Stability and performance of discrete-time switched linear systems

Systems & Control Letters 118 (2018) 1–7

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Stability and performance of discrete-time switched linear systems✩ Grace S. Deaecto a, *, José C. Geromel b a b

DMC/School of Mechanical Engineering, UNICAMP, 13083-860, Campinas, SP, Brazil DCA/School of Electrical and Computer Engineering, UNICAMP, 13083 - 852, Campinas, SP, Brazil

article

info

Article history: Received 24 April 2017 Received in revised form 29 January 2018 Accepted 15 May 2018

Keywords: Switched linear systems Stability analysis H2 control design

a b s t r a c t This paper deals with stability analysis and control design of discrete-time switched linear systems. The results are based on a new sufficient condition for exponential stabilizability. A performance index that falls in the context of H2 norm is considered in order to optimize the joint design of a state dependent switching function and a state feedback control law. All control design conditions are expressed through linear matrix inequalities (LMIs). Comparisons with other available design procedures are made by means of examples borrowed from the literature. The present procedure is more amenable for control synthesis purposes and simpler from both theoretical and numerical viewpoints. Three state feedback control switching strategies are presented. The complexity of the control law is discussed. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Switched systems have been widely studied in the last decades due to their intrinsic importance from both theoretical and practical viewpoints. The books [1], and [2] contain complete and useful results in the general framework of switched systems analysis and control design. In addition, the survey papers [3,4] and [5] and the references therein have put in evidence problems and new theoretical aspects that due to the lack of a complete solution needed to be further considered. In this general context, a switched system can be viewed as a dynamical system constituted of a number of subsystems and a rule that orchestrates the switching among them, in other words, a logical strategy that decides the activation of a specific subsystem at each instant of time. There are two main classes characterized by the nature of the switching rule. Indeed, it can be a perturbation or a control variable to be designed with specific goals of stability and performance enhancement. Several contributions on both classes are available in the literature. For the first class Refs. [6,7] provide sufficient conditions for stability based on multiple Lyapunov functions assuming arbitrary and bounded dwell time perturbations. This path was completely characterized by the necessary and sufficient conditions for stability and performance optimization provided in [8] and [9]. The main characteristic of these results is that they are well adapted ✩ This research was supported by the Brazilian National Research Council - CNPq (Grant numbers 303887/2014-1 and 306911/2015-9) and by São Paulo Research Foundation - FAPESP (Grant number 2017/20343-0). Corresponding author. E-mail addresses: [email protected] (G.S. Deaecto), [email protected] (J.C. Geromel).

*

https://doi.org/10.1016/j.sysconle.2018.05.006 0167-6911/© 2018 Elsevier B.V. All rights reserved.

for control systems purpose since they are expressed in terms of LMIs [10]. For the second class, almost the same has occurred. Sufficient conditions for the existence of a stabilizing switching rule (state or output dependent) have been proposed in Refs. [11] for switched affine systems, [12] for sampled-data switched linear systems, [13,14], and [15] for switched linear systems in discretetime, among many others. It is worth mentioning the importance of the sufficient conditions proposed in [15] where the role of the convex combination has been pointed out for the first time in the context of discrete-time switched linear systems. Naturally, in this vein, necessary and sufficient conditions for stabilizability [16] and [17] and LQR performance optimization [18–20], and [21] also appeared. This paper proposes new sufficient conditions for exponential stabilizability of discrete-time switched linear systems based on a quadratic but time-varying Lyapunov function. The conditions are expressed by linear matrix inequalities and, therefore, are simpler to solve than others available in the literature. The advantage of the present stabilizability conditions when compared to the cited ones is that, due to convexity, they apply to switching function and state feedback control joint design and can be generalized to cope with output feedback control and filtering. Moreover, on the contrary of [13], our approach is able to deal with performance optimization, being well adapted to handle H2 and H∞ indexes, see [22] for details. The notation used throughout is standard. For square matrices, tr(·) denotes the trace function. For real vectors or matrices, (′ ) refers to their transpose. The symbols R and N denote the sets of real and natural numbers, respectively. For any real symmetric matrix, X > 0 (X ≥ 0) denotes a positive (semi)definite matrix. The set composed of the N first positive natural numbers is denoted by

2

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7

K = {1, . . . , N }. The unit simplex composed by all nonnegative ∑ vectors λ ∈ RN such that j∈K λj = 1 is denoted by Λ. The ith column of any identity matrix is denoted square norm of ∑ by ei . The 2 2 a trajectory z(n), n ∈ N is ∥z ∥22 = n∈N ∥z(n)∥ where ∥z(n)∥ = z(n)′ z(n) is the square of the Euclidean norm. The greatest integer less or equal to a is ⌊a⌋. A square matrix is said Schur stable if its eigenvalues belong to the open region |z | < 1 of the complex plane.

