Event-Triggered Quantized-Data Feedback Control for Linear Systems Yanpeng Guan1 , Qing-Long Han1,∗ and Chen Peng1,2 1. Centre for Intelligent and Networked Systems and School of Engineering and Technology Central Queensland University, Rockhampton, QLD 4702, Australia 2. School of Mechatronic Engineering and Automation, Shanghai University, 200072, PR China ∗ Corresponding author: Tel. +61 7 4930 9270; E-mail:
[email protected] Abstract—This paper proposes an event-triggered quantizeddata feedback control scheme for linear systems. An eventtriggered communication scheme is introduced to select which sampled-data should be quantized and transmitted to the controller. The threshold is constructed by considering the difference between the current sampled data and the latest quantized data. A finite-level dynamical quantizer is developed based on the communication scheme and quantized data. An output feedback controller is designed to ensure that the state of the closed-loop is uniform ultimate bounded. A numerical example illustrates the effectiveness of the proposed approach.
I.
I NTRODUCTION
During the last decade, event-triggered control has attracted much attention. It is shown in some experimental results that the average sampling/transmission interval can be increased by an event-triggered sampling technique and thus limited resource may be saved [1]–[6]. In the framework of an eventtriggered mechanism, an event of sampling or transmission occurs only when some pre-determined threshold is violated. An event-triggered scheduler is proposed in [1] to trigger control tasks when the norm of the state measurement error achieves a state-dependent critical value. A self-triggered control strategy is presented in [7], where the next sampling instant is predicted by the current system state, and a tradeoff between the allocated resource and the required control performance is developed. Recently, several event-triggered mechanisms are developed in networked and distributed control systems [8]– [12] for efficient use of the limited transmission resources (e.g. battery power and/or network bandwidth) by reducing some unnecessary transmissions. In [10], a practical stabilization of nonlinear systems under asynchronous updates is established by using a decentralized event-triggered control strategy, where each component of state is triggered for sampling when the local error exceeds a specific positive scalar. More recently, based on output feedback, a decentralized event-triggered scheme is proposed in [11], where the local event-triggered threshold in a sensor node is established by the norm of the current values in the node and a nonnegative scalar and the corresponding event-triggered control system in [11] is modeled by an impulsive system. In [6], Kofman and Braslavsky proposes an event-triggered sampled-data feedback scheme based on hysteretic quantization, where the sampling of system output is triggered by the crossing of the signal through its quantization level. Signal quantization is usually an inevitable procedure in digital signal transmissions since analog signals have to be
quantized, and encoded by a finite number of bits before being transmitted through a digital communication channel. A quantizer is inherently a nonlinear device, which makes a partition of the signal space and maps each of the partition cells to one specific vector. Several quantizers have been designed to achieve different control objectives [13]–[19]. For instance, it is shown in [15] that the coarsest quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic and can be designed by solving a special linear quadratic regulator problem. Fu and Xie [16] generalize this result to a few quantized feedback design problems for linear systems by using a sector bound approach. To study the feedback stabilization problem of linear systems, a dynamic quantizer is proposed in [17], [18], where a dynamical adjustable parameter is introduced to zoom in or zoom out the quantization region at discrete instants of time. It is noted that in most of the existing results on eventtriggered control (e.g. [1]–[6]), system outputs have to be detected continuously by some special real-time detection hardware and the sampling is assumed to take place at exactly the moment when the event-triggered threshold is violated. This may pose a critical requirement for the real-time hardware although the sampled data in this case is convenient for further processing including quantization [20]. Another issue of this kind of continuous event-triggered sampling scheme is that a positive minimum inter-event time should be guaranteed to avoid the Zeno behavior. For example, a positive scalar need to be involved in the event-triggered threshold condition in [9], [11] to guarantee a nonzero