 and H∞ squared norms of the i-th subsystem transfer function Si (s), respectively. Let us now define the set S that contains all switching functions of the form σ(t) = g(x(t)) for some g : Rn → K ensuring global asymptotic stability of the origin and the set C that contains only the N time constant policies σ(t) = i ∈ K for all t ≥ 0. Clearly, C is a subset of S if and only if all subsystems are asymptotic stable. To ease the notation, in the sequel, α denotes an element of the set {2, ∞}. Definition 1: A switching strategy σα is consistent with respect to Jα if it belongs to S and guarantees that Jα (σα ) ≤ Jα (σ), ∀σ ∈ C. If this inequality is strict the strategy is said strictly consistent. A switching strategy is consistent whenever it imposes to the switched linear system a performance that is not worse than the one produced by each isolated subsystem. In other words, a strictly consistent switching strategy improves performance. Clearly, it is seen that the optimal state feedback switching strategy provided by σα∗ = arg inf σ∈S Jα (σ) is consistent. Indeed, if σ ∈ C also belongs to S then by definition Jα (σα∗ ) ≤ Jα (σ). On the contrary, if σ ∈ C but it does not belong to S then the same inequality holds because Jα (σ) is unbounded. Since the computation of this optimal strategy is a difficult task, suboptimal solutions are of great interest but only if they are strictly consistent. In this paper, our purpose is to provide suboptimal switching strategies that are strictly consistent with respect to Jα for both α ∈ {2, ∞}. They are formally expressed as σso (t) = arg min x(t)′ Pi x(t) i∈K
(5)
with Pi ∈ Rn×n being positive definite matrices for all i ∈ K that satisfy some conditions to be given afterwards. To this end, we need to introduce the following matrix sets, see [2] and [5]. The first, denoted X2 , is composed by positive definite matrices Pi ∈ Rn×n ,
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i ∈ K and a Metzler matrix Π = {πij } ∈ RN ×N satisfying the so called LyapunovMetzler inequalities
A′i Pi + Pi Ai + Ei
P
j∈K πji Pj
• −I
< 0, i ∈ K
(6)
whereas the second one denoted X∞ is composed by positive definite matrices Pi ∈ Rn×n , i ∈ K, a Metzler matrix Π = {πij } ∈ RN ×N and a positive scalar ρ ∈ R satisfying the so called Riccati-Metzler inequalities P ′ A P • • i + Pi Ai + i j∈K πji Pj Hi′ Pi −ρI • Ei Gi −I
< 0, i ∈ K
(7)
The elements of the Metzler matrices are such that πji ≥ 0, ∀i 6= j ∈ K × K and P j∈K πji = 0, ∀i ∈ K which implies that πii ≤ 0, ∀i ∈ K. Matrices satisfying these
requirements characterize the class M as, for instance, the null matrix Π = 0 and
Π = Θj with null elements except θii = −β and θji = β for all i 6= j ∈ K, for some j ∈ K and β ≥ 0. The feasibility of the sets Xα does not require that all matrices {A1 , · · · , AN } be Hurwitz. Indeed, in [5] it is proven that the inequalities (6) admit a solution whenever there exists λ ∈ Λ such that Aλ is Hurwitz which implies that the convex hull co{A1 , · · · , AN } must contain a Hurwitz matrix. The same holds for the Riccati-Metzler inequalities (7) because they reduce to the previous ones for ρ → +∞. III. C ONSISTENCY This section is devoted to presenting conditions to assure that the switching strategy based on the function (5) is consistent. From now on, it is assumed that Assumption 1: There exists at least one index ℓ ∈ K such that Aℓ is Hurwitz.