systems. It is well-defined whenever the switching function makes the closed-loop switched system exponentially stable, that is σ ∈ V. Definition 3. The H2 performance index associated with the closed-loop switched linear system (1)–(2) with initial condition x(−1) = 0 is given by J(σ ) =

2. Problem statement

x(n + 1) = Aσ x(n) + Eσ w (n)

(2)

where x(·) ∈ R , w (·) ∈ R and z(·) ∈ R , defined for all n ∈ N− = N ∪ {−1}, are the state, the exogenous input, and the controlled output, respectively. It is assumed that (1)–(2) evolves, for all n ∈ N− , from null1 initial condition x(−1) = 0. The control action is accomplished by means of a switching function σ (n) : N− → K which, in some instances, may be state dependent, that is, σ (n) = v (x(n)) for some function v (·) : Rnx → K. For a given performance index, the goal is to determine a suboptimal switching function to the control design problem inf J(σ )

(4)

where zr (n), n ∈ N− , is the output of the system corresponding to the impulsive input w (n) = er δ (n + 1), for all r = 1, . . . , nw . (1)

z(n) = Cσ x(n)

σ ∈V

∥zr ∥22

r =1

Consider a discrete-time switched linear system

nx

nw ∑

nw

nz

(3)

where V is the set of all switching policies assuring that the closedloop system (1)–(2) is globally stable. The following definition, presented in [13], will be useful in the sequel. Definition 1. The switched linear system x(n + 1) = Aσ x(n) is exponentially stabilizable if there exist constants c ≥ 0, 0 ≤ µ < 1 and for each x(0) ∈ Rnx a switching trajectory {σ (n)}n∈N such that ∥x(n)∥ ≤ c µn ∥x(0)∥ for all n ∈ N. It is important to keep in mind that in the context of exponential stability different switching functions {σ (n)}n∈N can be adopted for different initial conditions x(0) ∈ Rnx . The next definition is more stringent by imposing only one switching trajectory for all initial conditions. Definition 2. The switched linear system x(n + 1) = Aσ x(n) is Schur stabilizable if there exist constants c ≥ 0, 0 ≤ µ < 1 and a switching trajectory {σ (n)}n∈N such that ∥x(n)∥ ≤ c µn ∥x(0)∥ for all x(0) ∈ Rnx and all n ∈ N. If Schur stabilizability holds for some {σ (n)}n∈N then it also holds for the periodic switching law {σp (n)}n∈N with period κ ∈ N large enough, such that c µκ < 1 and σp (n) = σ (n) for n ∈ [0, κ ). Hence, in this case, periodic stabilizability as defined in [13] also holds. The validity of Definition 2 can be tested by x(n + 1) = Aσ x(n) with a periodic switching law. Consequently, Schur stabilizability holds if and only if there exists 0 < κ ∈ N finite such that for some {σ (n)}n=0,...,κ−1 the matrix product A(0,κ−1) = Aσ (κ−1) · · · Aσ (1) Aσ (0) is Schur stable. The fact that κ ≥ 1 may be taken arbitrarily large (but finite) puts in evidence that although not equivalent both stability concepts provided by Definitions 1 and 2 are closely related. We are now in position to introduce a performance index that is similar to the celebrated H2 norm of linear time-invariant 1 It is simple to verify that the effect of an impulsive input at n = −1 together with some given initial condition x(−1) and σ (−1) can be converted to an initial condition x(0). This adjustment is adopted with no loss of generality only to ease presentation. This is necessary because the system (1)–(2) is time-varying, see [14].

Simple algebraic manipulations put in evidence that this performance index can alternatively be calculated from (1)–(2) with null exogenous input w (n) = 0, ∀n ∈ N, and initial conditions x(0) = Eσ (−1) er for all r = 1, . . . , nw . As it was already commented, it is important to keep in mind that in Definition 3, the initial condition as well as the impulse perturbation have been displaced by one time interval in order to maintain this property true so as the performance index becomes simpler to be determined. It is immediate to see that the LQR index considered in [20] and [21] can be recast in the context of Definition 3. 2.1. Illustrative example 1 This is an example borrowed from Ref. [7] used to illustrate some relevant aspects of the system under consideration. It consists of two second order subsystems defined by matrices Ai = eAci T , i = 1, 2, with T = 0.1 and

[

Ac1

0 = 2

]

[

1 0 , Ac2 = −9 −2

]

