Networked Control under Communication Constraints ...

Networked Control under Communication Constraints: The Discrete-time Case Kun Liu, Xia Pan, Yuanqing Xia, Emilia Fridman and James Lam

Abstract— This paper considers discrete-time networked control systems in which distributed sensors, controllers, and actuators communicate through a shared communication medium that introduces large and bounded time-varying transmission delays. Access to the communication medium is orchestrated by a weighted try-once-discard protocol that determines which sensor node can access the network and transmit its corresponding data. The closed-loop system is modelled as a novel discrete-time hybrid system with time-varying delays in the dynamics and in the reset conditions. By Lyapunov method a new condition is derived for the exponential stability of the delayed hybrid systems with respect to the full state and not only to the partial state. An example of a discrete-time cartpendulum illustrates the efficiency of the time-delay approach.

Keywords: Discrete-time networked control systems, multiple sensors, discrete-time hybrid system, try-once-discard protocol, Lyapunov method. I. I NTRODUCTION Control of systems over communication networks with bandwidth limitations and interference channels is currently attracting a lot of attention in the control community. In such systems, when implemented, distributed sensor/actuator nodes will compete for access to the network. Only one node is allowed to obtain the access at each transmission instant. This leads to the so-called communication constraints that cannot be ignored in the analysis and synthesis of networked control systems (NCSs) [8], [22]. Therefore, protocols are necessary to orchestrate which node is given permission at each time instant. In the literature, three basic types of scheduling protocols have been presented: (i) Static protocols, such as the Round-Robin protocol where the nodes take turns transmitting its corresponding data in a periodic manner [7], [17], [18]; (ii) Quadratic protocols, such as try-once-discard (TOD) protocol, where the node that corresponds to the largest error between the current value and the last transmitted value has the highest priority to use the communication medium [7], [17], [18], [21]; (iii) Stochastic protocols, which determine the transmitted node through a Bernoulli or a Markov chain process [1], [3], [15], [20]. To understand the impact of scheduling protocols on control performance, several models have been developed. There are K. Liu, X. Pan and Y. Xia are with the School of Automation, Beijing Institute of Technology, Beijing 100081, China. E-mails: kunliubit, panxia, xia [email protected]. E. Fridman is with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail: [email protected]. J. Lam is with the Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: [email protected].

discretization-based model [2], [6], impulsive/hybrid formulation [7], [17] and time-delay approach [4], [5], [23], [24]. The time-delay approach was introduced for the stabilization of continuous-time NCSs with N = 2 sensor nodes under the Round-Robin protocol [13] and under the TOD protocol [12]. The closed-loop system was modeled as a switched system with multiple and ordered time-varying delays under Round-Robin scheduling or as a hybrid system with time-varying delays in the dynamics and in the reset equations under the TOD scheduling. For continuous-time NCSs, the extension from N = 2 to a general N ≥ 2 sensor nodes was recently presented in [14], in which a unified hybrid system model was given under both TOD and Round-Robin protocols for the closed-loop system that contains time-varying delays in the continuous dynamics and in the reset conditions. More recently, the time-delay approach was extended in [11] to the stability analysis of discrete-time NCSs with actuator constraints under the TOD scheduling protocol in the presence of N = 2 sensor nodes. A time-dependent Lyapunov functional was introduced and only partial stability of the resulting hybrid delayed system was guaranteed. The extension from N = 2 to a general N ≥ 2 sensor nodes is far from being straightforward. It has the following challenges: (1) The time-dependent Lyapunov functional of [11] is not applicable any more. (2) It is important to guarantee stability of the resulting closed-loop system with respect to the full state and not only to the partial state. In the present paper, we consider the exponential stability of discrete-time NCSs incorporating a general number N ≥ 2 of distributed sensor nodes, bounded time-varying transmission intervals, bounded time-varying transmission delays and TOD scheduling protocol. Following [14], we present a direct Lyapunov approach to guarantee stability of the resulting discrete-time hybrid system model with respect to the full state. The conditions are given in terms of linear matrix inequalities (LMIs). The efficiency of the presented approach is illustrated by a cart-pendulum system. Notation: Throughout the paper, the superscript ‘T ’ stands for matrix transposition. By Rn and Rn×m , we denote the n dimensional Euclidean space with vector norm | · | and the set of all n × m real matrices, respectively. For P ∈ Rn×n , the notation P > 0 means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by ∗, λmin (P ) denotes the smallest eigenvalue of matrix P . Z+ and N denote the set of nonnegative integers and positive integers, respectively.

