FEEDBACK control under data-rate constraints has been an

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1 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY Feedback Stabilization of Switched Linear Systems With Unknown Disturbances Under Data-Rate Constraints Guosong Yang, Student Member, IEEE, and Daniel Liberzon, Fellow, IEEE Abstract We study the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to counteract the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization. Index Terms Input-to-state stability ISS, Lyapunov methods, quantized feedback, switched systems. I. ITRODUCTIO FEEDBACK control under data-rate constraints has been an active research area for years, as surveyed in [1] and []. In many application-related scenarios, it is important to limit the information flow in the feedback loop due to cost concerns, physical restrictions, security considerations, etc. Besides these practical motivations, the question of how much information is needed to achieve a certain control objective is fundamental and intriguing from the theoretical viewpoint. In this work, a finite data transmission rate is achieved by generating the control input based on sampled and quantized state measurements, which is a standard modeling framework in the literature see, e.g., [3], [4], and [5, Ch. 5]. This paper studies the problem of feedback stabilization under data-rate constraints in the presence of external disturbances. In this context, the authors of [3] and [4] assumed known bounds on the disturbances and addressed asymptotic stabilization with Manuscript received ovember 9, 016; revised July 11, 017; accepted October 10, 017. Date of publication October 30, 017; date of current version June 6, 018. This work was supported by the ational Science Foundation under Grant CS and Grant ECCS Recommended by Associate Editor P. Bolzern. Corresponding author: Guosong Yang. D. Liberzon is with the Coordinated Science Laboratory, University of Illinois at Urbana Champaign, Urbana, IL USA liberzon@illinois.edu. G. Yang is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA USA guosongyang@ucsb.edu. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAC minimum data rates, while the authors of [6] and [7] avoided such assumptions by alternating between zooming-out and zooming-in stages and achieved input-to-state stability ISS [8]. See also [9] and [10] for related results in a stochastic setting. The study of switched and hybrid systems has attracted a lot of attention lately particularly relevant results are discussed in [5], [11], and [1] and many references therein. In stability and stabilization of switched systems, it is a standard technique to impose suitable slow-switching conditions, especially in the sense of dwell time [13] and average dwell-time ADT [14]. This approach also plays a crucial role in our analysis. Toward switched systems with disturbances, Hespanha and Morse [14] showed that ISS can be achieved under the same ADT condition as the one for stability in the disturbance-free case. Their result was made explicit only for the case of switched linear systems, and many similar results for switched nonlinear systems have been established since then see, e.g., [15] for ISS with a dwell time, [16] for ISS and integral-iss with an ADT, and [17] for input/output-to-state stability with an ADT. Early works on control under data-rate constraints in the context of switched systems were devoted to quantized control of Markov jump linear systems [18] [0]. However, the discrete modes in the results above were always known to the controller, which would remove a major difficulty in our problem setup, making the control problem essentially the same as the one in the case without switching. The problem of asymptotically stabilizing a switched linear system without disturbance using sampled and quantized state feedback was studied in [1], which also serves as the basis for this work. In [1], the controller was assumed to have a partial knowledge of the switching, that is, the active mode was unknown except at sampling times, and the switching was subject to mild slow-switching conditions characterized by a dwell time and an ADT. Assuming that the data rate was large enough but finite, asymptotic stability was achieved by propagating overapproximations of reachable sets of the state over sampling intervals. See [] for a related result using output feedback. This work generalizes the main result of [1] in the presence of a completely unknown disturbance. By extending the approach of reachable-set approximation and propagation from [1], we develop a communication and control strategy that achieves a variant of ISS with exponential decay. Due to the unknown disturbance, the state may be driven outside the approximation of reachable set at a sampling time after it has already been inside an earlier one i.e., the state escapes the range of quantization. Consequently, the closed-loop system may alternate multiple times between stabilizing and searching stages IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 108 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 An estimate of the disturbance bound is introduced in approximating reachable sets so that the state cannot escape unless the disturbance is larger than the estimate. A novel algorithm is designed to adjust the estimate and recover the state after escapes, so that the total length of searching stages is finite and the system eventually stays in a stabilizing stage, provided that the disturbance is globally essentially bounded by an unknown value. To the best of our knowledge, this work is the first result that combines switching, disturbances, and data-rate constraints, with the exception of its preliminary version [3] and our earlier result [4] for the easier case of disturbances with known bounds. This paper improves [3] by formulating continuous gain functions in the main theorem, removing unnecessary conditions, substantiating the results with complete proofs and clarifying remarks, and providing a detailed simulation study. This paper is organized as follows. Section II introduces the system definition, the information structure, and the basic assumptions. Our main result is stated in Section III. Section IV explains the communication and control strategy, assuming that suitable approximations of reachable sets are available. Such approximations are constructed in Section V. Section VI details the stability analysis with major steps summarized as technical lemmas. The simulation study is provided in Section VII. Section VIII concludes this paper with a summary and an outlook on future research topics. II. PROBLEM FORMULATIO A. System Definition We are interested in stabilizing a switched linear control system modeled by ẋ = A σ x + B σ u + D σ d, x0 = x 0 1 where x R n x is the state, u R n u is the control, and d R n d is the external disturbance. The set {A p,b p,d p :p P}denotes a collection of matrix triples defining the modes subsystems, where P is a finite index set. The function σ : R 0 P is a right-continuous piecewise constant switching signal, which specifies the active mode σt at each time t. The solution x is absolutely continuous and satisfies the differential equation 1 away from discontinuities of σ in particular, there is no state jump. An admissible disturbance d is a Lebesgue measurable and locally essentially bounded function. The switching signal σ is fixed but unknown to the sensor and the controller a priori. Discontinuities of σ are called switching times, orsim- ply switches. The number of switches on a time interval τ,t] is denoted by σ t, τ. Our first basic assumption is that the switching is slow in the sense of combined dwell time and ADT. Assumption 1 Switching: The switching signal σ admits 1 a dwell-time τ d > 0 such that σ t, τ 1 for all τ 0 and t τ,τ + τ d ]; an ADT τ a >τ d such that σ t, τ 0 + t τ t>τ 0 τ a with a constant 0 1. The notions of dwell time [13] and ADT [14] have become standard in the literature on switched systems. In Assumption 1, the dwell-time condition item 1 can be written in the form of with τ a = τ d and 0 =1; meanwhile, the ADT condition item would be implied by the dwell-time condition if τ a τ d. Fig. 1. Information structure. Switching signals satisfying Assumption 1 were referred to as hybrid dwell-time signals in [5]. Our second basic assumption is that every individual mode is stabilizable. Assumption Stabilizability: For each p P, the pair A p,b p is stabilizable, that is, there exists a state feedback gain matrix K p such that A p + B p K p is Hurwitz. In the following analysis, it is assumed that such a collection of stabilizing gain matrices K p,p P has been selected and fixed. However, even in the disturbance-free case, and when all individual modes are stabilized via state feedback or stable without feedback, stability of the switched system is not necessarily guaranteed see, e.g., [5, p. 19]. Throughout this work, denotes the -norm of a vector, or the induced -norm of a matrix, that is, v := v := max 1 i n v i for v =v 1,...,v n R n, and M := M = max n 1 i n j=1 M ij for M = M ij R n n. The left-sided limit of a piecewise absolutely continuous function z approaching t is denoted by zt := lim s t zs. We let δ d denote the essential supremum -norm of the disturbance d, that is, δ d := d := ess sup ds 3 s 0 and call it the disturbance bound. In the following analysis, it is assumed that δ d is finite as the state bound 6 in our main result below holds trivially when δ d =. However, its value is unknown to the sensor and the controller. B. Information Structure The feedback loop consists of a sensor and a controller. The sensor measures two sequences of data quantized measurements samples of the state xt k, and indices of the active modes σt k and transmits them to the controller at sampling times t k = kτ s, where k is a nonnegative integer and τ s > 0 is the sampling period. Each sample is encoded by an integer i k from 0 to n x, where is an odd integer so that the equilibrium at the origin is preserved. The controller generates the control input u to the switched linear system 1 based on the decoded data. As σt k Pand i k {0, 1,..., n x },the data transmission rate between the encoder and the decoder is R = log n x +1 + log P τ s bits per unit of time, where P is the cardinality of the index set P i.e., the number of modes. As illustrated in Fig. 1, this information structure allows us to separate the sensing and the control tasks in the following sense: the sensor does not have access to the exact control objective, and the controller does not