1 . 8

Matrices A1 and A2 are not Schur stable but the matrix product A(0,1) = A2 A1 of length κ = 2 is. The eigenvalues of A′(0,1) A(0,1) are {0.7586, 1.0792}. This means that the constraint A′(0,1) S A(0,1) < S is satisfied for some matrix S > 0 but it does not admit S = I ′ as a solution. The first integer ℓ ∈ N such that Aℓ(0,1) Aℓ(0,1) < I is ′ ℓ = 148. Hence, there exists A(0,295) such that A(0,295) A(0,295) < I which is a stability test similar to the one proposed in [13]. In this example, the effect of matrix S is very important to reduce the length of the matrix product. 3. Switching control design In this section, a switched linear system with null exogenous input w (n) = 0, ∀n ∈ N, and arbitrary initial condition x(0) = x0 ∈ Rnx is considered. In this case, the switched linear system (1)–(2) reduces to x(n + 1) = Aσ x(n), x(0) = x0 z(n) = Cσ x(n)

(5) (6)

for all n ∈ N. For a given κ ≥ 1, the sequence {mn }n∈N with generic term mn = κ⌊n/κ⌋ indicates the index of the first element of the (⌊n/κ⌋ + 1)th subsequence of length κ in the interval [0, n] for all n ∈ N. It is used to establish a sufficient condition that yields a switching function assuring exponential stability and norm bounded performance. 3.1. Stabilizing control design The stability analysis and control design conditions to be presented afterwards are based on the next theorem and corollary that constitute the main results of this paper.

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7

Theorem 1. Assume that there exist a positive integer κ ∈ N, vectors λ(n) ∈ Λ ⊂ RN , positive definite matrices P(n) ∈ Rnx ×nx and a positive definite matrix S ∈ Rnx ×nx satisfying ∑ (7) λi (n)A′i P(n + 1)Ai < P(n) i∈K

for all n = 0, 1, . . . , κ − 1 with the boundary conditions P(κ ) = P(0) = S > 0. Then, the switched linear system (5) is exponentially stable with the state dependent switching function σ (n) = v (x(n)) where

v (ξ ) = arg min ξ ′ A′i P(n + 1 − mn )Ai ξ . i∈K

This corollary has an important consequence as far as switched linear systems stabilizability is concerned. It can be verified that the discrete-time example given in [23] is such that A(0,3) = A2 A1 A1 A2 = 0 and hence it is Schur stabilizable. On the contrary of Proposition 3.1 of [23] this system admits a convex time-varying Lyapunov function V (ξ , n) since the conditions of Theorem 1 yield a stabilizing switching function. The time-varying case was not treated in [23] and so the identification of this important theoretical aspect is an additional contribution of this paper. 3.2. Norm bounded control design

(8)

Proof. Consider x(n) ̸ = 0, assume that (7) holds for some positive κ ∈ N and define a quadratic time-varying Lyapunov function candidate of the form V (ξ , n) = ξ ′ P(n)ξ . In the time interval n ∈ [0, κ ), the inequality (7) together with the switching function σ (n) = v (x(n)) given in (8), yield V (x(n), n) = x(n)′ P(n)x(n)

>

3

The next theorem shows how to determine an upper bound to the norm of the output trajectory, namely ∥z ∥22 . This result constitutes the theoretical basis for the evaluation of the performance index of interest previously defined. We believe that this result can also be applied to other indexes that depend on the norm of input and/or output trajectories as for instance, those related to H∞ theory.

i∈K

Theorem 2. Assume that there exist a positive integer κ ∈ N, vectors λ(n) ∈ Λ ⊂ RN , positive definite matrices P(n) ∈ Rnx ×nx and a positive definite matrix S ∈ Rnx ×nx satisfying

i∈K

∑

∑

λi (n)V (Ai x(n), n + 1)

≥ min V (Ai x(n), n + 1) = V (Aσ x(n), n + 1)

(9)

where we have used the fact that λ(n) ∈ Λ. This means that there exists ε > 0 small enough such that V (x(n + 1), n + 1) ≤ (1 − ε )V (x(n), n)

(10)

holds for all n ∈ [0, κ ). Now, taking into account the boundary conditions P(κ ) = P(0) = S > 0 and the periodic continuation of P(n) = P(n − mn ) the conclusion is by one hand that P(n) > 0 for all n ∈ N and by the other hand that the inequality V (x(n + 1), n + 1) ≤ µ2 V (x(n), n)

(11)

for all n ∈ N and for some µ2 = 1 − ε ∈ (0, 1) holds. By consequence, the quadratic time-varying function V (ξ , n) is a Lyapunov function associated with the switched linear system (5). Applying (11) recursively it follows that the inequality V (x(n), n) ≤ µ2n V (x(0), 0) holds for all n ∈ N and, consequently, with c 2 = 1/minn,m∈[0,κ )×[0,κ ) ∥P(n)∥∥P(m)−1 ∥ Definition 1 is verified. □ It is of great interest to determine under which conditions the inequalities (7) admit a solution. This matter is treated in the following corollary that goes beyond the result of [13]. Corollary 1. If the switched linear system (5) is Schur stabilizable then the conditions of Theorem 1 are feasible.

