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Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
CHAPTER
Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
17 Dan Ma, Zhuoyu Li
College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China
Chapter outline 1 Introduction....................................................................................... 349 2 Problem formulation............................................................................. 351 3 Event-triggered control ......................................................................... 354 3.1 Event generator and switching controller synthesis: The short network-induced delay case ........................................................ 354 3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case ........................... 357 4 Self-triggered control ........................................................................... 359 5 An example ....................................................................................... 361 6 Conclusions....................................................................................... 364 Acknowledgments .................................................................................. 364 References........................................................................................... 364
1 Introduction Networked control systems (NCSs), in which actuators, sensors, and controllers exchange information through the shared band-limited digital communication network, have received considerable attention in the last two decades [1–6]. Although NCSs allow for reduced wiring as well as for lower installation cost, the networkinduced delay and the packet dropout often occur inevitably. The reason is that the network bandwidth is limited and shared by multiple real-time tasks. In general, the network-induced delay is typically time varying, and in the extreme results in packet dropouts, even leads to the instability of the system. A challenging problem Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00017-2 © 2019 Elsevier Inc. All rights reserved.
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
herein is to cope with the negative impact of network-induced delays and packet dropouts. Meanwhile, for networked feedback control systems, how to guarantee the performance of the system by less traffic requirement and better resource utilization is of great importance. Event-triggered control is currently attracting more and more attention due to its abilities to significantly reduce the communication and computation resources in embedded control systems and distributed systems [6–15]. Unlike the traditional time-triggered control, in which the controller updates control law periodically, the event-triggered control is aperiodic. The key point is that the transmission instants in a networked feedback control loop are generated by the triggered events. Generally speaking, an event generator provides the even-triggered conditions, under which the ability, such as convergence and stability, of the systems can still be achieved. The controller updates as long as the event is violated. The aperiodic event-triggered control can be usually realized by the hardware and the software. The hardware and software realization are typically called event-triggering and self-triggering, respectively. Different from the event-triggered control, which needs to verify a specific condition continuously and determine whether the control task is triggered, the self-triggered control provides that the next triggered-time based on the receiving data of the systems. The related work can be found in Refs. [16–18]. In this chapter, we focus on the stabilization of networked feedback control systems with short network-induced delays and packet dropouts under the event- and self-triggered state feedback control, respectively. Distinct to the time-delay system approach adopted in Ref. [11], we model the NCS by using a switched system with one-step delay, whose distinctive advantage enables us to bypass the difficult construction of complex Lyapunov-Krasovskii functionals that are otherwise necessary. On the other hand, the switched system model can also reduce the conservatism of the robust control, which customarily uses the bound of delays instead of the timevarying characteristics of the network-induced delay [19]. Intuitively, the switched system model indicates the time-varying characteristics of the network-induced delay exactly by switchings of the subsystems. Furthermore, different from the most of the existing literature, we adopt the periodic event-triggered mechanism [10], under which the sampled state needs to be compared in the event generator. This kind of triggering mechanism both saves the computation and communication resources and ensures the strictly positive lower bound of the interevent times as well, which avoids the Zeno behavior. Finally, a self-triggered condition is provided as a more flexible way to adjust the triggered interval, which is related to the variation rate of the network-induced delay. Moreover, the triggered interval is monotonically increasing with the average dwell time, implying that with a slower rate in delay variation, the longer is the triggered interval. This chapter is organized as follows. Section 1 ends with the basic notations to be used in this chapter. Section 2 formulates the problem and presents a discretetime switched system model with limited subsystems to describe the NCS with short network-induced delays. Section 3 provides a codesign condition on the existence of the event-triggered mechanism and the switching controller to ensure the stabilization
2 Problem formulation
of the NCS. In the joint presence of short network-induced delays and packet dropouts, the NCS is also shown to be stabilized under a proper switching controller and an event-triggered mechanism. Section 4 extends the codesign method from the event-triggered control to the self-triggered control. Section 5 shows the feasibility and efficiency of the proposed method by using an example. At the end of this chapter, conclusion and remarks are given. Notation. R and N denote the sets of real and natural numbers. I is the identity matrix of any dimension. P > 0 is a symmetric positive definite matrix P. · stands for the Euclidian norm of vectors and its induced-spectral norm of matrices.
2 Problem formulation Consider an NCS depicted in Fig. 1. The plant is a continuous-time linear system in the state-space form x˙ (t) = Ap x(t) + Bp u(t),
(1)
where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, and Ap and Bp are the constant matrices with appropriate dimensions. The plant is controlled by a discrete-time controller. The information exchanges among sensors, controllers, and actuators through the shared communication network. In order to reduce the usage of computation and communication resources, an event generator is adopted to determine whether the current sampled-state measurement is transmitted to the controller via the networks or not. The gap between current state and past sampled state is as a measure novelty in feedback. The controller receives the sampled state through the networks as long as the gap exceeds a predesigned threshold.
Plant
Actuator
x(t )
Sensor
x( k )
Event generator
ZOH
x (bk ) Network
x(rk )
Discrete-time controller
FIG. 1 An NCS with an event generator.
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Without loss of generality, the sensor is time driven with a constant sampling period T. The sampling instants belong to Ω1 = {0, T, 2T, . . . , kT, . . .}, k ∈ N. The event generator is used to determine whether the sampled state is transmitted to the controller via the network. Denote the transmission instants of the event generator by Ω2 = {0, b1 T, b2 T, . . . , bk T, . . .} ⊆ Ω1 , bk ∈ N. The controller is event driven. Once the controller receives the latest sampled state, the controller updates immediately. The sampling instants received by the controller are denoted by Ω3 = {0, r1 T, r2 T, . . . , rk T, . . .} ⊆ Ω2 , rk ∈ N. The actuator is time driven, in which a buffer contains the most recent data sent by the controller. In order to describe the distribution characteristics of network-induced delays, the actuator reads the buffer periodically at a higher frequency than the sampling frequency. Moreover, we denote the reading buffer period of the actuator is T0 , so thus T0 = T/N. The short network-induced delay, that is, τk ≤ T and the bounded maximum allowable number of successive packet dropouts dMANSPD will be considered in this chapter. In particular, if all the sampled states are received by the controller, then we say Ω3 = Ω2 . Otherwise, Ω3 ⊂ Ω2 means the packet dropouts occur. Next, we give a timing diagram of the NCS in Fig. 2 in any sampling period [kT, (k + 1)T), k ∈ N. Since the network-induced delay τk is less than one sampling period T, there will exist u(k −1) and u(k) to control the plant in one sampling period [kT, (k + 1)T). As depicted in Fig. 2, the feedback control based on the sampled-state measurement x(k) arrived at the actuator at the instant t1 , which belongs to [kT + mT0 , kT + (m + 1)T0 ), m ∈ Z0 = {0, 1, . . . , N}. The actuator reads the buffer at time kT +(m+1)T0 and controller updates the control law by using the new arrival. Let us denote the control interval of u(k) and u(k − 1) by n0 (k)T0 and n1 (k)T0 , respectively. In the sequel, we know that
n0 (k), n1 (k) ∈ Z0 , n0 (k) + n1 (k) = N, n0 (k)T0 + n1 (k)T0 = NT0 = T,
where n1 (k)T0 indicates the size of the network-induced delay interval in one sampling period.
u (k )
... kT kT + T0
t1
(k + 1)T
u (k ) u (k – 1)
FIG. 2 A timing diagram of an NCS with short network-induced delays.
2 Problem formulation
By using a mapping: R2 → M = {0, 1, . . . , N}: [n0 (k) n1 (k)] → σ (k) to describe the relationship between the time-varying characteristics of the networkinduced delay and the switchings, where σ (k) is a nonnegative integer, we give the following indication: [n1 (k) n0 (k)] = [0 N] → σ (k) = 0, [n1 (k) n0 (k)] = [1 N − 1] → σ (k) = 1, .. . [n1 (k) n0 (k)] = [N 0] → σ (k) = N.
