1
Stabilizability of Discrete-Time Controlled Switched Linear Systems Donghwan Lee and Jianghai Hu
Abstract—The goal of this paper is to study the stabilizability of controlled discrete-time switched linear systems (CSLSs), where the discrete switching control input and the continuous control input coexist. For a given positive integer h, we propose a quantitative metric of stabilizability, called h-contraction rate (h-CR), and prove that it serves as an indicator of the stabilizability in the following sense: the h-CR is less than one for sufficiently large h if and only if the SLS is stabilizable. In addition, we develop computational tools to estimate upper and lower bounds of the h-CR. In particular, the upper bound estimation is expressed as a semidefinite programming problem (SDP). We derive its Lagrangian dual problem and prove that the dual problem with an additional rank constraint gives an exact characterization of the h-CR. An algorithm is developed to solve the dual problem with the rank constraint.
I. I NTRODUCTION The goal of this paper is to study the stabilization of discretetime controlled switched linear systems (CSLSs), where the discrete switching control input and the continuous control input coexist. CSLSs are different from switched linear systems (SLSs) where only the discrete switching control input is used to control the SLSs. Stability analysis and stabilization of SLSs/CSLSs have been studied extensively over the past decades [1] mainly based on Lyapunov theories, for instance, switched Lyapunov functions [2]–[5], polynomial Lyapunov functions [6], composite Lyapunov functions [7]– [11], and periodic Lyapunov functions (PLFs) [12]–[15]. Especially, the concept of PLFs was explored to study periodic systems [16]– [18] and the multirate sampled-data control systems [19]. It was not until recently that PLFs found their applications for general non-periodic systems. For example, for the robust stability and stabilization of uncertain linear systems [20], [21], it was revealed that, in general, PLFs provide less conservative stability analysis and control synthesis conditions, and lead to improved performances of the control systems such as robustness and H∞ performance. A periodic control Lyapunov function (PCLF) was proposed [12]–[15] to design stabilizing control policies and test stabilizability of SLSs. Beyond Lyapunov approaches, the concept of generating functions was used in [22] for the stability/stabilizability of SLSs. The joint spectral radius in [23] extends the concept of the spectral radius of linear systems to SLSs, and a new quantitative metric was developed in [24] for the resilient stabilization of CSLSs against adversarial switching by an adversary. In addition, a new stabilizability index was devised in [25] recently for the stabilization of SLSs. This paper attempts to extend the results in [14], [15] to the stabilization of CSLSs. In [14], stabilizability characterization of SLSs, computational tools, and their performance analysis were given based on the h-PCLF and the corresponding h-periodic control policy (h-PCP) for the switching input only, where h > 0 is the period. ∗ This material is based upon work supported by the National Science Foundation under Grant No. 1539527 D. Lee is with the Department of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign, IL 61801, USA
[email protected]. J. Hu is with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA
[email protected].
However, the existence of the continuous control input significantly increases the complexity of the analysis, and several results in [14] need to be reconsidered. For instance, is it possible to achieve the fastest exponential convergence rate, called the exponential stabilizing rate, by using the h-PCP for both the switching and continuous inputs? Can we derive a convex test to characterize the stabilizablility? If not, are there approximate convex relaxations? How conservative are these relaxations? Motivated by the above discussions, the main contributions of this paper are summarized as follows: (i) For a given positive integer h, we propose a quantitative metric of stabilizability called h-contraction rate (h-CR); (ii) A necessary and sufficient condition for stabilizability is developed by using the h-CR; (iii) It is proved that the h-CR can be used to compute an exponential stabilizing rate of the CSLS trajectories; (iv) Computational tools are developed to estimate upper and lower bounds of the h-CR. For contribution (i), the h-CR is defined as the smallest contraction constant of the state x(·) of the CSLS after h time steps with respect to any given norm k · k, i.e., the h-CR is the smallest ρ ∈ [0, ∞) such that kx(h)k ≤ ρkx(0)k. If the h-CR is less than one, then V (·) := k · k is a PCLF with period h. In this respect, we prove that the h-CR is less than one for some positive integer h if and only if the CSLS is stabilizable. The h-CR is a new quantitative metric of the stabilizability, which was not presented in [14]. Regarding (iii), we prove that as h tends to infinity, the h-th root of the h-CR converges to the