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This assures the sets Xα for α ∈ {2, ∞} are not empty. Otherwise Jα (σ) is unbounded for all σ ∈ C and based on the previous discussion strict consistency holds for all switched linear systems for which there exists A ∈ co{A1 , · · · , AN } that is Hurwitz. Theorem 1: Assume Gi = 0 and Hi are full column rank matrices for all i ∈ K. There exist matrices {P1 , · · · , PN , Π} ∈ X2 such that the switching function σso (t) = arg mini∈K x(t)′ Pi x(t) is consistent. Proof: Consider the switched system (1)-(2) with zero input w(t) = 0 and x(0) = x0 . Since X2 is not empty take any {P1 , · · · , PN , Π} ∈ X2 and adopt the Lyapunov function v(x) = mini∈K x′ Pi x. Following [5] it is seen that kzk22 < v(x0 ) = mini∈K x′0 Pi x0 . Hence, applying this to the switched linear system (1)-(2) with successive inputs w(t) = ek δ(t) and zero initial condition, using (3) we get J2 (σso )
0
(9)
holds. Taking the Metzler matrix Π = Θℓ , the constraints that define the set X2 are rewritten as
A′i Pi
+ Pi Ai + β(Pℓ − Pi ) Ei
• −I
< 0, i ∈ K
(10)
making simple to verify that for β > 0 large enough, matrices Pℓ = Q > 0 and Pi > Q > 0 arbitrary but fixed for i 6= ℓ ∈ K yield {P1 , · · · , PN , Π} ∈ X2 . Moreover, choosing Pi such that Hi′ Pi Hi > Hℓ′ QHℓ for all i 6= ℓ ∈ K which is always possible November 27, 2012
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due to the full column rank assumption, from (8) we obtain J2 (σso ) < min Tr(Hi′ Pi Hi ) i∈K
≤ inf {Tr(Hℓ′ QHℓ ) : A′ℓ Q + QAℓ + Eℓ′ Eℓ < 0} Q>0
≤ kSℓ (s)k22
(11)
where the last inequality above follows from (9). Hence, adopting the same reasoning for each ℓ ∈ K with Aℓ Hurwitz, we conclude that J2 (σso ) ≤ J2 (σ) for all σ ∈ C. Theorem 1 with Π = Θℓ puts in evidence that this feasible solution is well adapted to prove consistency but not strict consistency. Indeed, since Pi > Pℓ for all i 6= ℓ ∈ K the switching strategy (5) provides σso (t) = arg mini∈K x(t)′ Pi x(t) = ℓ for all t ≥ 0. This implies that no switching occurs and the H2 performance equals the one of the ℓ-subsystem. To get a strictly consistent strategy we need to go further and search for a solution in the entire set X2 , that is, by solving J2so =
inf
min Tr(Hi′ Pi Hi )
{P1 ,··· ,PN ,Π}∈X2 i∈K
(12)
and use the optimal solution Pi , ∀i ∈ K to build the switching strategy (5). This important feature will be fully addressed in the next section. For the moment, notice that problem (12) is nonconvex and may be difficult to solve directly (numerical methods based on polynomial optimization may be required). Finally, from (8), the cost J2so is the minimum upper bound to the true value J2 (σso ) which allows us to say that, in general, we will have J2 (σso ) ≪ J2so . These aspects are illustrated by means of the next simple example where the switched linear system (1)-(2) is given by 0 1 0 1 0 , A2 = , H1 = H2 = A1 = −2 −9 −2 −2 10
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(13)
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and E1 = [1 0], E2 = [0 1]. Since the matrices A1 and A2 are Hurwitz, it is immediate to verify that the best performance that we can obtain with a constant strategy is by adopting σ(t) = 1, ∀t ≥ 0 with the associated cost J2 (σ) = minℓ∈{1,2} kSℓ (s)k22 = 2.7778. Since N = 2, the Metzler matrix Π ∈ R2×2 , written as −p q Π= p −q
(14)