In light of the existing modeling method in Ref. [20] and combining with the earlier analysis of the network-induced delay, system (1) can be rewritten as a switched system Sσ (k) : x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1),
(2)
where A = eAp T , B0σ (k) =
n0 (k)−1
A0 = eAp T0 , Ai0 B0 ,
B0 =
B1σ (k) =
i=0
T0
0 n1 (k)−1
eAP τ Bp dτ , i+n0 (k)
A0
B0 .
i=0
It can be shown that system (2) is a switched system with N + 1 subsystems. For σ (k) = i, Si denotes the ith subsystem. We use Ω4 = {0, k1 T, . . . , kj T, . . .} ⊆ Ω1 , kj ∈ N to denote the set of switching instants. In what follows, we adopt multiple discrete-time event-triggered state feedback controller switchings to stabilize the NCS, which synchronously switch with the subsystems. This typically leads to better performance achieved by a single controller. Let us consider the following switching event-triggered feedback controller: u(k) = Kσ (k) x(rk ),
k ∈ [rk , rk+1 ),
(3)
where Kσ (k) is the switching state feedback controller gains to be designed later. The state-based event-triggered condition is given by rk+1 = min{k > rk | e(k) ≥ δ1 x(rk )},
(4)
where e(k) = x(k) − x(rk ), rk is the event-triggered instant, 0 < δ1 < 1 is the event threshold to be determined. Remark 1. Eq. (4) is a periodic event-triggered condition. The triggering instants are always integer times of the sampling period T, thus the Zeno behavior can be avoided. In the extreme case, δ1 = 0, rk+1 = rk +1, which leads to the time-triggered generator.
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Combining with Eqs. (2), (3), we have the closed-loop event-triggered switched system with one-step input delay
x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1), u(k) = Kσ (k) x(rk ), k ∈ [rk , rk+1 ).
(5)
Our goal is to design an event-triggered or a self-triggered switching controller to stabilize the NCS (5) simultaneously.
3 Event-triggered control In this section, we first provide a sufficient condition on the existence of the event generator and the switching controller to simultaneously stabilize the NCS (5) with short network-induced delays. Then, the stabilization of networked feedback control systems in the joint presence of short network-induced delays and packet dropouts is extended.
3.1 Event generator and switching controller synthesis: The short network-induced delay case Theorem 1. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0, and Ri with appropriate dimensions, such 0 < δ1 < 1, and matrices Pˆ i > 0, Q that the following inequalities ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
ˆ T + RT B T PA i 0i RTi BT1i
−Pˆ i
−λ2 Pˆ i
0
0
∗
ˆi −2λ2 Pˆ i + λ2 Q
0
∗
∗
−2ξ Pˆ i + ξ 2 I
∗
∗
∗
−RTi BT0i −Pˆ i
∗
∗
∗
∗
−δ1−2 I
∗
∗
∗
∗
∗ μPˆ i ≥ Pˆ j ,
ˆi ≥ Q ˆ j, μQ
i, j ∈ M,
0 Pˆ i 0
Pˆ i
⎤
⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ ˆ −Qi
(6)
(7)
hold, then under the event-triggered mechanism (4) and the discrete-time switched state feedback controller (3) with any switching signal satisfying the average dwell time τa > τa∗ :=
ln μ , 2 ln λ−1
system (5) is exponentially stabilized with the decay rate 1
α = λμ 2τa .
(8)
3 Event-triggered control
Moreover, the feedback control gains Ki = Ri Pˆ −1 i ,
i ∈ M.
(9)
Proof. Choose the multiple Lyapunov candidates for the NCS (5) as follows: Vi (k) = xT (k)Pi x(k) + xT (k − 1)Qi x(k − 1),
(10)
where Pi > 0 and Qi > 0, i ∈ M are symmetric positive definite matrices. For any sampling period [kT, (k + 1)T), in order to guarantee the exponential stabilization of system (5), the difference of Vi (k) along the ith subsystem of system (5) needs to be Vi (k + 1) − λ2 Vi (k) = xT (k + 1)Pi x(k + 1) + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1), = [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1).
Combining with the event-triggered condition (4), we get Vi (k + 1) − λ2 Vi (k) ≤ [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 xT (rk )x(rk ), = [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 [x(k) − e(k)]T [x(k) − e(k)], ⎡ ⎤T ⎡Σ (A + B K )T P B K −(A + B K )T P B K − δ 2 I ⎤ ⎡ ⎤ i 1i i i 0i i 1 0i i 0i i x(k) x(k) 1 ⎢ ⎥ = ⎣x(k − 1)⎦ ⎣ ∗ Σ2 −KiT BT1i Pi B0i Ki ⎦ ⎣x(k − 1)⎦ , e(k) e(k) ∗ ∗ Σ 3
where Σ1 = (A + B0i Ki )T Pi (A + B0i Ki ) − λ2 Pi + Qi + δ12 I, Σ2 = −λ2 Qi + KiT BT1i Pi B1i Ki , Σ3 = (δ12 − 1)I + KiT BT0i Pi B0i Ki .
Using the Schur complement three times, Eq. (6) is equivalent to ⎡
1
⎢ ⎣ ∗ ∗
Pˆ i (A + B0i Ki )T Pi B1i Ri
−Pˆ i (A + B0i Ki )T Pi B0i Ri − δ12 Pˆ i Pˆ i
2
−RTi BT1i Pi B0i Ri
∗
3
⎤ ⎥ ⎦ < 0,
(11)
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
where
1 = Pˆ i (A + B0i Ki )T Pi (A + B0i Ki )Pˆ i − λ2 Pˆ i + Pˆ i Qi Pˆ i + δ12 Pˆ i Pˆ i , ˆ i + Ri BT Pi B1i Ri ,
2 = −2λ2 Pˆ i + λ2 Q 1i
3 = −2ξ Pˆ i + ξ 2 I + δ12 Pˆ i Pˆ i + Ri BT0i Pi B0i Ri .
ˆ i − Pˆ i )Qi (Q ˆ i − Pˆ i ) > 0, which On the other hand, noticing that Qi > 0, we have (Q ˆ yields to −Pˆ i Qi Pˆ i < −2Pˆ i + Qi . Similarly, −Pˆ i Pˆ i < −2ξ Pˆ i + ξ 2 I holds for any arbitrary scalar ξ . Using the earlier two inequalities, multiplying by the diagonal matrix diag{Pi , Pi , Pi } and its transpose on both sides of Eq. (11), and setting Ri = Ki Pˆ i , we get Vi (k + 1) − λ2 Vi (k) < 0,
i ∈ M.
(12)
This indicates that each subsystem of Eq. (5) is exponentially stable. Hence, for any switching interval [kj , kj+1 ), we have Vσ (kj ) (k) < λ2(k−kj ) Vσ (kj ) (kj ),
j = 0, 1, . . . .
(13)
j = 0, 1, . . . .
(14)
Due to Vσ (k) (k) = Vσ (kj ) (k), we get Vσ (k) (k) < λ2(k−kj ) Vσ (kj ) (kj ),
Since the system state is continuous at any switching instant, from inequality (7), we know that Vσ (kj ) (kj ) ≤ μVσ (k− ) (kj− ), j
j = 0, 1, . . . .
(15)
According to Eqs. (14), (15), we can obtain Vσ (k) (k) = Vσ (kj ) (k) ≤ λ2(k−kj ) Vσ (kj ) (kj ) ≤ μλ2(k−kj ) Vσ (k− ) (kj− ) j
≤ μ2 λ2(k−kj−1 ) Vσ (kj−1 ) (kj−1 ) .. .