inter-event time for each sensor node. To overcome the two points, a class of discrete (or periodic) event-triggered transmission schemes are proposed in [8], [21]–[23] based on periodical sampling. In this case, the triggered sampled data is no longer bounded by the threshold and the data is difficult to be further analyzed. These issues motivate this paper. In this paper, the problem of event-triggered output feedback control for a class of linear time-invariant systems with signal quantization will be studied. An event-triggered communication scheme, a finite-level dynamical quantizer, and a static output feedback controller gain are designed in a unified framework. The output of the system is periodically sampled. An event-triggered communication scheme is proposed to determine whether or not the current sampled data should be further processed and used for feedback. The current sampled data is released to be quantized and transmitted to the controller when the difference between the current sampled data
and the latest quantized data exceeds a specific threshold. We construct a dynamical finite-level quantizer based on the eventtriggered communication scheme and the latest quantized data. An output feedback gain is developed to ensure that the state of the closed-loop system is uniform ultimate bounded. The main contribution of this paper is an integrated design method for event-triggered quantized feedback control scheme. The finite-level quantizer and the event-triggered communication scheme share the same parameters and they are closely correlated. Compared with the continuous eventtriggered sampling scheme, the real-time detection hardware is no longer needed. A static output feedback controller is also designed in this framework to guarantee the uniform ultimate boundedness of the system state. The organization of this paper is as follows. Section II presents an event-triggered communication scheme and formulates the quantized output feedback control problem. The uniform ultimate boundedness analysis is given for the eventtriggered closed-loop system with quantized-data in Section III. A static output feedback controller design method is developed in Section IV. The finite-level dynamic quantizer is given in Section V. A simulation example is given to demonstrate the effectiveness of the proposed approach in Section VI, and this paper is concluded in Section VII. Notation. Throughout this paper, superscripts ‘T’ and ‘+’ stand for matrix transposition and matrix M-P inverse, respectively. A real symmetric matrix 𝑃 > 0 denotes 𝑃 being a positive definite matrix. 𝐼 is used to denote an identity matrix with appropriate dimension. In symmetric block matrices, we use (∗) as an ellipsis for terms that can be induced by symmetry. For a given function 𝑓 (𝑡), 𝑓˙(𝑡) denotes the right derivative of 𝑓 (𝑡) with respect to 𝑡. II.
u (t )
y (t )
Plant
Sampler Quantizer
ZOH
Event-triggered
q ( y (ik h))
Controller
Fig. 1.
Decoder
^1, 2," , N k ` Digital channel
Encoder
A framework of quantized event-triggered feedback control
be transmitted is denoted as {𝑦(𝑖𝑘 ℎ)}∞ 𝑘=1 . To better convey our idea, the time delay during the process of signal sampling, quantization and transmission is not considered in this paper. ∞ Then one can see that {𝑖𝑘 ℎ}∞ 𝑘=1 is a subsequence of {𝑘ℎ}𝑘=1 and it is generated according to the following event-triggering communication logic 𝑖𝑘+1 ℎ = 𝑖𝑘 ℎ + min {𝑙ℎ ∣ ∣𝑦(𝑖𝑘 ℎ + 𝑙ℎ) − 𝑞(𝑦(𝑖𝑘 ℎ))∣2 + 𝑙∈ℤ
≥ 𝛿∣𝑞(𝑦(𝑖𝑘 ℎ))∣2 + 𝜖3 }
(2)
where 𝑞(⋅) is a finite-level dynamical quantizer to be designed later; 0 < 𝛿 < 1 and 𝜖 > 0 are two adjustable factors, which can be used to adjust the threshold of the communication scheme. Remark 1: The positive scalars involved in the eventtriggering mechanism conditions in [10], [11] are used to guarantee a nonzero minimum inter-event time. While in this paper, the main purpose of introducing 𝜖 > 0 in (2) is to design a finite-level quantizer in Section V. In this paper, we are interested in designing the following controller 𝑢(𝑡) = 𝐾𝑞(𝑦(𝑖𝑘 ℎ)),
P ROBLEM F ORMULATION
Consider the following linear time-invariant system { 𝑥(𝑡) ˙ = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡)
Actuator
𝑡 ∈ [𝑖𝑘 ℎ, 𝑖𝑘+1 ℎ)
(3)
where 𝐾 is the feedback gain to be determined. (1)
𝑛
where 𝑥(𝑡) ∈ ℝ , 𝑢(𝑡) ∈ ℝ and 𝑦(𝑡) ∈ ℝ are the system state, control input and the measurement output, respectively; 𝐴, 𝐵 and 𝐶 are constant matrices with appropriate dimensions; and the initial condition of the system is given by 𝑥(0) = 𝑥0 . In this paper, only limited information about the output 𝑦(𝑡) is available for feedback due to the following constraints:
For 𝑡 ∈ [𝑖𝑘 ℎ, 𝑖𝑘+1 ℎ), let 𝜂(𝑡)
=
𝑡 − max {𝑖𝑘 ℎ + 𝑙ℎ∣𝑖𝑘 ℎ + 𝑙ℎ ≤ 𝑡}, +
(4)
𝑒𝑘 (𝑡)
=
𝑦(𝑡 − 𝜂(𝑡)) − 𝑞(𝑦(𝑖𝑘 ℎ)).
(5)
𝑙∈ℤ
Then one can see that 𝑞(𝑦(𝑖𝑘 ℎ)) = 𝑦(𝑡 − 𝜂(𝑡)) − 𝑒𝑘 (𝑡).
