Analytical Analysis and Feedback Linearization Tracking Control of the General Takagi-Sugeno Fuzzy Dynamic Systems

Transcription

1 290 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO., FEBRUARY 999 In order to verify the erforance of the roosed ethod, exerients with the NIST nueral database have been carried out and the erforance of the roosed ethod has been coared with that of the revious ISR ethods. The exeriental results revealed that the roosed ethod had uch better discriination and generalization ower than the revious ISR ethods. REFERENCES [] J. S. Denker et al., Neural network recognizer for handwritten ZIP code digits, in Advances in Neural Inforation Processing Systes, D. S. Touretzky, Ed. San Mateo, CA: Morgan Kaufann, 989, vol., [2] Y. Le Cun et al., Constrained neural network for unconstrained handwritten digit recognition, in Proc. st Int. Worksho Frontiers Handwriting Recogn., Montreal, P.Q., Canada, Ar. 990, [3] G. L. Martin and J. A. Pittan, Recognizing hand-rinted letters and digits, in Advances in Neural Inforation Processing Systes, D. S. Touretzky, Ed. San Mateo, CA: Morgan Kaufann, 990, vol. 2, [4] O. Matan et al., Reading handwritten digits: A ZIP code recognition syste, IEEE Cout. Mag., vol. 25, no. 7, , July 992. [5] J. Keeler and D. E. Ruelhart, A self-organizing integrated segentation and recognition neural network, in Advances in Neural Inforation Processing Systes, J. E. Moody, S. J. Hanson, and R. P. Liann, Eds. San Mateo, CA: Morgan Kaufann, 992, vol. 4, [6] O. Matan, J. C. Burges, Y. Le Cun, and J. S. Denker, Multi-digit recognition using a sace dislaceent neural network, in Advances in Neural Inforation Processing Systes, J. E. Moody, S. J. Hanson, and R. P. Liann, Eds. San Mateo, CA: Morgan Kaufann, 992, vol. 4, [7] G. L. Martin, M. Rashid, and J. A. Pittan, Integrated segentation and recognition through exhaustive scans or learned saccadic jus, Int. J. Pattern Recognit. Artif. Intell., vol. 7, no. 4, , 993. [8] K. Iai, K. Gouhara, and Y. Uchikawa, Recognition of letters in lateral rinted strings using a three-layered BP odel with feedback connections, Syst. Cout. Jn., vol. 23, no. 2, , 992. [9] H. Fujisawa and Y. Nakano, Segentation ethods for character recognition: Fro segentation to docuent structure analysis, Proc. IEEE, vol. 80, , July 992. [0] R. M. Bo zinovi`c and S. N. Srihari, Off-line cursive word recognition, IEEE Trans. Pattern Anal. Machine Intell., vol., , Jan [] S.-W. Lee and J. S. Park, Nonlinear shae noralization ethods for the recognition of large-set handwritten characters, Pattern Recognit., vol. 27, no. 7, , 994. [2] K. Lang, A. Waibel, and G. Hinton, A tie-delay neural network architecture for isolated word recognition, Neural Networks, vol. 3, , 990. [3] D. Ruelhart, G. Hinton, and R. Willias, Learning internal reresentations by error roagation, in Parallel Distributed Processing, D. Ruelhart and J. McClelland, Eds. Cabridge, MA: MIT Press, 986. [4] Y. Yaashita, K. Higuchi, Y. Yaada, and Y. Haga, Classification of handrinted Kanji characters by the structured segent atching ethod, Pattern Recognit. Lett., vol., , 983. [5] S.-W. Lee, Off-line recognition of totally unconstrainted handwritten nuerals, IEEE Trans. Pattern Anal. Machine Intell., vol. 8, , June 996. Analytical Analysis and Feedback Linearization Tracking Control of the General Takagi-Sugeno Fuzzy Dynaic Systes Hao Ying Abstract Takagi Sugeno (TS) fuzzy odeling technique, a black-box discrete-tie aroach for syste identification, has widely been used to odel behaviors of colex dynaic systes. Analytical structure of TS fuzzy odels, however, is resently unknown, nor is its ossible connection with the traditional odels, causing at least two ajor robles. First, the fuzzy odels can hardly be utilized to design controllers for control of the hysical systes odeled. Second, there lacks a systeatic technique for designing a controller caable of controlling any given TS fuzzy odel to achieve desired tracking or setoint control erforance. In this aer, we rovide solutions to these robles. First of all, we have roved that a general class of TS fuzzy odels is nonlinear tievarying Auto-Regressive with the extra inut (ARX) odel. The fuzzy odels in this study are general because they use arbitrary continuous inut fuzzy sets, any tyes of fuzzy logic AND oerators, TS fuzzy rules with linear consequent and the generalized defuzzifier which contains the oular centroid defuzzifier as a secial case. Furtherore, we