(16)
≤ μNσ (k0 ,k) λ2(k−kj ) λ2(kj −kj−1 ) · · · λ2(k1 −k0 ) Vσ (k0 ) (k0 ) k−k0
≤ μ τa λ2(k−k0 ) Vσ (k0 ) (k0 ) = α 2(k−k0 ) Vσ (k0 ) (k0 ).
Denoting ε1 = min{λmin (Pi )}, ε2 = max{λmax (Pi )}, and ε3 = min{λmin (Qi )}, i∈M
i∈M
i∈M
Eq. (16) yields to ε1 x(k)2 ≤ ε1 x(k)2 + ε3 x(k − 1)2 ≤ Vσ (k) (k) < ε2 α 2(k−k0 ) x(k0 )2 .
(17)
3 Event-triggered control
Thus, x(k) ≤
ε2 (k−k0 ) x(k0 ) . α ε1
(18)
This completes the proof. Remark 2. Theorem 1 indicates that the exponential decay rate α is monotonically increasing with the parameter λ, and decreasing with the average dwell time τa . According to the modeling process of the NCS (5), the variation rate of the networkinduced delays can be reflected by the average dwell time τa . The smaller the variation rate of network-induced delay is, the larger the average dwell time τa is.
3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case Once the packet dropouts occur in the network transmission, the event-triggered mechanism (4) cannot be directly used to determine whether the sampled state is transmitted or not. The packet dropouts need to be incorporated to the eventtriggered mechanism. Toward this end, a novel event-triggered condition is provided as follows: bk+1 = min{k > bk | ˜e(k) ≥ δ2 x(bk )},
(19)
where e˜ (k) = x(k) − x(bk ), bk is the transmission instants of the event generator. The following theorem further gives a sufficient condition on the existence of a switching controller and an event generator to stabilize the NCS with short networkinduced delays and packet dropouts. Theorem 2. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist 1
0 ≤ δ2 ≤ ((1 + δ1 )(1 + ε)−dMANSPD ) dMANSPD +1 − 1,
(20)
0i Ki B1i Ki (1+δ1 ) + where 0 < δ1 < 1, ε = max A − I + B1−δ , and positive 1−δ 1 1 ∀i∈M
ˆ i > 0 and matrix Ri with appropriate dimensions such definite matrices Pˆ i > 0, Q that Eqs. (6), (7) hold, then under the event-triggered mechanism (19), the controller (3) and any switching signal with the average dwell time satisfying Eq. (8), system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the feedback control gains are given by 1 Eq. (9), and the exponential decay rate is α = λμ 2τa . Proof. Consider a successful transmission interval [rk , rk+1 ), in which there exists dk packet dropouts, that is, rk = b0 < b1 < b2 < · · · < bdk < bdk +1 = rk+1 . For all l = 1, . . . , dk , during the triggering interval, we have x(bl+1 − 1) − x(bl ) ≤ δ2 x(bl ) ,
(21)
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
which yields to x(bl+1 − 1) ≤ x(bl+1 − 1) − x(bl ) + x(bl ) ≤ (1 + δ2 ) x(bl ) .
(22)
Since x(rk ) = x(bl+1 − 1) − x(bl+1 − 1) + x(rk ) ≤ x(bl+1 − 1) + δ1 x(rk ) ,
(23)
then we have x(rk ) ≤
1 x(bl+1 − 1) . 1 − δ1
(24)
Consequently, we know that x(bl+1 − 2) ≤ x(bl+1 − 2) − x(rk ) + x(rk ) ≤ (1 + δ1 ) x(rk ) 1 + δ1 x(bl+1 − 1) . ≤ 1 − δ1
(25)
Consider system (5) at the instant k = bl+1 , then we have x(bl+1 ) = Ax(bl+1 − 1) + B0i Ki x(rk ) + B1i Ki x(bl+1 − 2),
i ∈ M.
Next, in view of the packet dropouts in the network transmission, for the interval [b1 , rk+1 ), combining with the earlier procedure, we know that x(bl+1 ) − x(bl+1 − 1) = Ax(bl+1 − 1) + B0i Ki x(rk )
+B1i Ki x(bl+1 − 2) − x(bl+1 − 1) ≤ (A − I)x(bl+1 − 1) + B0i Ki x(rk ) + B1i Ki x(bl+1 − 2) ≤ ε x(bl+1 − 1) .
(26)
In light of Eq. (22), it follows that x(bl+1 ) − x(bl+1 − 1) ≤ ε(1 + δ2 ) x(bl ) ,
(27)
which further leads to x(bl+1 ) ≤ x(bl+1 ) − x(bl+1 − 1) + x(bl+1 − 1) − x(bl ) + x(bl ) ≤ [(1 + ε)(1 + δ2 )] x(bl ) ≤ [(1 + ε)(1 + δ2 )]l+1 x(rk ) .
(28)
4 Self-triggered control
For any k ∈ [bdk , bdk +1 ), combining with Eqs. (21), (26), (28), we can get k −1 d x(k) − x(rk ) ≤ x(k) − x(bdk ) + (x(bl+1 ) − x(bl+1 − 1)) l=0 dk −1 + (x(bl+1 − 1) − x(bl )) l=0 ≤
dk
δ2 x(bl ) +
d k −1
l=0
ε(1 + δ2 ) x(bl )
l=0
⎛ ⎞ dk d k −1 l l δ2 ((1 + ε)(1 + δ2 )) + ε(1 + δ2 )((1 + ε)(1 + δ2 )) ⎠ x(rk ) ≤⎝ l=0
l=0
= ((1 + δ2 )dk +1 (1 + ε)dk − 1) x(rk ) . (29)
According to Eqs. (20), (29), it leads to x(k) − x(rk ) ≤ δ1 x(k) ,
(30)
where δ1 = (1 + δ2 )dk +1 (1 + ε)dk − 1. The earlier analysis indicates the novel event-triggered condition (19) can guarantee (4). Combining with the existence conditions on the switching controller in Theorem 1, the proof is completed. Remark 3. From Eq. (20), we can see that the proposed event-triggered threshold depends on the maximum allowable number of successive packet dropouts dMSNSPD . Since dMSNSPD ≥ 0, this implies that δ2 ≤ δ1 . Thus we conclude that the packet dropouts increase the number of the event-triggered instants in order to guarantee the stabilization of the NCS with delays and packet dropouts. In the extreme that, if dMSNSPD = 0 and δ2 = δ1 , that is, no packet dropout occurs in the network transmission, Theorem 2 reduces to Theorem 1.
4 Self-triggered control In this section, we study the stabilization problem of the NCS subject to short network-induced delays and packet dropouts under a self-triggered mechanism. The following theorem gives a sufficient condition on the stabilization of the NCS with short network-induced delays and packet dropouts under the self-triggered control and the switching controller simultaneously. Theorem 3. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0 and matrix Ri with appropriate dimensions positive definite matrices Pˆ i > 0, Q such that Eqs. (6), (7), (20) hold, then under the switching controller (3) with the average dwell-time satisfying Eq. (8), and the self-triggered mechanism
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
bk+1 = bk + max 1,
δ (1 − A) x(bk ) , lnA 1 − 2 Θ
(31)
where Θ = ψ1
bk −1 ε2 λμ 2τa x(k0 ) + ψ2 x(bk ) , ε1
ψ1 = max {B1i Ki } , ψ2 = max {B0i Ki − I} , ∀i∈M
∀i∈M
system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the exponential decay rate is 1 α = λμ 2τa , and the feedback control gains are given by Eq. (9). Proof. For any interval [bk , bk+1 ), combining with the dynamics of the NCS (5), we compute the state error e˜ (k + 1) = x(k + 1) − x(bk ) = Ax(k) + B0i Ki x(bk ) + B1i Ki x(k − 1) − x(bk ) = A˜e(k) + B1i Ki x(k − 1) + (B0i Ki + A − I)x(bk ),
(32) i ∈ M.
