Robust Quantized Approach to Fuzzy Networked ... - IEEE Xplore

IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS, VOL. 2, NO. 1, MARCH 2012

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Robust Quantized Approach to Fuzzy Networked Control Systems Magdi S. Mahmoud, Senior Member, IEEE, and Abdul-Wahid A. Saif

Abstract—This paper investigates the problem of robust control for uncertain discrete-time Takagi–Sugeno (T-S) fuzzy networked control systems (NCSs) with state quantization. A new model of network-based control with simultaneous consideration of network induced delays and packet dropouts is proposed. Using fuzzy Lyapunov–Krasovskii functional, we derive a less conservative delay-dependent stability condition for the closed fuzzy controller is developed for the asympNCSs. Robust totic stabilization of the NCSs and expressed in linear matrix inequality-based conditions. Numerical simulation examples show the feasibility applications of the developed technique. Index Terms—Discrete time-varying delay, fuzzy systems, linear control. matrix inequality (LMI), networked

I. INTRODUCTION N RECENT years, it has been recognized that fuzzy system models are qualified to represent a certain class of nonlinear dynamic systems following the Takagi–Sugeno (T-S) fuzzy model [1]. Since then there have been several approaches for the study of stability analysis and robust controller synthesis using the so-called parallel distributed compensation (PDC) method for uncertain nonlinear systems [2], [3]. Sufficient conditions have been derived based on the feasibility testing of a linear matrix inequality (LMI) in [4]–[7] and extended for classes of nonlinear discrete-time systems with time delays in [8]–[10] via different approaches. Recently, much attention has been paid to the stability issue of network-based control systems [11]. Several results pertaining to the analysis and design of networked control systems (NCSs) enhanced their wide benefits such as reducing system wiring, ease of system diagnosis and maintenance, and increasing system agility, to name a few. However, communication network in the control loops gave rise to some new issues, especially the intermittent losses or delays of the communicated information due to use of a network, which imposes a challenge to system analysis and design. To address this challenge, many results have been developed in consideration of network-induced delay and packet dropout, [12]–[18], with focus on stability analysis and controller design with random delays.

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Manuscript received October 07, 2011; revised December 14, 2011; accepted January 27, 2012. Date of publication March 15, 2012; date of current version April 11, 2012. This work was recommended by Guest Editor N. Verma. The authors are with the Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JETCAS.2012.2187403

Further consideration of the communication of the NCSs over the channel emphasized the importance of signal quantization, which has significant impact on the performance of NCSs. In this regard, the problem of guaranteed cost control and quantized controller design were discussed in [17] by using two quantizers in the network both from sensor to controller and from controller to actuator, and the network-induced delay and data dropped were considered as well. Recent advances converted the quantized feedback design problem into a robust control problem with sector bound uncertainties, [11] and [16]–[18]. Parallel investigations to the class of switched discrete-time systems with interval time-delays were developed in [19]–[21], [22], and [23]. Despite the potential of these developments, the problem of how to analyze the stability of nonlinear NCSs with data drops still open. On the other hand, most industrial plants have severe nonlinearities, which lead to additional difficulties for the analysis and design of control systems. Though some issues on nonlinear NCSs have been investigated [23], [24], limited work state feedback controller design has been found on robust of networks for fuzzy systems with consideration of both network conditions and signal quantization. The guaranteed cost networked control and robust problem based on the T-S fuzzy model was treated in [25]. The results were derived by using a single Lyapunov function (SLF) method, which in general leads to a conservative result. Designing fuzzy controllers for a class of nonlinear networked control systems was considered in [26]–[28] by solving approximate uncertain linear networked T-S models with both network induced-delay and packet dropout. However, they do not quantize the signals. The foregoing facts motivate the present study. state feedIn this research work, we address the robust back control problem for discrete-time networked systems with state quantization and disturbances. The T-S fuzzy systems with norm-bounded uncertainties are utilized to characterize the nonlinear NCSs. Since the computation available is often limited, the quantized feedback controller is designed under consideration of effect of network-induced delay and data dropout, the employed quantizer is time-varying. By using a new fuzzy Lyapunov–Krasovskii functional (LKF), we provide a sufficient LMI-based condition for the existence of a fuzzy controller. Two simulation examples show the feasibility of the developed technique. Notations and Facts: In the sequel, the Euclidean norm is used for vectors. We use and to denote the transpose and the inverse of any square matrix , respectively. We use to denote a symmetric positive definite (positive semidefinite, negative, negative semidefinite matrix