exponential stabilizing rate. Nevertheless, it is also proved that the exponential stabilizing rate is not always achieved by the corresponding h-PCP. For contribution (iv), the upper bound estimate is expressed as the optimal value of a semidefinite programming problem (SDP) [26]. We prove that the SDP-based bound is not tight in some cases via examples. In addition, we derive its Lagrangian dual SDP problem and prove that the optimal value of the dual SDP problem with an additional rank constraint provides an exact characterization of the h-CR. Therefore, the conservatism of the SDP relaxation is due to the rank constraint. Compared to [14], the duality analysis, exact characterization of the h-CR, and conservatism analysis are new. In particular, the duality analysis gives rise to new algorithms for stabilizability analysis. To compute a locally optimal solution, the exact optimization problem is converted into another one with non-convex constraints, and an algorithm based on the DC programming [27] is developed to calculate a lower bound of the h-CR. II. P RELIMINARIES AND P ROBLEM F ORMULATION The adopted notation is as follows: N and N+ : sets of nonnegative and positive integers, respectively; Rn : n-dimensional Euclidean space; Rn×m : set of all n × m real matrices; AT : transpose of matrix A; A ≻ 0 (A ≺ 0, A 0, and A 0, respectively): symmetric positive definite (negative definite, positive semi-definite, and negative semi-definite, respectively) matrix A; for any matrix M , M † denotes its pseudoinverse; In : n × n identity matrix; I: identity matrix with an appropriate dimension; 0n×m : n × m zero matrix; k · k: any vector norm or the corresponding induced matrix norm; for a set S, |S| denotes the cardinality of the set; Sn : symmetric n × n matrices; Sn + : set of symmetric n × n positive semi-definite
2
matrices; Sn ++ : set of symmetric n × n positive definite matrices; tr(A) and det(A): trace and determinant of matrix A, respectively; k mod h: remainder of k divided by h for k, h ∈ N+ ; ⌊x⌋: the maximum integer less than x ∈ R; ∆k with k ∈ N+ : unit simplex defined as ∆k := {(α1 , . . . , αk ) : α1 + · · · + αk = 1, αi ≥ 0, ∀i ∈ {1, . . . , k}}. Consider the discrete-time controlled switched linear system (CSLS) described by x(k + 1) = Aσ(k) x(k) + Bσ(k) u(k),
x(0) = z ∈ Rn ,
(1)
where k ∈ N is the time step, x : N → Rn is the state, u : N → Rm is the continuous control, and σ : N → M := {1, 2, . . . , N } is the mode. For each i ∈ M, Ai ∈ Rn×n and Bi ∈ Rn×m are constant matrices, and the pair (Ai , Bi ) is called a subsystem or subsystem matrices. Both u and σ are controlled, as compared to SLSs which have only the mode as a control input, i.e., Bi = 0, ∀i ∈ M. Denote the pair of continuous and discrete control inputs ξ := (u, σ) : k−1 N → Rm × M. Then, the sequence πk := (ξ(i))i=0 or π∞ := ∞ (ξ(i))i=0 is called the hybrid-control sequence [10]. In this paper, the state driven by πk or π∞ with an initial state z ∈ Rn will be denoted by x(·; z, πk ) or x(·; z, π∞ ), respectively. In the case that the hybrid-control sequence πk (or π∞ ) depends on the initial state z, it will be denoted by πk, z (or π∞, z ). The definition of exponential stabilizability of CSLSs is given below. Definition 1 (Exponential stabilizability). The CSLS (1) is called exponentially stabilizable with the parameters a ≥ 1 and c ∈ [0, 1) if starting from any initial state x(0) = z ∈ Rn , there exists a hybrid-switching sequence π∞, z such that for all k ∈ N, kx(k; z, π∞, z )k ≤ ack kzk.
Proof. If ρ(Th ) ≤ 1 for some h ≥ 1, then by the definition of π ˜h , we have kx(h; z, π ˜h )k ≤ ρ(Th ) · kzk, ∀z ∈ Rn , which implies kx(ht + h; z, π ˜h,∞ )k ≤ ρ(Th ) · kx(ht; z, π ˜h,∞ )k, ∀t ∈ N, and by induction kx(ht; z, π ˜h,∞ )k ≤ ρ(Th )t · kzk,
Problem 1. Determine whether or not a given CSLS is stabilizable. III. h-C ONTRACTION R ATE AND S TABILIZABILITY OF CSLS S Define the hybrid-control policy (3)
πh ∈Πh
which is optimal in the sense that it generates the hybrid-control sequence of length h to minimize the state norm after h time steps. Here, Πh is the set of all admissible hybrid-control sequences of length h, i.e., θk (z) ∈ M and ωk (z) is finite for all k ∈ {0, . . . h − 1}, ωk (z) ∈ Rm and θk (z) ∈ M are the continuous and discrete control inputs at time k, respectively. If there are multiple minimizers in (3), any choice will suffice. Concatenating π ˜h at time instants k = ht, t ∈ N, an infinite-horizon hybrid-control policy can be defined as (4)
∀t ∈ N.
(5)
Noting that k = h ⌊k/h⌋ + (k mod h), we have kx(k; z, π ˜h,∞ )k
= kx(h ⌊k/h⌋ + (k mod h); z, π ˜h,∞ )k
≤ kx(h ⌊k/h⌋ + (k mod h); z, π ˜h,∞ ) − x(h ⌊k/h⌋ ; z, π ˜h,∞ )k
+ kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k kx(h ⌊k/h⌋ + (k mod h); z, π ˜h,∞ ) − x(h ⌊k/h⌋ ; z, π ˜h,∞ )k = kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k
× kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k + kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k max
sup
k=0,..., h−1 ξ∈Rn , kξk=1
× kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k
The goal of this paper is stated in the following problem.