( ) λi (n) A′i P(n + 1)Ai + Ci′ Ci < P(n)

for all n = 0, 1, . . . , κ − 1 with the boundary conditions P(κ ) = P(0) = S > 0. Then, the switched linear system (5)–(6) is exponentially stable and ∥z ∥22 < x′0 Sx0 , for any x0 ̸ = 0, with the state dependent switching function σ (n) = v (x(n)) where

( ) v (ξ ) = arg min ξ ′ A′i P(n + 1 − mn )Ai + Ci′ Ci ξ .

S > A(0,κ−1) S A(0,κ−1) + ε0 I ′

(12)

with ε0 > 0. Hence, choosing ε0 > 0, . . . , ε(κ−1) > 0 accordingly, simple calculations show that P(κ ) = P(0) = S and P(j) = A′(j,κ−1) S A(j,κ−1) + εj I

(13)

for j = κ − 1, . . . , 1 satisfy inequalities (7) whenever λi (n) ∈ Λ is chosen such as λi (n) = 1 if i = σ (n) and λi (n) = 0 otherwise, for each n = 0, . . . , κ − 1. □

(15)

i∈K

Proof. The proof has the same pattern as that of Theorem 1. It is clear that (14) together with the switching function (15) imply that the switched linear system under consideration is exponentially stable. Moreover the quadratic time-varying function V (ξ , n) = ξ ′ P(n)ξ and the periodic continuation provides V (x(n + 1), n + 1) < V (x(n), n) − z(n)′ z(n)

(16)

for all n ∈ N. Hence, the summation for all n ∈ N implies that ∥z ∥22 < V (x0 , 0) = x′0 Sx0 whenever 0 ̸= x0 ∈ Rnx , proving thus the claim. □ This result has some implications that can be analyzed in the context of Bellman’s Principle of Optimality. Let us denote V (x, n) a generic cost-to-go function from the state x(n) = x ∈ Rnx at time n ∈ N to n = +∞. The dynamic programming recursive inequality V (x, n) ≥ min x′ Cσ′ Cσ x + V (Aσ x, n + 1)

{

Proof. Suppose that the switched linear system (5) satisfies Definition 2. In this case, there exists κ ≥ 1 such that A(0,κ−1) is Schur stable which implies that there exists S > 0 satisfying the Lyapunov inequality

(14)

i∈K

}

σ ∈K

= min

∑

λ(n)∈Λ

( ) λi (n) x′ Ci′ Ci x + V (Ai x, n + 1)

(17)

i∈K

characterizes a suboptimal solution to problem (3) with J(σ ) = ∥z ∥22 . Moreover, optimality is attained whenever (17) holds with equality. Inequality (14) is just (17) for V (ξ , n) = ξ ′ P(n)ξ and some λ(n) ∈ Λ and n ∈ [0, κ ). The dynamic programming recursive inequality (17) for a time period of finite length κ ≥ 1 becomes V (x, ℓκ ) ≥ min

σ (·)∈K

{ℓκ+κ−1 ∑

} z(k) z(k) + V (x(ℓκ + κ ), ℓκ + κ ) ′

(18)

k=ℓκ

for all ℓ ∈ N. Taking into account the initial and final boundary conditions, imposing the quadratic positive definite function

4

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7

ϑ (ξ ) = ξ ′ S ξ , it follows that V (ξ , ℓκ ) = ϑ (ξ ) for all ℓ ∈ N in which

being appropriate to measure the complexity by the cardinality

case (18) reduces to

|S |. In fact, for the implementation of the optimal switching law N |S | symmetric matrices must be stored. The previous results of Theorems 1 and 2 do not suffer of this limitation. Denoting Vi (n) = A′i P(n)Ai + Ci′ Ci for all i ∈ K and all n ∈ [0, κ ) then κ N matrices

ϑ (x) ≥ min

{ℓκ+κ−1 ∑

σ (·)∈K

} z(k) z(k) + ϑ (x(ℓκ + κ )) ′

(19)

k=ℓκ

where x(ℓκ ) = x for each ℓ ∈ N. This inequality whenever admits a time invariant solution ϑ (ξ ) puts in clear evidence that the subsequence {x(ℓκ )}ℓ∈N , and by consequence the entire sequence {x(n)}n∈N , converge uniformly to zero and ∥z ∥22 < V (x0 , 0) = ϑ (x0 ) = x′0 Sx0 . Following [20] and [21] the stationary solution ϑ (x) is not a quadratic function which implies that, in general, enforcing it to be quadratic, the equality in (19) cannot be reached. However, for a given initial condition x0 ∈ Rnx and a periodic stabilizing switching sequence with the first period given by σ (n) = π (n) for all n ∈ [0, κ ), simple calculations show that ∥z ∥22 = x′0 Sx0 where S > 0 solves A′π (n) P(n + 1)Aπ (n) + Cπ′ (n) Cπ (n) = P(n)

Corollary 2. Under the conditions of Theorem 2, with the state dependent switching function σ (n) = v (x(n)) given in (15), the switched linear system (1)–(2) is exponentially stable and the associated H2 cost satisfies ′ J(σ ) < tr(Em SEm )

(21)

whenever m = σ (−1) is given.