III. A DISCRETE - TIME HYBRID SYSTEM MODEL u k!

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y !k "

Consider (1) under the static output feedback control. In the following, we propose a hybrid system model for the closed-loop system of NCS provided above.

yN -k . !

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.%&/0$1 yN ! s p "

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Fig. 1.

The architecture of NCSs with N sensors

II. NCS

A. A discrete-time hybrid time-delay model T Denote by yˆ(sp ) = [ˆ y1T (sp ) · · · yˆN (sp )]T ∈ Rny the output information submitted to the scheduling protocol. At each sampling instant sp , one of the outputs yi (sp ) ∈ Rni is transmitted over the network, that is, one of the yˆi (sp ) values is updated with the recent state yi (sp ). Let i∗p ∈ I = {1, . . . , N } denote the active output node at the sampling instant sp . Then { yi (sp ), i = i∗p , yˆi (sp ) = (4) yˆi (sp−1 ), i ̸= i∗p .

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MODEL AND PRELIMINARIES

In this section, we demonstrate the discrete-time description of NCS model and provide some preliminaries. Consider the networked control scheme depicted in Figure 1, where a linear discrete-time plant, N distributed sensors, a controller node and an actuator node are all connected via communication networks. The linear time-invariant discretetime plant is given by { x(k + 1) = Ax(k) + Bu(k), k ∈ Z+ , (1) yi (k) = Ci x(k), i = 1, . . . , N, where x(k) ∈ Rn denotes the state of the plant, u(k) ∈ Rnu the control input, and yi (k) ∈ Rni (i = 1, . . . , N ) the measurement outputs of the plant. The matrices A and B may be certain or uncertain. The given by [ initial condition ]is T T x(0) = x0 . We denote C = C1T · · · CN , y(k) = ]T [ T ∑N T (k) y1 (k) · · · yN ∈ Rny and i=1 ni = ny . The sequence of sampling instants 0 = s0 , s1 , s2 , . . . is strictly increasing in the sense that sp+1 − sp ≤ MATI, where {sp } is a subsequence of Z+ and MATI denotes the maximum allowable transmission interval. Denote by tp the updating time instant of the zero-order holder (ZOH). Suppose that the updating data at the instant tp on the actuator side has experienced an uncertain transmission delay ηp = tp − sp as it is transmitted through the network. The delays may be either smaller or larger than the sampling interval provided that the transmission order of data packets is maintained [16]. Assume that the network-induced delay ηp and the time span between the updating and the current sampling instants are bounded:

We denote e(k) by the error between the system output y(sp ) and the latest available information yˆ(sp−1 ): e(k) = col{e1 (k), · · · , eN (k)} ≡ yˆ(sp−1 ) − y(sp ), k ∈ [tp , tp+1 − 1]), k ∈ Z+ , yˆ(s−1 ) , 0, e(k) ∈ Rny . (5) The choice of i∗p will depend on the transmission error and will be chosen according to TOD scheduling protocol which is defined below. The controller and the actuator are supposed to be eventdriven. The most recent output information on the controller side is denoted by yˆ(sp ). Assume that there exists a matrix K = [K1 · · · KN ], Ki ∈ Rnu ×ni such that A + BKC is Schur stable. Consider the static output feedback controller u(k) = K yˆ(tp − ηp ), k ∈ [tp , tp+1 − 1], k ∈ Z+ .

(6)

From (4), it follows that the controller (6) can be rewritten as ∑N u(k) = Ki∗p yi∗p (tp − ηp ) + i=1,i̸=i∗p Ki yˆi (tp−1 −ηp−1 ), (7) for k ∈ [tp , tp+1−1], where i∗p is the index of the active node at sp and ηp is the transmission delay. Therefore, from (1) and (4)–(7), we obtain the following closed-loop system: ∑N x(k + 1) = Ax(k) + A1 x(tp −ηp )+ i=1,i̸=i∗p Bi ei (tp ), e(k + 1) = e(k), k ∈ [tp , tp+1 − 2], k ∈ Z+ , (8) with the delayed reset system for k = tp+1 − 1,

(3)

x(tp+1 ) = Ax(tp+1 − 1) + A1 x(tp − ηp ) ∑N + i=1,i̸=i∗p Bi ei (tp ), ei (tp+1 ) = Ci [x(tp − ηp ) − x(tp+1 − ηp+1 )], i = i∗p , ei (tp+1 ) = ei (tp )+Ci [x(tp −ηp )−x(tp+1 −ηp+1 )], i ̸= i∗p . (9) where A1 = BKC, K = [K1 · · · KN ], Bi = BKi , i = 1, . . . , N.