π ˜h, ∞ = {˜ πh , π ˜h , π ˜h , . . .}.
Proposition 1. If there exists h ∈ N+ such that 0 < ρ(Th ) ≤ 1, then the inequality (2) holds with the parameters a = φh /ρ(Th ), c = ρ(Th )1/h under π ˜h,∞ in (4), where φh := maxk=0,..., h−1 supξ∈Rn , kξk=1 (kx(k; ξ, π ˜h,∞ )−ξk+1). Moreover, if there exists h ∈ N+ such that ρ(Th ) > 1, then the inequality (2) holds with the parameters a = φh and c = ρ(Th )1/h .
≤
Assumption 1. Each subsystem (Ai , Bi ) is not stabilizable.
z∈Rn , kzk=1
z∈Rn , kzk6=0
second equality follows from the homogeneity of Th . Since the set {z ∈ Rn : kzk = 1} is compact and kTh (z)k is continuous in z, the supremum is attained and can be replaced with max. Now, we claim that if ρ(Th ) < 1, then the CSLS (1) is exponentially stabilizable.
(2)
For CSLSs, any c ∈ R+ satisfying (2) for some a ∈ [1, ∞) will be called an exponential convergence rate. The exponential stabilizing rate, denoted by c∗ ∈ R+ , is the infimum of all such exponential convergence rates. Note that c∗ provides a quantitative metric of the CSLS’s stabilizability. We refer to the notion of exponential stabilizability simply as stabilizability if there is no confusion. Clearly, the CSLS (1) is stabilizable if one of the subsystems is stabilizable. A nontrivial problem is to stabilize the system when none of the subsystems is stabilizable [10]. In this respect, the following assumption is imposed.
π ˜h (z) := (ωk (z), θk (z))h−1 k=0 = arg min kx(h; z, πh )k,
Note that the hybrid-control sequence of length h generated by (3) only depends on z. However, the infinite-horizon hybrid-control policy (4) is constructed in such a way that the state at every h samples is utilized periodically. Specifically, the initial state x(0) is first used to determine the inputs σ(0), u(0), . . . , σ(h − 1), u(h − 1); under these inputs, the resulting state x(h) is then used to determine σ(h), u(h), . . . , σ(2h − 1), u(2h − 1); and so on. In this sense, (4) can be considered as a (generalized) closed-loop control policy that is invoked every h steps rather than every step. Define the homogeneous operator Th : Rn → Rn by Th (z) := x(h; z, π ˜h (z)). Motivated by [24], we introduce a quantity called the h-contraction rate (h-CR) of the CSLS (1) with respect to norm k · k defined kTh (z)k = sup kTh (z)k, where the as ρ(Th ) := sup kzk
(kx(k; ξ, π ˜h,∞ ) − ξk + 1)
= φh kx(h ⌊k/h⌋ ; z, π ˜h,∞ )k.
(6)
It can be proved that φh is bounded for finite h ∈ N+ because kx(k; ξ, π ˜h,∞ )k is continuous in ξ and ξ takes values in a compact set (the unit sphere). Using ⌊k/h⌋ ≥ k/h − 1, ρ(Th ) ≤ 1, and (5) and (6) yield ||x(k; z, π ˜h,∞ )|| ≤ φh ||x(h ⌊k/h⌋ ; z, π ˜h,∞ )|| ≤ φh φh ρ(Th )k/h−1 · kzk = ρ(T (ρ(Th )1/h )k · kzk, and the desired h) result follows by Definition 1. If ρ(Th ) > 1 for some h ≥ 1, then using ⌊k/h⌋ ≤ k/h, ρ(Th ) > 1, (5) and (6) result in kx(k; z, π ˜h,∞ )k ≤ φh (ρ(Th )1/h )k · kzk. Remark 1. If there exists h ∈ N+ such that ρ(Th ) = 0, then it can be proved that for any c > 0, one can find a sufficiently large a ≥ 1 such that (2) holds. Indeed, the CSLS is finite-time stabilizable to the origin by the hybrid-control policy π ˜h . The next two results provide the converse arguments of Proposition 1, i.e., if the CSLS (1) is stabilizable, then ρ(Th ) < 1 holds for some h ∈ N+ .
3
Proposition 2. If the CSLS (1) is exponentially stabilizable with the parameters a ≥ 1 and c ∈ [0, 1), then it holds that ρ(Th ) ≤ ach ,
∀h ∈ N.