enables us to solve problem (12) by gridding the objective function for all (p, q) in the box [0, 2] × [0, 2]. The global optimum provides the pair (p, q) = (0.45, 0.00) and J2so = 2.1929. Finally, from time simulation the true cost J2 (σso ) = 1.6357 is calculated. This confirms that the σso strategy is strictly consistent and presents a cost 40% smaller than the best cost produced by any constant strategy σ ∈ C. This is a significative improvement due, exclusively, to switching. Theorem 2: Assume Gi = G for all i ∈ K. There exist matrices {P1 , · · · , PN , Π} ∈ X∞ such that the switching function σso (t) = arg mini∈K x(t)′ Pi x(t) is consistent. Proof: Consider the switched linear system (1)-(2) with zero initial condition x(0) = 0. Since X∞ is not empty take any {P1 , · · · , PN , Π, ρ} ∈ X∞ and adopt the Lyapunov function v(x) = mini∈K x′ Pi x. Following the same reasoning as in [2] we conclude that kzk22 − ρkwk22 < 0 for all w ∈ L2 . Hence, we get J∞ (σso ) < ρ. As before, from Assumption 1, matrix Aℓ is Hurwitz for some ℓ ∈ K and consequently, with the Metzler matrix Π = Θℓ , the constraints that define the set X∞ are rewritten as ′ • Ai Pi + Pi Ai + β(Pℓ − Pi ) • Hi′ Pi −ρI • < 0, i ∈ K Ei Gi −I
(15)
and we verify that for β > 0 large enough, matrices Pℓ = Q > 0, Pi > Q > 0 arbitrary but fixed for i 6= ℓ ∈ K the inequalities (15) reduce to the one for i = ℓ and to ρI > G′i Gi November 27, 2012
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for i 6= ℓ ∈ K which are redundant provided that Gi = G for all i ∈ K. As a result, since {P1 , · · · , PN , Π, ρ} ∈ X∞ , we get J∞ (σso ) ≤ kSℓ (s)k2∞ . This being true for all ℓ ∈ K such that Aℓ is Hurwitz allows the conclusion that J∞ (σso ) ≤ J∞ (σ) for all σ ∈ C. The same remarks valid for the H2 performance index are also valid for H∞ performance. In particular, to search for a strictly consistent strategy one has to solve so J∞ =
inf
{P1 ,··· ,PN ,Π,ρ}∈X∞
ρ
(16)
and use the optimal matrices Pi , ∀i ∈ K to build the switching function (5). This feature will be treated in the next section. The Riccati-Metzler inequalities are difficult to handle due to the nonconvexity inherited by the product of variables. This fact certainly makes so more demanding the solution of problem (16). Since J∞ is still only an upper bound, so it can be expected that the true value of the cost satisfies J∞ (σso ) ≪ J∞ .
Following the proofs of Theorem 1 and Theorem 2 an important difference between them becomes apparent. Suppose all isolated subsystems are asymptotically stable, choose the Metzler matrix Π = 0 which decouples the inequalities that define the set X2 and determine the associated cost, yielding J2 (σso ) < min inf {Tr(Hi′Pi Hi ) : A′i Pi + Pi Ai + Ei′ Ei < 0} i∈K Pi >0
≤ min kSi (s)k22 i∈K
(17)
where it is to be noticed that the optimal Pi > 0 is the observability gramian of the i-th subsystem and consistency of this particular switching strategy holds. Unfortunately, the same reasoning is not valid for H∞ performance. The main reason is that for Π = 0 the inequalities defining X∞ are not decoupled (because the scalar variable ρ does not
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depend on i ∈ K) and the associated cost ′ • • Ai Pi + Pi Ai J∞ (σso ) < inf ρ : Hi′ Pi −ρI • Pi >0,ρ Ei G −I ≤ max kSi (s)k2∞
< 0, i ∈ K
i∈K
(18)
can not be used to establish consistency. These aspects are now illustrated by the following example where the subsystems are defined as before and G1 = G2 = G = 1. Both isolated subsystems are asymptotically stable and present approximatively the same H∞ norm. With the constant strategy σ(t) = 2, ∀t ≥ 0 we get the associated cost J∞ (σ) = minℓ∈{1,2} kSℓ (s)k2∞ = 35.9356. Adopting the Metzler matrix as in (14) and searching a solution of problem (16) in the box [0, 5] × [0, 5] we found so (p, q) = (5.00, 4.50) and the corresponding cost J∞ = 18.0677 which represents almost
50% of cost reduction. From Theorem 2, the switching strategy σso is strictly consistent so and its actual cost is even smaller than J∞ . Even though we can not guarantee that the