Let us move our attention to the following discrete-time switched linear system x(n + 1) = Aσ x(n) + Bσ u(n) + Eσ w (n) z(n) = Cσ x(n) + Dσ u(n)

∥zr ∥22

inf J(σ , L)

<

inf

P(·),λ(·)∈Λ,S ,L

er Eσ′ (−1) SEσ (−1) er

= tr Em′ SEm

(

′ tr(Em SEm )

(27)

subject to

)

(22)

As it has been commented before, more accurate results are obtained whenever this upper bound is minimized with respect to the choice of matrix S > 0 that defines the initial and the final boundary conditions of inequality (14). From this observation the minimum upper bound is calculated by solving ′ tr(Em SEm ) : (14), P(κ ) = P(0) = S > 0

{

∑

( ) λi (n) A′Li P(n + 1)ALi + CLi′ CLi < P(n)

(28)

i∈K

which leads to the upper bound (21). □

inf

(26)

′

r =1

P(·),λ(·)∈Λ,S

(25)

where L = (L1 , . . . , LN ) is the matrix of gains. Clearly, the closedloop system reduces to (1)–(2) with the obvious replacements ALi = Ai + Bi Li → Ai and CLi = Ci + Di Li → Ci for all i ∈ K. Let κ ≥ 1 be given. From the previous results, in particular, those of Theorem 2 and Corollary 2, the problem to be solved is

r =1 nw ∑

(24)

where we have included the control input u(·) : N− → Rnu . All assumptions adopted to system (1)–(2) remain valid. The goal now is to determine a switching function σ (n) = v (x(n)) and a linear state feedback of the general form u(n) = Lσ x(n) such as to preserve the closed-loop system exponential stability and minimize a suitable upper bound to the control design problem (σ ,L)∈V

Proof. From Theorem 2, exponential stability is assured by the switching function (15) and ∥z ∥22 < x′0 Sx0 . On the other hand, from Definition 3 we have nw ∑

4. State feedback control design

(20)

for all n ∈ [0, κ ) subject to P(0) = P(κ ) = S > 0. Hence, in this particular case, the cost can be exactly evaluated by a quadratic function. From Theorem 2, in the general case, the quantity ϑ (x(0)) = x′0 Sx0 is a guaranteed cost that must be minimized from the feasible choices of matrix S > 0. Furthermore, it is expected that by the increasing of κ ∈ N more accurate solutions are obtained.

J(σ ) =

need to be stored for the implementation of the switching law (15). Under the additional assumption that Ai for all i ∈ K are nonsingular matrices and supposing that the constraints (14) are arbitrarily tight (which is always true whenever P(n) for all n ∈ [0, κ ) solve problem (23)) then once the matrix P(0) = S > 0 has been calculated, P(n) for all n ∈ [0, κ ) and, by consequence Vi (n) = A′i P(n)Ai + Ci′ Ci for all i ∈ K, are calculated at each time n ∈ N in complete synchronism with the switched system. Therefore, proceeding in this way, the number N + 1 of matrices to be stored becomes independent of κ .

}

(23)

for some given κ ≥ 1. It is immediate to see that this problem is not simple to solve. The difficulty stems from the fact that the constraints represented by inequalities (14) are jointly nonconvex with respect to the variables P(·) and λ(·) ∈ Λ. This point will be addressed afterwards. 3.3. Complexity An important issue is the controller complexity. The optimality of problem (3) is characterized by the stationary solution of (17). Following [20] and [21] it has the form ϑ (ξ ) = minS ∈S ξ ′ S ξ

for all n = 0, 1, . . . , κ − 1 and the boundary conditions P(0) = P(κ ) = S > 0. Once again, this is a hard to solve nonconvex problem. However, from (17), a simplification is possible by noticing that, with no loss of generality, the variable λ(n) can be constrained to belong to the set of vertices Λv of Λ instead of Λ ⊂ RN . Doing this, problem (27)–(28) can be reformulated. For a given κ ≥ 1 denote C(κ ) the set obtained from the cartesian product of K by itself κ times, that is, C(n + 1) = K × C(n) starting from C(1) = K. This set has N κ elements each one being a list of κ elements of K. The notation Cℓ (κ ) indicates the ℓth element of C(κ ) for each 1 ≤ ℓ ≤ N κ . Moreover, i(n) ∈ Cℓ (κ ) denotes the nth element of Cℓ (κ ) for each 0 ≤ n ≤ κ − 1. The next lemma gives an equivalent problem to (27)–(28) that can be split in a series of convex subproblems. Theorem 3. Let κ ≥ 1 be given. The switching control design problem (27)–(28) is equivalent to min