Moreover, since τM = MATI +ηM − 1, it holds that ηm < τM .

For k ∈ [tp , tp+1 − 1] we represent tp − ηp = k − τ (k), where τ (k) = t − tp + ηp . Therefore, (8)–(9) can be considered as a discrete-time hybrid system with time-varying

tp+1 − 1 − tp + ηp ≤ τM , 0 ≤ ηm ≤ ηp ≤ ηM , p ∈ Z+ , (2) where τM , ηm and ηM are known non-negative integers. Then we have (tp+1 − 1) − sp tp+1 − tp

= ≤ ≤

sp+1 − sp + ηp+1 − 1 MATI +ηM − 1 = τM , τM − ηm + 1.

interval delays. Furthermore, the delays τ (k) exist both in the dynamics (8) and in the reset conditions (9). Even for ηp = 0 we have the delayed state x(tp ) = x(k − τ (k)) with τ (k) = k − tp . B. Weighted TOD scheduling protocol Let Qi > 0, i = 1, . . . , N, be some weighting matrices. At the sampling instant sp , the weighted TOD protocol is a protocol for which the active output node with the index i∗p is defined as any index that satisfies √ √ | Qi∗p ei∗p (k)|2 ≥ | Qi ei (k)|2 , (10) where k ∈ [tp , tp+1 − 1], p ∈ Z+ , i = 1, . . . , N. √ In other words, the norms of weighted errors Qi ei (k) are compared and the output node i∗p corresponds to the largest network-induced error will be granted the access to the network. A possible choice of i∗p is given by √ i∗p = min{arg max | Qi (ˆ yi (sp−1 ) − yi (sp )) |2 }. i∈{1,...,N }

The conditions for computing the weighting matrices Q1 , . . . , QN will be given in Theorem 1 below. Remark 1 The packet dropouts under scheduling protocols could be accommodated, if there is an additional perfect (without packet dropouts) feedback channel to send a reception/dropout acknowledgement to the active sensor, and if this acknowledgement is completed within one sampling period. Then as in [7] and [14], packet dropouts can be modeled as prolongations of the transmission interval. In the present paper, the objective is to derive LMI condition for exponential stability of the hybrid system (8)– (10) via direct Lyapunov method. IV. M AIN RESULTS : EXPONENTIAL STABILITY OF NCS S UNDER TOD SCHEDULING PROTOCOL Definition 1 The hybrid system (8)–(10) is said to be exponentially stable if there exist constants b > 0 and 0 < κ < 1 such that, for initial condition xt0 ∈ Rn × · · · × Rn , the | {z } τM +1 times

solutions of the hybrid system (8)–(10) satisfy |x(k)|2 ≤ bκ2(k−t0 ) {∥xt0 ∥2c + |e(t0 )|2 }, and {|e(k)|2 } ≤ bκ2(k−t0 ) {∥xt0 ∥2c + |e(t0 )|2 }, where ∥xt0 ∥c = supt0 −τM ≤s≤t0 |x(s)|. We apply the following discrete-time Lyapunov functional to system (8)–(10): ∑N Ve (k) = V (k) + i=1 eTi (tp )Qi ei (tp ), (11) V (k) = V˜ (k) + VG (k),