have established a sile necessary and sufficient condition for analytically deterining local stability of the general TS fuzzy dynaic odels. The condition can also be used to analytically check quality of a TS fuzzy odel and invalidate the odel if the condition warrants. More iortantly, we have develoed a feedback linearization technique for systeatically designing an outut tracking controller so that outut of a controlled TS fuzzy syste, which ay or ay not be stable, of the general class achieves erfect tracking of any bounded tie-varying trajectory. We have investigated stability of the tracking controller and established a necessary and sufficient condition, in relation to stability of noniniu hase systes, for analytically deciding whether a stable tracking controller can be designed using our ethod for any given TS fuzzy syste. Three nuerical exales are rovided to illustrate the effectiveness and utility of our results and techniques. Index Ters AR odels, feedback linearization, fuzzy control, fuzzy odeling, stability. I. INTRODUCTION Nuerous successful industrial alications have shown the ower of Takagi Sugeno (TS) fuzzy odeling aroach [0], which is a black-box discrete-tie odeling aroach develoed for odeling colex dynaic systes [], [5], [6], [7], [9]. Coared with the conventional black-box odeling techniques [7] that can only utilize nuerical data, TS odeling aroach allows one to take advantage of both qualitative and quantitative inforation [8]. This advantage is ractically iortant and even crucial in any circustances. Qualitative inforation, such as exert/oerator knowledge and exerience about a hysical syste to be odeled, can readily be incororated into TS fuzzy odels in the for of fuzzy sets, fuzzy logic, or fuzzy rules. Virtually all the TS fuzzy odels in the literature use linear functions of inut variables as consequent of the fuzzy rules. Many learning schees have been develoed to autoatically configure one or ore coonents of TS fuzzy odels so that a TS fuzzy odel can quickly be established when qualitative/quantitative inforation is available [6], [5]. Desite of Manuscrit received Noveber 0, 996; revised Seteber 7, 998. This work was suorted in art by Texas Higher Education Coordinating Board Grant The author is with the Deartent of Physiology and Biohysics, Bioedical Engineering Center, The University of Texas Medical Branch, Galveston, TX USA. Publisher Ite Identifier S (99) /99$ IEEE

2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY success of TS fuzzy systes in ractice, analytical study has been scarce [24], [25], coared with the existing analytical results on Madani fuzzy systes (e.g., [8], [20] [23]). TS fuzzy odeling schee will be ore owerful if several ajor robles associated with it are overcoe. First, alost all the TS fuzzy odels develoed so far were used erely for eirically iicking easured inut-outut data sets of the hysical systes odeled. However, a fuzzy odel that can iic a nuber of easured inut-outut data sets does not necessary ean the odel is a valid one. More rigorous ethods are needed to ensure odel quality. Yet, there exists no analytical eans for theoretically checking quality of a TS fuzzy odel and ossibly invalidate it. Second, TS fuzzy odels were seldo utilized as control odels (the only excetions aear to be the studies [], [2], [4] in which soe TS fuzzy controllers were develoed to control one tye of TS fuzzy odels. Stability and robustness of the closed-loo fuzzy control syste were the ain subjects in these research). This is in shar contrast to the conventional black-box odels, like Auto-Regressive (AR), Moving-Average (MA), and Auto-Regressive with the extra inut (ARX) odels [7], which were develoed to facilitate design of a controller for controlling the hysical syste odeled. Third, analytical structure of TS fuzzy odels is currently unknown, letting alone ossible connection between the TS fuzzy odels and the conventional odels. The fuzzy odels have always been treated and used as black boxes. Finally, there exist no systeatic ethod that can be used to design a controller to control a given TS fuzzy odel and achieve not only syste stability but user-desired tracking or setoint control erforance. The objectives of this research were to solve these robles for a general class of TS fuzzy dynaic systes that use arbitrary continuous inut fuzzy sets, any tyes of fuzzy logic AND oerators, fuzzy rules with linear consequent and the generalized defuzzifier which