3 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 109 have access to the exact state. The communication and control strategy is explained in detail in Section IV. The sampling period τ s is assumed to be no larger than the dwell time τ d in Assumption 1, that is, τ s τ d 4 so that there is at most one switch on each sampling interval t k,t k+1 ]. Due to the ADT τ a >τ d in Assumption 1, switches actually occur less often than once per sampling period. Our last basic assumption is a lower bound on the data rate. Assumption 3 Data rate: The sampling period τ s satisfies Λ p := e A p τ s < p P. 5 The inequality in 5 can be interpreted as a lower bound on the data rate R since it requires τ s to be sufficiently small with respect to. This bound is the same as the one for the disturbance-free case in [1, Assumption 3], and similar datarate bounds appeared in [3], [4], and [6] for stabilizing nonswitched linear systems; see [7, Sec. V] and [1, Sec..] for more discussions on their relation. III. MAI RESULT The control objective is to stabilize the system defined in Section II-A under the data-rate constraint described in Section II-B in a robust sense. More precisely, we intend to establish the following ISS-like property. Theorem 1 Exponential decay: Consider the switched linear control system 1. Suppose that Assumptions 1 3 and inequality 4 hold. Then, there is a communication and control strategy that yields the following property: provided that the ADT τ a is large enough, there exist a constant λ > 0 and gain functions g, h : R 0 R >0 such that for all initial states x 0 R n x and disturbances d : R 0 R n d,wehave xt e λt g x 0 +h d t 0. 6 The communication and control strategy is described in Section IV. The lower bound on τ a is given by 49 in Section VI-A3. The exponential decay rate λ is given by 64, and the nonlinear gain functions g and h are given by 65, both in Section VI-C3. From the proof, it will be clear that both g and h can be made continuous and strictly increasing. However, g0 > 0 due to the sampling and quantization, h0 > 0 due to the unknown disturbance, and both gs and hs have superlinear growth rates as s, which is consistent with [7, Cor..3]. Consequently, the state bound 6 does not give the standard ISS [8], but rather the input-to-state practical stability [8] with exponential decay, that is, xt e λt γ x x 0 +γ d d +C t 0 with the gain functions γ x,γ d K defined by 1 γ x s :=gs g0, γ d s :=hs h0 7 and the constant C := g0 + h0 > 0. 8 Remark 1: Along the lines of [9, Sec. VI], the state bound 6 can be restated as ISS with respect to a set. More specifically, 6 implies that the uniform asymptotic gain UAG property 1 A function f : R 0 R 0 is of class K if it is continuous, positive definite, strictly increasing, and unbounded. [9] holds for the set A := {v R n x : v h0}, that is, for each pair of constants ε, δ > 0, there exists a time T ε,δ := max{lngh0 + δ/ε/λ, 0} such that if x 0 A δ, then xt A γ d d +ε for all t T ε,δ with the gain function γ d K defined in 7, where v A := inf v A v v is the Chebyshev distance from a point v to the set A. Inthe context of nonswitched systems, it has been shown that if UAG holds for A, then the system is ISS with respect to the closure of the reachable set from A with d 0 [9, Lemma VI.]. The state bound 6 implies the following stability property. Corollary Practical stability: In particular, the communication and control strategy in Theorem 1 yields the following property: provided that the ADT τ a is large enough, the system 1 is practically stable, that is, for each ε>0, there exists a small enough δ>0such that for all initial states x 0 R n x and disturbances d : R 0 R n d,wehave x 0, d δ sup xt ε + C 9 t 0 with the constant C defined by 8. Corollary shows that, if the initial state and the disturbance are both small, then the solution is confined within a neighborhood of the hypercube of radius C centered at the origin. In Section VI-D, we will establish practical stability with a smaller constant C through a more direct approach. IV. COMMUICATIO AD COTROL STRATEGY In this section, we describe the communication and control strategy in detail, assuming that suitable approximations of reachable sets of the state are available at all sampling times. Such approximations are derived in the next section. The initial state x 0 is unknown. At t 0 =0, the sensor and the controller are both provided with x 0 =0 and arbitrarily selected initial estimates E 0 > 0 and δ 0 > 0 for x 0 and the disturbance bound δ d defined in 3, respectively. Starting from t 0 =0, at each sampling time t k, the sensor determines if the state xt k is inside the hypercube of radius E k centered at x k denoted by S k := {v R n x : v x k E k } or equivalently, if xt k x k E k. 10 The hypercube S k is the approximation of the reachable set at t k, which is also used as the range of quantization. If 10 holds i.e., if xt k S k, we say that the state is visible, and the system is in a stabilizing stage described in Section IV-A. Otherwise, the state is lost, and the system is in a searching stage described in Section IV-B. To compensate for the unknown disturbance, we introduce an estimate δ k of the disturbance bound δ d in calculating E k+1. ote that if δ k <δ d, then it is possible that xt k S k but xt k+1 / S k+1 unlike in the disturbance-free case, where xt k S k implies that xt l S l for all l k. If the state is visible at t k, then the system is in a stabilizing stage until the first sampling time t j >t k such that xt j / S j ; in this case, we say that the state escapes at t j. Likewise, if the state is lost at t k, then the system is in a searching stage until the first sampling time t i >t k such that xt i S i ;in this case, we say that the state is recovered at t i. Due to the unknown disturbance, the system may alternate multiple times between stabilizing and searching stages. The rules for adjusting

4 110 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 the estimate δ k so that there are only a finite number of escapes are explained in Section IV-C. A. Stabilizing Stage At each sampling time t k in a stabilizing stage, the encoder divides the hypercube S k into n x equal hypercubic boxes, per dimension, encodes each box by a unique integer index from 1 to n x, and transmits the index i k of the box containing xt k to the decoder, along with the active mode σt k. The controller learns that 10 holds upon receiving i k {1,..., n x }.The decoder follows the same predefined indexing protocol as the encoder, so that it is able to reconstruct the center c k of the hypercubic box containing xt k from i k. Simple calculation shows that xt k c k 1 E k, c k x k 1 E k. 11 The controller then generates the control input ut =K σ tk ˆxt for t [t k,t k+1, where K σ tk is the state feedback gain matrix in Assumption, and ˆx is the state of the auxiliary system ˆx = A σ tk ˆx + B σ tk u =A σ tk + B σ tk K σ tk ˆx 1 with the boundary condition ˆxt k =c k. 13 In particular, the auxiliary state ˆx is reset to c k at each sampling time t k in a stabilizing stage. Both the sensor and the controller then use two functions F and G to calculate x k+1 := F σt k,σt k+1,c k E k+1 := Gσt k,σt k+1,x k,e k,δ k 14 for the next sampling time t k+1 without further communication. The functions F and G are designed so that xt k+1 x k+1 Gσt k,σt k+1,x k,e k,δ d 15 and G is strictly increasing in the last argument, which is δ k in 14 and δ d in 15. Hence, the state may escape at t k+1 only if δ k <δ d. However, xt k+1 S k+1 does not imply that δ k δ d. The formulas for F and G are derived in Section V-A. B. Searching Stage At each sampling time t k in a searching stage, there is an unknown ˆD k such that E k < xt k x k ˆD k. 16 For example, if the state escapes at t j, then 15 implies that ˆD j = Gσt j 1,σt j,x j 1,E j 1,δ d, while if it is lost at t 0 =0, then ˆD 0 = x 0. The encoder sends i k =0, the overflow symbol, to the decoder. Upon receiving i k =0, the controller learns the state is lost and sets the control input to be u 0 on [t k,t k+1. Both the sensor and the controller then use a function Ĝ to calculate x k+1 := x k E k+1 := Ĝx k, 1 + ε E E k,δ k 17 for the next sampling time t k+1 without further communication, where ε E > 0 is an arbitrary design parameter. The function Ĝ is designed so that xt k+1 x k+1 Ĝx k, ˆD k,δ d 18 and it is strictly increasing in the last two arguments. ote that the second argument of Ĝ in 18 is ˆD k, whereas the one in 17 is 1 + ε E E k. With the additional coefficient 1+ε E,itis ensured that the growth rate of E k dominates that of ˆD k ; thus, the state will be recovered in a finite time, as shown in Section V-B following the derivation of Ĝ. C. Adjusting the Estimate of the Disturbance Bound When the state escapes at a sampling time t j, the sensor and the controller learn that δ j 1 <δ d and adjust the estimate by enlarging it to δ j =1+ε δ δ j 1, where ε δ > 0 is an arbitrary design parameter. The estimate remains unchanged in all other cases; in particular, it is adjusted once per searching stage. Thus, it is ensured that there is a finite number of searching stages in total, as the estimate becomes greater than or equal to the disturbance bound δ d after finitely many adjustments, and the state cannot escape after that. V. APPROXIMATIO OF REACHABLE SETS In this section, we derive the recursive formulas needed to implement the communication and control strategy. In Section V-A, we consider a stabilizing stage and formulate the functions F and G in 14 so that 15 holds. In Section V-B, we consider a searching stage, formulate the function Ĝ in 17 so that 18 holds, and prove that the state will be recovered in a finite time. A. Stabilizing Stage Suppose that the state is visible at a sampling time t k, that is, 10 holds. 1 Sampling Interval With o Switch: When σt k =p = σt k+1 19 for some p P,thereisnoswitchont k,t k+1 ] due to 4. Combining the switched linear system 1 and the auxiliary system 1, we obtain that ẋ = A p x + B p u + D p d ˆx = A p ˆx + B p u. The error e := x ˆx satisfies that ė = A p e + D p d, et k = xt k c k 1 E k on [t k,t k+1, where the boundary condition follows from 11 and 13. Hence tk et k+1 = +1 e A p τ s et k + e A p t k +1 τ D p dτdτ t k τ s e A p τ s et k + e A p s D p ds δ d 0 Λ p E k +Φ p τ s δ d =: ˆD k+1 with the constant Λ p in 5 and the increasing function Φ p : [0,τ s ] R defined by Φ p t := t 0 e A p s D p ds. 0