where k ∈ [tp , tp+1 − 1], k ∈ Z+ , p ∈ Z+ , and ∑N VG (k) = i=1 (τM − ηm )× ∑k−1 k−s−1 T η (s)CiT Gi Ci η(s), s=tp −ηp λ ∑ k−1 V˜ (k) = xT (k)P x(k) + s=k−ηm λk−s−1 xT (s)S0 x(s) ∑k−ηm −1 k−s−1 T + s=k−τM λ x (s)S1 x(s) ∑−1 ∑k−1 +ηm j=−ηm s=k+j λk−s−1 η T (s)R0 η(s) ∑−ηm −1 ∑k−1 k−s−1 T η (s)R1 η(s) +(τM − ηm ) j=−τ s=k+j λ M with η(k) = x(k + 1) − x(k), P > 0, Si > 0, Ri > 0, Gj > 0, Qj > 0, 0 < λ < 1, i = 0, 1, j = 1, . . . , N. The term VG is to deal with the delays in the reset conditions ∑N VG (t[p+1 ) − λVG (tp+1 − 1) = (τM − ηm ) i=1 ∑tp+1 −1 tp+1 −s−1 T η (s)CiT Gi Ci η(s) s=tp+1 −ηp+1 λ ] ∑tp+1 −2 tp+1 −s−1 T T − s=t λ η (s)C G C η(s) i i i −η p p ∑ N ≤ (τM − ηm ) i=1 η T (tp+1 − 1)CiT Gi Ci η(tp+1 − 1) ∑N ∑tp+1 −ηp+1 −1 T −(τM −ηm ) i=1 λτM s=t η (s)CiT Gi Ci η(s) p −ηp ∑N T ≤ (τM − ηm ) i=1 η (tp+1 − 1)CiT Gi Ci η(tp+1 − 1) √ ∑N − i=1 λτM | Gi Ci [x(tp+1 − ηp+1 ) − x(tp − ηp )]|2 . (12) The following lemma gives sufficient conditions for the stability of (8)–(10): Lemma 1 Assume that there exist scalar 0 < λ < 1, integers 0 ≤ ηm < τM , matrices 0 < Qi ∈ Rni ×ni , 0 < Ui ∈ Rni ×ni , 0 < Gi ∈ Rni ×ni , i = 1, . . . , N, and Ve (k) of (11) such that along (8)–(10) the following inequality ∑N V˜ (k + 1)−λV˜ (k)+(τM −ηm ) i=1 η T (k)CiT Gi Ci η(k) √ ∑N + i=1,i̸=i∗p | Wi ei (tp )|2 , Ψ(k) ≤ 0 (13) and the LMIs [ ] Γi Qi , Ωi < 0, i = 1, . . . , N, (14) ∗ Qi − λτM Gi hold for k ∈ [tp , tp+1 − 1], where 1 Wi = − τM −η Ui + (1 − λ)Qi , m λ−(1−λ)(τM −ηm ) Γi = − Qi + (1 + N −1

1 τM −ηm )Ui .

(15)

Then Ve (k) satisfies the following inequalities along (8)– (10): √ ∑N 1 Ve (k + 1)−λVe (k)− τM −η | Ui ei (tp )|2 i=1,i̸=i∗ m p √ −(1−λ)| Qi∗p ei∗p (tp )|2 , Θ1 (k) ≤ 0, k ∈ [tp , tp+1 −2] (16) and √ ∑N Ve (tp+1 ) − λVe (tp+1 − 1) + i=1,i̸=i∗p | Ui ei (tp )|2 √ +(1 − λ)(τM − ηm )| Qi∗p ei∗p (tp )|2 , Θ2 ≤ 0. (17) Moreover, the following bounds λmin (P )|x(k)|2 ≤ V (k) ≤ Ve (k) ≤ λk−t0 Ve (t0 ), ∑N √ Ve (t0 ) = V (t0 ) + i=1 | Qi ei (t0 )|2 , k ≥ t0 , k ∈ N, (18)

and

∑N

√ 2 ˜λk−t0 Ve (t0 ), i=1 | Qi ei (k)| ≤ c

tp+1 − 1 imply (19)

with c˜ = λ−(τM −ηm ) , are valid for the solutions of (8)–(10) initialized by xt0 ∈ Rn × · · · × Rn , e(t0 ) ∈ Rny . Thus, the | {z }

Ve (tp+1 ) ≤

−(1 − λ)(τM

τM +1 times

≤

λtp+1 −tp Ve (tp ) √ ∑N −(1 − λ) i=1,i̸=i∗p | Ui ei (tp )|2 √ −(1 − λ)2 (τM − ηm )| Qi∗p ei∗p (tp )|2

≤

λtp+1 −tp Ve (tp ).