contains the oular centroid defuzzifier as a secial case. II. A GENERAL CLASS OF TS FUZZY DYNAMIC SYSTEM MODELS A TS fuzzy odel is coosed of inut fuzzy sets, fuzzy logic AND oerators, fuzzy rules with linear functions of inut variables, and a defuzzifier. For the general TS fuzzy odels studied in this aer, we denote the jth TS fuzzy rule as R j ( j ; being the total nuber of the fuzzy rules) R j : IF y(n) is A0j AND y(n 0 ) is A j AND AND y(n 0 ) is A j THEN y(n +)=a 0jy(n) +a jy(n 0 ) + + a j y(n 0 ) +b 0j u(n) + b j u(n 0 ) + + b j u(n 0 ) = a ij y(n 0 i) + b kj u(n 0 k) () where y(n) and u(n) are, resectively, odel outut and inut at tie n (n; a ositive integer, reresents saling tie nt where T is saling eriod). Here, a ij and b kj are constant araeters. A ij is a fuzzy set fuzzifying y(n 0 i); and we denote its ebershi function as ij (y(n 0 i)) which ay be any shae but required to be continuous. We suose that there are P i different fuzzy sets for fuzzification of y(n 0 i) in all the fuzzy rules. Subsequently, there exist =P 0 2 P 2P different cobinations of the fuzzy sets and that any fuzzy rules are needed to cover all the cobinations. To cobine the ebershi values of the fuzzy sets in the rule antecedent, any tyes of fuzzy logic AND oerators ay be used and different tyes of AND oerators ay be used in different rules. Using as a sybol to reresent an arbitrary tye of fuzzy logic AND oerator, the cobined ebershi for y(n +)in the rule consequent in the jth rule is = 0j (y(n)) j (y(n 0 )) j (y(n 0 )) where ~y(n) =(y(n);y(n 0 ); ;y(n 0 )): We reresent y(n) =y(n 0 ) = = y(n 0 ) =0by~y(n) =0 and in such a case is exressed as j (0): To roduce cris odel outut, the generalized defuzzifier [3] is used and the odel outut is y(n +) = a ijy(n 0 i) + j (~y(j)) b kj u(n 0 k) where (0 <+) is a design araeter. Different defuzzification strategies can be realized by using different values for : The oular centroid defuzzifier and ean of axiu defuzzifier are just two secial cases when = and ; resectively [3]. III. ANALYTICAL STRUCTURE AND LOCAL STABILITY OF THE GENERAL TS FUZZY DYNAMIC SYSTEM MODELS In this section, we first reveal the analytical structure of the abovedefined general TS fuzzy dynaic syste odels and relate the resulting structure to ARX odel. We then establish a necessary and sufficient condition for analytically judging local stability of the fuzzy dynaic syste odels at the equilibriu oint (i.e., origin). Finally, we show how to use the condition to check and ossibly invalidate a TS fuzzy odel. Theore : The general TS fuzzy dynaic syste odels are nonlinear tie-varying ARX dynaic odels. Proof: We rewrite (2) as where y(n +) = = a ij y(n 0 i) + i (~y(n))y(n 0 i) + a ij i (~y(n)) = b kj b kj u(n 0 k) (2) ' k (~y(n))u(n 0 k) (3) and ' k (~y(n)) = : (4)

3 292 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO., FEBRUARY 999 The result of (3) can be exressed as y(n +)0 = i(~y(n))y(n 0 i) 'k(~y(n))u(n 0 k): (5) Recall that the linear tie-invariant ARX dynaic odel [7] is y(n +)+ ciy(n 0 i) = dku(n 0 k) +e(n +) (6) where ci and dk are constant araeters and e(n +)reresents rando error. Coaring (5) with (6), one sees that the general TS fuzzy dynaic syste odels are nonlinear tie-varying ARX odels without the e(n+) ter. The nonlinearity and tie-variation of the fuzzy odels are due to i(~y(n)) and 'k(~y(n)); whose values are deterined by s that change with ~y(n) and hence with tie. This theore shows that, in the context of traditional dynaic syste odeling, TS fuzzy odeling is indeed a rational and viable way to construct nonlinear tie-varying dynaic odels. A significant and unique advantage of the fuzzy odeling aroach is that both qualitative inforation (e.g., knowledge and exerience of syste exert/oerator) and quantitative inforation (e.g., easured nuerical data) can be utilized during odeling. Further, traditional odeling ainly focuses on linear tie-invariant dynaic systes whereas, as we show here, TS fuzzy odeling schee can handle nonlinear tie-varying dynaic systes. Therefore, TS fuzzy odeling aroach ay be ore desirable and effective when dealing with colex systes. Disclosure of the analytical structure of the general TS fuzzy dynaic syste odels akes it ossible to theoretically and recisely investigate various asects of the fuzzy syste odels. In resent work, we focused on stability of the fuzzy odels, which characterizes one of the ost iortant asects of hysical systes. There exist two tyes of syste stability: global stability and local stability, and one tye cannot relace the other as each has its distinctive advantages and disadvantages. Generally seaking, global stability conditions for nonlinear systes are, in ost cases, sufficient conditions, and necessary ones are uncoon. Excet for linear systes, it is rare that a global stability condition is a necessary and sufficient condition. The ost widely used ethodology for global stability deterination is the one develoed by Lyaunov, which requires a Lyaunov function to be found for the syste involved. Regardless of ethodologies, their foreost assution/requireent is that the colete and analytical exression of the syste is exlicitly available. This is iractical to the general TS fuzzy systes as their colete structures are usually not analytically derivable. A TS fuzzy syste odel is ade u of several interrelated nonlinear coonents: inut fuzzy sets, fuzzy rules, fuzzy logic AND oerators and a defuzzifier. As such, the structures of ost of the fuzzy systes are inherently colex and can be any nonlinear and tie-varying fors, aking analytical derivation of the colete structures for the whole inut sace virtually iossible. Aside fro the structure availability, even when the assution/requireent is et, roerly deterining global stability is still very difficult or likely iossible. Constructing Lyaunov functions is ore an art than science and heavily involves trial and error. Due to the structural and araetric colexity of the general fuzzy systes, finding a roer Lyaunov function for all the systes is ractically iossible. In view of these difficulties as well as nonlinear and tie-varying nature of the TS fuzzy syste odels and generality of their coonents, we decided to concentrate on local stability. Deterining local stability requires uch less inforation. For the general TS fuzzy systes, we only need to know: ) the structure of the fuzzy syste around the equilibriu oint (i.e., ~y(n) = 0); 2) the linearizability of the fuzzy syste at the equilibriu oint. Furtherore, the stability condition that we develoed is a necessary and sufficient one, aking it ractically useful. We now establish the local stability condition. Since stability is an inherent roerty of a syste, it is unrelated to syste inut. In other words, stability of fuzzy dynaic syste odels (3) is deterined by the following nonlinear tie-varying difference equation: y(n +)0 i(~y(n))y(n 0 i) =0: (7) If (7) is linearizable at ~y(n) = 0, then Lyaunov s linearization ethod [9] can be utilized to judge local stability of the resulting linear difference equation, which will rovide stability inforation about nonlinear systes (7) around the equilibriu oint. Local stability can ractically be deterined by the following sile necessary and sufficient condition. Theore 2: If nonlinear difference equation (7) for a TS fuzzy dynaic syste odel of the general class (3) is linearizable at the equilibriu oint, the fuzzy odel is locally stable at the equilibriu oint if and only if its corresonding linearized syste y(n +)0 i(0)y(n 0 i) =0 (8) is stable, where according to (4) j (0) aij i(0) = : (9) j (0) Proof: If nonlinear difference equation (7) of a TS fuzzy syste odel is linearizable at the equilibriu oint, that is, 0 i)) = unique 0 i) ~y(n)=0 for all i (0) then the linearized syste odel is y(n +)0 i(0)y(n 0 i) =0: Using Lyaunov s linearization ethod, the establishent of Theore 2 iediately follows. The easiest way to deterine stability of (8) is to use the z- transfor. That is, (8) is stable if and only if all the roots of the corresonding z-transfor equation z 0 i(0)z 0i =0 are inside the unit circle. Later in this aer, we will use an unstable TS fuzzy syste odel as an exale (see Exale ) to show how easily Theore 2 can be eloyed for the deterination of local stability. In addition to the local stability deterination, another use of Theore 2 is to qualitatively check quality of a TS fuzzy syste odel. If the hysical syste odeled is known to be stable at the equilibriu oint, the result of alying Theore 2 to the fuzzy syste odel should confir it. If the confiration occurs, the odel