5 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 111 Therefore, we set E k+1 = Gp, p, x k,e k,δ k := Λ p E k +Φ p τ s δ k. 1 As x is continuous, 15 holds with x k+1 set as the auxiliary state ˆx approaching t k+1, that is, x k+1 = F p, p, c k :=ˆxt k+1 =S pc k with the matrix S p := e A p +B p K p τ s. Sampling Interval With a Switch: When σt k =p q = σt k+1 3 for some p, q P, there is exactly one switch on t k,t k+1 ] due to 4. Let t k + t with t 0,τ s ] denote the unknown switching time. Then { p, t [tk,t k + t σt = q, t [t k + t, t k+1 ]. Before the switch, mode p is active on [t k,t k + t. Following essentially the calculations for the case with no switch, we see that the error e = x ˆx satisfies that et k + t ea p t E k +Φ p tδ d with the function Φ p defined by 0. As t k + t is unknown, we estimate xt k + t by comparing it with ˆxt k + t = e A p +B p K p t c k of the auxiliary system 1 at an arbitrarily selected time t k + t [t k,t k+1 ] via the triangle inequality. First ˆxt k + t ˆxt k + t e A p +B p K p t e A p +B p K p t c k e A p +B p K p t e A p +B p K p t x k + 1 E k where the last inequality follows partially from 11. Then xt k + t ˆxt k + t ˆxt k + t ˆxt k + t + et k + t e A p +B p K p t e A p +B p K p t x k + 1 E k + ea p t E k +Φ p tδ d =: ˆD k+1t, t, δ d. 4 After the switch, mode q is active on [t k + t, t k+1 ]. Combining the switched linear system 1 and the auxiliary system 1 with u = K p ˆx, we obtain that ż = Āpqz + D q d for z := x, ˆx R n x with the matrices [ ] [ ] Aq B q K p Dq Ā pq :=, Dq =. 0 n x n x A p + B p K p 0 n x n d Combining it with a second auxiliary system ẑ = Āpqẑ, ẑt k + t =ˆxt k + t, ˆxt k + t 5 we obtain that with the boundary condition zt k + t ẑt k + t ż = Āpqz + D q d ẑ = Āpqẑ = max{ xt k + t ˆxt k + t, ˆxt k + t ˆxt k + t } ˆD k+1t, t, δ d where the first inequality follows from the property that the -norms of two vectors v, w and their concatenation satisfy v,w = max{ v, w }. 6 Hence zt k+1 ẑt k+1 t + t tk +1 = eā pqτ s t zt k + t+ eā pqt k +1 τ Dq dτdτ t k + t eā pqτ s tẑt k + t eā pqτ s t zt k + t ẑt k + t τ s t + eā pqs Dq ds 0 eā pqτ s t ˆD k+1t, t, δ d + Φ pq τ s tδ d with the increasing function Φ pq :[0,τ s ] R defined by Φ pq t := t 0 δ d eā pqs Dq ds. Again, we estimate zt k+1 by comparing it with ẑt k + t = eā pqt t ẑt k + t of the second auxiliary system 5 at an arbitrarily selected time t k + t [t k,t k+1 ] via the triangle inequality. First ẑt k+1 t + t ẑt k + t eā pqτ s t eā pqt t ẑt k + t eā pqτ s t eā pqt t e A p +B p K p t x k + 1 E k where the last inequality follows partially from 11 and 6. Then zt k+1 ẑt k + t zt k+1 ẑt k+1 t + t + ẑt k+1 t + t ẑt k + t eā pqτ s t ˆD k+1t, t, δ d + eā pqτ s t eā pqt t e A p +B p K p t x k + 1 E k + Φ pq τ s tδ d =: ˆD k+1t,t, t, δ d. 7

6 11 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 To remove the dependence on the unknown t, we take the supremum over t with fixed t and t and obtain that zt k+1 ẑt k + t sup ˆD k+1t,t, t, δ d =: ˆD k+1. t 0,τ s ] Therefore, we set E k+1 by first replacing the disturbance bound δ d in ˆD k+1 t,t, t, δ d with the estimate δ k and then taking the maximum over t, that is, E k+1 = Gp, q, x k,e k,δ k := sup t 0,τ s ] ˆD k+1t,t, t, δ k. 8 Clearly, the design parameters t and t should be selected so that E k+1 is minimized. However, their optimal values cannot be determined without imposing further constraints on the matrices {A p,b p,d p,k p : p P}. As x is continuous, 15 holds with x k+1 set as the projection of the second auxiliary state ẑ approaching t k + t onto the x-component, that is, x k+1 = F p, q, c k :=I n x n x with the matrix H pq := I n x n x 0 n x n x eā pqt t 0 n x n x ẑt k + t =H pq c k 9 In x n x I n x n x e A p +B p K p t. In the remainder of this subsection, we derive a simpler but more conservative bound for E k+1, which is more useful for computations. First, the norm of the difference of two matrix exponentials can be simplified via the following lemma. Lemma 1: For all square matrices X and Y,wehave e X +Y e X e X + Y Y. Proof: See Appendix A. Based on Lemma 1 and the property that e Ms e M s M R n n s R the definition 8 implies that with the constants E k+1 α pq x k + β pq E k + γ pq δ k 30 α pq := e Ā pq τ s e A p +B p K p max{τ s, t } A p + B p K p max{τ s t,t } + e Ā pq max{τ s, t t,τ s +t t } Āpq max{t t,τ s + t t } e A p +B p K p t β pq := 1 α pq + 1 e Ā pq + A p τ s γ pq := e Ā pq τ s Φ p τ s + Φ pq τ s. 31 Remark : If we set t = t =0, then 30 becomes E k+1 α 0 pq x k + β 0 pqe k + γ pq δ k Using Lemma 1 instead of the inequality that M I M +1 for all square matrices M as in [1, eq. 0] ensures that α pq 0 as τ s 0, a property we will use in the comparison to [14] in Remark 4. However, for a large enough τ s, it is possible that the bound in Lemma 1 is worse. with the constants α 0 pq := e Ā pq τ s e A p +B p K p τ s A p + B p K p τ s + Āpq τ s βpq 0 := 1 α0 pq + 1 e Ā pq + A p τ s. 3 Although this choice of t and t considerably simplifies the formula of the bound, it does not necessarily minimize E k+1. B. Searching Stage Suppose that the state is lost at a sampling time t k, that is, 16 holds. 1 Reachable-Set Approximation: Let p = σt k, and consider an arbitrary t t k,t k+1 ].Ifσt =p, then there is no switch on t k,t] due to 4; thus xt x k = ea p t t k xt k + t t k e A p t τ D p dτdτ x k e A p t t k I x k + e A p t t k xt k x k t tk + e A p s D p ds 0 Γ x k + Λ ˆD k + Φδ d with the constants Γ := max t [0,τ s ],p P ea p t I Λ := max t [0,τ s ],p P ea p t 1 Φ := max Φ pt = max Φ pτ s. t [0,τ s ],p P p P δ d 33 If σt =q p, then there is exactly one switch on t k,t] due to 4; thus xt x k = ea q t t k t xt k + t+ t t k + t e A q t τ D q dτdτ x k e A q t t k t I x k + e A q t t k t xt k + t x k t tk t + e A q s D q ds 0 δ d Γ x k + Λ xt k + t x k + Φδ d Γ x k + Λ Γ x k + Λ ˆD k + Φδ d + Φδ d Λ+1 Γ x k + Λ ˆDk + Λ+1 Φδ d where t k + t denotes the unknown switching time. As Λ 1, the bound for the second case holds for both cases, that is, xt x k ᾱ x k + β ˆD k + γδ d =: ˆD k+1 34 for all t t k,t k+1 ] with the constants ᾱ := Λ+1 Γ, β := Λ, γ := Λ+1 Φ.