This leads to ˜e(k + 1) ≤ A ˜e(k) + B1i Ki x(k − 1) + B0i Ki − I x(bk ) ≤ A ˜e(k) + ψ1 x(k − 1) + ψ2 x(bk ) .
(33)
Based on Theorem 2, we know that the system state satisfies x(k − 1) ≤
ε2 (k−1−k0 ) x(k0 ) ≤ α ε1
ε2 (bk −1) x(k0 ) . α ε1
(34)
x(k0 ) + ψ2 x(bk ) .
(35)
Combining with Eqs. (33), (34), we have ˜e(k + 1) ≤ A ˜e(k) + ψ1
ε2 ε1 λμ
bk −1 2τa
Thus, for k ∈ [bk , bk+1 ), ˜e(k) ≤ Ak−bk ˜e(bk ) + (1 − A)−1 (1 − Ak−bk )Θ = (1 − A)−1 (1 − Ak−bk )Θ.
If the interevent interval satisfying Eq. (31) holds, we have (1 − A)−1 (1 − Ak−bk )Θ ≤ δ2 x(bk ) ,
k ∈ [bk , bk+1 ).
Define (1 − A)−1 (1 − ATk )Θ = δ2 x(bk ), which implies that δ2 (1 − A) x(bk ) Tk = bk+1 − bk = lnA 1 − . Θ
(36)
5 An example
This indicates that the self-triggered condition (31) can guarantee the eventtriggered condition (19), which completes the proof. Remark 4. The interevent interval Tk is related to the average dwell time τa , which implies that the self-triggered mechanism depends on the variation rate of the network-induced delay. The relationship can be summarized as the following two aspects: (i) Θ is decreasing with the increasing of the average dwell time τa . If A < 1, then the interevent interval Tk is monotonically increasing, that is, δ2 (1 − A) x(bk h) ↓→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ (ii) Θ is decreasing with the increasing of the average dwell time τa . If A ≥ 1, the interevent interval Tk is also monotonically increasing, that is, δ2 (1 − A) x(bk h) ↑→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ Therefore, the intertrigger interval Tk is monotonically increasing with the average dwell time τa , which implies that with a slower rate in network-induced delay variation, the longer is the triggered interval.
5 An example In this section, we give a numerical example to illustrate the effectiveness of the −8 1 , Bp = main results. Consider the plant (1) with the parameters Ap = 0.1 0.01 −0.2 0 . Set the sensor sampling period T = 0.2 s, and the actuator sampling 0 0.1 period T0 = 0.1 s, which implies that N = 2. Hence, the NCS (5) has three subsystems. Suppose that the number of successive packet dropouts in the network transmission is bounded by dMANSPD = 2. Next, according to the modeling process of the NCS (5), we can obtain
0.2019 1.2214 , 1.0202 1.0020 0 0 , B01 = B10 = 0 0 0 0 , B12 = B02 = 0 0 A=
−0.0422 0.0263 , −0.0541 0.0306 0.0138 0.0105 −0.0284 , B11 = −0.0201 0.0100 −0.0340 −0.0422 0.0263 . −0.0541 0.0306 B00 =
0.0158 0.0206
,
361
CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Choosing μ = 1.04, λ = 0.9539, ξ = 4 and solving Eqs. (6), (7) by Theorem 2, we get δ1 = 0.0023, δ2 = 0.00017; 5420.50 −3442.71 ˆ ˆP0 = , Q0 = −3442.71 2191.19 1,088,218.66 −691,480.11 ; R0 = 1,857,449.10 −1,180,425.81 5428.46 −3444.67 ˆ ˆP1 = , Q1 = −3444.67 2190.40 216,135.40 −123,120.83 ; R1 = 367,021.72 −209,072.76 5347.72 −3399.09 ˆ ˆP2 = , Q2 = −3399.09 2165.14 −14,626.41 8125.68 . R2 = −25,047.14 13,914.91
10,506.45 −6681.07
−6681.07 4256.42
10,560.95 −6720.51
−6720.51 4284.49
10,644.80 −6765.70
−6765.70 4308.14
,
,
,
Furthermore, we can have the feedback control gain
156.62 −69.48 K0 = , 245.63 −152.78
1991.96 3076.38 K1 = , 3382.56 5224.03
and the average dwell time τa > τa∗ = 1 2τa
ln μ 2 ln λ−1
−163.53 −252.98 K2 = , −280.04 −433.22
= 0.4155. Taking τa = 0.5, we have
α = λμ = 0.9921. Fig. 3 gives a switching signal σ (k) with the average dwell time τa = 0.5 >τa∗ . Fig. 4 shows the trajectories of the system state converges to zero. Figs. 5 and 6 give the event-triggered instants and the time-triggered instants, respectively.
2
1.5
Switching signal
362
1
0.5
0
0
1
FIG. 3 Switching signal.
2
3
4
5 Time (s)
6
7
8
9
10
5 An example
1.5
System state 1 System state 2
x (k)
1 0.5 0 –0.5 0
1
2
3
4
5 Time (s)
FIG. 4 System state.
FIG. 5 Event-triggered instants.
FIG. 6 Time-triggered instants.
6
7
8
9
10
363
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
The simulation results indicate that the event-triggered mechanism reduces the computation of the controller and saves the communication resources with respect to the traditional periodic sampling control.
6 Conclusions A codesign method of the event-triggered generator and the switching feedback controller for NCSs with short network-induced delays and packet dropouts is investigated in this chapter. Both the event threshold and the switching frequency depend on the variation rate of the network-induced delays and the maximal number of successive packet dropouts to stabilize the networked feedback system. Moreover, a self-triggered condition as the flexible software realization is also developed. Finally, a numerical example shows that the updating frequency of the control law can be reduced and simultaneously maintain the system performance.
Acknowledgments The authors gratefully acknowledge the support of the Natural Science Foundation of China under Grants 61603079 and 61773098.
References [1] G.C. Walsh, H. Ye, L.G. Bushnell, Stability analysis of networked control systems, IEEE Trans. Control Syst. Technol. 10 (3) (2002) 438–446. [2] P. Antsaklis, J. Baillieul, Special issue on technology of networked control systems, Proc. IEEE 95 (1) (2007) 5–8. [3] H. Zhang, Z. Zhang, Z. Wang, New results on stability and stabilization of networked control systems with short time-varying delay, IEEE Trans. Cybern. 46 (12) (2015) 2772–2781. [4] C. Tan, L. Li, H. Zhang, Stabilization of networked control systems with both network-induced delay and packet dropout, Automatica 59 (2015) 194–199. [5] A. Selivanov, E. Fridman, Observer-based input-to-state stabilization of networked control systems with large uncertain delays, Automatica 74 (2016) 63–70. [6] D. Liu, G.H. Yang, Robust event-triggered control for networked control systems, Inf. Sci. 459 (2018) 186–197, https://doi.org/10.1016/j.ins.2018.02.057. [7] K.E. Årzén, A simple event-based PID controller, in: Proceedings of IFAC World Congress, 1999, pp. 8687–8692. [8] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control 52 (9) (2007) 1680–1685. [9] M.C.F. Donkers, W.P.M.H. Heemels, Output-based event-triggered control with guaranteed L∞ -gain and improved and decentralized event-triggering, IEEE Trans. Autom. Control 57 (6) (2012) 1362–1376.