4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY builder can be ore confident about the quality of the fuzzy syste odel, at least about its behaviors around the equilibriu oint. Otherwise, the fuzzy syste odel is incorrect and a new fuzzy syste odel needs to be established. This sile qualitative odel verification can ractically be iortant and useful for the TS fuzzy syste odeling technique as there reviously did not exist any analytical eans for checking and invalidating a TS fuzzy dynaic syste odel. At resent, the coon ractice on validation of a fuzzy syste odel is using couter siulation, which is not only tie-consuing but, ost iortantly, can lead to erroneous validation because fuzzy systes are nonlinear and tie-varying and no siulation can be corehensive enough to cover all ossible situations. Syste identification and controller design are two closely related issues in theory and ractice of conventional control and odeling [3]. Knowing the analytical structure of the general TS fuzzy dynaic syste odels enabled us to develo a design technique to systeatically design an outut tracking controller for the in achieving erfect tracking of any desired trajectory that is bounded and tie-varying. Without the derivation of the above analytical structure of the fuzzy dynaic odels, it is iossible to develo the design technique resented below. IV. SYSTEMATIC DESIGN OF OUTPUT TRACKING CONTROLLERS BASED ON FEEDBACK LINEARIZATION In this section, we develo a systeatic controller design technique for outut tracking control of the general TS fuzzy dynaic systes (3). We assue that () the fuzzy syste odel is a true reresentation of the hysical syste to be controlled; and (2) the fuzzy syste can either be stable or unstable. Our control objective is to ake outut of the general TS fuzzy dynaic systes achieve erfect tracking of any bounded tie-varying trajectories. We denote such a trajectory as r(n). Our another requireent is that the outut of the controller that we design ust always be bounded, including when n!: The objective of our controller design is to roduce such controller outut that y(n) = r(n) all the tie (i.e., erfect tracking). The rincile underlying our design ethod is feedback linearization, a well-established nonlinear controller design technique [4], [9]. The essence of this technique is using feedback to cancel internal nonlinearities of the syste to be controlled and aking the closed-loo control syste linear so that linear controller design techniques can be used. Note that at tie n; we know the values of r(n +);u(n 0 ); ; u(n 0 ); y(n); ; y(n 0 ); and we can calculate the values of 0 (~y(n)); ; (~y(n));' 0 (~y(n)); ;' (~y(n)): To be general, suosedly we do not know exlicit exressions of 0(~y(n)); ; (~y(n));'0(~y(n)); ;' (~y(n)) (indeed, one will not be able to obtain the in any cases). We assue that ' 0 (~y(n)) 6= 0 for any n; eaning outut of the fuzzy systes always deends uon inut of the systes. Using feedback linearization, we choose tracking controller for the general TS fuzzy dynaic systes (3) as follows: u(n) = ' 0(~y(n)) 0 i (~y(n))y(n 0 i) 0 ' k (~y(n))u(n 0 k) +r(n +) : () k= Substituting () into (5), we obtain the outut of the closed-loo fuzzy control systes y(n +)=r(n +); for any n which eans that we have achieved erfect tracking. The erfect tracking always starts fro the beginning of the control, i.e., fro tie n = 0. Practically, whether a controller so designed can achieve erfect tracking for the actual hysical syste reresented by the fuzzy odel used in the design deends on how accurate the odel is. The erfect tracking can be achieved if the odel accurately describes the hysical syste. Otherwise, the erfect tracking control erforance will not be guaranteed, owing to incolete cancellation of the syste nonlinearities. The extent of the erforance degradation relates to the degree of isatch between the fuzzy odel and the real syste. The issue here is about the robustness of the resulting hysical control syste. Obviously, this issue is not eculiar to the control of the fuzzy systes; rather it is a general, difficult and still oen issue to nonlinear syste control as whole. It has hardly been addressed [2]. A controller designed by our feedback linearization technique can always achieve erfect tracking, starting at tie n = 0, for any given desired trajectory. However, the controller outut ay or ay not be bounded, i.e., the controller is not guaranteed to be stable. A controller is ractically eaningless if its outut is not bounded, because such a controller cannot hysically be realized. We now study what deterines stability of the controller and under what conditions the controller is stable or unstable. For better resentation, we divide general fuzzy systes (3) into two grous: the general TS fuzzy systes with = 0 and the general TS fuzzy systes with, and