solution found inside that box is the global optimum, it improves H∞ performance. IV. S TRICT
CONSISTENCY CERTIFICATION
In this section, we investigate strict consistency. For α ∈ {2, ∞}, the relevant issue is to quantify the performance improvement due to a switching strategy σ ∈ S defined as ∆α = minσ∈C Jα (σ) − Jα (σso ). However, using the fact that the suboptimal switching σso satisfies Jα (σso ) ≤ Jαso , we will evaluate the more conservative performance improvement given by δα = minσ∈C Jα (σ) − Jαso , which clearly satisfies ∆α ≥ δα . Since ∆α is difficult to determine, the value of δα is used to verify consistency. If δα > 0 strict consistency is assured. On the other hand if δα = 0 the switching strategy σso still may be strictly consistent if ∆α > 0. It is also possible to have ∆α > δα > 0 but with these values being arbitrarily close to zero. In this case, once again, no performance November 27, 2012
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improvement occurs, see [8]. Hence, the evaluation of δα is an important certification for a given sub-optimal switching strategy. Remembering that the set M is composed by all Metzler matrices with nonnegative off diagonal elements πji ≥ 0, ∀i 6= j ∈ K × K satisfying the normalization constraint P πii = − j6=i∈K πji ≤ 0, ∀i ∈ K, from the Gershgorin Circle Theorem it can be seen
that each eigenvalue of Π ∈ M is located inside a circle with center (πii , 0) and radius P j6=i∈K |πji | = |πii | for all i ∈ K. Hence, all eigenvalues belong to the closed left right hand side of the complex plane. On the other hand, the Frobenius-Perron Theorem
indicates that the null eigenvalue exhibits the maximum real part and associated with it we have e′ Π = 0, where e′ = [1 1 · · · 1] ∈ RN and Πλ = 0 for some λ ∈ Λ. This class of Metzler matrices has an important converse property: Given any λ ∈ Λ, the matrix Π = −I + λe′ ∈ M and Πλ = 0. Furthermore, for given symmetric matrices P Rj , ∀j ∈ K and λ ∈ Λ such that Rλ = j∈K λj Rj = 0, it is always possible to find Π ∈ M and positive definite matrices Wj , ∀j ∈ K such that Πλ = 0 and Ri =
X
πji Wj , ∀i ∈ K
(19)
j∈K
Indeed, for Π = −I +λe′ this equality constraints reduce to Ri =
P
j∈K
λj Wj −Wi , ∀i ∈
K, which admit the solution Wi = WN + (RN − Ri ), ∀i 6= N ∈ K with WN arbitrary. Taking WN > 0 large enough, we get Wi > 0, ∀i ∈ K. For simplicity and comparison purposes with the recent results in [1], it is assumed that each subsystem has the state space representation defined by matrices (Ai , H, E, G), for all i ∈ K, there exists λ ∈ Λ such that Aλ is Hurwitz and Sλ (s) is the transfer function of the system (Aλ , H, E, G). Theorem 3: The switching strategy σso provided by the optimal solution of problem (12) or problem (16) is such that δα ≥ mini∈K kSi (s)k2α − minλ∈Λ kSλ (s)k2α . Proof: Since the proof of both cases have almost the same pattern, only the proof for α = 2 is presented. The claim follows if J2so ≤ minλ∈Λ kSλ (s)k22 . To this end, we
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search a feasible solution in X2 of the form Pi = P + µ−1 Wi with P > 0, Wi > 0 for all i ∈ K, µ > 0 and a Metzler matrix Π = µ(−I + λe′ ) ∈ M with λ ∈ Λ. Since X
πji Pj =
j∈K
X
λj Wj − Wi , ∀i ∈ K
(20)
j∈K
the previous discussion assures that this solution is feasible whenever µ > 0 is taken large enough and
A′i P
+ P Ai + Ri E
• −I
0 must satisfy the inequality A′λ P + P Aλ + E ′ E < 0. Finally, the objective function of problem (12) yields J2so ≤ min Tr H ′ P + µ−1 Wi H i∈K
≤ Tr(H ′ P H) + µ−1 min Tr(H ′ Wi H) i∈K
(22)
where the right hand side approaches to Tr(H ′ P H) as µ goes to infinity. As a consequence, the limit solution obtained by making µ → ∞ is such that J2so ≤ inf {Tr(H ′ P H) : A′λ P + P Aλ + E ′ E < 0} P >0
