2012 Australian Control Conference
15-16 November 2012, Sydney, Australia
Event-triggered L2 Controller Design of Networked Control Systems with Quantized Measurement Yanpeng Guan, Chen Peng and Qing-Long Han∗ Abstract— This paper considers L2 control for a networked control system with quantized measurement. The system state is periodically sampled and quantized. An event-triggered transmitter is introduced to determine whether or not the current quantized measurement should be transmitted through the communication channel to the controller. In this eventtriggered networked control system framework, a networked controller is designed to guarantee the L2 stability of the closedloop system. Two numerical examples are given to demonstrate the effectiveness of the proposed method. It is shown that the average transmission interval could be increased substantially while the control performance is maintained.
I. INTRODUCTION Networked control systems (NCSs) are feedback control systems where signals are transmitted through communication networks. Due to some advantages such as flexible installation, low cost and easy maintenance, NCSs have been applied to a broad range of areas including aviation, mobil sensor networks, industry process control, remote surgery [1]. In the past decade, considerable efforts have been made towards the stability analysis and controller design of NCSs. Because of the communication networks in the NCSs, there are several issues which need to be dealt with, including network-induced delays [2]–[4], data-packet dropouts [5]– [7], which may lead to deterioration of the system performance. Signal quantization is usually an inevitable procedure in an NCS since analog signals have to be quantized before being transmitted through a communication channel under limited bandwidth. 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 [8]. For instance, it is shown in [9] that the coarsest quantizer that quadratically stabilizes a single input linear discrete 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 Grant 61074024, the Natural Science Foundation of Jiangsu Province of China under Grant BK2010543. ∗ Corresponding author, Tel: +61 7 4930 9270; Fax: +61 7 4930 9729; E-mail:
[email protected] Y. Guan and Q.-L. Han are with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Rockhampton, QLD 4702, Australia
[email protected] C. Peng is with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Rockhampton, QLD 4702, Australia, and also with the School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China
[email protected]
ISBN 978-1-922107-63-3
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time invariant system is logarithmic and can be designed by solving a special linear quadratic regulator problem. This result is generalized to a few quantized feedback design problems for linear systems by using a sector bound approach [10]. To study the feedback stabilization problem of linear systems, a dynamic quantizer is provided in [11], which can been adjusted online by updating the parameter of the quantizer at discrete instants of time. Recently, event-triggered control has received increasing attention due to some experimental results suggesting that the average sampling interval could be increased by eventtriggered sampling technique [12]–[14]. An event-triggered control scheme is developed in [13] to study distributed networked control systems, where the current state of the subsystem is released only when the local error of the subsystem exceeds a specified state-dependent threshold. A self-triggered control strategy is presented in [15], where the next sampling instant is determined by the current system state and a tradeoff between allocated resources and required control performance is developed. It is noted that in most of the existing results on event-triggered control, system outputs have to be measured continuously by some special real-time detection hardware and thus the output errors are always assumed to be within a specified threshold. Another point is that it is difficult to design a desired controller for a system with the continuous event-triggered sampling technique and the control law u = Kx in [13]–[16] is pre-given. This paper will propose a new framework for an eventtriggered NCS. Inspired by the periodic (or discrete) eventtriggered mechanisms [17], [18], we firstly employ a sensor to periodically sample the state of the NCS. Then the sampled-data is quantized for transmission through the communication channel to the controller. An event-triggered transmitter is employed