hybrid system (8)–(10) is exponentially stable. Proof: First, from (11) and (13), it holds that Θ1 (k) ≤ Ψ(k) for k ∈ [tp , tp+1 − 2]. Therefore, Θ1 (k) ≤ 0 of (16) holds if Ψ(k) ≤ 0 for k ∈ [tp , tp+1 − 2]. Furthermore, from (11)–(14), we have Θ2 ≤ V˜ (tp+1 ) − λV˜ (tp+1 − 1) ∑N +(τM − ηm ) i=1 η T (tp+1 −1)CiT Gi Ci η(tp+1 −1) √ ∑N − i=1 [λτM | Gi Ci [x(sp+1 ) − x(sp )]|2 ] ∑N √ √ + i=1 | Qi ei (tp+1 )|2 − λ| Qi ei (tp )|2 √ ∑N + i=1,i̸=i∗p | Ui ei (tp )|2 √ +(1 − λ)(τM − ηm )| Qi∗p ei∗p (tp )|2 √ ∑N 1 = Ψ(tp+1 − 1) + τM −η | Ui ei (tp )|2 i=1,i̸=i∗ m p ∑N √ −(1 − λ) i=1,i̸=i∗p | Qi ei (tp )|2 ∑N τ M √ | Gi Ci [x(sp+1 ) − x(sp )]|2 ] − i=1 λ ∑N [ √ √ + i=1 | Qi ei (tp+1 )|2 − λ| Qi ei (tp )|2 √ ∑N + i=1,i̸=i∗p | Ui ei (tp )|2 √ +(1 − λ)(τM − ηm )| Qi∗p ei∗p (tp )|2 . Note that under TOD protocol √ √ ∑N −| Qi∗p ei∗p (tp )|2 ≤ − N 1−1 i=1,i̸=i∗p | Qi ei (tp )|2 . From (13) and (14), we have Ψ(tp+1 − 1) ≤ 0 and λτM Gi∗p − Qi∗p Ci∗p > 0, respectively. Denote ζi = col{ei (tp ), Ci [x(sp+1 ) − x(sp )]}. Then employing (9) we arrive at √ Θ2 ≤ Ψ(tp+1 −1)−| λτM Gi∗p −Qi∗p Ci∗p [x(sp )−x(sp+1 )]|2 ∑N + i=1,i̸=i∗p ζiT Ωi ζi ≤ 0, that yields (17). The next step is to prove (18) and (19). By the comparison principle, for k ∈ [tp , tp+1 − 1] the inequality (16) implies Ve (k) ≤ λk−tp Ve (tp ) +

∑N

{| i=1,i̸=i∗ p

√ Ui ei (tp )|2 }

√ +(1 − λ)(τM − ηm )| Qi∗p ei∗p (tp )|2 .

(20)

Note that (14) guarantees 0 < (1 − λ)(τM − ηm ) < λ < 1 1 M −ηm ) and Ui < (1 + τM −η )Ui < λ−(1−λ)(τ Qi ≤ Qi , i = N −1 m 1, . . . , N . Hence, V (k) ≤ λk−tp Ve (tp ), k ∈ [tp , tp+1 − 1].

(21)

On the other hand, the inequalities (17) and (20) with k =

√ ∑N | Ui ei (tp )|2 i=1,i̸=i∗ p √ − ηm )| Qi∗p ei∗p (tp )|2

λVe (tp+1 − 1) −

Then Ve (tp+1 ) ≤ λtp+1 −tp−1 Ve (tp−1 ) ≤ λtp+1 −t0 Ve (t0 ).

(22)

Substiting in (22) p+1 for p and taking into account (21), we arrive at (18), which yields exponential stability of (8)–(10) since λmin (P )|x(k)|2 ≤ V (k), V (t0 ) ≤ δ∥xt0 ∥2c for some scalar δ > 0. Moreover, (22) with p + 1 replaced by p implies (19) since for k ∈ [tp , tp+1 − 1] λtp −t0 = λk−t0 λtp −k ≤ c˜λk−t0 .

Remark 2 For discrete-time NCS under TOD scheduling protocol, Lemma 2 of [11] guarantees only partial stability of the closed-loop system with N = 2 sensor nodes. While Lemma 1 in this paper guarantees that (18) gives a bound not only on x(k) but also on ei (k), i = 1, . . . , N . That is why Lemma 1 provides stability of system (8)–(10) with respect to the full state. The following theorem on the exponential stability of (8)– (10) is derived from Lemma 1 and the standard arguments for the delay-dependent analysis.