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are the premise variables, each are the fuzzy sets, is the number of is the state vector, if-then rules and is the control input, is the output, is indicates the disturbance input which belongs to the maximum allowable signal transmission delay and is the known initial state condition. The uncertain matrixes are represented by

where

(2) describe the nominal dywhere the matrixes are known constant real namics and are matrixes with appropriate dimensions. The matrixes unknown time-varying and satisfying . Following [1] and using a center average defuzzifier, product inference, and incorporating fuzzy “blending,” the fuzzy system under consideration can be cast into the form Fig. 1. Networked control system.

and to denote the identity matrix. Matrixes, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. In symmetric block matrixes or complex matrix expressions, we use the symbol to represent a term that is induced by symmetry. Fact 1: For any real matrixes and with appropriate dimensions and , it follows that

(3) where

Sometimes, the arguments of a function will be omitted when no confusion can arise.

(4) where is the grade of membership of . In the sequel, we assume that

II. PROBLEM SETUP A typical networked control system is depicted in Fig. 1, in which the sampler is clock-driven and the quantizer, the controller, the zero-order hold (ZOH) are event-driven. The samwith the sampling instants pling period is assumed to be as . The plant belongs to class of uncertain discrete-time systems where the parametric uncertainties are norm-bounded. In what follows, we consider that this class is represented by T-S fuzzy model composed of a set of fuzzy implications, and each implication is expressed by a linear system model. The th rule of this T-S model has the following form: is and is , then Rule j: If

(1)

in

and therefore

Our objective in this paper is to design a fuzzy state feedback controller with state quantization for the system in Fig. 1. III. CLOSED-LOOP FUZZY SYSTEM In what follows, we proceed to consider establish the main result for the uncertain discrete-time fuzzy networked control systems described by (3) and design the quantized fuzzy

MAHMOUD AND SAIF: ROBUST QUANTIZED APPROACH TO FUZZY NETWORKED CONTROL SYSTEMS

state feedback controller. We consider a limited capacity communication channel and for reducing the amount of data rate of transmitting in the network, which led to the increase quality of service of the network, we assume that the state vector is measurable. The state signal from sensor to the controller is quantized via a quantizer, and then transmitted with a single packet. To reflect realty, network-induced time delay is modeled as an input delay and the packet dropout will be considered.

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where is the initial state of the quantizer and is . In this regard, a parameter associated with the quantizer a particular characterization of the quantizer is given by

where any

. It follows from [19] that, for , a sector bound expression can be expressed as

A. State-Feedback Control In effect, we seek to design the state-feedback controller (5) where quel and

is the feedback law to be defined in the seare some integers such that . Introduce which contains the information of packet dropouts and improper packet sequence in the control signal. Note that . , It has been pointed out in [19] that when there would be no packets dropout and the case represents continuous packets lost. In addition, when , the new packet reaches the destination before the old one. This case might lead to a less conservative result. In the sequel, and it is readily seen that we assume that

For simplicity in exposition, we use to denote . can be written as . Thus, Remark III.1: In the sequel, we assume that the updating signal at the instant has experienced signal transmission delay , however the delay between the sensor and quantizer is neglected. In view of the limited capacity in communication channel, the state signal from sensor to the controller is quanfor reducing the amount tized via a logarithmic quantizer of data rate of transmitting in the network. When the static and , the state feedback controller time-invariant quantizer would be in the form of , which is the same as a traditional one. Incorporating the notion of parallel distributed compensation, the following fuzzy state-feedback stabilizing control law is used. is and is , then Rule j: If (7) is the control gain for rule where ingly, the overall fuzzy control law is expressed by

It should be observed that accounts for the time from the when sensor nodes sample the sensor data from instant the plant to the instant when actuator transfer data to the plant. Extending on this, we remark that

. Accord-

(8) Applying controller (8) to system (3) with some mathematical manipulations, the resulting closed-loop system can be cast into the form

Consequently, we define

where

are known finite integers.