(6)
The error-dependent closed-loop system can be obtained as 𝑥(𝑡) ˙ = 𝐴𝑥(𝑡) + 𝐵𝐾𝐶𝑥(𝑡 − 𝜂(𝑡)) − 𝐵𝐾𝑒𝑘 (𝑡).
∙
𝑦(𝑡) is periodically sampled with a sampling period ℎ > 0;
∙
For efficient use of resources, an event-triggering mechanism is employed to select which sampled data should be further processed and used for feedback;
∙
The selected sampled data is quantized, and represented by a finite number of bits before being transmitted to the controller through a digital channel.
We introduce the following definition of uniform ultimate boundedness as in [24], [25] and Lemma 1 as a analysis tool of the uniform ultimate boundedness. The proof of Lemma 1 is similar to that in [24], and thus is omitted here.
A conceptual framework of the event-triggered feedback control systems with quantized data is shown in Figure 1. For notational convenience, we denote the sampling time instants sequence as {𝑘ℎ}∞ 𝑘=1 . The selected sampled data to
Definition 1: The state of system (7) is said to be uniform ultimate bounded, if there exist a compact set 𝑈 ∈ ℝ𝑛 and a scalar 𝜀 > 0 such that for all 𝑥(𝜃) = 𝜑(𝜃) ∈ 𝑈, 𝜃 ∈ [−ℎ, 0], there is a 𝑇 > 0 such that ∥𝑥(𝑡)∥ < 𝜀, ∀𝑡 ≥ 𝑇 .
(7)
We supplement the initial condition of the system on [−ℎ, 0] as 𝑥(𝑡) = 𝜑(𝑡), 𝑡 ∈ [−ℎ, 0], where 𝜑(𝑡) is a continuous function on [−ℎ, 0] and 𝜑(0) = 𝑥0 .
Lemma 1: Let 𝑉 (𝑡, 𝑥𝑡 ) be a Lyapunov functional of system (7) which satisfies { 𝜁1 (∣𝜑(0)∣) ≤ 𝑉 (𝑡, 𝑥𝑡 ) ≤ 𝜁2 (max−ℎ≤𝜃≤0 ∣𝜑(𝜃)∣) (8) 𝑉˙ (𝑡, 𝑥𝑡 ) ≤ −𝜁3 (∣𝜑(0)∣) + 𝜁3 (𝜀)
Then for all 𝑒𝑘 (𝑡) in (5), we have that
where 𝜀 > 0 is a positive constant, 𝜁1 (⋅) and 𝜁2 (⋅) are continuous, strictly increasing functions, and 𝜁3 (⋅) is a continuous, nondecreasing function. If
Choose the following Lyapunov-Krasovskii functional ∫ 𝑡 𝑉 (𝑡, 𝑥(𝑡)) = 𝑥𝑇 (𝑡)𝑃 𝑥(𝑡) + (ℎ − 𝜂(𝑡)) 𝑥˙ 𝑇 (𝑠)𝑄𝑥(𝑠)𝑑𝑠 ˙ 𝑡−𝜂(𝑡) ] [ +(ℎ − 𝜂(𝑡)) 𝑥𝑇 (𝑡) 𝑥𝑇 (𝑡 − 𝜂(𝑡)) ] [ ][ 𝑅1 𝑅2 − 𝑅1 𝑥(𝑡) . (16) × 𝑥(𝑡 − 𝜂(𝑡)) ∗ 𝑅1 − 𝑅2 − 𝑅2𝑇
𝑉˙ (𝑡, 𝑥𝑡 ) < 0, when ∣𝑥(𝑡)∣ > 𝜀
(9)
then the state of system (7) is uniform ultimate bounded. The purpose in what follows is to design a controller in the form of (3) and a finite-level quantizer 𝑞(⋅) such that the state of system (7) is uniform ultimate bounded under the proposed event-triggering communication scheme (2). III.