Theorem 1 For any given scalar 0 < λ < 1, integers 0 ≤ ηm < τM , and Ki , i = 1, . . . , N , the solutions of the hybrid system (8)–(10) satisfy the bound (18) and (19) and thus, are exponentially stable, if there exist n × n matrices P > 0, S0 > 0, R0 > 0, S1 > 0, R1 > 0, S12 , ni × ni matrices Qi > 0, Ui > 0, Gi > 0, i = 1, . . . , N, such that (14) and the following LMIs are feasible for i = 1, . . . , N : [ Φ=

R1 ∗

S12 R1

] ≥ 0,

(23)

Σi + (F0i − F1i )T H(F0i − F1i ) − λτM (F i )T ΦF i < 0, (24)

where = [A 0n×n A1 0n×n F˜0i ], = [B2 · · · BN ], i = 1, = [B1 · · · BN −1 ], i = N, = [B1 · · · Bj |j̸=i · · · BN ], i = 2, . . . N − 1, F1i = [In 0n×(3n+ny −ni ) ], F2i = [ [In − In 0n×(2n+ny −ni ) ], ] 0n×n In −In 0n×n 0n×(ny −ni ) i F = , 0n×n 0n×n In −In 0n×(ny −ni ) i T i ηm i T i Σi = (F0 ) P F0 + Υi − λ (F2 ) R0 F2 , Υi = diag{S0 −λP, −ληm (S0 −S1 ), 0n×n , −λτM S1 , ϕi }, ϕi = diag{W2 , . . . , WN }, i = 1, ϕi = diag{W1 , . . . , WN −1 }, i = N, ϕi = diag{W1 , . . . , Wj |j̸=i , . . . , WN }, i = 2, . . . N − 1, ∑ T 2 R0 +(τM − ηm )2 R1 +(τM −ηm ) N H = ηm l=1 Cl Gl Cl . (25) F0i F˜0i F˜0i F˜0i

with Wi (i = 1, . . . , N ) given by (15). Furthermore, if the above inequalities hold with λ = 1, then they are feasible for λ = 1 − ε, where ε > 0 is small enough. Proof: Consider k ∈ [tp , tp+1 − 1], p ∈ Z+ and define ξi (k) = col{x(k), x(k − ηm ), x(k − τ (k)), x(k − τM ), ξ¯i (k)}, i = 1, . . . , N, where ξ¯i (k) = col{e1 (k), . . . , ej (k)|j̸=i , . . . , eN (k)}, i = 2, . . . , N − 1, ξ¯i (k) = col{e2 (k), . . . , eN (k)}, i = 1, ξ¯i (k) = col{e1 (k), . . . , eN −1 (k)}, i = N, Let i∗p = i ∈ N. Considering V˜ (k) along (8) and applying the Jensen inequality (e.g., [10]), we have ∑k−1 ηm s=k−ηm η T (s)R0 η(s) ≥ ξiT (k)(F2i )T R0 F2i ξi (k), and ∑k−ηm −1 T −(τM − ηm ) s=k−τ η (s)R1 η(s) ∑k−ηM m −1 T = −(τM − ηm ) s=k−τ (k) η (s)R1 η(s) ∑k−τ (k)−1 −(τM − ηm ) s=k−τM η T (s)R1 η(s) [ ]T T i M −ηm ≤ − ττ(k)−η ξ (k) [I 0 F R1 [In 0n×n ]F i ξi (k) n n×n] i m [ ]T −ηm T i − ττMM−τ R1 [0n×n In ]F i ξi (k) (k) ξi (k) [0n×n In ]F T i T i ≤ −ξi (k)(F ) ΦF ξi (k). The latter inequality holds for ηm < τ (k) < τM if (23) is feasible [19]. For τ (k) = ηm or τ (k) = τM , the inequality ∑k−ηm−1 T −(τM −ηm ) s=k−τ η (s)R1η(s) ≤−ξiT (k)(F i )T ΦF i ξi (k) M is still valid by the Jensen inequality. Then ∑N V˜ (k + 1)−λV˜ (k)+(τM −ηm ) i=1 η T (k)CiT Gi Ci η(k) √ ∑N + l=1,l̸=i | Wl el (tp )|2 ≤ ξiT (k)[Σi + (F0i )T HF0i − λτM (F i )T ΦF i ]ξi (k) ≤ 0. Hence, from Lemma 1, inequalities (14), (23) and (24) imply (18) and (19). This completes the proof. Remark 3 The conditions of Theorem 1 possess one of 2n× 2n, N of 2ni × 2ni , N of (4n + ny − ni ) × (4n + ny − ni ), 2 ni (ni +1) ,i∈ i ∈ I, LMIs, and have the number 7n +5n+3N 2 I, of decision variables.

m, a

l

! F M

x

Fig. 2.

Inverted pendulum system.