B. Quantizer In this work, we denote the quantizer as (9) where is a symmetric, static and timeinvariant quantizer and the associated set of quantization levels is expressed as

which belongs to the class of switched time-delay system [15], where

(6) Note that the quantization regions are quite arbitrary. In case of logarithmic quantizer, the set of quantization levels becomes (10)

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Remark III.2: With regards to the work published in [26]–[28] compared to the present work, the common denominator is to deal with fuzzy networked control systems. We note however, that the system representation in [26]–[28] was in continuous-time while the model setup (1) is in discrete-time, which constitutes a basic difference. Therefore, it is a difficult task to their results with ours. IV. QUANTIZED FUZZY CONTROL DESIGN In this section, we seek to establish a sufficient condition for control problem. This condithe solvability of the robust tion will be expressed in an LMI framework to facilitate the design of the desired fuzzy state feedback controllers. Based on the so-called parallel distributed compensation scheme, with reference to the NCSs in Fig. 1, the following theorem establishes a delay-dependent stabilization condition for the closed-loop fuzzy networked control system (9). Theorem IV.1: Consider system (9). Given the bounds and a scalar constant , there exists a fuzzy controller in the form of (8), such that the uncertain disturbance atclosed-loop fuzzy system (9) with an tention level is asymptotically stable, if there exist matrixes , and scalars matrixes , satisfying

(11)

(14) Proof: See the Appendix. Remark IV.1: It is significant to observe that Theorem IV.1 provides a delay-dependent condition for the design of robust for fuzzy NCS in terms of feasibility testing of a family of . strict LMIs with a total number of LMI-variables as The key feature is that the matrix gain is treated as a direct LMI variable. This will eventually lessen the conservatism in robust fuzzy control design. Remark IV.2: It is worthy to note that the number of LMIs increases linearly with the number of rules which limits the applicability of the method for very large values of . Had we used

then Theorem IV.1 reduces to the following corollary. and a scalar conCorollary IV.1: Given the bounds , there exists a fuzzy controller in the form of (8), stants such that the uncertain closed-loop fuzzy system (9) with an disturbance attention level is asymptotically stable, if there , exist matrixes and scalars matrixes , satisfying (15)

(12)

(13)

(16)


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B. Case of NCS Without Quantizer (17) and the number of LMI variables would be paid is that the LKF becomes nonfuzzy.

In this case, the resulting closed-loop fuzzy system can be expressed as

. The price

V. SPECIAL CASES In this section, we seek to derive a sufficient condition for the control problem for two relevant solvability of the robust special cases: the first is the nominal case where the uncertainties are absent and the second pertains to the NCS without quantizer. A. Nominal Case In the absence of uncertainties closed-loop fuzzy system becomes

, the

(21) The corresponding control design is given by the following corollary. and a scalar conCorollary V.2: Given the bounds , there exists a fuzzy controller in the form of (8), stants such that the uncertain closed-loop fuzzy system (21) with an disturbance attention level is asymptotically stable, if there exist matrixes , matrixes and scalars , satisfying (22)

(18) where

(19) The following control design holds. Corollary V.1: Consider system (18). Given the bounds and a scalar constants , there exists a fuzzy controller in the form of (8), such that the uncertain disturbance attenclosed-loop fuzzy system (18) with an tion level is asymptotically stable, if there exist matrixes , maand scalars trixes , satisfying (20) are given by (12). where Proof: Following from Theorem IV.1 by setting .