study the controller stability accordingly. A. Controller Stability for the General TS Fuzzy Dynaic Systes with =0 According to (3), the general TS fuzzy dynaic systes with = 0 are described by y(n +)0 i (~y(n))y(n 0 i) ='0(~y(n))u(n): (2) This class of fuzzy dynaic systes is widely used in theory and ractice of fuzzy control and odeling. According to (), controllers designed using our ethod for these fuzzy systes are u(n) = '0(~y(n)) 0 i(~y(n))y(n 0 i) +r(n +) : (3) Because a desired trajectory r(n) is always bounded and y(n 0 i) = r(n 0 i) for i =0; ; ;;y(n 0 i) are bounded, too. Thus, u(n) is always bounded and the controllers are always stable. B. Controller Stability for the General TS Fuzzy Dynaic Systes with We now study the controller stability for the general TS fuzzy dynaic systes with in (3). For this grou of fuzzy systes, if desired trajectory constantly varies, the controller outut will, too. Because of the tie-varying and nonlinear nature of the fuzzy systes, it is difficult to analyze the controller stability if desired trajectory endlessly changes. A related iortant question is: if a desired trajectory does not change forever, say it is only a ste function, will the controller designed be guaranteed always stable? The answer, as we will show now, is no. Assue a desired trajectory has a final and fixed osition. Our tracking control task is to ake outut of the general TS fuzzy dynaic systes with follow a desired trajectory to reach a final and fixed osition within a finite eriod of tie. One exale of such tracking control is to ark a car while another one is to reach a still object by a robot ar. Without loss of generality, we

5 294 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO., FEBRUARY 999 assue that the desired trajectory varies with tie before tie N and becoes unchanged thereafter. Matheatically, a desired trajectory is described by a tie series: r(0);r(); ;r(n );r f ;r f ; ; where r f is the final fixed osition and r(n) =r f when n>n: Since a controller designed using our ethod always achieves erfect tracking, outut of the fuzzy systes is always r f after tie N: This eans that y(n) = = y(n 0 ) = r f when n>n+ : Additionally, when n>n+ ; ' k (~y(n)) and i(~y(n)) in () becoe constants because ~y(n) becoes constant ~r f (i.e., y(n) = = y(n 0 ) = r f ): We denote ' k (~r f ) and i(~rf ) as resective values of ' k (~y(n)) and i (~y(n)) when n>n + : Using all these facts, nonlinear tie-varying controller () becoes a linear tie-invariant controller when n>n + ' k (~r f )u (n 0 k) = 0 i(~rf ) r f ; k : (4) k= In order for the controller to be stable (i.e., u(n) is bounded), all the roots of the z-transfor equation of (4) ' k (~r f )z 0k =0 (5) k= ust be inside the unit circle. One sees that whether a controller is stable deends on ' k (~r f ); which are the araeter values of the fuzzy dynaic syste to be controlled when ~y(n) =~r f : The controller stability deends not only on the araeters of the fuzzy syste but also on the final fixed osition of the desired trajectory, r f : For the sae fuzzy syste, it is ossible that the controller is stable for one final osition but unstable for another one. We will show this in Exales 2 and 3 in next section. If a controller is stable, the controller outut corresonding to r f ; designated as u f ; can be couted by letting u(n 0 ) = = u(n 0 ) =u f in (4), which yields u f = 0 i (~r f ) ' k (~r f ) k= r f : (6) It can easily be roved that if the denoinator of (6) is relaced by ' k (~r f );u f couted will be for the general TS fuzzy systes with = 0, eaning (6) contains the steady-state controller outut for those fuzzy systes as a secial case. Controller outut will reach and stay at u f after tie N + for the fuzzy systes with = 0. For the fuzzy systes with, controller outut will reach and stay at u f after tie N + ; where >:According to (4), how large is deends on how stable the controller is, which is deterined by ' k (~r f ): The ore stable the controller, the saller the : Requiring all the roots of (5) be inside the unit circle is equivalent to requiring the general TS fuzzy dynaic systes be iniu hase systes when ~y(n) =~r f (note that the fuzzy systes becoe linear tie-invariant systes when ~y(n) = ~r f ): A discrete-tie syste that has oen-loo zeros outside the unit circle is a noniniu hase syste [9]. All the fuzzy systes with = 0 are always iniu hase systes, regardless of the desired trajectory. As such, controllers designed using our feedback