(7)
Proof. Suppose that the CSLS (1) is exponentially stabilizable with the parameters a ≥ 1 and c ∈ [0, 1). Then, for any initial state x(0) = z ∈ Rn , there exists a hybrid-control sequence π∞, z such that kx(k; z, π∞, z )k ≤ ack kzk holds for all k ∈ N, which implies ρ(Th ) ≤ ach . Corollary 1. The CSLS (1) is stabilizable if and only if there exists h ∈ N, h ≥ 1, such that ρ(Th ) < 1. Proof. Straightforward from Proposition 1 and Proposition 2. In the next proposition, we prove that if the CSLS is stabilizable, then ρ(Th )1/h converges to c∗ as h → ∞. Proposition 3. The CSLS (1) is exponentially stabilizable if and only if limh→∞ ρ(Th )1/h =: ρ∗ < 1. Moreover, if ρ∗ < 1, then ρ∗ = c∗ . Proof. If the CSLS is stabilizable, then by Proposition 2, ρ(Th )1/h ≤ a1/h c, where c is any exponential convergence rate. By taking the limit h → ∞ and using the definition of c∗ , we have lim suph→∞ ρ(Th )1/h ≤ c∗ + ε, where ε > 0 is arbitrary. Thus, one gets lim suph→∞ ρ(Th )1/h ≤ c∗ . To prove the reversed inequality, note that by Proposition 1, ρ(Th )1/h ≥ c∗ , which implies lim inf h→∞ ρ(Th )1/h ≥ c∗ . Therefore, ρ∗ = c∗ , and the first result follows since c∗ < 1. Conversely, if limh→∞ ρ(Th )1/h < 1, then there exists h ∈ N+ such that ρ(Th ) < 1. By Proposition 1, the CSLS is exponentially stabilizable. Assume that the CSLS (1) under (4) is stabilized with a given h ∈ N+ . Then, its h-periodic exponential stabilizing rate is defined as c∗h := inf {c ≥ 0 : there exists a < ∞ such that
o kx(k; z, π ˜h,∞ (z))k ≤ ack kzk; ∀z ∈ Rn , ∀k ∈ N .
The following proposition is sometimes useful to calculate rough overestimates of ρ(Th ) and c∗h . Proposition 4. For any given (i1 , . . . , ih ) ∈ Mh and Fi ∈ Rm×n , i ∈ M, we have ρ(Th ) ≤ k(Ai1 + Bi1 Fi1 ) · · · (Aih + Bih Fih )k and c∗h ≤ k(Ai1 +Bi1 Fi1 ) · · · (Aih +Bih Fih )k1/h , where k · k is the induced matrix norm corresponding to the vector norm used throughout this paper. Proof. Since Fi is a state-feedback gain matrix in mode i, by the definition of h-CR, the right-hand side of the first inequality is larger than or equal to ρ(Th ). The second inequality is obtained by using the first inequality and Proposition 1. Obviously, c∗ ≤ c∗h holds. A natural question is whether the exponential stabilizing rate c∗ defined in Section II can be achieved by using (4), i.e., whether c∗ = c∗h for some h ≥ 1. The next result shows that asymptotically this is indeed the case. Proposition 5. Assume that ρ(Th ) < 1 holds for some h ∈ N+ . Then, limh→∞ c∗h = c∗ . Proof. Since c∗h is only defined for those h ≥ 1 where the CSLS is stabilizable by the h-periodical policy in (4), it is not clear whether the limit limh→∞ c∗h is well defined. We first prove that ˜ ≥ 1 such that c∗h is well defined for all h ≥ h. ˜ By there exists h hypothesis, ρ(Th ) < 1 holds for some h ∈ N+ . By Proposition 1, the CSLS is stabilizable with the parameters a ≥ 1 and c ∈ [0, 1). ˜ such that ρ(Th ) ≤ By Proposition 2, there exists a sufficiently large h
˜ This implies that the CSLS is stabilzable by ach < 1, ∀h ≥ h. ˜ and thus, c∗h is well the h-periodical policy in (4) for all h ≥ h, ˜ ˜ By Proposition 1, defined for all h ≥ h. Now, assume h ≥ h. 1 ∗ h ch ≤ ρ(Th ) ; and by Proposition 2, ρ(Th ) ≤ ach for any parameters a ≥ 1 and c ∈ [0, 1) of exponential stabilizability. Combining the two inequalities, we have c∗h ≤ a1/h c. From the definition of c∗ , for any ε > 0 such that c∗ + ε < 1, c∗ + ε is an exponential convergence rate. Thus, we have c∗h ≤ a1/h (c∗ + ε). Taking the limit h → ∞ and noting that c∗h ≥ c∗ and that ε > 0 is arbitrary, we obtain the desired conclusion. Despite the above result, it is possible that c∗h > c∗ for any finite h ∈ N+ , i.e., the exponential stabilizing rate c∗ cannot be exactly achieved by (4). An example of SLSs with this property is given in [25], and a further SLS example is presented in [14]. An example of CSLSs, which was inspired by the one in [25], is given below.