to check the novelty in the quantized state measurement. The quantized-data is released only when a pre-designed threshold of the novelty is violated. With the quantization error taken into account, a sufficient condition is given to ensure the L2 stability of the NCS. A networked controller is also designed to guarantee the the L2 stability of the closed-loop system. There are two aspects worth mentioning about our eventtriggered NCS framework. On one hand, the sampling and event-triggered transmission scheme are executed separately. Therefore, the real-time detection hardware is no longer needed compared with the continuous event-triggered sampling technique. On the other hand, the event-triggered transmitter is introduced to trigger the release of the quantizeddata. This structure is expected to be more effective than the
© 2012 Engineers Australia
case where the event-triggered scheme is implemented before the quantizer in the sense of saving the limited network resources. In the latter case, the average novelty of the eventtriggered sampled-data may become small due to the effect of quantization, which could reduce the effect of the eventtriggered sampling technique. The organization of this paper is as follows. Section 2 gives an event-triggered NCS model and formulates the L2 control problem. The L2 stability analysis and controller design are given in Section 3. Two numerical examples are given to demonstrate the effectiveness of the proposed approach in Section 4, and the paper is concluded in Section 5. Notation. Throughout this paper, a superscript ‘T ’ stands for matrix transposition, a real symmetric matrix P > 0 denotes P being a positive definite matrix. I denotes an identity matrix with appropriate dimension. In symmetric block matrices, we use (∗) as an ellipsis for terms that can be induced by symmetry. The notation L2 [0, ∞) refers to the space functions with the norm ∞ of square integrable 1 v2 = [ 0 v T (s)v(s)ds] 2 for any v ∈ L2 [0, ∞). II. PROBLEM FORMULATION Consider the following linear time-invariant system x(t) ˙ = Ax(t) + Bu(t) + Bω ω(t), t ≥ t0
(1)
where x(t) ∈ Rn , u(t) ∈ Rm and ω(t) ∈ L2 [0, ∞) are the system state, control input and the exogenous disturbance, respectively; A, B and Bω are constant matrices with appropriate dimensions; and the initial condition of the system (1) is given by x(t0 ) = x0 . It is assumed that the system state x(t) is fully measured and is periodically sampled with a sampling period h > 0. Figure 1 gives a framework of an NCS, where a network is used to transmit system output to the controller. Before being transmitted through the network, the sampled-data x(kh) is quantized by a quantizer q(·). For efficient use of the network resources, an event-triggered transmitter is proposed to determine whether or not the current quantized measurement q(x(kh)) is to be transmitted to the controller. To better convey our idea of the event-triggered transmitter, the network-induced delays and data dropouts are not taken into consideration in this paper.
" !
" !
"!
Fig. 1.
ik+1 h = ik h + min{lh|eT (rk,l h)Φe(rk,l h) l≥1
≥ δq T (x(ik h))Φq(x(ik h))}
(2)
where Φ > 0 is a weighting matrix; 0 < δ < 1 is an adjustable factor and the threshold of the event-triggered communication scheme; and e(rk,l h) = q(x(rk,l h)) − q(x(ik h)),
(3)
rk,l h = ik h + lh,
(4)
l = 1, 2, 3, . . . .
Remark 1: The event-triggered transmission scheme (2) is used to save the limited network resources by reducing the number of data packets that are required to be transmitted through the network. Different to the general event-triggered sampling technique, our event-triggered mechanism and the sampling are executed separately. Therefore, the real-time detection hardware is no longer needed while the network traffic load can still be reduced. Remark 2: One can see from (2) that as the value of δ increases, the number of data packets triggered for transmission will be reduced. On the contrary, if δ → 0+, the event-triggered transmitter will gradually degenerate toward a time-triggered transmitter. We are interested in designing the following controller u(t) = Kq(x(ik h)),
t ∈ [ik h, ik+1 h)
(5)
where K ∈ Rm×n is to be determined. The quantizer q(·) is assumed to be in the form of q(·) = [q1 (·) . . . qn (·)]τ . Each qj (·) is a logarithmic quantizer with quantization levels (j)
(j)
Uj = {±ui , ui
(j)
= ρij u0 , i = 0, ±1, ±2, . . .} ∪ {0}, (j)
0 < ρj < 1, u0 > 0 (j)
