IEEE论文-An approach to adaptive control of fuzzy dynamic systems.pdf

268IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002 An Approach to Adaptive Control of Fuzzy Dynamic Systems Gang Feng AbstractThis paper discusses adaptive control for a class of fuzzy dynamic models. The adaptive control law is first designed in each local region and then constructed in global domain. It is shown that the resulting fuzzy adaptive control system is globally stable. Robustness issues of the adaptive control system are also addressed. A simulation example is given for demonstration of the application of the approach. Index TermsAdaptive control, fuzzy modeling, nonlinear sys- tems, stability. I. INTRODUCTION S INCE the first paper on fuzzy sets 1 was published, fuzzy logic control has attracted a great attention from both the academic and industrial communities. Many people have de- voted a great deal of time and effort to both theoretical research and implementation techniques for fuzzy logic controllers. Much progress has been made in successfully applying FLC in industrial control systems 25. Duringthepastacoupleofyears,manysystematicfuzzycon- troller design methods have been developed 612 based on the TakagiSugeno (TS) model, or the fuzzy dynamic models. The basic idea of these methods is: i) to represent the complex nonlinear system in a family of local linear models, each linear model represents the dynamics of the complex system in one local region; ii) to construct a global nonlinear model by aggre- gating all the local models through the fuzzy membership func- tions. The primary advantage of this model is that the controller design can be mainly based on each local model, which is much easier than that for nonlinear systems in the global region, and then the global controller can beconstructed from the local con- trollers. It has also been shown that fuzzy systems can approximate any nonlinear functions over a convex compact region 13. Basedonthisobservation,anumberofattemptshavebeenmade to use fuzzy logic for adaptive control of nonlinear systems 1417. The basic idea of most of these works is to use fuzzy basisfunctionstoapproximatetheunknownnonlinearfunctions and update the constant parameters of the function on line and then implement adaptive control using the conventional control technology. Theauthorin18 recentlyproposedamodel-based fuzzy control as well as adaptive control method. In this paper, we will develop an adaptive control design method for a class of fuzzy dynamic models. The basic idea is Manuscript received April 18, 2001; revised August 27, 2001 and October 3, 2001. This work was supported by the City University of Hong Kong (SRG 7001245). The author is with the Department of MEEM, City University of Hong Kong, Kowloon, Hong Kong (e-mail: .hk). Publisher Item Identifier S 1063-6706(02)02971-5. to design an adaptive controller in each local region and then construct the global adaptive controller by suitably integrating the local adaptive controllers together in such a way that the global closed-loop adaptive control system is stable. Therestofthepaperisorganizedasfollows.SectionIIformu- lates the fuzzy system modeling, Section III presents the fuzzy adaptive control design and stability proof. Robustness issues are also discussed in the section. One example of simulation is presented in Section IV, which is followed by concluding re- marks in Section V. II. FUZZYSYSTEMMODELING Many physical systems are very complex in practice so that their rigorous mathematical models can be very difficult to obtain if not impossible. However, many physical systems can indeed be expressed in some form of mathematical models locally, or those systems can be expressed as an aggregation of a set of mathematical models. Various fuzzy models have been proposed in the last few years, see, for example, 3, 6. Here, we consider using the following fuzzy model to represent a complex single-inputsingle-output (SISO) system that includes both fuzzy inference rules and local analytic linear models: IFisANDis THEN (2.1) wheredenotes the -th fuzzy inference rule,the number of inference rules,are fuzzy sets, the system input variable,the output of the system, are coefficients of theth subsystem, and some measurable system variables. The model (2.1) can also be described in the state space form. Let . . . (2.2) then, the model (2.1) becomes IFisANDis THEN (2.3) 1063-6706/02$17.00 2002 IEEE IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002269 where . . . . . . . . . . . . . Letbe the normalized membership function of the in- ferred fuzzy setwhere (2.4) and (2.5) It is noted that the model in (2.1) only represents the properties ofthesysteminalocalregionandthusisreferredtoasthefuzzy dynamic local model. In order to clarify the concept of the local dynamic models, we introduce the following definition. Definition 2:1: The th fuzzy local dynamic model is a bi- nary set defined as FLDM or FLDM whereorrepresentsthecrispinputoutput relationship or dynamic properties of the system at the crisp point. By using a standard fuzzy inference method, that is, using a singleton fuzzifier, product fuzzy inference and center-average defuzzifier, the following dynamic global fuzzy model can be obtained: (2.6) which can also be rewritten as (2.7) It should be noted that the membership functionss are in generalthefunctionofthesystemoutputanditsderivatives,and thus the system described in (2.6) or (2.7) is a nonlinear system in general. The state-space form of (2.7) can be expressed as (2.8) It is noted thatare the same. In this paper, we are going to discuss the design of control systemsfor(2.7)whenthecoefficientsoftheplantareunknown. It is assumed that the membership functions have been chosen a priori based on the experts knowledge or the plant data. The objective of the fuzzy adaptive control is to find an adap- tive control law so that the output of the system tracks a given bounded reference signal. That is, the output error (2.9) approaches zero as time goes to infinity. It is assumed that the derivatives ofare also bounded. III. ADAPTIVECONTROLDESIGN In this section, we will present adaptive control design for the fuzzy dynamic systems discussed in the previous section. A. Adaptation Algorithm It is assumed that the derivatives of the output are available for measurement in this paper. If this is not the case, a stable filter is needed for obtaining the information of derivatives of the output 1922. Define then the plant (2.7) can be rewritten as (3.1) This is a regression form withas a plant parameter vector and as a regressor vector. It should be noted that the plant (3.1) is in general nonlinear but it is linear with respect to its unknown parameters. Therefore, all the parameter adaptation algorithms developed for linear plants can be employed for the estimation of the unknown parameters in (3.1). Here, we consider the fol- lowing least squares algorithm: (3.2) where With the above update law, we have the following conver- gence results 19, 20. Lemma 1: The parameter update law (3.2), when applied to the fuzzy dynamic model (3.1) has the following properties. E1)is continuous and bounded. E2) . E3). In this paper, we will use the certainty equivalence principle to design adaptive controllers. Thus we will first consider the con- 270IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002 troller design when the parameters of the fuzzy dynamic model are known a priori. B. Controller Design With Known Parameters Define then we can design the following local fuzzy control law: IFisANDis THEN (3.3) The global control law can be obtained as follows: (3.4) where (3.5) are coefficients of a stable polynomial defined as which specifies the desired output tracking error dynamics, and are assumed to be nonsingular. Substituting the control law (3.4) into the fuzzy dynamic model (2.7) leads to the following closed-loop system (3.6) This in turn will guarantee that the tracking error will approach zero as time goes to infinity and the error dynamics will be de- termined by the coefficients. C. Adaptive Control Design Based on the certainty equivalence principle, we choose the following local adaptive control law: IFisANDis THEN (3.7) where Then, the global control law can be obtained as follows: (3.8) where (3.9) Remark 3.1: It is noted that in order to guarantee the nonsin- gularityofthecontrollaw,hastobeensured.Thereare anumberofmethodstoachievethisinadaptivecontrolcommu- nity such as projection to the known convex region 1922. Substituting the control law (3.8) into the fuzzy dynamic model (2.7) leads to the following closed-loop system: (3.10) Define (3.11) the above closed-loop system can be expressed in a state-space form as (3.12) that is (3.13) where . . . . . . . . . . It should be noted that the matrixhas all its eigenvalues in the left-hand side of the -plane. Then we have the following main