(23) where the various terms are as in (12)–(14). VI. SIMULATION EXAMPLES In what follows, we illustrate the applicability of our method by two examples: the first example deals with a representative numerical example of uncertain system and the second treats a nominal water-quality system. A. Example 1 A typical simulation example is considered to illustrate the fuzzy controller design procedure developed in Theorem IV.1.

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A class of discrete-time fuzzy networked control systems model with state quantization is described by. Rule 1: If is , then

Fig. 2. Controlled-output trajectory.

Rule 2: If

is

, then The membership functions for the rules 1, 2, 3 are

For the purpose of implementation, we consider the fuzzy system to be controlled through a network. A quantizer is selected to be of of logarithmic type with , leading to . The bounds on data packet dropout are selected as , respectively. Using the solver Scilab 5.0, the statefeasible solution of Theorem IV.1 yields the fuzzy feedback controller gains of the form Rule 3: If

is

, then

The simulation results of the state and controlled-output trajectories are plotted in Figs. 2–4. It is quite evident that all the state and output variables of the fuzzy system settle at the equilibrium level within 20 s. B. Example 2 A discrete water-pollution model of some reaches along the River Nile with multiple operating points is considered. The model represents two aggregate bio-strata, the first one is for algae and the other is for ammonia products. The data values are taken from [31]. The purpose here is to show the applicability of our design approach without uncertainties. We wish to design a fuzzy feedback controllers for this system


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Rule 2: If

is

, then

Rule 3: If

is

, then

Fig. 3. First state trajectory.

Note in the current situation that a single control action (either proportional to change in stream velocity or pretreated waste water) is used in simulation experiments to reflect seasonal water plans. The bounds on data packet dropout are selected as , respectively. Invoking the solver Scilab 5.0, the feasible solution of LMIs (20) yields the state-feedback gains

Fig. 4. Second state trajectory.

based on Corollary V.1. In this case, the associated fuzzy sets are characterized by

These results come in agreement with our theoretical developments. To further show the validity of our design method, we simulate the closed-loop water quality system using the distura randomly generbance ated switching signal from a uniform distribution in the interval (0, 1). The obtained state trajectories from 500 samples under state and dynamic output feedback are plotted in Figs. 5 and 6, respectively. VII. CONCLUSION

The transition among sets is allowed by centralized signaling center through supervisory agents and based on data observation collected from remote sensing stations. The corresponding model matrixes are given by is , then Rule 1: If

In this paper, we have addressed the problem of robust state feedback controller design for discrete-time T-S fuzzy networked control systems including state quantization. We have constructed a novel uncertain T-S fuzzy system model of network-based control to approximate nonlinear networked control systems. A quantized feedback fuzzy controller has been designed under consideration of effect of network-induced delay and data dropout, and the time-varying quantizer has

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(24) where are matrixes

of

appropriate

dimensions and are fuzzy weighting matrixes that are directly include the membership functions instead of a single matrix, a fact that aims at relaxing the conservatism. For simplicity in notation, we let

Fig. 5. Algae and Ammonia trajectories.

(25) In terms of the state increment , we consider the LKF the time-span

and

Fig. 6. State feedback control trajectories.

been selected to be logarithmic. By employing a fuzzy Lyapunov-Krasovskii functional, we have derived some LMI-based sufficient conditions for the existence of fuzzy controller. Numerical simulation examples have been presented to illustrate the efficiency of the theoretic results. (26) APPENDIX PROOF OF THEOREM IV.1 In what follows, we adopt a parameter-dependent approach and define [15]. Consider system (9) with

Remark VIII.1: Note in the Lyapunov functional (26) that the first term is standard to the delay-less nominal systems while the second term and the first part of the fifth term together correspond to the delay-dependent conditions. The second part of the third term and the fourth terms are added to compensate for the enlargement in the time interval from to . The introduction of and plus


appropriate free-weighting matrixes (to be introduced later on) serve in reducing the number of manipulated variables, a feature which improves the performance of the developed delay-dependent stability criterion. This is quit evident upon comparison with the LKFs in [29] and [30]. . A straightforward We focus initially on the case computation gives the first-difference of along the solutions of (25) with the help of (9) and (10) as

for some matrixes

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, and proceed to get

(29)