References
[10] D.P. Borgers, V.S. Dolk, W.P.M.H. Heemels, Dynamic periodic event-triggered control for linear systems, IEEE Trans. Autom. Control 58 (4) (2013) 847–861. [11] D. Yue, E. Tian, Q.L. Han, A delay system method to design of event-triggered control of networked control systems, in: IEEE Conference on Decision and Control and European Control Conference, December, 2011, pp. 1668–1673. [12] X. Wang, M.D. Lemmon, Event-triggering in distributed networked control systems, IEEE Trans. Autom. Control 56 (3) (2011) 586–601. [13] X.M. Zhang, Q.L. Han, Event-triggered dynamic output feedback control for networked control systems, IET Control Theory Appl. 8 (4) (2014) 226–234. [14] M. Cao, F. Xiao, L. Wang, Event-based second-order consensus control for multi-agent systems via synchronous periodic event detection, IEEE Trans. Autom. Control 60 (9) (2015) 2452–2457. [15] W. Hu, L. Liu, G. Feng, Output consensus of heterogeneous linear multi-agent systems by distributed event-triggered/self-triggered strategy, IEEE Trans. Cybern. 47 (8) (2017) 1914–1924. [16] X. Wang, M.D. Lemmon, Self-triggered feedback control systems with finite-gain stability, IEEE Trans. Autom. Control 54 (3) (2009) 452–467. [17] Y. Tang, H. Gao, J. Kurths, Robust H∞ self-triggered control of networked systems under packet dropouts, IEEE Trans. Cybern. 46 (12) (2015) 3294–3305. [18] C. Peng, Q.L. Han, On designing a novel self-triggered sampling scheme for networked control systems with data losses and communication delays, IEEE Trans. Ind. Electron. 63 (2) (2016) 1239–1248. [19] C. Peng, T.C. Yang, Event-triggered communication and H∞ control co-design for networked control systems, Automatica 49 (5) (2013) 1326–1332. [20] W.A. Zhang, L. Yu, New approach to stabilisation of networked control systems with time-varying delays, IET Control Theory Appl. 2 (12) (2008) 1094–1104.
365
Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
17 Dan Ma, Zhuoyu Li
College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China
Chapter outline 1 Introduction....................................................................................... 349 2 Problem formulation............................................................................. 351 3 Event-triggered control ......................................................................... 354 3.1 Event generator and switching controller synthesis: The short network-induced delay case ........................................................ 354 3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case ........................... 357 4 Self-triggered control ........................................................................... 359 5 An example ....................................................................................... 361 6 Conclusions....................................................................................... 364 Acknowledgments .................................................................................. 364 References........................................................................................... 364
1 Introduction Networked control systems (NCSs), in which actuators, sensors, and controllers exchange information through the shared band-limited digital communication network, have received considerable attention in the last two decades [1–6]. Although NCSs allow for reduced wiring as well as for lower installation cost, the networkinduced delay and the packet dropout often occur inevitably. The reason is that the network bandwidth is limited and shared by multiple real-time tasks. In general, the network-induced delay is typically time varying, and in the extreme results in packet dropouts, even leads to the instability of the system. A challenging problem Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00017-2 © 2019 Elsevier Inc. All rights reserved.
349
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
herein is to cope with the negative impact of network-induced delays and packet dropouts. Meanwhile, for networked feedback control systems, how to guarantee the performance of the system by less traffic requirement and better resource utilization is of great importance. Event-triggered control is currently attracting more and more attention due to its abilities to significantly reduce the communication and computation resources in embedded control systems and distributed systems [6–15]. Unlike the traditional time-triggered control, in which the controller updates control law periodically, the event-triggered control is aperiodic. The key point is that the transmission instants in a networked feedback control loop are generated by the triggered events. Generally speaking, an event generator provides the even-triggered conditions, under which the ability, such as convergence and stability, of the systems can still be achieved. The controller updates as long as the event is violated. The aperiodic event-triggered control can be usually realized by the hardware and the software. The hardware and software realization are typically called event-triggering and self-triggering, respectively. Different from the event-triggered control, which needs to verify a specific condition continuously and determine whether the control task is triggered, the self-triggered control provides that the next triggered-time based on the receiving data of the systems. The related work can be found in Refs. [16–18]. In this chapter, we focus on the stabilization of networked feedback control systems with short network-induced delays and packet dropouts under the event- and self-triggered state feedback control, respectively. Distinct to the time-delay system approach adopted in Ref. [11], we model the NCS by using a switched system with one-step delay, whose distinctive advantage enables us to bypass the difficult construction of complex Lyapunov-Krasovskii functionals that are otherwise necessary. On the other hand, the switched system model can also reduce the conservatism of the robust control, which customarily uses the bound of delays instead of the timevarying characteristics of the network-induced delay [19]. Intuitively, the switched system model indicates the time-varying characteristics of the network-induced delay exactly by switchings of the subsystems. Furthermore, different from the most of the existing literature, we adopt the periodic event-triggered mechanism [10], under which the sampled state needs to be compared in the event generator. This kind of triggering mechanism both saves the computation and communication resources and ensures the strictly positive lower bound of the interevent times as well, which avoids the Zeno behavior. Finally, a self-triggered condition is provided as a more flexible way to adjust the triggered interval, which is related to the variation rate of the network-induced delay. Moreover, the triggered interval is monotonically increasing with the average dwell time, implying that with a slower rate in delay variation, the longer is the triggered interval. This chapter is organized as follows. Section 1 ends with the basic notations to be used in this chapter. Section 2 formulates the problem and presents a discretetime switched system model with limited subsystems to describe the NCS with short network-induced delays. Section 3 provides a codesign condition on the existence of the event-triggered mechanism and the switching controller to ensure the stabilization
2 Problem formulation
of the NCS. In the joint presence of short network-induced delays and packet dropouts, the NCS is also shown to be stabilized under a proper switching controller and an event-triggered mechanism. Section 4 extends the codesign method from the event-triggered control to the self-triggered control. Section 5 shows the feasibility and efficiency of the proposed method by using an example. At the end of this chapter, conclusion and remarks are given. Notation. R and N denote the sets of real and natural numbers. I is the identity matrix of any dimension. P > 0 is a symmetric positive definite matrix P. · stands for the Euclidian norm of vectors and its induced-spectral norm of matrices.
2 Problem formulation Consider an NCS depicted in Fig. 1. The plant is a continuous-time linear system in the state-space form x˙ (t) = Ap x(t) + Bp u(t),
(1)
where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, and Ap and Bp are the constant matrices with appropriate dimensions. The plant is controlled by a discrete-time controller. The information exchanges among sensors, controllers, and actuators through the shared communication network. In order to reduce the usage of computation and communication resources, an event generator is adopted to determine whether the current sampled-state measurement is transmitted to the controller via the networks or not. The gap between current state and past sampled state is as a measure novelty in feedback. The controller receives the sampled state through the networks as long as the gap exceeds a predesigned threshold.
Plant
Actuator
x(t )
Sensor
x( k )
Event generator
ZOH
x (bk ) Network
x(rk )
Discrete-time controller
FIG. 1 An NCS with an event generator.