′ inf tr(Em SEm )

1≤ℓ≤N κ P(·),S ,L

(29)

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7

subject to ′ CLi(n) < P(n) A′Li(n) P(n + 1)ALi(n) + CLi(n)

(30)

for all n = 0, 1, . . . , κ − 1, i(n) ∈ Cℓ (κ ) and the boundary conditions P(0) = P(κ ) = S > 0. Proof. Restricting, with no loss of generality, λ(n) ∈ Λv ⊂ Λ for all n = 0, 1, . . . , κ − 1, the evaluation of the objective function of problem (27)–(28) is done from the generation of all vertices of Λv . It can be verified that, for a given κ ≥ 1, they constitute the set Cℓ (κ ). The set C(κ ) is just a collection of such elements including all possibilities obtained by varying the counter ℓ in the interval 1 ≤ ℓ ≤ N κ . The minimum value of the criterion calculated in this way provides the optimal solution. □ The importance of this theorem is twofold. First, if one wants to design the switching function only, then imposing Li = 0 for all i ∈ K, the constraint (30) are κ coupled LMIs. In addition, if the goal is to design the switching function and the state feedback control law jointly then (30) becomes κ coupled LMIs provided that a suitable one-to-one change of variables is performed. This enables the evaluation of the actual cost from the solution of a convex programming problem that can be approached by methods available in the literature. Second, problem (29)–(30) is a mixed problem involving a set of continuous variables P(n), S and L and a set of discrete ones i(n) ∈ Cℓ (κ ) with the key property that it reduces to a convex problem whenever a sequence i(n) ∈ Cℓ (κ ) is given. For a moderate value of κ , direct enumeration can be adopted. In the general case, this is an ideal situation to apply a Branch and Bound algorithm, see [24] for a complete review on this topic. Corollary 3. Let κ ≥ 1 be given. Consider the joint design problem min

′ −1 inf tr(Em R Em )

(31)

1≤ℓ≤N κ X (·),R,Y

subject to

⎡

Gi(n) + G′i(n) − X (n) ⎣ Ai(n) Gi(n) + Bi(n) Yi(n) Ci(n) Gi(n) + Di(n) Yi(n)

• X (n + 1) 0

⎤ • •⎦ > 0

(32)

I

for all n = 0, 1, . . . , κ − 1, i(n) ∈ Cℓ (κ ) and the boundary conditions X (0) = X (κ ) = R > 0. Matrices P(n) = X (n)−1 for all n = 1 0, 1, . . . , κ − 1, S = R−1 and Li(n) = Yi(n) G− i(n) for each i(n) ∈ Cℓ (κ ) constitute a suboptimal solution to the problem (29)–(30). Proof. Starting from n = 0, inequality (32) together with the boundary condition X (0) = R > 0 imply that X (1) > 0 and consequently X (n) > 0 for all n in the entire time interval [0, κ]. Then, the well known inequality, [25] G′i(n) X (n)−1 Gi(n) ≥ Gi(n) + G′i(n) − X (n) > 0

(33)

holds and Gi(n) is nonsingular. Defining the state feedback gain 1 Li(n) = Yi(n) G− i(n) , inequality (32) indicates that

⎡

X (n)−1 ⎣ Ai(n) + Bi(n) Li(n) Ci(n) + Di(n) Li(n)