2 0 0 Example 1. Consider the CSLS (1) with A1 = 0 2 1, B1 = 0 0 1 0 1 0 0 1 0 0, A2 = 0 0 2, B2 = 0, x(0) = e := 1. In this 0 0 0 2 0 1
example, the norm is the Euclidean norm for vectors and the Tspectral norm for matrices. With the initial state x(0) = 0 0 1 , x3 (k) is non-decreasing; hence the CSLS is not stabilizable, i.e., c∗ ≥ 1. For any h ∈ N+ , h ≥ 2, let (i1 , . . . , ih ) = (1, 1, . . . , 1, 2) and F1 = 0 0 0 , F2 = −1 0 0 . By Proposition 4, we
0
0
0
√
k = have k(A2 + B2 F2 )Ah−1 1
0 0 2 = 2 2. Therefore,
0 0 2 √ c∗h ≤ (2 2)1/h , and c∗ = limh→∞ c∗h ≤ 1. Combining, we have shown that c∗ = 1. On the other hand, it is easy to see that, starting from x(0) = e, (3) produces the switching sequence (θk (e))hk=0 = (1, 1, . . . , 1, 2) and the continuous control input (ωk (e))hk=0 = (0, 0, . . . , 0), under which we have x(h) = 2e. By induction and the homogeneity of (3), we have x(ℓh) = 2ℓ e, ∀ℓ ∈ N; hence c∗h ≥ 21/h > 1.
IV. C OMPUTATION Define the value function Vt : Rn → R, t ∈ {0, 1, . . . , h}, as Vt (z) := inf πt ∈Πt kx(t; z, πt )k2 , t ∈ {0, 1, . . . , h}, with V0 (z) := kzk2 . By standard results of dynamic programming [28], for any finite integer h ∈ N+ , the value function can be computed recursively using the one-stage value iteration Vt+1 (z) = inf u∈Rm , σ∈M Vt (Aσ z + Bσ u), for all z ∈ Rn , t ∈ {0, 1, . . . , h − 1}. Throughout the section, we will consider the Euclidean norm for vectors and spectral norm for matrices. The following result states that the value function can be characterized by the switched Riccati sets (SRSs) proposed in [10], [11]. Definition 2 (Switched Riccati set). For any P ∈ Sn +, define ρi (P ) := (Ai − Bi (P 1/2 Bi )† P 1/2 Ai )T P (Ai − Bi (P 1/2 Bi )† P 1/2 Ai ). The sequence of sets {Hk }hk=0 generated iteratively by Hk+1 = {ρi (X) ∈ Sn + : i ∈ M, X ∈ Hk } with H0 = {In } is called the switched Riccati sets (SRSs). Using SRSs and [11, Theorem 1], the value function Vk can be represented as a piecewise quadratic function. Lemma 1 ( [11, Theorem 1]). For t ∈ {0, 1, . . . , h}, the value function Vt (z) is represented by Vt (z) = minX∈Ht z T Xz. By the definition of the h-CR, we can write ρ(Th )2 = supz∈Rn , kzk=1 Vh (z) = supz∈Rn , kzk=1 minX∈Hh z T Xz. Therefore, to estimate ρ(Th ) for a given h ∈ N+ , we need to solve the following problem.
4
Problem 2. Given h ∈ supz∈Rn , kzk=1 minX∈Hh z T Xz.
N+ ,
γh
compute
:=
Note that γh = ρ(Th )2 , and hence, the computation of γh is equivalent to computing ρ(Th ). A. SDP Approach to Problem 2 Motivated by computational algorithms developed in [10], [22], we propose a semidefinite programming problem (SDP) [26] to compute an overestimation of γh . Problem 3. Solve γhp :=
min
γ
γ, α1 ,...,αk ∈R
subject to
k X i=1
(8) |Hh |
αi Xi γIn ,
X
αi = 1,
αi ≥ 0,
i=1
Lemma 3. If any X ∈ Hh is of the form
where det(H) ≥ 1, then γhp ≥ 1.
Once a solution to (8) is obtained, it can be used to estimate c∗h .
Proof. Although proofs have been given in [10, Corollary 1] or [8, Theorem 3], another proof will be presented here to give insights for subsequent developments. Note that γh in ! Problem 2 can be |H Ph | T αi Xi z, where ∆|Hh | expressed as sup min z i=1
is the unit simplex. Switching the sup and min ! gives the upper bound |H Ph | T λh ≤ min sup z αi Xi z, where the right hand i=1
Example 2. Consider the CSLS (1) with A1 = 1 , A2 = 2
1.5 0
1 , B2 = 1.5
γ1p
2 0
γhd :=
1
T
. The
Example 3. Consider the CSLS (1) with A1 :=
1/2 0
γhp
≥ 1.