351
352
CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Without loss of generality, the sensor is time driven with a constant sampling period T. The sampling instants belong to Ω1 = {0, T, 2T, . . . , kT, . . .}, k ∈ N. The event generator is used to determine whether the sampled state is transmitted to the controller via the network. Denote the transmission instants of the event generator by Ω2 = {0, b1 T, b2 T, . . . , bk T, . . .} ⊆ Ω1 , bk ∈ N. The controller is event driven. Once the controller receives the latest sampled state, the controller updates immediately. The sampling instants received by the controller are denoted by Ω3 = {0, r1 T, r2 T, . . . , rk T, . . .} ⊆ Ω2 , rk ∈ N. The actuator is time driven, in which a buffer contains the most recent data sent by the controller. In order to describe the distribution characteristics of network-induced delays, the actuator reads the buffer periodically at a higher frequency than the sampling frequency. Moreover, we denote the reading buffer period of the actuator is T0 , so thus T0 = T/N. The short network-induced delay, that is, τk ≤ T and the bounded maximum allowable number of successive packet dropouts dMANSPD will be considered in this chapter. In particular, if all the sampled states are received by the controller, then we say Ω3 = Ω2 . Otherwise, Ω3 ⊂ Ω2 means the packet dropouts occur. Next, we give a timing diagram of the NCS in Fig. 2 in any sampling period [kT, (k + 1)T), k ∈ N. Since the network-induced delay τk is less than one sampling period T, there will exist u(k −1) and u(k) to control the plant in one sampling period [kT, (k + 1)T). As depicted in Fig. 2, the feedback control based on the sampled-state measurement x(k) arrived at the actuator at the instant t1 , which belongs to [kT + mT0 , kT + (m + 1)T0 ), m ∈ Z0 = {0, 1, . . . , N}. The actuator reads the buffer at time kT +(m+1)T0 and controller updates the control law by using the new arrival. Let us denote the control interval of u(k) and u(k − 1) by n0 (k)T0 and n1 (k)T0 , respectively. In the sequel, we know that
n0 (k), n1 (k) ∈ Z0 , n0 (k) + n1 (k) = N, n0 (k)T0 + n1 (k)T0 = NT0 = T,
where n1 (k)T0 indicates the size of the network-induced delay interval in one sampling period.
u (k )
... kT kT + T0
t1
(k + 1)T
u (k ) u (k – 1)
FIG. 2 A timing diagram of an NCS with short network-induced delays.
2 Problem formulation
By using a mapping: R2 → M = {0, 1, . . . , N}: [n0 (k) n1 (k)] → σ (k) to describe the relationship between the time-varying characteristics of the networkinduced delay and the switchings, where σ (k) is a nonnegative integer, we give the following indication: [n1 (k) n0 (k)] = [0 N] → σ (k) = 0, [n1 (k) n0 (k)] = [1 N − 1] → σ (k) = 1, .. . [n1 (k) n0 (k)] = [N 0] → σ (k) = N.
In light of the existing modeling method in Ref. [20] and combining with the earlier analysis of the network-induced delay, system (1) can be rewritten as a switched system Sσ (k) : x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1),
(2)
where A = eAp T , B0σ (k) =
n0 (k)−1
A0 = eAp T0 , Ai0 B0 ,
B0 =
B1σ (k) =
i=0
T0
0 n1 (k)−1
eAP τ Bp dτ , i+n0 (k)
A0
B0 .
i=0
It can be shown that system (2) is a switched system with N + 1 subsystems. For σ (k) = i, Si denotes the ith subsystem. We use Ω4 = {0, k1 T, . . . , kj T, . . .} ⊆ Ω1 , kj ∈ N to denote the set of switching instants. In what follows, we adopt multiple discrete-time event-triggered state feedback controller switchings to stabilize the NCS, which synchronously switch with the subsystems. This typically leads to better performance achieved by a single controller. Let us consider the following switching event-triggered feedback controller: u(k) = Kσ (k) x(rk ),
k ∈ [rk , rk+1 ),
(3)
where Kσ (k) is the switching state feedback controller gains to be designed later. The state-based event-triggered condition is given by rk+1 = min{k > rk | e(k) ≥ δ1 x(rk )},
(4)
where e(k) = x(k) − x(rk ), rk is the event-triggered instant, 0 < δ1 < 1 is the event threshold to be determined. Remark 1. Eq. (4) is a periodic event-triggered condition. The triggering instants are always integer times of the sampling period T, thus the Zeno behavior can be avoided. In the extreme case, δ1 = 0, rk+1 = rk +1, which leads to the time-triggered generator.
353
354
CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Combining with Eqs. (2), (3), we have the closed-loop event-triggered switched system with one-step input delay
x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1), u(k) = Kσ (k) x(rk ), k ∈ [rk , rk+1 ).
(5)
Our goal is to design an event-triggered or a self-triggered switching controller to stabilize the NCS (5) simultaneously.
3 Event-triggered control In this section, we first provide a sufficient condition on the existence of the event generator and the switching controller to simultaneously stabilize the NCS (5) with short network-induced delays. Then, the stabilization of networked feedback control systems in the joint presence of short network-induced delays and packet dropouts is extended.
3.1 Event generator and switching controller synthesis: The short network-induced delay case Theorem 1. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0, and Ri with appropriate dimensions, such 0 < δ1 < 1, and matrices Pˆ i > 0, Q that the following inequalities ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
ˆ T + RT B T PA i 0i RTi BT1i
−Pˆ i
−λ2 Pˆ i
0
0
∗
ˆi −2λ2 Pˆ i + λ2 Q
0
∗
∗
−2ξ Pˆ i + ξ 2 I
∗
∗
∗
−RTi BT0i −Pˆ i
∗
∗
∗
∗
−δ1−2 I
∗
∗
∗
∗
∗ μPˆ i ≥ Pˆ j ,
ˆi ≥ Q ˆ j, μQ
i, j ∈ M,
0 Pˆ i 0
Pˆ i
⎤
⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ ˆ −Qi
(6)
(7)
hold, then under the event-triggered mechanism (4) and the discrete-time switched state feedback controller (3) with any switching signal satisfying the average dwell time τa > τa∗ :=
ln μ , 2 ln λ−1
system (5) is exponentially stabilized with the decay rate 1
α = λμ 2τa .
(8)
3 Event-triggered control
Moreover, the feedback control gains Ki = Ri Pˆ −1 i ,
i ∈ M.
(9)
Proof. Choose the multiple Lyapunov candidates for the NCS (5) as follows: Vi (k) = xT (k)Pi x(k) + xT (k − 1)Qi x(k − 1),
(10)
where Pi > 0 and Qi > 0, i ∈ M are symmetric positive definite matrices. For any sampling period [kT, (k + 1)T), in order to guarantee the exponential stabilization of system (5), the difference of Vi (k) along the ith subsystem of system (5) needs to be Vi (k + 1) − λ2 Vi (k) = xT (k + 1)Pi x(k + 1) + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1), = [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1).
Combining with the event-triggered condition (4), we get Vi (k + 1) − λ2 Vi (k) ≤ [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 xT (rk )x(rk ), = [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 [x(k) − e(k)]T [x(k) − e(k)], ⎡ ⎤T ⎡Σ (A + B K )T P B K −(A + B K )T P B K − δ 2 I ⎤ ⎡ ⎤ i 1i i i 0i i 1 0i i 0i i x(k) x(k) 1 ⎢ ⎥ = ⎣x(k − 1)⎦ ⎣ ∗ Σ2 −KiT BT1i Pi B0i Ki ⎦ ⎣x(k − 1)⎦ , e(k) e(k) ∗ ∗ Σ 3
where Σ1 = (A + B0i Ki )T Pi (A + B0i Ki ) − λ2 Pi + Qi + δ12 I, Σ2 = −λ2 Qi + KiT BT1i Pi B1i Ki , Σ3 = (δ12 − 1)I + KiT BT0i Pi B0i Ki .
Using the Schur complement three times, Eq. (6) is equivalent to ⎡
1
⎢ ⎣ ∗ ∗
Pˆ i (A + B0i Ki )T Pi B1i Ri
−Pˆ i (A + B0i Ki )T Pi B0i Ri − δ12 Pˆ i Pˆ i
2
−RTi BT1i Pi B0i Ri
∗
3
⎤ ⎥ ⎦ < 0,
(11)
355
356
CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
where
1 = Pˆ i (A + B0i Ki )T Pi (A + B0i Ki )Pˆ i − λ2 Pˆ i + Pˆ i Qi Pˆ i + δ12 Pˆ i Pˆ i , ˆ i + Ri BT Pi B1i Ri ,
2 = −2λ2 Pˆ i + λ2 Q 1i
3 = −2ξ Pˆ i + ξ 2 I + δ12 Pˆ i Pˆ i + Ri BT0i Pi B0i Ri .