≤ kSλ (s)k22
(23)
which holds for all λ ∈ Λ with Aλ Hurwitz, then the claim follows. This result deserves some remarks. The first remark is related to the fact that the lower bound on δα has been obtained from a suboptimal solution of problem (12). Even though, in general, we have δ2 ≫ 0, unless in some pathological cases, the most evident being characterized by equal subsystems Ai = A, ∀i ∈ K for which switching obviously makes no sense. The second one concerns the switching strategy associated with the limit
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solution provided in the proof of Theorem 3. From (5) we obtain bl σso (t) = arg min x(t)′ P + µ−1 Wi x(t) i∈K
= arg min x(t)′ (WN + (RN − Ri )) x(t) i∈K
= arg min x(t)′ (A′i P + P Ai ) x(t)
(24)
i∈K
which is exactly the switching rule proposed in [1]. This fact allows the conclusion that the design procedure proposed in [1] yields to a particular feasible solution of problem (12). Hence, solving it we expect to get a better switching strategy as far as performance improvement is concerned. In order to illustrate the result of Theorem 3, we have considered the switched linear system (1)-(2) defined by matrices 1 0 1 0 0 0 0 A1 = 0 0 1 , A2 = 0 0 1 , H = 0 −1 −2 −5 −9 −9 −1.5 5
and E = [1 0.5 1], G = 0. It has been verified that matrix Aλ is Hurwitz for all λ ∈ Λ. Taking into account that the subsystems are Hurwitz, it is seen that the best performance considering a constant strategy is σ(t) = 1, ∀t ≥ 0 with the associated cost J2 (σ) = minℓ∈{1,2} kSℓ k22 = 7.2917. For this switched system minλ∈Λ kSλ (s)k22 = 3.3834 occurs at λ∗ = [0.752 0.248]′ and, consequently, from Theorem 3, we have δ2 ≥ 3.9083 which implies strict consistency. Adopting the Metzler matrix Π ∈ R2×2 given in (14), solving problem (12) by gridding the objective function for all (p, q) in the box [0, 500] × [0, 500], we have obtained the plot indicated in Figure 1 where the plane surface represents the value minσ∈C J2 (σ). The optimal solution obtained for (p, q) large enough is J2so = minλ∈Λ kSλ (s)k22 = 3.3834, while restricting the search in the previous box, we have obtained J2so = 3.3932 for (p, q) = (160, 490). By numerical simulation and considering the same Metzler matrix, we have calculated the actual cost J(σso ) = 2.7842
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25 20 15 10 5 0 0
0 100
100
200
200 300
300 400
400 500
500
p Figure 1.
q
J2so cost against the elements (p, q) of the Metzler matrix.
associated with the switching strategy (5), which implies the performance improvement ∆2 = 4.5075. V. C ONCLUSION In this paper, the concept of consistency for switched linear systems has been introduced. It is used to construct a quality certificate for suboptimal solutions of problems involving H2 and H∞ performance indexes. It has been shown how to construct min-type switching strategies that are, in general, strictly consistent with respect to the mentioned indexes. The difficulty of computing a strictly consistent switching strategy stems from the need to solve a nonconvex problem with a particular structure. This feature may be relevant for the development of efficient algorithms to cope with this class of optimization problems. Theoretical results have been illustrated by numerical examples. R EFERENCES [1] F. Blanchini, D. Casagrande, P. Gardonio, and S. Miani, “On optimal damping of vibrating structures”, Preprints of the 18th IFAC World Congress, pp. 10268–10273, 2011. [2] G. S. Deaecto, and J. C. Geromel, “H∞ control for continuous-time switched linear systems”, ASME Journal of Dyn. Syst., Measur., and Contr., vol. 132, number 041013, 2010.
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