result. Theorem 1: Given the fuzzy dynamic system (2.7), if the adaptive control law is chosen as (3.7) or equivalently (3.8), and if the adaptation law of the parameters is chosen as in (3.2), thentheclosed-loopadaptivecontrolsystemwillbestableinthe sense that all the signals in the loop are bounded. Furthermore, IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002271 the output tracking errorwill approach zero as time goes to infinity. Proof: Consider (3.13) and it follows that its autonomous system, that is,is globally uniformly exponentially stable.Thenthereexistpositiveconstantsandsuch that (3.14) Squaring both sides and using Schwarzs inequality, (3.14) im- plies that (3.15) where which is bounded due to the property E3) of the estimator. It is noted that we have the following result based on the definitions ofand, (3.16) for some positive constantsand. Then, it follows from (3.15) and (3.16) that there exists some constantssuch that: (3.17) Since, it follows from (3.17) using BellmanGronwall Lemma19,20thatisbounded.Itimpliesthat,and thusare bounded. Furthermore, it follows from the results of Lemma 1 and the definition ofthatis bounded. This, in turn,impliesthatastimegoestoinfinity.Thenitfollows from (3.13) thatas time goes to infinity and therefore as time goes to infinity. D. Robustness Issues The fuzzy dynamic models discussed in the previous sec- tions are assumed to be ideal, that is, the systems are assumed to be modeled exactly by the fuzzy dynamic models. However in practice, the systems are always subject to various kinds of uncertainties such as unmodeled dynamics and/or bounded dis- turbances. As shown for the linear adaptive control systems, even a small uncertainty could lead to unstable adaptive control system.Asaresult,varioustypesofrobustadaptationalgorithm and robust adaptive control algorithms have been developed to cope with these various sorts of uncertainties. These techniques include dead zones, relative dead zones,-modification among many others 1922. In this section, we will use the relative dead zones algorithm for demonstration. In this case, we con- sider the following fuzzy dynamic model: IFisANDis THEN (3.18) whererepresents the unmodeled dynamics and bounded dis- turbances. By using a standard fuzzy inference method, that is, using a singleton fuzzifier, product fuzzy inference and center-average defuzzifier, the following dynamic global fuzzy model can be obtained, (3.19) It is assumed that there exists an upper bound functionfor the uncertainties, that is (3.20) wherecan be expressed as (3.21) for two unknown small constantsand. Remark 3.2: It is noted that the above assumption is in fact weaker than the standard assumption as in the ordinary robust adaptive control designs such as 19 and 20, where the two small constants are also assumed to be known. With the same definition of the regressor and the model pa- rameter vector, the same certainty equivalence control law (3.7) or equivalently (3.8), we obtain the following closed-loop con- trol system (3.22) or in a state-space form as (3.23) In this case, the following least-squares algorithm with relative dead zone will be used for robust parameter estimation: (3.24) 272IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002 where, and the termis a dead-zone func- tion, which ensures that the parameter estimator isnot disrupted by small errors. The dead zone is defined as follows: if otherwise (3.25) with if if if where, andis calculated by (3.26) whereis an update rate parameter, and . It should be noted thatandwill be always nonnegative and nondecreasing. Then, we have the following convergence properties for the above robust parameter estimation algorithm. Lemma 2: The parameter update law (3.24)(3.26), when applied to the fuzzy dynamic model (3.19) has the following properties. E1)is continuous and bounded. E2)andare continuous and bounded, andand converge to constants, sayand, respectively. E3) . E4). Proof: Please refer to 21. Then, we have our main result in the following theorem. Theorem 2: Given the fuzzy dynamic model (3.19), if the adaptive control law is chosen as in (3.7) or equivalently (3.8), the adaptation law of the parameters is chosen as in (3.24)(3.26), and, then the closed-loop adaptive control system will be stable in the sense that all the signals in the loop are bounded. Furthermore, the steady state performance satisfies Proof: Consider (3.23) and it follows that its