• X (n + 1) 0

⎤ • •⎦ > 0

5

The optimal joint design problem (31)–(32) provides the set C∗ℓ (κ ) from which there is no difficulty to extract the switching dependent state feedback gains Li for all i ∈ K provided that each element of K appears in C∗ℓ (κ ) at least once. If this is not the case, subsystems that never will be activated by the control policy are readily identified. There are two other situations covered by Corollary 3. Although of less interest in the context of switched linear systems control they are important in the theoretical and practical perspective. First, replacing (Gi(n) , Yi(n) ) = (G, Y ) by constant matrices yields switching independent state feedback gains L = YG−1 which simplifies the implementation of the control law. Second, replacing (Gi(n) , Yi(n) ) = (G(n), Y (n)) by time-varying matrices yields switching independent time-varying state feedback gains of the form L(n) = Y (n)G(n)−1 . As it can be seen the second situation is theoretically important because problems appearing in Theorem 3 and Corollary 3 become equivalent. Actually, in this case, the problem stated in Corollary 3 for state feedback control synthesis is expressed through LMIs without introducing any conservatism with respect to the nonconvex design problem of Theorem 3. In the next section several illustrative examples are solved. 5. Examples In this section illustrative examples borrowed from the literature are discussed. Whenever only the switching function has to be designed, Theorem 3 has been applied directly to the open-loop system. On the other hand, for the joint switching function and state feedback control design, the conditions of Corollary 3 have been solved. In both cases, explicit enumeration of all vertices of the feasible set has been adopted. 5.1. Illustrative example 2 In this subsection three examples are presented. For each one we have considered the open-loop matrices provided in references appearing in Table 1, Ci = I and Ei = I of compatible dimensions for all i ∈ K. The H2 costs are also shown in Table 1. It is interesting to observe that the mentioned costs may not present a uniform behavior with respect to κ . Indeed, a periodic sequence of length κ2 is able to reproduce a generic periodic sequence of length κ1 only if κ2 = ακ1 for some integer α ≥ 2. Under this condition, the associated costs are non increasing. This is illustrated by comparing the costs in columns corresponding to κ1 = 2 and κ2 = 4 (costs are non increasing) and corresponding to κ1 = 2 and κ2 = 3 (costs are not comparable). Moreover, in the second example (Fiacchini, and Jungers [16]), the complexity of our switching law corresponding to κ = 2 is much lower. Besides, it imposes to the closed-loop system a performance as good as the one of the switching law associated to κ = 10. It is clear that, in general, the effect of the switching controller may be important for cost reduction mainly if its complexity, measured by the value of κ , is allowed to increase. Fig. 1 presents the time simulation of the second example (Fiacchini, and Jungers [16]) with κ = 2 and C∗ℓ = [1 2]. It is clear that the controller is effective in terms of stability and performance. 5.2. Illustrative example 3

(34)

I

holds and the Schur Complement with respect to the last two rows and columns provides (30) with P(n) = X (n)−1 satisfying the boundary conditions P(0) = P(κ ) = S = R−1 . Observing that the objective function of problems (29) and (31) are equal under the same change of variables S = R−1 > 0, the proof is concluded. □

In this subsection we have considered three examples with the open-loop matrices given in the references appearing in Table 2. In all cases, and accordingly to the LQR problem, we have considered (Ci , Di ) orthogonal and such that Ci′ Ci = I and D′i Di = I of compatible dimensions for all i ∈ K. Moreover, to define the H2 cost we have adopted Ei = I for all i ∈ K. Table 2 shows the minimum guaranteed costs as a function of κ ∈ {1, 3, 4, 5}

6

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7 Table 1 H2 performance cost.

κ Examples taken from

1

2

3

4

10

Geromel, and Colaneri [7] Fiacchini, and Jungers [16] Ex. 7 Fiacchini et al. [13] Ex. 28

∞ ∞ ∞

762.73 20.95

∞

∞

∞

154.73 20.95 140.25

42.81 20.68 95.49

39.92

Table 2 State feedback control and H2 performance cost.

κ 1

3

4

5

Zhang et al. [21] Ex. A

L Li(n) L(n)

8.83 8.83 8.83

8.83 7.06 7.06

8.83 7.05 7.04

8.83 7.04 7.04

L Li(n) L(n)

∞ ∞ ∞

∞

∞

∞

Zhang et al. [17] Ex. 1

10.06 9.08

9.05 9.05

10.04 9.05

Zhang et al. [17] Ex. 2

L Li(n) L(n)

22.24 22.24 22.24

22.24 22.24 17.50

22.24 17.55 16.91

22.24 17.86 16.91

Fig. 2. State trajectories, control law, and switching sequence.

interesting to observe that the subsystem 4 is Schur stable but the minimum guaranteed cost policy passes through unstable subsystems in order to provide a better solution. These examples put in evidence the flexibility and accuracy of our jointly switching function and control design method. The structure of the state feedback gain matrices can be chosen as well as the complexity of the switching function measured in terms of the parameter κ ≥ 1 provided by the designer. Whenever a solution is found, the closed-loop system is assured to be exponentially stable and exhibits a guaranteed H2 performance. Fig. 1. State trajectories and switching sequence.

6. Conclusion and the control structure, namely, a constant gain (L), switching dependent gains (Li(n) ) and switching independent time-varying gains (L(n)). Keeping in mind that the first (Zhang et al. [21] Ex. A) and the third (Zhang et al. [17] Ex. 2) examples present, at least, a Schur stable closed-loop subsystem whenever the optimal LQR gain is used then the minimum guaranteed cost is bounded above by the minimum LQR cost appearing in the column corresponding to κ = 1. In both cases, improvements are observed for κ > 1 meaning that the proposed switching law is consistent, see [26]. All subsystems of the second example (Zhang et al. [17] Ex. 1) are not stabilizable in accordance to the unbounded costs given in the column corresponding to κ = 1. A switching law with low complexity κ = 3 is able to stabilize the system. We have solved the conditions of Corollary 3 for the third example (Zhang et al. [17] Ex. 2), with κ = 5. The minimum cost 17.86 corresponding to the optimal sequence C∗ℓ = [3 2 3 4 4] and the optimal switching dependent state feedback gains