7 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 113 From β = Λ 1, it follows that ˆD k+1 ˆD k. In order to dominate the growth rate of ˆD k+1,weset E k+1 = Ĝx k, 1 + ε E E k,δ k := ᾱ x k +1+ε E βe k + γδ k 35 with the arbitrary design parameter ε E > 0. Recovery in a Finite Time: Suppose that the state escapes at a sampling time t j oritislostatt j = t 0 =0 and remains lost at t j+1,...,t k 1. Then, the disturbance estimate satisfies that δ k 1 = = δ j+1 = δ j =1+ε δ δ j 1. From the recursive formulas 34 and 35, it follows that ˆD k = β k j ˆDj + β k j 1 β 1 ᾱ x j + γδ d E k = ˆβ k j E j + ˆβ k j 1 ˆβ 1 ᾱ x j + γδ j 36 with the constant 3 ˆβ := 1 + εe β > β. Let c β := ˆβ 1/ β 1, and consider the integer-valued functions η E,η δ : R 0 Z 0 defined by { log1+εe s, s > 1 η E s := 0, 0 s 1 { log1+εe c β s, s > 1 η δ s := 37 0, 0 s 1 where : R Z denotes the ceiling function, that is, s := min{m Z : m s}. Consider the integer k := j + max{η E ˆD j /E j,η δ δ d /δ j }. First, it holds that ˆβ k j E j β k j 1 + ε E η E ˆD j /E j E j β k j ˆDj. Second, if δ d δ j, then ˆβ k j 1 ˆβ 1 δ j β k j 1 β 1 δ d due to ˆβ > β and k j; otherwise ˆβ k j 1 ˆβ 1 δ j > β k j 1 β 1 β 1 ˆβ ε E k j δ j β k j 1 β 1 β 1 ˆβ ε E η δ δ d /δ j δ j β k j 1 β 1 δ d. Hence, E k ˆD k, that is, the state is recovered no later than t k. Denote by t i the sampling time of recovery. Then 4 i j max{η E ˆD j /E j,η δ δ d /δ j } From 33, it follows that β = Λ 1, and Λ =1only if all eigenvalues of all A p have nonpositive real parts. In the following analysis, we assume that β >1 so that the first formula in 36 is well defined, which can be achieved by letting β =max{ Λ, 1+ε} for an arbitrary ε>0 if necessary. The special case where β =1can be treated using similar arguments and is omitted here for brevity. 4 The function η δ is piecewise defined since if δ d δ j in 38 which is possible as the escape only implies δ d >δ j 1 = δ j /1 + ε δ then the second However, δ d being unknown implies that neither the sensor nor the controller is able to predict how long it will take to recover the state. VI. STABILITY AALYSIS In this section, we show that the communication and control strategy described in Section IV fulfills the claims of Theorem 1. In Section VI-A, we formulate a Lyapunov-based bound with exponential decay for stabilizing stages. Then, we derive its exponential growth for searching stages in Section VI-B. In Section VI-C, we calculate the maximum number of searching stages and prove the ISS-like property in Theorem 1. A stronger version of Corollary is established in Section VI-D. A. Stabilizing Stage 1 Sampling Interval With o Switch: Consider a sampling interval t k,t k+1 ] such that 19 holds, as in Section V-A1. As A p + B p K p is Hurwitz, for S p in, there exist positivedefinite matrices P p,q p R n x n x such that S p P p S p P p = Q p < Let λm and λm denote the largest and smallest eigenvalues of a matrix M, respectively, and define χ p := n x S p P p S p λq p + n x S p P p S p. 40 Due to the inequality in 5, there exists a sufficiently small constant φ 1 > 0 such that 1 + φ 1 Λ p < for all p P. Then, for each p, there exists a sufficiently large constant ρ p > 0 such that 1 χ p φ 1Λ p ρ p < Consider a family of positive-definite functions V p : R n x R 0 R 0 defined by V p x, E :=x P p x + ρ p E, p P. 4 For the sampling interval t k,t k+1 ] with no switch, the following lemma provides a bound for V σ tk +1 x k+1,e k+1 in terms of V σ tk x k,e k and the disturbance estimate δ k. Lemma : Consider a sampling interval t k,t k+1 ] such that 10 and 19 hold. Then V p x k+1,e k+1 νv p x k,e k +ν d δ k 43 term on the right-hand side of the second formula in 36 is larger than or equal to that of the first formula for all k j. Similarly, the function η E is piecewise defined since if ˆD 0 = x 0 E 0 in 57, then there is no searching stage at the beginning.