autonomous system, that is,is globally uniformly exponentially stable.Thenthereexistpositiveconstantsandsuch that (3.27) From the definition of dead-zone function, we have (3.28) It follows from (3.27) and (3.28) that: (3.29) Squaring both sides and using Schwarzs inequality, (3.29) im- plies that (3.30) whereand (3.16) has been used. Since the right-hand side of (3.30) is monotonic nonde- creasing, we obtain Thus, if, then one can show that (3.31) where. IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002273 Then, it follows from (3.16) and (3.31) that there exists some constantsandsuch that (3.32) Since, it follows using BellmanGronwall Lemma 19, 20 thatis bounded. It implies that, and thusare bounded. Furthermore, sinceis bounded,implies. Therefore, the fact thatis uniformly continuous implies that tends to zero as time goes to infinity 19, 20. It then follows from the (3.22) and (3.28) that whenapproaches infinity: The proof is thus completed. If only bounded disturbances are present, then we can have the following stronger results. Corollary 1: With the same conditions as in Theorem 2 ex- cept that there is no unmodeled dynamics, i.e., then the adaptive control system with the simpler update law for (3.33) is stable in the same sense defined in Theorem 2. Proof: It follows directly from the proof of Theorem 2. IV. ANEXAMPLE OFSIMULATION This section presents a simulation example to demonstrate theapplication oftheproposedadaptive controlalgorithms, that is, adaptive balancing of an inverted pendulum on a cart. The dynamic equations of motion of the pendulum are given as (4.1) wheredenotes the angle of the pendulum from the vertical axis.m/s is the gravity constant, m is the mass of the pendulum,isthelengthofthependulum, is the mass of the cart, andis the force applied to the cart. In this simulation, the pendulum parameters are chosen as kg,kg, andm. In the context of this paper, it is assumed that the operating points and the corresponding fuzzy membership functions have been determined. The operating points are chosen asbeing around 0 and. The fuzzy membership functions are chosen as Gaussians shown in Fig. 1. The desired reference model is chosen as (4.2) whereis a reference input, chosen as Fig. 1.Membership functions. Fig. 2.Response with adaptive control. The design parameters are chosen in this case. The initial parameter is chosen. A number of simula- tions with different starting angles of the pendulum have been conducted and one typical result, the initial position , is recorded in Fig. 2, where the angle is in radian. For comparison, the response of the nonadaptive control system with the same initial conditions are also obtained and shown in Fig. 3 where the steady-state tracking error can be observed. In order to show the adaptation capability of the proposed adaptive control system, the following case is also considered. That is, the mass of the pendulum is changed from 2 to 8 kg at the times. The result is shown in Fig. 4. It can be seen that the adaptive control algorithm can cope withthevariationoftheplantdynamicswell.Similarlyforcom- parison,theresponseofthenonadaptivecontrolsystemwiththe same initial conditions are also obtained and shown in Fig. 5 wherethesignificantsteadystatetrackingerrorcanbeobserved. 274IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002 Fig. 3.Response with nonadaptive control. Fig. 4.Response for adaptive control with? ? ?kg jumping to? ? ?kg. Ithasbeendemonstratedthroughthesimulationsthatthepro- posedfuzzyadaptivecontrolschemescanbeusedforthecontrol of unknown nonlinear pendulum-cart system. V. CONCLUSION This paper presents a new fuzzy adaptive control system for a class of nonlinear systems represented by the fuzzy dynamic models. The basic idea of the approach is to design the local linear adaptive controller in each local region and construct the global fuzzy adaptive controller in such a way that the stability of the closed-loop adaptive control system is guaranteed. This Fig.5.Response for nonadaptive control with?kg jumping to?kg. paper only addresses a limited class of fuzzy dynamic models. However,it is believed that the idea can be extended to the more general cases, which though requires much more effort and will be our future research topics. ACKNOWLEDGMENT The author is grateful to the reviewers for a number of con- structive comments that have improved the presentation of this paper. REFERENCES 1 L. A. Zadeh, “Outline o