[ ∗] L2 L∗3 L∗4

0.1903 0.1412 −0.0716

[ =

0.1673 0.0288 −0.3098

−0.3560 −0.1541 0.1525

] −0.3268 −0.0483 0.0131

were obtained. Fig. 2 presents the state and control trajectories and the corresponding state dependent switching sequence. It is

In this paper, we have provided a new method to cope with switching function and state feedback control design. New sufficient conditions for exponential stabilizability and H2 guaranteed cost optimization have been proposed. The existence of a solution to these conditions has been analyzed. A special two points boundary value problem was the key issue to obtain a stationary solution assuring exponential stability and guaranteed performance that can be improved by allowing controllers of higher complexity. The main features of the reported results were illustrated by examples borrowed from the recent literature. The optimal control design problem can be split in several uncoupled convex subproblems which constitute an ideal structure to be solved by a branch and bound algorithm. This aspect together with the consideration of dynamic output feedback controllers and filters design are open problems to be faced in future research efforts. References [1] D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, USA, 2003. [2] Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design, Springer, London, 2005. [3] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag. 19 (1999) 59–70. [4] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control 54 (2009) 308–322.

G.S. Deaecto, J.C. Geromel / Systems & Control Letters 118 (2018) 1–7 [5] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, SIAM Rev. 49 (2007) 545–592. [6] J. Daafouz, J. Bernussou, Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties, Systems Control Lett. 43 (2001) 355–359. [7] J.C. Geromel, P. Colaneri, Stability and stabilization of discrete time switched systems, Int. J. Control 79 (2006) 719–728. [8] J.W. Lee, G.E. Dullerud, Uniform stabilization of discrete-time switched and Markovian jump linear systems, Automatica 42 (2006) 205–218. [9] J.W. Lee, G.E. Dullerud, Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems, SIAM J. Control Optim. 45 (2006) 1329–1358. [10] S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [11] G.S. Deaecto, Dynamic output feedback H∞ control of continuous-time switched affine systems, Automatica 71 (2016) 44–49. [12] G.S. Deaecto, M. Souza, J.C. Geromel, Discrete-time switched linear systems state feedback design with application to networked control, IEEE Trans. Automat. Control 60 (2015) 877–881. [13] M. Fiacchini, A. Girard, M. Jungers, On the stabilizability of discrete-time switched linear systems: novel conditions and comparisons, IEEE Trans. Automat. Control 61 (2016) 1181–1193. [14] J.C. Geromel, P. Colaneri, P. Bolzern, Dynamic output feedback control of switched linear systems, IEEE Trans. Automat. Control 53 (2008) 720–733. [15] G. Zhai, Quadratic stabilizability of discrete-time switched systems via state and output feedback, in: Proceedings of the 40-th IEEE Conference on Decision and Control, 2001, pp. 2165–2166.

7

[16] M. Fiacchini, M. Jungers, Necessary and sufficient condition for stabilizability of discrete-time linear switched system: A set-theory approach, Automatica 50 (2014) 75–83. [17] W. Zhang, A. Abate, J. Hu, M.P. Vitus, Exponential stabilization of discrete-time switched linear systems, Automatica 45 (2009) 2526–2536. [18] D. Antunes, W.P.M. Heemels, Linear quadratic regulation of switched systems using informed policies, IEEE Trans. Automat. Control 62 (2017) 2675–2688. [19] D. Gorges, M. Izák, S. Liu, Optimal control and scheduling of switched systems, IEEE Trans. Autom. Control 56 (2017) 135–140. [20] B. Lincoln, A. Rantzer, Relaxing dynamic programming, IEEE Trans. Autom. Control 51 (2006) 1249–1260. [21] W. Zhang, J. Hu, A. Abate, Infinite-horizon switched LQR problems in discretetime: a suboptimal algorithm with performance analysis, IEEE Trans. Automat. Control 57 (2012) 1815–1821. [22] H.R. Daiha, L.N. Egidio, G.S. Deaecto, J.C. Geromel, H∞ state feedback control design of discrete-time switched linear systems, in: Proc. of the IEEE Conference on Decision and Control, 2017, pp. 5882–5887. [23] F. Blanchini, C. Savorgnan, Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions, Automatica 44 (2008) 1166–1170. [24] D.P. Bertsekas, Dynamic Programming and Optimal Control, second ed. Athena Scientific, 2001. [25] M.C. de Oliveira, J. Bernussou, J.C. Geromel, A new discrete-time robust stability condition, Systems Control Lett. 37 (1999) 261–265. [26] G.S. Deaecto, A.R. Fioravanti, J.C. Geromel, Suboptimal switching control consistency analysis for discrete-time switched linear systems, Eur. J. Control 19 (2013) 214–219.