ˆ i − Pˆ i )Qi (Q ˆ i − Pˆ i ) > 0, which On the other hand, noticing that Qi > 0, we have (Q ˆ yields to −Pˆ i Qi Pˆ i < −2Pˆ i + Qi . Similarly, −Pˆ i Pˆ i < −2ξ Pˆ i + ξ 2 I holds for any arbitrary scalar ξ . Using the earlier two inequalities, multiplying by the diagonal matrix diag{Pi , Pi , Pi } and its transpose on both sides of Eq. (11), and setting Ri = Ki Pˆ i , we get Vi (k + 1) − λ2 Vi (k) < 0,
i ∈ M.
(12)
This indicates that each subsystem of Eq. (5) is exponentially stable. Hence, for any switching interval [kj , kj+1 ), we have Vσ (kj ) (k) < λ2(k−kj ) Vσ (kj ) (kj ),
j = 0, 1, . . . .
(13)
j = 0, 1, . . . .
(14)
Due to Vσ (k) (k) = Vσ (kj ) (k), we get Vσ (k) (k) < λ2(k−kj ) Vσ (kj ) (kj ),
Since the system state is continuous at any switching instant, from inequality (7), we know that Vσ (kj ) (kj ) ≤ μVσ (k− ) (kj− ), j
j = 0, 1, . . . .
(15)
According to Eqs. (14), (15), we can obtain Vσ (k) (k) = Vσ (kj ) (k) ≤ λ2(k−kj ) Vσ (kj ) (kj ) ≤ μλ2(k−kj ) Vσ (k− ) (kj− ) j
≤ μ2 λ2(k−kj−1 ) Vσ (kj−1 ) (kj−1 ) .. .
(16)
≤ μNσ (k0 ,k) λ2(k−kj ) λ2(kj −kj−1 ) · · · λ2(k1 −k0 ) Vσ (k0 ) (k0 ) k−k0
≤ μ τa λ2(k−k0 ) Vσ (k0 ) (k0 ) = α 2(k−k0 ) Vσ (k0 ) (k0 ).
Denoting ε1 = min{λmin (Pi )}, ε2 = max{λmax (Pi )}, and ε3 = min{λmin (Qi )}, i∈M
i∈M
i∈M
Eq. (16) yields to ε1 x(k)2 ≤ ε1 x(k)2 + ε3 x(k − 1)2 ≤ Vσ (k) (k) < ε2 α 2(k−k0 ) x(k0 )2 .
(17)
3 Event-triggered control
Thus, x(k) ≤
ε2 (k−k0 ) x(k0 ) . α ε1
(18)
This completes the proof. Remark 2. Theorem 1 indicates that the exponential decay rate α is monotonically increasing with the parameter λ, and decreasing with the average dwell time τa . According to the modeling process of the NCS (5), the variation rate of the networkinduced delays can be reflected by the average dwell time τa . The smaller the variation rate of network-induced delay is, the larger the average dwell time τa is.
3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case Once the packet dropouts occur in the network transmission, the event-triggered mechanism (4) cannot be directly used to determine whether the sampled state is transmitted or not. The packet dropouts need to be incorporated to the eventtriggered mechanism. Toward this end, a novel event-triggered condition is provided as follows: bk+1 = min{k > bk | ˜e(k) ≥ δ2 x(bk )},
(19)
where e˜ (k) = x(k) − x(bk ), bk is the transmission instants of the event generator. The following theorem further gives a sufficient condition on the existence of a switching controller and an event generator to stabilize the NCS with short networkinduced delays and packet dropouts. Theorem 2. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist 1
0 ≤ δ2 ≤ ((1 + δ1 )(1 + ε)−dMANSPD ) dMANSPD +1 − 1,
(20)
0i Ki B1i Ki (1+δ1 ) + where 0 < δ1 < 1, ε = max A − I + B1−δ , and positive 1−δ 1 1 ∀i∈M
ˆ i > 0 and matrix Ri with appropriate dimensions such definite matrices Pˆ i > 0, Q that Eqs. (6), (7) hold, then under the event-triggered mechanism (19), the controller (3) and any switching signal with the average dwell time satisfying Eq. (8), system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the feedback control gains are given by 1 Eq. (9), and the exponential decay rate is α = λμ 2τa . Proof. Consider a successful transmission interval [rk , rk+1 ), in which there exists dk packet dropouts, that is, rk = b0 < b1 < b2 < · · · < bdk < bdk +1 = rk+1 . For all l = 1, . . . , dk , during the triggering interval, we have x(bl+1 − 1) − x(bl ) ≤ δ2 x(bl ) ,
(21)
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
which yields to x(bl+1 − 1) ≤ x(bl+1 − 1) − x(bl ) + x(bl ) ≤ (1 + δ2 ) x(bl ) .
(22)
Since x(rk ) = x(bl+1 − 1) − x(bl+1 − 1) + x(rk ) ≤ x(bl+1 − 1) + δ1 x(rk ) ,
(23)
then we have x(rk ) ≤
1 x(bl+1 − 1) . 1 − δ1
(24)
Consequently, we know that x(bl+1 − 2) ≤ x(bl+1 − 2) − x(rk ) + x(rk ) ≤ (1 + δ1 ) x(rk ) 1 + δ1 x(bl+1 − 1) . ≤ 1 − δ1
(25)
Consider system (5) at the instant k = bl+1 , then we have x(bl+1 ) = Ax(bl+1 − 1) + B0i Ki x(rk ) + B1i Ki x(bl+1 − 2),
i ∈ M.
Next, in view of the packet dropouts in the network transmission, for the interval [b1 , rk+1 ), combining with the earlier procedure, we know that x(bl+1 ) − x(bl+1 − 1) = Ax(bl+1 − 1) + B0i Ki x(rk )
+B1i Ki x(bl+1 − 2) − x(bl+1 − 1) ≤ (A − I)x(bl+1 − 1) + B0i Ki x(rk ) + B1i Ki x(bl+1 − 2) ≤ ε x(bl+1 − 1) .
(26)
In light of Eq. (22), it follows that x(bl+1 ) − x(bl+1 − 1) ≤ ε(1 + δ2 ) x(bl ) ,
(27)
which further leads to x(bl+1 ) ≤ x(bl+1 ) − x(bl+1 − 1) + x(bl+1 − 1) − x(bl ) + x(bl ) ≤ [(1 + ε)(1 + δ2 )] x(bl ) ≤ [(1 + ε)(1 + δ2 )]l+1 x(rk ) .
(28)
4 Self-triggered control
For any k ∈ [bdk , bdk +1 ), combining with Eqs. (21), (26), (28), we can get k −1 d x(k) − x(rk ) ≤ x(k) − x(bdk ) + (x(bl+1 ) − x(bl+1 − 1)) l=0 dk −1 + (x(bl+1 − 1) − x(bl )) l=0 ≤
dk
δ2 x(bl ) +
d k −1
l=0
ε(1 + δ2 ) x(bl )
l=0
⎛ ⎞ dk d k −1 l l δ2 ((1 + ε)(1 + δ2 )) + ε(1 + δ2 )((1 + ε)(1 + δ2 )) ⎠ x(rk ) ≤⎝ l=0
l=0
= ((1 + δ2 )dk +1 (1 + ε)dk − 1) x(rk ) . (29)
According to Eqs. (20), (29), it leads to x(k) − x(rk ) ≤ δ1 x(k) ,
(30)
where δ1 = (1 + δ2 )dk +1 (1 + ε)dk − 1. The earlier analysis indicates the novel event-triggered condition (19) can guarantee (4). Combining with the existence conditions on the switching controller in Theorem 1, the proof is completed. Remark 3. From Eq. (20), we can see that the proposed event-triggered threshold depends on the maximum allowable number of successive packet dropouts dMSNSPD . Since dMSNSPD ≥ 0, this implies that δ2 ≤ δ1 . Thus we conclude that the packet dropouts increase the number of the event-triggered instants in order to guarantee the stabilization of the NCS with delays and packet dropouts. In the extreme that, if dMSNSPD = 0 and δ2 = δ1 , that is, no packet dropout occurs in the network transmission, Theorem 2 reduces to Theorem 1.