8 114 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 with the constants 5 ν := max p P ν p { 1 ν p := max ν d := max p P χ p φ 1Λ p ρ p, 1 λq } p λp p 1+ 1φ1 ρ p Φ p τ s. 44 Proof: See Appendix B. Sampling Interval With a Switch: Consider a sampling interval t k,t k+1 ] such that 3 holds, as in Section V-A. Let h pq be the largest singular value of H pq in 9, that is, h pq := λhpqh pq. Consider the functions V p and V q defined by 4. For the sampling interval t k,t k+1 ] with a switch, the following lemma provides a bound for V σ tk +1 x k+1,e k+1 in terms of V σ tk x k,e k and the disturbance estimate δ k. Lemma 3: Consider a sampling interval t k,t k+1 ] such that 10 and 3 hold. Then V q x k+1,e k+1 μv p x k,e k +μ d δk 45 with the constants μ := max μ pq p,q P { λpq h pq μ pq := max λp p 1 n x λp q h pq μ d := max p,q P ρ p + + φ α pqρ q λp p + + φ β pqρ q ρ p 1+ φ ρ q γ pq 46 where φ > 0 is an arbitrary design parameter. Proof: See Appendix C. Remark 3: From the definitions of ν in 44 and the inequality 41, it follows that ν<1. Meanwhile, if we set t = t =0 in 9, then h pq =1for all p, q P, and from the definition of μ in 46, it follows that μ 1 >ν. While this may not hold for general t,t [0,τ s ], we are able to ensure μ>ν 47 by letting μ =1if all μ pq < 1. Meanwhile, the relation between μ d and ν d depends on the values of φ 1 and φ. Since 45 holds for all φ > 0, given an arbitrary φ 1, a sufficiently small φ e.g., φ = φ 1 can be selected so that μ d ν d. 48 Alternatively, we can simply replace μ d with max{μ d,ν d } if necessary. In the following analysis, we assume the inequalities 47 and 48 hold. Consequently, the bound in 45 holds for all 5 The denominator in the second term of the maximum in the definition of ν p in 44 is reduced to 1/n x of the corresponding term in [1, eq. 34]. This improvement is due to the more suitable inequalities 67 and 68 from linear algebra. The first numerator in the first term of the maximum in the definition of μ pq in 46 is reduced to 1/n x of the corresponding term in [1, eq. 37] for the same reason. } sampling intervals in stabilizing stages, regardless of whether there is a switch. 3 Combined Bound at Sampling Times: Combining the bounds 43 and 45, we derive a condition on the ADT τ a in that ensures a bound with exponential decay for V σ tk x k,e k at sampling times t k in a stabilizing stage. Lemma 4: Consider a sequence of consecutive sampling times t i,...,t k 1 in a stabilizing stage. Suppose that the ADT τ a satisfies τ a > 1+ ln μ τ s. 49 ln1/ν There exists a sufficiently small constant φ 3 0, 1 such that V σ tk x k,e k < Θ 0 θ k i V σ ti x i,e i +Θ d δi 50 with the constants 0 in and θ := μ + φ 31 νμ d /ν d τ s /τ a ν + φ 3 1 ν τ s /τ a 1 < 1 Θ:= μ + φ 31 νμ d /ν d > 1 ν + φ 3 1 ν μ Θ d := φ 3 1 ν ν d + μ d. 51 Proof: See Appendix D. Remark 4: In [14], the authors considered switched linear systems with inputs disturbances and derived a lower bound on the ADT that ensured a variant of ISS with exponential decay. 6 The lower bound 49 on the ADT τ a in Lemma 4, in the absence of sampling and quantization, is consistent with the one in [14, Th. ]. More specifically, the case without sampling and quantization can be approximated by letting τ s 0 and. Consequently, S p I +A p + B p K p τ s in, H pq I in 9, and α pq,β pq 0 in 31; thus λq p ν 1 min p P λp p, μ max λp q p,q P λp p in 44 and 46 with large enough ρ p for all p P. Moreover, the first-order approximation in τ s of the Lyapunov equation 39 is given by Ap + B p K p P p + P p A p + B p K p τ s = Q p. As the index set P is finite, from Assumption, it follows that there exists a constant λ 0 > 0 such that all A p + B p K p + λ 0 I are Hurwitz; thus, the approximated Lyapunov equation above holds with P p satisfying A p + B p K p + λ 0 I P p + P p A p + B p K p + λ 0 I= I and Q p =λ 0 P p + Iτ s. Then, 49 can be approximated by τ a > lnμ λpp min p P λp p λ λp p with μ := max p,q P,p q λp q /λp p which is in a similar form as the lower bound τ a τa ln μ > λ 0 6 More precisely, the result in [14] is stated in terms of input-to-state e λt - weighted, L -induced norm, which ensures an exponential decay rate.

9 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 115 in [14] see [5, p. 59, eq. 3.10] for an explicit bound on τ a. The additional terms result from the more complex Lyapunov functions 4 we used due to the sampling and quantization. In particular, the additional coefficients in the numerator and the first term of the denominator are generated when completing the squares. Meanwhile, we can made λp p arbitrarily large and thus the second term of the denominator arbitrarily small by selecting a sufficiently small λ 0. B. Searching Stage 1 Recovery: Suppose that the state escapes at a sampling time t j and is recovered at t i, as in Section V-B. Then, [t j,t i is a searching stage, while the sampling period [t j 1,t j belongs to a stabilizing stage. At t j, from 15 and 16, it follows that with E j < xt j x j ˆD j ˆD j = Gσt j 1,σt j,x j 1,E j 1,δ d E j = Gσt j 1,σt j,x j 1,E j 1,δ j 1. From formulas 1 and 8 of G, it follows that ˆD j /E j <δ d /δ j 1 =1+ε δ δ d /δ j. Let c ε := max{1+ε δ, ˆβ 1/ β 1} and define an integer-valued function η : R 0 Z 0 by ηs := max{η E 1 + ε δ s, η δ s} log 1+εE c ε s, s > 1 = log 1+εE 1 + ε δ s, 1 + ε δ 1 <s 1 0, 0 s 1 + ε δ 1. 5 From 38, it follows that which, combined with 36, implies that i j ηδ d /δ j 53 E i = ˆβ i j E j + ˆβ i j 1 ˆβ 1 ᾱ x j + γδ j < ˆβ ᾱ i j ˆβ 1 x j + E j + < ˆβ η δ d /δ j ᾱ ˆβ 1 x j + E j + γ ˆβ 1 δ j γ ˆβ 1 δ j. 54 For the searching stage [t j,t i, the following lemma provides a bound for V σ ti x i,e i at the recovery in terms of V σ tj x j,e j at the escape and δ d,δ j. Lemma 5: Suppose that the state escapes at a sampling time t j and is recovered at t i. Then V σ ti x i,e i ˆβ η δ d /δ j ωv σ tj x j,e j +ω d δj 55 with ω := max ω pq p,q P { λpq ω pq := max λp p + + φ 4ᾱ ρ q ˆβ 1 λp p, + φ } 4ρ q ρ p ω d := max 1+ γ ρ q q P φ 4 ˆβ 56 1 where φ 4 > 0 is an arbitrary design parameter. Proof: See Appendix E. Initial Capture: The case where the state is lost at t 0 =0 and is recovered at t i0 for the first time can be analyzed in a similar manner. From 38 with j =0and ˆD 0 = x 0, it follows that i 0 η E x 0 /E 0 +η δ δ d /δ 0 57 which, combined with 36 and x 0 =0, implies that E i0 < ˆβ η E x 0 /E 0 +η δ δ d /δ 0 E 0 + γ ˆβ 1 δ 0. For the searching stage [0,t i0, the following lemma provides a bound for V σ ti 0 0,E i0 at the first recovery in terms of V σ 0 0,E 0 =ρ σ 0 E0 at t =0and x 0,E 0,δ d,δ 0. Lemma 6: Suppose that the state is lost at t 0 =0 and is recovered at t i0. Then V σ ti 0 0,E i0 ˆβ η E x 0 /E 0 +η δ δ d /δ 0 ω 0 V σ 0 0,E 0 +ω d δ0 58 with ω 0 := max 1+ φ 4 ρq 1 q P ρ σ 0 ω where φ 4 is the design parameter in 56. Proof: The proof is essentially the same as the one of Lemma 5 and is omitted here. C. Exponential Decay 1 umber of Searching Stages: As explained in Section IV-C, the closed-loop system alternates between a finite number of searching and stabilizing stages and eventually stays in a stabilizing stage. Let 0=j 0 i 0 <j 1 <i 1 < <j s <i s be such that [t jm,t im denotes a searching stage and [t im,t jm +1 denotes a stabilizing stage for each m {0,..., s }. 7 As the estimate is enlarged by a factor of 1+ε δ every time the state escapes, it satisfies δ k =1+ε δ m δ 0 k {j m,...,j m +1 1}. Hence, s d δ d with the integer-valued function d : R 0 Z 0 defined by { log1+εδ s/δ 0, s > δ 0 d s := 59 0, 0 s δ 0. 7 There is a searching stage at the beginning i.e., i 0 > 0 if and only if x 0 >E 0 ; for the final stabilizing stage to be well defined, we let j s +1 := and t j s +1 :=.