0 , A2 := 2
0 , taken from [14]. 0
In [15], it was proved that the SLS is stabilizable by an appropriate switching policy. However, any element X of Hh is of the form (Ai1 · · · Aih )T (Ai1 · · · Aih ), (i1 , . . . , ih ) ∈ Mh . Since det(A1 ) = det(A2 ) = 1, we have det(X) = 1 for all X ∈ Hh . Therefore, by Lemma 2, γhp ≥ 1 for all h ∈ N+ . In other words, the SLS is an example of stabilizable SLSs such that γhp ≥ 1 for all h ∈ N+ , i.e., the stabilizability is not identifiable via Problem 3. In the following, we provide an example of CSLSs which is stabilizable but γhp ≥ 1 for all h ∈ N+ . To this end, we need the following lemma, whose proof will be given in Proposition 9.
max
s∈R, Z∈Sn
s
subject to
Proposition 7. Problem 4 is the Lagrangian dual problem of Problem 3, and there is no duality gap, i.e., γhp = γhd . Proof. For Lagrangian multipliers Z ∈ Sn ≥ 0, i ∈ + , µi {1, . . . , |Hh |}, η ∈ R, define the Lagrangian function of Problem 3. L(Z, µ, η, α, γ)
|Hh |
:= γ + tr Z |Hh |
γ2p
Lemma 2 ( [14, Prop. 7]). If det(X) ≥ 1, ∀X ∈ Hh , then
0
We claim that Problem 4 is a dual problem of Problem 3, and γhp = γhd for all h ∈ N+ .
1 , taken from [10, Example 1]. 0
− sin(π/20) , B1 = B2 = cos(π/20)
0 0, A2 = 1
tr(ZX) ≥ s, ∀X ∈ Hh , Z 0, tr(Z) = 1.
0 , B1 = 2
Problem 2 is a convex optimization, which can be solved efficiently. We also know that γhp < 1 implies the stabilizability. A natural question is whether limh→∞ γhp < 1 also holds when the SLS is stabilizable. In the following, we will show that the answer to this question is in general negative via a counterexample. It was already proved for SLSs, i.e., Bi = 0, ∀i ∈ M in [14].
cos(π/20) sin(π/20)
0 2 0
Problem 4. Solve
Solving Problem 3 yields = 2.0187 and = 0. Therefore, the CSLS is stabilizable (to the origin in two steps). For CSLSs in Example 1, γ1p = 4.7701 and γhp = 8 for all h ∈ {2, . . . , 7}.
first two states x1 (k) and x2 (k) are not affected by the continuous control input and only controlled by the switching sequence, and their dynamics follows those of the SLS in Example 3. The third state x3 (k) can be always counteracted by the continuous control input. Therefore, the CSLS is stabilizable. On the other hand, using Definition 2 and direct calculations, we can prove that any element of Hk has the form given in Lemma 3. Therefore, by Lemma 3, γhp ≥ 1 for all h ∈ N+ .
side is easily seen to be the optimal value of Problem 3.
0(n−r)×n
1/2 Example 4. Consider the CSLS (1) with A1 = 0 1 cos(π/20) − sin(π/20) 0 sin(π/20) cos(π/20) 0, and B1 = B2 = 0 0 0 1
Proposition 6. It holds that γh ≤ γhp , and hence, ρ(Th ) ≤ (γhp )1/2 1 and c∗h ≤ (γhp ) 2h hold.
α∈∆|H | z∈Rn , kzk=1 h
0n×(n−r) , 0r×r
H
Since Problem 3 is a convex optimization, we will next investigate its dual form for new insights and computational benefits.
|H |
where {Xi }i=1h is an enumeration of Hh .
z∈Rn ,kzk=1 α∈∆|Hh |
+
X
X i=1
αi Xi − γIn +
|Hh |
X i=1
αi − 1 η
µi (−αi ).
(9)
i=1
The corresponding dual function is given by d(Z, µ, η) = =
=
inf
α∈Rn , γ∈R
(
−η
−∞
inf
α∈Rn , γ∈R
L(Z, µ, η, α, γ)
|Hh |
X i=1
αi (tr(ZXi ) + η − µi ) − η + γ(1 − tr(Z))
if tr(ZXi ) + η − µi = 0, tr(Z) = 1 otherwise.
(10)
Therefore, the dual problem supZ0, µ≥0, η∈R d(Z, µ, η) can be written by sup
Z∈Sn , µi , η∈R
−η
subject to
tr(ZXi ) + η − µi = 0, µi ≥ 0, tr(Z) = 1, ∀i ∈ {1, . . . , |Hh |}. Problem 4 is obtained with appropriate change of variables. There is no duality gap because Problem 3 is a convex optimization and the Slater’s condition [29, Sec. 5.2.3] holds. In some cases, Problem 4 has computational benefits compared to Problem 3 in terms of the number of decision variables: Problem 3 has |Hh | + 1 decision variables, where |Hh | could be large for large h, while Problem 4 has (n2 + n)/2 + 1 decision variables. In addition, Problem 4 provides another proof of Lemma 2. Due to the zero duality gap, the statement of Lemma 2 is equivalent to γhd ≥ 1
5
whenever det(X) ≥ 1, ∀X ∈ Hh . This is proved in the following proposition.