4 Self-triggered control In this section, we study the stabilization problem of the NCS subject to short network-induced delays and packet dropouts under a self-triggered mechanism. The following theorem gives a sufficient condition on the stabilization of the NCS with short network-induced delays and packet dropouts under the self-triggered control and the switching controller simultaneously. Theorem 3. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0 and matrix Ri with appropriate dimensions positive definite matrices Pˆ i > 0, Q such that Eqs. (6), (7), (20) hold, then under the switching controller (3) with the average dwell-time satisfying Eq. (8), and the self-triggered mechanism
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
bk+1 = bk + max 1,
δ (1 − A) x(bk ) , lnA 1 − 2 Θ
(31)
where Θ = ψ1
bk −1 ε2 λμ 2τa x(k0 ) + ψ2 x(bk ) , ε1
ψ1 = max {B1i Ki } , ψ2 = max {B0i Ki − I} , ∀i∈M
∀i∈M
system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the exponential decay rate is 1 α = λμ 2τa , and the feedback control gains are given by Eq. (9). Proof. For any interval [bk , bk+1 ), combining with the dynamics of the NCS (5), we compute the state error e˜ (k + 1) = x(k + 1) − x(bk ) = Ax(k) + B0i Ki x(bk ) + B1i Ki x(k − 1) − x(bk ) = A˜e(k) + B1i Ki x(k − 1) + (B0i Ki + A − I)x(bk ),
(32) i ∈ M.
This leads to ˜e(k + 1) ≤ A ˜e(k) + B1i Ki x(k − 1) + B0i Ki − I x(bk ) ≤ A ˜e(k) + ψ1 x(k − 1) + ψ2 x(bk ) .
(33)
Based on Theorem 2, we know that the system state satisfies x(k − 1) ≤
ε2 (k−1−k0 ) x(k0 ) ≤ α ε1
ε2 (bk −1) x(k0 ) . α ε1
(34)
x(k0 ) + ψ2 x(bk ) .
(35)
Combining with Eqs. (33), (34), we have ˜e(k + 1) ≤ A ˜e(k) + ψ1
ε2 ε1 λμ
bk −1 2τa
Thus, for k ∈ [bk , bk+1 ), ˜e(k) ≤ Ak−bk ˜e(bk ) + (1 − A)−1 (1 − Ak−bk )Θ = (1 − A)−1 (1 − Ak−bk )Θ.
If the interevent interval satisfying Eq. (31) holds, we have (1 − A)−1 (1 − Ak−bk )Θ ≤ δ2 x(bk ) ,
k ∈ [bk , bk+1 ).
Define (1 − A)−1 (1 − ATk )Θ = δ2 x(bk ), which implies that δ2 (1 − A) x(bk ) Tk = bk+1 − bk = lnA 1 − . Θ
(36)
5 An example
This indicates that the self-triggered condition (31) can guarantee the eventtriggered condition (19), which completes the proof. Remark 4. The interevent interval Tk is related to the average dwell time τa , which implies that the self-triggered mechanism depends on the variation rate of the network-induced delay. The relationship can be summarized as the following two aspects: (i) Θ is decreasing with the increasing of the average dwell time τa . If A < 1, then the interevent interval Tk is monotonically increasing, that is, δ2 (1 − A) x(bk h) ↓→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ (ii) Θ is decreasing with the increasing of the average dwell time τa . If A ≥ 1, the interevent interval Tk is also monotonically increasing, that is, δ2 (1 − A) x(bk h) ↑→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ Therefore, the intertrigger interval Tk is monotonically increasing with the average dwell time τa , which implies that with a slower rate in network-induced delay variation, the longer is the triggered interval.
5 An example In this section, we give a numerical example to illustrate the effectiveness of the −8 1 , Bp = main results. Consider the plant (1) with the parameters Ap = 0.1 0.01 −0.2 0 . Set the sensor sampling period T = 0.2 s, and the actuator sampling 0 0.1 period T0 = 0.1 s, which implies that N = 2. Hence, the NCS (5) has three subsystems. Suppose that the number of successive packet dropouts in the network transmission is bounded by dMANSPD = 2. Next, according to the modeling process of the NCS (5), we can obtain
0.2019 1.2214 , 1.0202 1.0020 0 0 , B01 = B10 = 0 0 0 0 , B12 = B02 = 0 0 A=
−0.0422 0.0263 , −0.0541 0.0306 0.0138 0.0105 −0.0284 , B11 = −0.0201 0.0100 −0.0340 −0.0422 0.0263 . −0.0541 0.0306 B00 =
0.0158 0.0206
,
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
Choosing μ = 1.04, λ = 0.9539, ξ = 4 and solving Eqs. (6), (7) by Theorem 2, we get δ1 = 0.0023, δ2 = 0.00017; 5420.50 −3442.71 ˆ ˆP0 = , Q0 = −3442.71 2191.19 1,088,218.66 −691,480.11 ; R0 = 1,857,449.10 −1,180,425.81 5428.46 −3444.67 ˆ ˆP1 = , Q1 = −3444.67 2190.40 216,135.40 −123,120.83 ; R1 = 367,021.72 −209,072.76 5347.72 −3399.09 ˆ ˆP2 = , Q2 = −3399.09 2165.14 −14,626.41 8125.68 . R2 = −25,047.14 13,914.91
10,506.45 −6681.07
−6681.07 4256.42
10,560.95 −6720.51
−6720.51 4284.49
10,644.80 −6765.70
−6765.70 4308.14
,
,
,
Furthermore, we can have the feedback control gain
156.62 −69.48 K0 = , 245.63 −152.78
1991.96 3076.38 K1 = , 3382.56 5224.03
and the average dwell time τa > τa∗ = 1 2τa
ln μ 2 ln λ−1
−163.53 −252.98 K2 = , −280.04 −433.22
= 0.4155. Taking τa = 0.5, we have
α = λμ = 0.9921. Fig. 3 gives a switching signal σ (k) with the average dwell time τa = 0.5 >τa∗ . Fig. 4 shows the trajectories of the system state converges to zero. Figs. 5 and 6 give the event-triggered instants and the time-triggered instants, respectively.
2
1.5
Switching signal
362
1
0.5
0
0
1
FIG. 3 Switching signal.
2
3
4
5 Time (s)
6
7
8
9
10
5 An example
1.5
System state 1 System state 2
x (k)
1 0.5 0 –0.5 0
1
2
3
4
5 Time (s)
FIG. 4 System state.
FIG. 5 Event-triggered instants.
FIG. 6 Time-triggered instants.
6
7
8
9
10
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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout
The simulation results indicate that the event-triggered mechanism reduces the computation of the controller and saves the communication resources with respect to the traditional periodic sampling control.
6 Conclusions A codesign method of the event-triggered generator and the switching feedback controller for NCSs with short network-induced delays and packet dropouts is investigated in this chapter. Both the event threshold and the switching frequency depend on the variation rate of the network-induced delays and the maximal number of successive packet dropouts to stabilize the networked feedback system. Moreover, a self-triggered condition as the flexible software realization is also developed. Finally, a numerical example shows that the updating frequency of the control law can be reduced and simultaneously maintain the system performance.
Acknowledgments The authors gratefully acknowledge the support of the Natural Science Foundation of China under Grants 61603079 and 61773098.
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