In terms of

we cast (29) with

into the form

(27) (30) To facilitate the delay-dependence analysis, we invoke the following identities: where are given by (14). If for all admissible uncertainties satisfying (2), then by Schur complements it , for any guaranfollows from (30) that teeing the internal stability. Proceeding further and to assure the closed-loop stability with -disturbance attenuation, we follow [15] to get

(28)

(31)

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when

where

REFERENCES

(32) Next, by applying Fact 1, we obtain

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H

(33) for some scalars correspond to after deleting the last element, and

. Note that . The quantities given by (12)

H

(34)

(35) where tion of

are given by (14). Further convexificain (33) yields

(36) By Schur complements using the algebraic inequality for any matrix , the desired stability condition can then be cast into the LMI (11), which concludes the proof. ACKNOWLEDGMENT The authors would like to thank the deanship of scientific research (DSR) at KFUPM for support through research group project RG1105-1.

H H

H

H

H

H


[27] L. Zhang and H. Fang, “Fuzzy controller design for networked control system with time-variant delays,” J. Syst. Eng. Electron., vol. 17, no. 1, pp. 172–176, 2006. [28] X. Jiang and Q. Han, “On designing fuzzy controllers for a class of nonlinear networked control systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 4, pp. 1050–1060, Aug. 2008. [29] V. F. Montagner, V. J. S. Leite, S. Tarbouriech, and P. L. D. Peres, “Stability and stabilization of discrete-time linear systems with statedelay,” in Proc. Am. Control Conf., Portland, Oregon, Jun. 8–11, 2005, pp. 3806–3811. output-feedback control for [30] L. Zhang, P. Shi, and E. K. Boukas, “ switched linear discrete-time systems with time-varying delays,” Int. J. Control, vol. 80, pp. 1354–1365, 2007. [31] M. S. Mahmoud and S. J. Saleh, “Regulation of water quality standards in streams by decentralized control,” Int. J. Control, vol. 41, pp. 525–540, 1985.

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Magdi S. Mahmoud (SM’83) received the Ph.D. degree in systems engineering from Cairo University, Cairo, Egypt, in 1974. He has been a Professor of Engineering since 1984. He is now a Distinguished University Professor at KFUPM, Saudi Arabia. He worked at different universities world-wide including Egypt, Kuwait, UAE, U.K., USA, Singapore, and Australia. He has given invited lectures in Venezuela, Germany, U.K., and USA. He has been actively engaged in teaching and research in the development of modern methodologies to computer control, systems engineering and information

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technology. He is the principal author of 26 books, inclusive book-chapters and the author/co-author of more than 475 peer-reviewed papers. Dr. Mahmoud is the recipient of two national, one regional, and two university prizes for outstanding research in engineering. He is a fellow of the IEE, the CEI (U.K.), and a registered consultant engineer of information engineering and systems (Egypt).

Abdul-Wahid A. Saif received the B.Sc. degree from the Physics Department and the M.Sc. degree from the Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, and the Ph.D. degree from Control and Instrumentation Group, Department of Engineering, Leicester University, Leicester, U.K. He is currently an Associate Professor of Control and Instrumentation in Systems Engineering Department (SE) at King Fahd University of Petroleum and Minerals. He worked as a Research Assistant in SE Department and as a Lecturer in Electrical Engineering Department and a Lecturer in Physics Department in the same University. After finishing the Ph.D. degree, he joined the Systems Engineering. His research interest is simultaneous and strong stabilization, robust control and H8-optimization, instrumentation and computer control. He taught several courses in modeling and simulation, digital control, digital systems, microprocessor and microcontrollers in automation, optimization, numerical methods, PLC’s, process control and control system design. He has published more than 35 papers in reputable journals and conferences.