γh =
Proposition 8. If det(X) ≥ 1, ∀X ∈ Hh , then γhd ≥ 1.
is equivalent to the dual function in the proof of Proposition 7, and L(Z, µ, η, α, γ) is the Lagrangian defined in (9). The proof is concluded by using the explicit form of the dual function in (10).
n×n
Sn +,
there exists M ∈ R such that Proof. Since Z ∈ Z = M T M , e.g., the Cholesky decomposition. From the inequality of arithmetic and geometric means, we have tr(ZX) = tr(M T M X) = tr(M XM T ) ≥ n det(M XM T )1/n = n det(M )1/n det(X)1/n det(M T )1/n = n det(Z)1/n det(X)1/n , where we used the fact that M XM T ∈ Sn + . Using det(X) ≥ 1, ∀X ∈ Hh , and choosing Z = (1/n)In lead to tr(ZXi ) ≥ n det(Z)1/n ≥ n((1/n)n )1/n = 1. Thus, s ≥ 1 is a feasible solution, and the desired result follows. Again, a similar argument of the proof of Proposition 8 can be used to prove Lemma 3. Proposition 9. If any X ∈ Hh is of the form given in Lemma 3, then γhd ≥ 1. 1 In−r 0n×(n−r) and use the inequality Proof. Choose Z = n−r 0(n−r)×n 0r×r of arithmetic and geometric means to have tr(ZX) = (1/(n − r))tr(H) ≥ (1/(n − r))(n − r)det(H)1/(n−r) ≥ 1.
γhe =
Problem 5. Solve max
s∈R, Z∈Sn
Z 0,
s
subject to
tr(Z) = 1,
tr(ZX) ≥ s, ∀X ∈ Hh ,
rank(Z) = 1.
Proposition 10. Problem 5 is an exact representation of Problem 2, i.e., γhe = γh . Proof. Problem 2 can be represented by sup
min
z∈Rn , kzk=1 α∈∆|Hh |
=
T
z
|Hh |
X i=1
The equality z T z = 1 can be replaced with the convex constraint z z ≤ 1 without affecting the optimal value because whenever z T z < 1, z can be scaled so that the optimal value increases unless one of elements of Hh is a zero matrix. Using the Schur complement [29, Sec. A.5.5.], we obtain another optimization form. T
Problem 6. Solve subject to 1 zT 0. z T Xz ≥ s, ∀X ∈ Hh , z In
min
sup
Z0, tr(Z)=1 γ∈R, α∈∆|Hh | rank(Z)=1
(t)
(sh , ∆z ∗ ) := arg max
s∈R, ∆z∈Rn T
αi Xi )) − γ + γ |Hh |
=
sup
min
Z0, tr(Z)=1 γ∈R, α∈∆|Hh | rank(Z)=1
γ + tr(Z(
s
subject to
z T Xz + z T X∆z + ∆z Xz ≥ s,
|Hh | i=1
s
Algorithm 1 DC programming for Problem 6. 1: Set h ∈ N+ and t = 0; initialize z ∈ Rn . 2: repeat 3: Solve
αi X i z
tr(Z(
max
s∈R, z∈Rn
We note that Problem 6 is an equivalent but different form of Problem 2 and can be derived directly from Problem 2. Nevertheless, the procedure presented in this paper provides new insights and gives rise to new algorithms for stabilizability analysis. Potential applications are summarized in the last paragraph. The nonconvex constraints are the scaler inequalities, which has a favorable structure of difference of convex functions (DC functions) [27]. Its local solution can be effectively found by using the DC programming techniques [27].
X
s subject to z T Xz ≥ s, ∀X ∈ Hh , z T z = 1.
max
s∈R, z∈Rn
γhe =
Another benefit of Problem 4 is that the rank of Z provides information on the quality of the estimates. In particular, adding the rank constraint rank(Z) = 1 to Problem 4, one can obtain an exact (although nonconvex) optimization formulation of Problem 2. In other words, the gap between the SDP-based overestimation γhp = γhd and γh is induced only by dropping the rank constraint rank(Z) = 1 from the exact optimization formulation.
L(Z, µ, η, α, γ), where the infimum
Note that Problem 4 is a SDP relaxation of Problem 5 obtained by dropping the rank constraint. Problem 5 is still difficult to solve because the rank constrained optimization is a non-convex problem, and known to be NP-hard [26]. Nevertheless, a local solution to Problem 5 provides a lower bound on γh , and it sometimes provides useful information on the quality of the estimate γhd = γhp . By replacing Z with Z = zz T , where z ∈ Rn , Problem 5 can be converted into the following nonconvex optimization problem
B. Exact Characterization of Problem 2
γhe :=
inf
sup
n Z0, tr(Z)=1 γ∈R, α∈R rank(Z)=1 η∈R, µ≥0
X i=1
αi Xi − γIn )),
(11)
1 z + ∆z
(z + ∆z)T In
∀X ∈ Hh ,
0.