10 116 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 Global Bound at Sampling Times: Combining the bound in Lemma 4 for stabilizing stages and the ones in Lemmas 5 and 6 for searching stages, we establish a global bound for V σ tk x k,e k in stabilizing stages in terms of ρ σ 0 at t =0and x 0,E 0,δ d,δ 0. Lemma 7: Consider a sampling time t k such that 10 holds. Then V σ tk x k,e k Θ 0 Ψ d δ d ψ L d δ d θ k ψ L x x 0 ω 0 ρ σ 0 E0 + ω d δ0 +C d δ d δ0 with the functions L x,l d : R 0 Z 0 and C d : R 0 R >0 defined by 8 L x s :=η E s/e 0 d s L d s :=η δ s/δ 0 + η1 + ε δ l s/δ 0 l=1 d s C d s :=Θ d +Θ d + ω d and the constants ψ := ˆβθ 1/, Ψ:=ωΘ 0, ψ d := 1 + ε δ /Ψ where η E and η δ are defined by 37, η by 5, and d by 59. Proof: See Appendix F. Remark 5: The gain functions d,l x,l d, and C d in Lemma 7 are piecewise constant and satisfy that L x s =0for all 0 s E 0 and that d s =L d s =0for all 0 s δ 0. A more conservative bound that depends continuously on x 0 and δ d can be established by replacing them with continuous strictly increasing gain functions. First, d s d s for all s 0 with d K defined by { 1 + log1+εδ s/δ 0, s > δ 0 d s := s/δ 0, 0 s δ 0. Second, L x s L x s for all s 0 with L x K defined by { 1 + log1+εe s/e 0, s > E 0 L x s := s/e 0, 0 s E 0. Third, L d s L d s for all s 0 with L d K defined by L d s := log 1+εE c β s/δ 0 + d s 1 log 1+εE c ε s/δ 0 l=1 + log 1+εE s/δ 0 + d s+1 d s d s+1/ 1 log 1+εE 1 + ε δ for s>δ 0 ; and L d s := + log 1+εE c β s/δ 0 for 0 s δ 0. Finally, C d s C d s for all s 0 with the continuous strictly increasing function C d : R 0 R >0 defined by 9 ψ l d C d s :=Θ d + 1 ψ d s d ψ d Θ d + ω d. 1 ψ d 8 The sum L x x 0 +L d δ d gives a bound for the total length of all searching stages in terms of sampling intervals. 9 For C d to be well defined, the design parameter ε δ should be selected so that ψ d 1. The special case where ψ d =1can be treated via similar arguments and is omitted here for brevity cf., footnote 3. 3 Intersample Bound: First, consider an arbitrary t in a stabilizing stage, that is, t [t k,t k+1 ] such that 10 holds. Following similar arguments as in Section V-A with t = t = 0, we estimate xt by comparing it with c k in 11, the center of the hypercubic box containing xt k, via the triangle inequality. Ifthereisnoswitchont k,t], then 4 holds with t t k in place of t; thus xt c k = xt ˆxt k ˆD k+10,t t k. Otherwise, there is exactly one switch on t k,t] due to 4, and 7 holds with t in place of t k+1 and t 0,t t k ] denoting the unknown switching time; thus xt c k zt ẑt k ˆD k+10, 0,t t k where the first inequality follows from 6. Comparing the corresponding coefficients in 4, 7, 31, and 3, we see that in both cases xt c k α 0 x k + β 0 E k + γδ d with α 0 := max p,q P α0 pq, β 0 := max p,q P β0 pq, γ := max pq. p,q P 60 Applying the triangle inequality, we obtain xt c k + xt c k α 0 +1 x k + β E k + γδ d where the second inequality follows from 11. Let λ P min := min p P λp p, ρ min := min ρ p, p P and define λ P := max p P λp p ρ := max p P ρ p 61 ξ := λp min β / ρ min α 0 +1 α 0 +1 Ξ:= + β0 +1 1/. ρ min λ P min From Young s inequality with ξ, it follows that 10 α 0 +1 x k + β E k 1 + ξα 0 +1 x k + =Ξ λ P min x k + ρ min E k Ξ V σ tk x k,e k ξ β Ek Hence xt Ξ V σ tk x k,e k +γδ d which, combined with Lemma 7 and Remark 5, implies that xt ΞΘ 0 / Ψ d δ d / ψ L d δ d θ k/ ψ L x x 0 ω 0 ρe 0 + ω d δ 0 + Cd δ d δ 0 + γδ d 10 For an ε>0, Young s inequality with ε states that ab εa /+b /ε for all a, b R. Whenε =1, the term with ε is omitted for brevity.

11 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 117 where the last inequality follows partially from the property a + b a + b a, b 0. 6 Moreover, due to t [t k,t k+1 ] and θ<1, it holds that θ k/ θ 1/ θ t/τ s. Second, consider an arbitrary t [t jm,t im.from34,it follows that xt x j m ˆD im E im. Following similar arguments as in the first case, we obtain xt x j m + E im α 0 +1 x i m + β E im Ξ V σ ti m x i m,e im ΞΘ 0 / Ψ d δ d / ψ L d δ d θ i m / ψ L x x 0 ω 0 ρe 0 + ω d δ 0 + Cd δ d δ 0 + γδ d in which θ i m / θ t/τ s <θ 1/ θ t/τ s. Combining the results above, we obtain xt ΞΘ 0 / Ψ d δ d / ψ L d δ d θ t/τ s ψ L x x 0 θ ω 0 ρe 0 + ω d δ 0 + Cd δ d δ 0 + γδ d 63 for all t 0. From Young s inequality with an arbitrary design parameter φ>0, it follows that xt 1 φ ψ L x x 0 + φ Ψ d δ d ψ L d δ d ΞΘ 0 / θ θ t/τ s ω 0 ρe 0 + ω d δ 0 +ΞΘ 0 / Ψ d δ d / ψ L d δ d Cd δ d δ 0 + γδ d = 1 ΞΘ 0 / ω 0 ρe 0 + ω d δ 0 ψ L x x 0 θ t/τ s φ θ + φ ΞΘ 0 / θ ω 0 ρe 0 + ω d δ 0 Ψ d δ d ψ L d δ d +ΞΘ 0 / Cd δ d δ 0 Ψ d δ d / ψ L d δ d + γδ d for all t 0. Remark 6: Here, Young s inequality is applied to restate the state bound 63 in the standard ISS form 6. However, besides increasing the value of the state bound, it also has the following consequence. If there is no disturbance and the sensor and the controller know that, then δ d = δ 0 =0; thus, 63 becomes xt ΞΘ 0 / ω 0 ρ/θe 0 θ t/τ s ψ L x x 0, that is, it reduces to a similar form as the one for the disturbance-free case [1, eq. 5]. Meanwhile, 6 cannot be reduced to the same form since hδ d =φξθ 0 / ω 0 ρ/θe 0 / > 0 even if δ d = δ 0 =0. Hence, 6 holds with the exponential decay rate λ := ln θ > 0 64 τ s and the gain functions g, h : R 0 R >0 defined by gs := 1 φ hs := φ ΞΘ 0 / ω 0 ρe 0 + ω d δ 0 ψ L x s θ ΞΘ 0 / ω 0 ρe 0 + ω d δ 0 Ψ d s ψ L d s θ +ΞΘ 0 / Cd sδ 0 Ψ d s/ ψ L d s + γs. 65 D. Practical Stability Following essentially the calculations from [1, Sec. 5.5] and Sections V-A, VI-A3, and VI-C3, we establish a stronger version of Corollary with a smaller constant C. Proposition 3 Practical stability: Consider the switched linear control system 1. Suppose that Assumptions 1 3 and inequality 4 hold. Then, the communication and control strategy in Theorem 1 yields the following property: provided that the ADT τ a satisfies 49, for each ε>0, there exists a small enough δ>0such that 9 holds for all initial states x 0 R n x and disturbances d : R 0 R n d with the constant C =ΞΘ 0 / Θ d δ Proof: See Appendix G. In particular, from g0 + h0 =ΞΘ 0 / 1 φ + φ ω0 ρe 0 + ω d δ 0 + Θ d δ 0 θ > ΞΘ 0 / Θ d δ 0 it follows that the constant C in Proposition 3 is smaller than the one in Corollary. Proposition 3 also improves the practical stability result in [3, Th. 1]. Moreover, from the proof, it will be clear that the additional bound in [3, eq. 39] on the ADT τ a is not necessary for establishing practical stability. VII. SIMULATIO STUDY Our communication and control strategy is simulated with the following data: P = {1, }, [ ] [ ] A 1 :=, B =, 0 K 1 =[ 0] [ ] [ ] A :=, B 1 0 =, 1 K =[0 1 ]. and τ s =0.5, =5,τ d =1.05, τ a =7.55, and 0 =5 so that the basic Assumptions 1 3 hold. We set t = t =0 in 8, ε E =0.8 in 17 and ε δ =1in 59. The disturbance d is kept 0 most of the time and turned on for two sampling intervals with the constant value 10 when the state stays small more specifically, when x < for ten consecutive sampling