Each ui corresponds to a segment such that qj (·) can map the whole segment to this quantization level and all the segments form a partition of the real set R. The associated quantizer qj (·) is defined as follows [9], [10]: ⎧ (j) (j) (j) 1 1 ui , if 0 < 1+σ ui < v ≤ 1−σ ui ⎨ j j qj (v) = 0, if v=0 ⎩ −qj (−v), if v 0, that is, under zero initial condition, x(t)2 < γ ω(t)2 for any nonzero ω(t) ∈ L2 [0, ∞). III. MAIN RESULTS In this section, we will first analyze the L2 stability of the event-triggered NCS. Then a controller will be designed to ensure the L2 stability of the closed-loop system. Theorem 1: Given a scalar γ > 0, with communication scheme (2), the closed-loop system (12) is finite-gain L2 stable from ω to x with a gain less than γ, if there exist real matrices P > 0, Q > 0, R1 = R1T , R2 , Y1 , Y2 , Y3 , Z1 , Z2 with appropriate dimensions and a scalar ε > 0 such that P + hR1 hR2 − hR1 >0 (13) ∗ hR1 − hR2 − hR2T ⎤ ⎡ Γ11 Γ12 Γ13 hY1T Γ15 ⎢ ∗ Γ22 Y2T hY2T Γ25 ⎥ ⎥ ⎢ ⎢ ∗ ∗ Γ33 hY3T Γ35 ⎥ (14) ⎥ 0 and γ > 0, ˜ Z˜ −1 , under the communication scheme (2) with Φ = Z˜ −T Φ the closed-loop system (12) is finite-gain L2 stable from ω to x with a gain less than γ, if there exist real matrices ˜T , R ˜ 2 , Y˜1 , Y˜2 , Y˜3 , Z, ˜ K ˜ with ˜ > 0, Φ ˜ > 0, R ˜1 = R P˜ > 0, Q 1 appropriate dimensions and real constants κ and ε > 0 such that ˜ 2 − hR ˜1 ˜1 hR P˜ + hR (26) ˜ 1 − hR ˜ 2 − hR ˜T > 0 ∗ hR 2
224
⎡
Υ11 ∗ ∗ ∗ ∗
Υ11 ⎢ ∗ ⎢ ⎣ ∗ ∗
Υ12 Υ22 ∗ ∗ ∗
Υ13 Y˜2T
Υ33 ∗ ∗ ˜1 Υ12 + hR ˜ Υ22 + hQ ∗ ∗
hY˜1T hY˜2T hY˜3T ˜ −hQ ∗ Υ13 Υ23 Υ33 ∗
Υ15 Υ25 Υ35 0 Υ55
⎤ ⎥ ⎥ ⎥ 0, Q > 0, R1 = R1T , R2 , Y1 , Y2 , Y3 , Z1 , Z2 with appropriate dimensions and a scalar ε > 0 such that inequalities (13)-(15) hold, where Γ33 , Γ15 , Γ25 , Γ35 , Γ55 are ˜ 15 , Γ ˜ 25 , Γ ˜ 35 , Γ ˜ 55 , respectively, and ˜ 33 , Γ replaced by Γ
1.2 1 0.8 0.6 0.4 0.2 0 0
5
10 15 20 Release time (Second)
Fig. 3.
25
Release time in Example 1.
Within the simulation period Ts = 25s, it is shown that the system state is sampled and quantized 2500 times, respectively, while only 112 quantized state measurement
TABLE II M AXIMUM ALLOWABLE SAMPLING PERIOD hmax IN E XAMPLE 2 δ 0.1 0.2 0.3 0.4 0.5 0.6
hmax with Theorem 1 1.0562 0.8487 0.6874 0.5496 0.4269 0.3135
hmax with Theorem 3 1.0549 0.8453 0.6800 0.5430 0 0
is triggered to be transmitted to the controller through the network. The average release time interval is 0.2232s, which is much longer than the sampling period 0.01s. Compared with the periodic transmission scheme (when δ = 0), it is clear that a large proportion of the required network resources may be saved, which shows the effectiveness of the approach. Table I shows the corresponding transmitted packets number Nt within the simulation time T = 25s when different value of δ is used in the event-triggered transmitter. One can see that as the value of δ increases, the transmitted packets number Nt decreases. Example 2: Consider the following system [23] 0 1 0 x(t) ˙ = x(t) + u(t). (35) 0 −0.1 0.1 A non-networked controller u = [−3.75 − 11.5]x(t) is used to compute the maximum allowable sampling period hmax . The quantization density is chosen as ρ1 = ρ2 = 0.85. We consider the system first in the framework where the event-triggered mechanism is executed after the signal quantization, then in the framework where the event-triggered mechanism is executed before the signal quantization. Table II shows the value of hmax when the system is studied in the two different frameworks, respectively. One can see that, in the first framework, the value of hmax is larger than the one in the other framework by around 10% with the same value of the event-triggering parameter δ. It can also be seen that the range of δ in the two frameworks is different. And as the value of δ increases, the state of the system is required to be sampled more frequently, although not all the measurement is to be transmitted. V. CONCLUSION A new event-triggered networked control system framework where the quantized measurements are event-triggered to be released for transmission through the network channel to the controller is proposed. It is shown that by this way, the average transmission interval could be increased substantially, which implies the required network resources for an NCS can be reduced while maintaining the desired control performance. In addition, the event-triggered threshold is checked for each quantized measurement and the real-time detection hardware is no longer needed compared with the continuous event-triggered sampling technique. R EFERENCES [1] J. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,” Proceedings of the IEEE, vol. 95, no. 1, 2007, pp. 138–162.
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