U NIFORM U LTIMATE B OUNDEDNESS A NALYSIS
In this section, we will give a sufficient condition which ensures that the state of system (7) is uniform ultimate bounded. Theorem 1: With communication scheme (2) and controller (3), the state of system (7) is uniform ultimate bounded, if there exist real matrices 𝑃 > 0, 𝑄 > 0, 𝑅1 = 𝑅1𝑇 , 𝑅2 , 𝑌1 , 𝑌2 , 𝑌3 , 𝑍1 , 𝑍2 with appropriate dimensions such that ] [ 𝑃 + ℎ𝑅1 ℎ𝑅2 − ℎ𝑅1 >0 (10) ∗ ℎ𝑅1 − ℎ𝑅2 − ℎ𝑅2𝑇 ⎤ ⎡ Γ11 Γ12 Γ13 ℎ𝑌1𝑇 −𝑍1𝑇 𝐵𝐾 ⎢ ∗ Γ22 Γ23 ℎ𝑌2𝑇 −𝑍2𝑇 𝐵𝐾 ⎥ ⎥ ⎢ (11) ⎢ ∗ −𝛿𝐶 𝑇 ⎥ < 0 ∗ Γ33 ℎ𝑌3𝑇 ⎦ ⎣ ∗ ∗ ∗ −ℎ𝑄 0 ∗ ∗ ∗ ∗ 𝛿−1 ⎡ ⎤ Γ13 −𝑍1𝑇 𝐵𝐾 Γ11 Γ12 + ℎ𝑅1 𝑇 ⎢ ∗ Γ22 + ℎ𝑄 Γ23 + Ω −𝑍2 𝐵𝐾 ⎥ (12) ⎣ ⎦ ˜𝑇 , 𝑅 ˜ 2 , 𝑌˜1 , 𝑌˜2 , 𝑌˜3 , 𝑍˜ and 𝑈 with appropriate ˜ > 0, 𝑅 ˜1 = 𝑅 0, 𝑄 1 dimensions such that [ ] ˜1 ˜ 2 − ℎ𝑅 ˜1 𝑃˜ + ℎ𝑅 ℎ𝑅 (24) ˜ 1 − ℎ𝑅 ˜ 2 − ℎ𝑅 ˜𝑇 > 0 ∗ ℎ𝑅 2 ⎤ ⎡ Υ13 ℎ𝑌˜1𝑇 Υ15 Υ11 Υ12 𝑇 ⎢ ∗ Υ22 𝑌˜2 + 𝐵𝑋 ℎ𝑌˜2𝑇 Υ25 ⎥ ⎥ ⎢ ⎢ ∗ (25) ∗ Υ33 ℎ𝑌˜3𝑇 Υ35 ⎥ ⎥ 0 in (2) is replaced by a variable scalar 𝜖(𝑘) > 0 as in [10], where 𝜖(𝑘) is a decreasing sequence and lim𝑘→0 𝜖(𝑘) = 0, then the asymptotic stability could be attained for system (7).
+∥𝑒𝐴ℎ − 𝐼∥ ⋅ ∣𝑦(𝑖𝑘+1 ℎ − ℎ) − 𝑞(𝑦(𝑖𝑘 ℎ)) + 𝑞(𝑦(𝑖𝑘 ℎ))∣ (31) ≤ 𝑀𝑘 3
where 𝑀𝑘 = 𝑀1 ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ + 𝑀2 𝜖 2 with ∫ ℎ √ √ 𝑒𝐴𝑠 𝑑𝑠 ⋅ 𝐵𝐾∣ + ∥𝑒𝐴ℎ − 𝐼∥(1 + 𝛿) 𝑀1 = 𝛿 + ∣𝐶 0
𝑀2 = 1 + ∥𝑒𝐴ℎ − 𝐼∥. On the other hand, the communication scheme (2) implies that √ ∣𝑦(𝑖𝑘+1 ℎ) − 𝑞(𝑦(𝑖𝑘 ℎ))∣ ≥ 𝛿∣𝑞(𝑦(𝑖𝑘 ℎ))∣. (32) One can see that ∣𝑦(𝑖𝑘+1 ℎ)∣ must lie in one of the following regions: √ 3 3 𝑟1 = [0, 𝜖 2 ], 𝑟2 (𝜖 2 , (1 − 𝛿)∣𝑞(𝑦(𝑖𝑘 ℎ))∣) √ 𝑟3 = ((1 + 𝛿)∣𝑞(𝑦(𝑖𝑘 ℎ))∣, ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ + 𝑀𝑘 ]. We denote the number of quantization levels in the three regions as 𝐿1 , 𝐿2 and 𝐿3 , respectively. It can be seen from (30) that 𝐿1 = 2. Let √ 𝐿2 3 ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ ⋅ 𝜌 2 (1 − 𝛿) ≤ 𝜖 2 . One can find that 𝐿2 ≥ 2⌈log𝜌 [(1 −
√
(33)
3
𝛿)𝜖 2 ∣𝑞(𝑦(𝑖𝑘 ℎ))∣]⌉
(34)
VI.