12 118 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 state converge to the origin by halving the estimate δ k every ten sampling intervals, as shown in Fig. 4. We conjecture that, for general disturbances, a similar modification to our communication and control strategy can be made to establish ISS with respect to the origin. Fig.. Simulation example. Fig. 3. With a constant estimate δ k δ 0, the state x 0 even if d 0. Fig. 4. d 0. With a converging estimate δ k 0, the state x 0 when intervals. The initial estimate is δ 0 =. Fig. plots a typical behavior of the first component x 1 of the continuous state in orange solid line and the corresponding component ˆx 1 of the auxiliary state in blue dash-dot line. Switching times are denoted by vertical gray dotted lines, and sampling times when the disturbance is turned on by vertical yellow dashed lines; captures are marked by red circles, and escapes by green crosses. Observe the searching stages at t =0the state is lost due to x 0 >E 0 and t =0.5 and 31 the state escapes due to the disturbance, and the nonsmooth behavior of x when ˆx experiences a jump. The value of τ a is empirically selected to be large enough to provide consistent convergence in simulations. For this example, the theoretical lower bound 49 on the ADT τ a is approximately 8.13, which is rather conservative. However, our result is significantly less conservative than the one in [1, Sec. 6] for the disturbance-free case, which generated a theoretical bound of τ a 85.5 while consistent convergence was observed with τ a =7.55 as well. The improvement is due to the more careful calculations in the stability analysis, such as the ones explained by footnotes and 5. Fig. 3 exhibits the case where the unknown disturbance d is transient or d 0, so that once the state is captured it will never escape. Due to the nonzero initial estimate δ 0, the state x will converge to the set A = {v R n x : v h0} visualized by the shaded area instead of the origin. Following essentially the idea of zooming-in from [6], we are able to make the VIII. COCLUDIG REMARKS In this paper, we studied the feedback stabilization of a switched linear system with a completely unknown disturbance under data-rate constraints. A finite data transmission rate was achieved via sampled and quantized state measurements. We extended the approach of reachable-set approximation and propagation from [1] by introducing an estimate of the disturbance bound to compensate for the disturbance. A communication and control strategy was designed to achieve a variant of ISS with exponential decay, based on a novel algorithm for adjusting the estimate and recovering the state when it escapes the range of quantization. As discussed in Section VII, we intend to advance our result via the zooming-in technique from [6] to establish ISS with respect to the origin. However, reducing the estimate in stabilizing stages can lead to an unbounded number of searching stages, and further work is needed to establish convergence for the communication and control strategy. For a nonswitched linear control system, the minimum data rate for feedback stabilization equals the topological entropy [30] of the open-loop system [3], [4]. In the context of switched systems, neither the topological entropy nor the minimum data rate for feedback stabilization has been well established. See [31, Ch. 6] for some initial results on the topological entropy of switched linear systems. These two notions and their relation could become intriguing topics for future research. APPEDIX A PROOF OF LEMMA 1 For all square matrices X and Y,wehave e X +Y e X = 1 m! X + Y m X m m =0 = 1 m m X m i Y i m! i m =1 i=1 m = 1 1 m 1 X m 1 j Y j+1 m 1! j j +1 m =1 m =1 1 m 1! = e X + Y Y. j=0 m 1 j=0 m 1 j X m 1 j Y j+1 APPEDIX B PROOF OF LEMMA We first recall the following useful facts from linear algebra. From the definition of -norm, it follows that v v v, v 1 v n v 1 v 67

13 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 119 for all vectors v, v 1,v R n.also λs v Sv v v λs v Rn \{0} 68 for all symmetric matrices S R n n i.e., S = S. At t k+1, from 19, it follows that V p x k+1,e k+1 =x k+1 P p x k+1 + ρ p E k+1 with x k+1 given by and E k+1 given by 1. First, can be rewritten as x k+1 = S pc k = S p x k +Δ k with Δ k := c k x k. Then x k+1 P p x k+1 =x k S p P p S p x k +x k S p P p S p Δ k +Δ k S p P p S p Δ k x k P p Q p x k +n x x k S p P p S p Δ k + n x S p P p S p Δ k where the last inequality follows from 39 and 67. Moreover, 67 and 68 imply that x k Q p x k λq p x k x k λq p λp p x k P p x k x k Q p x k λq p x k x k λq p x k. Combining the inequalities above and completing the square, we obtain that x k+1 P p x k+1 1 λq p x k P p x k 1 λp p λq p x k +n x x k Sp P p S p Δ k + n x Sp P p S p Δ k 1 λq p x k P p x k + χ p Δ k λp p 1 λq p x k 1 λq p x k P p x k + λp p nx Sp P p S p Δ k λqp 1 χ p Ek where the last inequality follows partially from 11. Second, from 1 and Young s inequality with φ 1, it follows that E k+1 = Λp E k +Φ p τ s δ k 1 + φ 1Λ p E k φ1 Φ p τ s δ k. Therefore V p x k+1,e k+1 1 λq p λp p φ 1Λ p which in turn implies 43. x k P p x k + ρ p E k + APPEDIX C PROOF OF LEMMA 3 At t k+1, from 3, it follows that 1 χ p ρ p 1+ 1φ1 ρ p Φ p τ s δ k V q x k+1,e k+1 =x k+1 P q x k+1 + ρ q E k+1 with x k+1 given by 9 and E k+1 given by 8. First, 9 can be rewritten as x k+1 = H pqc k = H pq x k +Δ k with Δ k = c k x k. Then x k+1 P q x k+1 λp q h pqx k +Δ k x k +Δ k λp q h pqx k x k +n x λp q h pq Δ k λp q h pq x k P p x k + λp p 1 n x λp q h pqek where the inequalities follows from 11, 67, 68, and Young s inequality. Second, from 30 and Young s inequality with φ, it follows that Ek+1 α pq x k + β pq E k + γ pq δ k + φ αpq x k + βpqe k + 1+ φ γpqδ k for every φ > 0, in which x k x k x k x k P p x k /λp p due to 67 and 68. Therefore V q x k+1,e k+1 λpq h pq λp p φ α pqρ q λp p n x λp q h pq ρ p 1+ φ ρ q γ pqδ k which in turn implies 45. x k P p x k + + φ βpqρ q ρ p Ek ρ p APPEDIX D PROOF OF LEMMA 4 First, consider the function ζ :[0, 1 R defined by ζs =1+ lnμ + s1 νμ d/ν d ln1/ν + s1 ν. From 47 and 48, it follows that Θ > 1 in 51, and that ζ is continuous and increasing. Moreover, as ζ0 = 1 +