Remark 4 The stability analysis of discrete-time systems with time-varying delay in the state can be alternatively analyzed by substituting the switched system transformation approach for the Lyapunov-Krasovskii method. More details can be found in [9]. V. I LLUSTRATIVE E XAMPLE In this section, we will illustrate how the derived conditions can be verified through the widely used inverted pendulum system. The dynamics of the inverted pendulum on a cart shown in Figure 2 can be described in the following as in e.g., [25]:   0 1 0 x˙ (a+ml2 )b m2 gl2  x   ¨ 0 −  2 a(M +m)+M ml a(M +m)+M ml2    θ˙ =  0 0  0 mgl(M +m) mlb θ¨ 0 − a(M +m)+M ml2 a(M +m)+M ml2     0 x 2 a+ml   x˙   a(M +m)+M ml2  u  θ +   0 ml θ˙ 

0 0 1 0



  × 

a(M +m)+M ml2

(26)

with M = 1.096kg, m = 0.109kg, l = 0.25m, g = 9.8m/s2 , a = 0.0034kg · m2 and b = 0.1N/m/sec. In the model, x, θ, a and b represent cart position coordinate, pendulum angle from vertical, the friction of the cart and inertia of the pendulum, respectively. We choose a time Ts = 0.01s to discretize system (26) and obtain the following discrete-time system model: 

 1 0.01 0 0 0 0.9991 0.0063 0   x(k + 1) =  x(k) 0 0 1.0014 0.01  0 −0.0024 0.2784 1.0014  0  0.0088  + u(k), k ∈ Z+ . 0.0001  0.0236

(27)

The pendulum can be stabilized by a state feedback u(k) = Kx(k) with the gain K = [K1 K2 ] [ ] K = [K [ 1 K2 ], K1 = 7.7606] 14.6847 , (28) K2 = −86.7306 −26.3029 , which leads to the closed-loop system having eigenvalues {0.5374, 0.9860 + 0.0177i, 0.9860 − 0.0177i, 0.9924}. Suppose that the spatially distributed components of the

state of the cart-pendulum system (27) are not accessible simultaneously. We start with the case of N = 2 and consider two measurements yi (k) = Ci x(k), k ∈ Z+ , where [ ] [ ] 1 0 0 0 0 0 1 0 C1 = , C2 = . (29) 0 1 0 0 0 0 0 1 By Theorem 1 with λ = 1, Table I lists the maximum values of MATI that preserve the stability of hybrid timedelay system (8)–(10) for the different values of ηm . It is observed that the presented TOD protocol stabilizes the system for smaller maximum values of τM than the RoundRobin protocol. The latter protocol studied in [11] leads to a switched closed-loop system and possesses more decision variables in the LMI conditions. TABLE I E XAMPLE (N = 2):

MAX . VALUE OF τM DIFFERENT

τM \ ηm [11] (Round-Robin) Theorem 1 (TOD)

0 29 17

1 30 18

= MATI + ηM

FOR

ηm

2 30 19

3 31 20

6 33 23

8 34 25

11 36 27

We proceed next with the case of N = 4, where C1 , . . . , C4 are the rows of I4 and K1 , . . . , K4 are the entries of K given by (28). Table II shows the maximum values of τM = MATI + ηM that preserve the exponential stability of the hybrid system (8)–(10). Moreover, when ηm > MATI (ηm = 2, . . . , 5), the proposed condition of Theorem 1 is still feasible (communication delays are larger than the sampling intervals). TABLE II E XAMPLE (N = 4):

MAX . VALUE OF τM DIFFERENT

τ M \ ηm Theorem 1 (TOD)

0 1

1 2

= MATI + ηM

FOR

ηm 2 3

3 4

4 5

5 6

VI. C ONCLUSIONS This paper has considered the stability analysis of discretetime NCSs with multiple sensors under TOD protocol, variable delays and variable sampling intervals. The closedloop system model is formulated as a discrete-time hybrid system with time-varying delays in the dynamics and in the reset conditions. Sufficient conditions via an appropriate Lyapunov functional are derived for the exponential stability of the delayed hybrid system with respect to the full state. Extending the current model of NCSs so as to include the effect of quantization is essential in future work. VII. ACKNOWLEDGEMENT This work was partially supported by the National Natural Science Foundation of China (grant no. 61503026), the Foundation of Beijing Institute of Technology (grant no. 20150642003), the Israel Science Foundation (grant no. 1128/14), and HKU CRCG 201411159139.

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