4: z ← z + ∆z ∗ , t ← t + 1, tf = t 5: until k∆z ∗ k/kzkis sufficiently small (tf )
6: Return sh
.
|H |
where {Xi }i=1h is an enumeration of Hh . For any fixed Z 0 such that tr(Z) = 1, rank(Z) = 1, the minimization in (11) is a constrained convex optimization which satisfies the Slater’s condition [29, Sec. 5.2.3]. Given multipliers µi ≥ 0, i ∈ {1, . . . , |Hh |}, and η ∈ R, its !! Lagrangian function ! is ˜ L(η, µ, γ, α) := γ +tr
Z
|H Ph | i=1
|H Ph | i=1
αi Xi − γIn
+
|H Ph | i=1
αi − 1
η+
µi (−αi ). Due to the Slater’s condition, the duality gap is
zero, and the minimization in (11) can be replaced with the dual ˜ µ, γ, α). Thus, we have problem supη∈R, µ≥0 inf γ∈R, α∈Rn L(η,
Remark 2. Note that Problem 6 and Problem 5 are equivalent. (2) (1) (t) By [27], the sequence (sh )∞ t=1 is nondecreasing, i.e., sh ≤ sh ≤ (t) e · · · , and sh ≤ γh for all t ≥ 1. Example 5. We provide several examples to illustrate the validity and usefulness the CSLS (1) approaches. Consider of the proposed 1
0 with A1 = 0
0
B2 = 0
0
0 1 0 0
0
0 0 −1 0
0
T
1 0 −1 0 0 2 , B = , A2 = 0 0 1 1 0 1 2
0 1 0 0
1 0 0 0
0 1 , 1.5 0
. Note that neither subsystem is stabilizable.
6
1
0.8
3
3
γd, s(t)
0.6
0.4 (t)
s3
0.2
γd 3
0
−0.2
1
2
3
4 Iteration t
5
(t)
Fig. 1. Time history of the optimal value s3
6
7
in Algorithm 1 with h = 3.
Solving Problem 4 with h = 3, we have γ3d = 0.8205. The time (t) history of s3 defined in Algorithm 1 is shown in Figure 1. A locally (t ) optimal value of Problem 5 found by Algorithm 1 is s3 f = 0.5917, e and this proves 0.5917 ≤ γ3 = γ3 ≤ 0.8205. For another 2
0
example, consider the CSLS (1) with A1 = −1
A2
−2 0 = 2 0
0 −1.5 0 0
1 0 1.5 0
0 1.5 0 0
0 0 0 1 , B1 = 0, and B2 0 1.5 −1 1
0 1 0 0 , −2 0 0 2 0 0 = 1. 0
Solving Problem 4 with h = 3, one gets γ3d = 1.5711, while using (t ) Algorithm 1 we have s3 f = 1.4194 (optimal value at iteration step tf = 6). Therefore, 1.4194 ≤ γ3 = γ3e ≤ 1.5711. Finally, for the CSLS in Example 1, γ1d = 4.7701 and γhd = 8 for h ∈ {2, . . . , 7}. Underestimates obtained by using Algorithm 1 are identical to γhd for all h ∈ {1, . . . , 7}. Therefore, the SDP solution γhd is exact in this case. For the CSLSs in Example 3 and Example 4, we have γhd = 1 for h ∈ {1, . . . , 7}. In addition, Algorithm 1 produces the underestimates 1 for h ∈ {1, . . . , 7}. Thus, one concludes that γh = 1 for h ∈ {1, . . . , 7}. There are additional applications of the analysis given in this paper. First, the dual problem in Problem 4 with rank minimization heuristics [30], [31] potentially provides tighter upper bounds. Secondly, Problem 4 provides gaps between upper bounds on the minimum of quadratics function in Problem 2 and the individual quadratic functions. Therefore, it can be used to develop a new tree pruning algorithm as in [10] by removing inactive inequality constraints. Finally, for large-scale problems, distributed optimization methods [32] can be applied to solve Problem 4 in parallel. C ONCLUSION In this paper, we have developed a quantitative metric of stabilizability of CSLSs, called the h-CR, and the corresponding computational algorithms. Using Lagrangian duality of convex optimizations, we have developed algorithms to compute upper and lower bounds of the h-CR. A future research direction is to study the conditions under which the SDP relaxation is exact. Another possible direction is to study the connection between the result in this paper and those in [5], where an asymptotically exact necessary and sufficient condition was given in terms of Lyapunov theorem and SDP problems. R EFERENCES
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