linearization ethod are always stable. For any fuzzy syste with, it belongs to one of the following three situations: ) it is a iniu hase syste for any value of r f ; 2) it is a noniniu hase syste for any value of r f ; 3) it is a iniu hase syste for soe values of r f and is a noniniu hase syste for the reaining values. We suarize these controller stability results in the for of theore as follows. Fig.. Illustrative ebershi function definition of the six fuzzy sets used in Exales, 2, and 3. The atheatical definitions are given in (7) and (8) and the values of the araeters are listed in Table I. TABLE I VALUES OF THE PARAMETERS IN THE SIX MEMBERSHIP FUNCTIONS USED IN EXAMPLES, 2, AND 3. THE MATHEMATICAL DEFINITIONS ARE GIVEN IN (7) AND (8) Theore 3: Controller () designed for general TS fuzzy dynaic systes (3) with = 0 is always stable for any bounded tie-varying trajectory. Controller () designed for a fuzzy syste with is stable at a given value of r f if and only if the fuzzy syste is a iniu hase syste at that value. Since the designed fuzzy control syste always achieves erfect tracking, the controller can be regarded stable between tie 0 and N + : If the syste satisfies Theore 3, it is stable for the rest of the tie. Therefore, the fuzzy control syste is stable in a global sense, not in a local sense (i.e., around the origin only). For any given fuzzy syste with, before utilizing our design ethod, one should use (5) to check stability of the controller to be designed. If the controller is deterined to be stable, then design it. Otherwise, the desired erfect tracking is not achievable for the given fuzzy syste odel because the stability condition stated by Theore 3 is a necessary and sufficient one. V. NUMERICAL EXAMPLES We now deonstrate three exales that are related to each other to show how to use our new results and ethods. The first exale dislays how to use Theore 2 to analytically deterine local stability of a TS fuzzy dynaic syste odel. We urosely use an unstable fuzzy syste. Exale : Suose that we have identified a hysical syste using the TS fuzzy odeling technique. Assue the resulting TS

6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY fuzzy syste odel has the following eight TS fuzzy rules: R: IF y(n) is P0 AND y(n 0 ) is P AND y(n 0 2) is P2 THEN y(n +)=02y(n) 0 y(n 0 ) 0 3:y(n 0 2) + u(n) +0:5u(n0 ) R2: IF y(n) is P0 AND y(n 0 ) is P AND y(n 0 2) is N2 THEN y(n +)=04y(n) 0 8y(n 0 ) 0 :8y(n 0 2) +9u(n) +0:4u(n 0 ) R3: IF y(n) is P0 AND y(n 0 ) is N AND y(n 0 2) is P2 THEN y(n +)=07y(n) 0 2:3y(n 0 ) 0 :9y(n 0 2) 0 2u(n) +4:6u(n0 ) R4: IF y(n) is P0 AND y(n 0 ) is N AND y(n 0 2) is N2 THEN y(n +)=03y(n) 0 8:5y(n 0 ) 0 :y(n 0 2) + u(n) +0:5u(n0 ) R5: IF y(n) is N0 AND y(n 0 ) is P AND y(n 0 2) is P2 THEN y(n +)=07:5y(n) 0 2:6y(n 0 ) 0 2y(n 0 2) + 0:5u(n) +0:7u(n0 ) R6: IF y(n) is N0 AND y(n 0 ) is P AND y(n 0 2) is N2 THEN y(n +)=03y(n) 0 5:5y(n 0 ) 0 :2y(n 0 2) + :2u(n) 0 u(n 0 ) R7: IF y(n) is N0 AND y(n 0 ) is N AND y(n 0 2) is P2 THEN y(n +)=0:7y(n) 0 4:2y(n 0 ) 0 :8y(n 0 2) + 4:2u(n) +0:2u(n0) R8: IF y(n) is N0 AND y(n 0 ) is N AND y(n 0 2) is N2 THEN y(n +)=0:3y(n) 0 5:8y(n 0 ) 0 2:9y(n 0 2) 0 3:6u(n) +0:2u(n 0 ): Here, P i and N i (i = 0,, 2 and P stands for Positive whereas N stands for Negative ) are six fuzzy sets (Fig. illustrates the definitions). The ebershi functions of P i are described by P (y 3 )= 0; y 3 a P k P y 3 + d P ; a P <y 3 b P (7) ; y 3 >b P whereas the ebershi functions of N i are defined by N (y 3 )= ; y 3 a N k Ny 3 + d N; a N <y 3 b N (8) 0; y 3 >b N where y 3 is y(n);y(n 0 ) or y(n 0 2): The other araeters define the shae of the ebershi functions and their values are listed in Table I. Product AND fuzzy logic is used for all the AND s in the rules. Also, the oular centroid defuzzifier is used (i.e., = ). The question is: is this TS fuzzy dynaic syste odel stable around ~y(n) =0? Solution: According to Theore, this TS fuzzy dynaic syste odel is a nonlinear tie-varying ARX syste y(n +)0 2 i(~y(n))y(n 0 i) = ' k (~y(n))u(n 0 k): In order to obtain the corresonding linearized syste y(n +)0 0(0)y(n) 0 (0)y(n 0 ) 0 2(0)y(n 0 2) = '0(0)u(n)+'(0)u(n 0 ) we first need to deterine whether the nonlinear syste is linearizable at ~y(n) =0. For this secific syste, we can find it out without the exlicit exressions of i(~y(n)) and ' k (~y(n)): The nonlinear syste is linearizable because the conditions exressed in (0) hold. This is due to: ) all the ebershi functions for y(n);y(n 