14 10 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 63, O. 7, JULY 018 lnμ/ ln1/ν <τ a /τ s due to 49, there exists a sufficiently small constant φ 3 0, 1 such that ζφ 3 <τ a /τ s ; thus, θ<1 in 51. The remaining proof follows in principle from the arguments in [17] and [3]. If there is an integer l {i,...,k 1} such that then 43 implies that V σ tl x l,e l > 1 φ 3 1 ν ν dδ l 69 V σ tl +1 x l+1,e l+1 < ν + φ 3 1 νv σ tl x l,e l if σt l+1 =σt l, whereas 45 implies that V σ tl +1 x l+1,e l+1 < μ + φ 3 1 νμ d /ν d V σ tl x l,e l if σt l+1 σt l. Hence, for two integers l,l such that i l <l k and that 69 holds for all l {l,...,l 1}, we have V σ tl x l,e l < μ + φ 3 1 νμ d /ν d σ t l,t l ν + φ 3 1 ν l l σ t l,t l V σ tl x l,e l =ν + φ 3 1 ν l l Θ σ t l,t l V σ tl x l,e l, < ν + φ 3 1 ν l l Θ 0 +l l τ s /τ a V σ tl x l,e l = θ l l Θ 0 V σ tl x l,e l where σ t l,t l denotes the number of switches on t l,t l ], and the last inequality follows from Θ > 1 and the ADT condition. Therefore, if 69 holds for all l {i,...,k 1}, then V σ tk x k,e k <θ k i Θ 0 V σ ti x i,e i. Otherwise, for k := max { l k 1:V σ tl x l,e l ν d δ l /φ 31 ν } it holds that V σ tk +1 x k +1,E k +1 μv σ tk x k,e k +μ dδ k see also Remark 3; thus μ φ 3 1 ν ν dδ k + μ dδ k =Θ dδ k V σ tk x k,e k <θ k k 1 Θ 0 V σ tk +1 x k +1,E k +1 Θ 0 Θ d δ k as 69 holds for all l {k +1,...,k 1}. The proof of Lemma 4 is completed by combining the bounds for the two cases and noticing that δ l = δ i for all l {i,...,k 1}. APPEDIX E PROOF OF LEMMA 5 Let p, q Pdenote the active modes at sampling times t j,t i, respectively. At the sampling time t i of recovery, we have V q x i,e i =x i P q x i + ρ q E i with x i = x j and E i bounded by 54. First, from 68, it follows that x i P q x i λp q λp p x j P p x j. Second, following 54 and Young s inequality with φ 4,we obtain Ei ˆβ η δ d /δ j ᾱ ˆβ 1 x j + E j + γ ˆβ 1 δ j ˆβ η δ d /δ j ᾱ + φ 4 ˆβ 1 x j + Ej + 1+ φ4 γ ˆβ 1 δ j for every φ 4 > 0, in which x j x j x j x j P p x j /λp p due to 67 and 68. Therefore V q x i,e i ˆβ η δ d /δ j λpq λp p + + φ 4ᾱ ρ q ˆβ 1 λp p + + φ 4ρ q ρ p ρ p E k + which, in turn, implies 55. x j P p x j 1+ φ 4 γ ρ q ˆβ 1 δ j APPEDIX F PROOF OF LEMMA 7 Let [t im,t jm +1 denote the stabilizing stage containing t k, that is, i m k j m Substituting 50 with i = i m and k = j m +1 into 55 with j = j m +1 and i = i m +1, we obtain V σ ti m +1 x i m +1,E im +1 ˆβ η δ d /δ j m +1 ωv σ tj m +1 x j m +1,E jm +1 +ω d δj m +1 < ˆβ η δ d /δ i m +1 ωθ 0 θ j m +1 i m V σ ti m x i m,e im +Θ d 1 + ε δ m δ0 + ωd 1 + ε δ m +1 δ0 =Ψˆβ η δ d /δ i m +1 θ j m +1 i m V σ ti m x i m,e im +Θ d + ψ d ω d 1 + ε δ m δ0 in which θ j m +1 i m θ i m +1 i m θ η δ d /δ j m +1 due to 53 and θ<1. Hence V σ ti m +1 x i m +1,E im +1 < Ψ ˆβ η δ d /δ i m +1 θ i m +1 i m θ η δ d /δ j m V σ ti m x i m,e im +Θ d + ψ d ω d 1 + ε δ m δ0 Ψψ η δ d /δ i m +1 θ i m +1 i m V σ ti m x i m,e im +Θ d + ψ d ω d 1 + ε δ m δ 0.

15 YAG AD LIBERZO: FEEDBACK STABILIZATIO OF SWITCHED LIEAR SYSTEMS WITH UKOW DISTURBACES 11 Based on this recursive bound, it is straightforward to derive V σ ti m x i m,e im < Ψψ η δ d /δ i m θ i m i m 1 Ψψ η δ d /δ i m 1 θ i m 1 i m V σ ti m x i m,e im +Θ d + ψ d ω d 1 + ε δ m δ0 +Θd + ψ d ω d 1 + ε δ m 1 δ0 Ψ ψ η δ d /δ i m +ηδ d /δ i m 1 θ i m i m V σ ti m x i m,e im +Θ d + ψ d ω d 1 + ψ d 1 + ε δ m δ0 < < Ψ m ψ m l =1 η δ d /δ i l θ i m i 0 V σ ti 0 0,E i0 +Θ d + ψ d ω d 1 + ψ d + + ψ m 1 d δ0, < Ψ m ψ m l =1 η δ d /δ i l θ i m i 0 V σ ti 0 0,E i0 +Θ d + ψ d ω d δ 0 m 1 l=0 ψ l d which, combined with 57 and 58, implies that V σ ti m x i m,e im Ψ m ψ m l =1 η δ d /δ i l ω 0 V σ 0 0,E 0 +ω d δ0 +Θ d + ψ d ω d δ 0 m 1 l=0 θ i m i 0 ˆβη E x 0 /E 0 +η δ δ d /δ 0 ψ l d Ψ m ψ η δ δ d /δ 0 + m l =1 η δ d /δ i l θ i m ψ η E x 0 /E 0 ω 0 ρ σ 0 E 0 + ω d δ 0 +Θ d + ψ d ω d δ 0 m 1 l=0 ψ l d. Finally, substituting the previous bound into 50 with i = i m, we obtain V σ tk x k,e k < Θ 0 θ k i m V σ ti m x i m,e im +Θ d δi m < Θ 0 θ k i m Ψ m ψ η δ δ d /δ 0 + m l =1 η δ d /δ i l θ i m ψ η E x 0 /E 0 ω 0 ρ σ 0 E0 + ω d δ0 m 1 +Θ d + ψ d ω d δ0 ψd l +Θ d 1 + ε δ m δ0 l=0 Θ 0 Ψ m ψ η δ δ d /δ 0 + m l =1 η δ d /δ i l θ k ψ η E x 0 /E 0 ω 0 ρ σ 0 E0 + ω d δ0 + Θ d m l=0 ψ l d + ω d m ψd l l=1 δ 0. The proof of Lemma 7 is completed by replacing m with its upper bound d δ d. APPEDIX G PROOF OF PROPOSITIO 3 First, suppose x 0 δ E 0 and δ d δ δ 0. Then, the system is always in the stabilizing stage, and the estimate of the disturbance bound is always δ 0. Suppose also that there is an integer k 1 1 such that c k =0i.e., the state xt k is inside the central hypercubic box for all k k 1 1. Then, similar arguments as in Sections V-A1 and V-A show that u 0 on [0,t k0 and x k =0for all k {0,...,k 1}; thus E k+1 Λ min E k k {0,...,k 1 1} 70 with Λ min := min p P Λ p due to 1 and 8. Second, following similar analysis on state bounds when u 0 as in Section V-B, for each k k 1,wehave xt β k x 0 + β k 1 β 1 γδ d t t k. 71 Third, following similar arguments as in Section VI-C3, for each k k 1,wehave xt Ξ V σ tk x k,e k +γδ d t [t k,t k+1 ] which, combined with 50 for i =0and 6, implies that xt ΞΘ 0 / θ k 1 ρe 0 + Θ d δ 0 +γδ d t t k1 7 with ρ in 61. Finally, the proof of Lemma 3 is completed via the following three steps. First, given an arbitrary ε>0, from 66 and 7, it follows that if ΞΘ 0 / θ k 1 ρe 0 + γδ d ε then xt ε + C for all t t k1. Second, taking E 0 as fixed, calculate a sufficiently large k 1 such that ΞΘ 0 / θ k 1 ρe 0 ε/. Third, calculate a sufficiently small δ such that γδ ε/ and β k 1 + β k 1 1 β 1 γ δ ε which, combined with 71, implies that xt ε for all t t k1 ; and that δ δ 0 and β k β k γ β 1 δ Λmin k1 1 E 0 which, combined with 70 and 71, implies that c k =0for all k k 1 1 and that the systems is always in the stabilizing stage, making the analysis above valid.

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Guosong Yang S 11 received the B.Eng. degree in electronic engineering from Hong Kong University of Science and Technology, Kowloon, Hong Kong, in 011, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana Champaign, Urbana, IL, USA, in 013 and 017, respectively. He is currently a Postdoctoral Scholar with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA. His research interests include switched and hybrid systems, control with limited information, and nonlinear control theory. Daniel Liberzon M 98 SM 04 F 13 was born in the former Soviet Union in He did his undergraduate studies in the Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia, from 1989 to 1993, and received the Ph.D. degree in mathematics under Prof. R. W. Brockett of Harvard University from Brandeis University, Waltham, MA, USA, in Following a postdoctoral position with the Department of Electrical Engineering, Yale University, from 1998 to 000, he joined the University of Illinois at Urbana Champaign, Urbana, IL, USA, where he is currently a Professor with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory. He has authored the books entitled Switching in Systems and Control Boston, MA, USA: Birkhäuser, 003 and Calculus of Variations and Optimal Control Theory: A Concise Introduction Princeton, J, USA: Princeton Univ. Press, 01. His research interests include nonlinear control theory, switched and hybrid dynamical systems, control with limited information, and uncertain and stochastic systems. Dr. Liberzon has received several recognitions, including the 00 IFAC Young Author Prize and the 007 Donald P. Eckman Award. He delivered a plenary lecture at the 008 American Control Conference. He is a Fellow of IFAC and an Editor for Automatica nonlinear systems and control area.