We take the sampling period ℎ = 0.01𝑠. The parameters of the communication scheme (2) are given as: 𝛿 = 0.1, 𝜖 = 0.01. By using Theorem 2, we can get a feasible controller 𝐾 = −1.35. For simulation purpose, the initial condition is assumed to be 𝑥(0) = [0.5 − 0.5]𝑇 . With the obtained controller, communication scheme (2) and quantizer (30), the state response of the closed-loop system is plotted in Figure 2, which shows that the state is well bounded. The release time distribution of the quantized data is illustrated in Figure 3. Within the simulation time 𝑇𝑠 = 25𝑠, the output 𝑦(𝑡) is sampled 2500 times, while only 53 sampled measurements are quantized and transmitted to the controller for feedback, which implies a certain proportion of resources may be saved. The average data quantization and transmission time interval is 0.472𝑠, which is much longer than the sampling period ℎ = 0.01𝑠. The desired control performance (uniform ultimate boundedness) is guaranteed, which shows the effectiveness of the proposed approach. 0.5 x1 x2
(35)
State responses
3
which leads to 𝜌
𝐿3 2
(1 +
√
3
𝛿)
𝑀2 𝜖 2 ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ 3 𝑀2 𝜖 2
≥
𝑀1 + 1 +
≥
𝑀1 + 1 +
=
𝑀1 + 2𝑀2 + 1.
A S IMULATION E XAMPLE
In this section, a numerical example taken from [11] is employed to demonstrate the effectiveness of the proposed approach. Consider the following system ⎧ ] [ ] [ ⎨ 0 0 1 𝑢(𝑡) 𝑥(𝑡) + 𝑥(𝑡) ˙ = 1 −2 3 (39) ⎩ 𝑦(𝑡) = [ −1 4 ] 𝑥(𝑡).
where ⌈⋅⌉ is the ceiling function, which is the integer obtained by rounding up. Let √ 𝐿3 ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ ⋅ 𝜌 2 (1 + 𝛿) ≥ ∣𝑞(𝑦(𝑖𝑘 ℎ))∣ + 𝑀𝑘 = (𝑀1 + 1)∣𝑞(𝑦(𝑖𝑘 ℎ))∣ + 𝑀2 𝜖 2
0
1 32 2𝜖
(36)
−0.5 0
5
Then, we have 𝐿3 ≥ 2⌈log𝜌
(38)
𝑀1 + 2𝑀2 + 1 √ ⌉. 1+ 𝛿
(37)
The results can be summarized in the following Theorem 3: The quantizer (30) proposed for system (7) with communication scheme (2) always has a finite number of
Fig. 2.
10 15 Time (Second)
20
25
State response of the closed-loop system.
VII.
C ONCLUSION
This paper gives a novel event-triggered feedback control scheme with quantized output measurement. The current
Release interval (Second)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
Fig. 3.
5
10 15 20 Release time (Second)
25
Release time intervals of the system.
sampled-data is used for feedback only when the difference between the current and latest quantized sampled-data exceeds a specific threshold according to an event-triggered communication scheme. A dynamical finite-level quantizer is designed based on the communication scheme. The uniform ultimate boundedness analysis and the corresponding output feedback controller design are also presented. A numerical example shows that with the proposed approach, the number of transmission could be substantially reduced while maintaining the required control performance, which shows the potential of its application in practical sampled-data control systems and networked control systems. ACKNOWLEDGMENT This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780; the Research Advancement Award Scheme (January 2010 December 2012) at Central Queensland University, Australia; and the Natural Science Foundation of China under Grants 61074024 and 61273114. R EFERENCES [1] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, vol. 52, no. 9, 2007, pp. 1680–1685. [2] C. Fiter, L. Hetel, W. Perruquetti, and J.-P. Richard, A state dependent sampling for linear state feedback, Automatica, vol. 48, no. 8, 2012, pp. 1860–1867. [3] X. Jia, X. Chi, Q.-L. Han, and Zheng, N.-N, Event-triggered fuzzy control for a class of nonlinear networked control systems using a membership function deviation approach, accepted for publication in Information Sciences. [4] J. Lunze and D. Lehmann, A state-feedback approach to event-based control, Automatica, vol. 46, no. 1, 2010, pp. 211–215. [5] L. Li, M. Lemmon, and X. Wang, Stabilizing bit-rates in quantized event triggered control systems, in Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control, Beijing, China, 2012, pp. 245–254. [6] E. Kofman and J. Braslavsky, Level crossing sampling in feedback stabilization under data-rate constraints, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 4423–4428. [7] A. Anta and P. Tabuada, To sample or not to sample: self-triggered control for nonlinear systems, IEEE Transactions on Automatic Control, vol. 55, no. 9, 2010, pp. 2030–2042.
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