0 ); and y(n 0 2) are differentiable at ~y(n)) = 0; 2) yielded by roduct AND fuzzy logic are differentiable at ~y(n) =0. As a result, i(~y(n)) and ' k (~y(n)) are differentiable at ~y(n) =0. The values of i (0) and ' k (0) can easily be couted using (4), and the resulting linearized syste at ~y(n) =0 is y(n + ) + 3:4555y(n) + 4:462y(n 0 ) + :968y(n 0 2) =0:669u(n)+0:0459y(n 0 ): (9) The corresonding z-transfor equation is z 3 +3:4555z 2 +4:462z +:968 = 0: The three roots are z = and z = 0: :794i: The last two roots are outside the unit circle. Hence, the given TS fuzzy dynaic syste is unstable at ~y(n) =0. Fig. 2 shows the syste outut when a very sall initial value is given (y(0) = 0.000). The syste outut diverges with tie, clearly deonstrating instability of the syste and confiring our analytical result. In the second exale below, we exhibit how to use our feedback linearization design ethod to design a stable controller for the unstable TS fuzzy syste given in Exale and achieve erfect outut tracking erforance. Exale 2: Using the feedback linearization ethod resented in this aer, design a tracking controller for the TS fuzzy dynaic syste in Exale so that the outut of the fuzzy syste erfectly follows the following trajectory (Fig. 3): r(n) = Is the controller so designed stable? 0:8 sin(3n=00); 0 n 50 0:4; 5 n 00:

7 296 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO., FEBRUARY 999 Fig. 2. Siulated outut of the TS fuzzy dynaic syste given in Exale, confiring local instability of the fuzzy syste, which is deterined analytically by our necessary and sufficient stability condition. The initial syste outut is set Fig. 4. Outut of the outut tracking controller designed using our feedback linearization technique in Exale 2. The controller is stable, confiring the result of the analytical deterination. The steady-state outut of the controller is.846, the sae as the value couted using (6). syste when ~y(n) = r f = 0.4, for n>50: Thus, the tracking controller to be designed will be stable. The tracking controller is u(n) = '0(~y(n)) 0 2 i (~y(n))y(n 0 i) 0 '(~y(n))u(n 0 ) + r(n +) : According to (6), the steady-state outut of the designed controller at r f is u f = 0 2 i(~r f ) ' k (~r f ) r f : Fig. 3. Outut of the unstable TS fuzzy dynaic syste controlled by an outut tracking controller in Exale 2, which is designed using our feedback linearization technique. Sign reresents the desired outut trajectory whereas sign + reresents the fuzzy syste outut. The figure shows that erfect tracking is achieved. Note that the final fixed osition of the desired trajectory, r f ; is 0.4. Solution: Before designing the controller, we should use Theore 3 to deterine whether the controller to be designed will be stable. According to the given desired trajectory, the final fixed osition is: r f = 0.4. According to (5), the z-transfor equation for the controller stability deterination is whose root is 0 (~r f )+'(~r f )z 0 =0 z = 0 ' (~r f ) '0(~r f ) : It can be calculated easily fro the given TS fuzzy syste that '0(~r f ) = :464 and '(~r f ) = :2623; and hence the root is z = , indicating that the fuzzy syste is a iniu hase Fro the given fuzzy syste, we coute the values of i (~r f ) and ' k (~r f ) as: 0(~r f ) =03:985;(~r f ) =05:5249;2(~r f ) = 02:0732;'0(~r f ) = :464; and '(~r f ) = Consequently, u f =.846. Fig. 3 dislays the syste outut along with the desired trajectory. The trajectory is always erfectly tracked. The corresonding controller outut is exhibited in Fig. 4. The controller is stable and indeed the steady-state outut is.846, as exected. In the last exale, we show that the controller designed in Exale 2 becoes unstable for the sae fuzzy syste at a different value of r f : Exale 3: In Exale 2, if the final fixed osition of the desired trajectory is 0.7 instead of 0.4, for 5 n 00, will the designed controller still be stable? Solution: Now r f = 0.7. One can calculate that '0(~r f ) =.208 and '(~r f ) =.4867, and hence the root is z = (outside the unit circle). This eans that the fuzzy syste becoes a noniniu hase syste when ~y(n) =~r f = 0.7 and consequently the designed controller becoes unstable for the new final osition of the trajectory. Although the erfect tracking is still achieved, as shown in Fig. 5, the controller outut grows without bound and the controller is unusable (Fig. 6), as redicted.

8 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY that whether the designed controller is stable deends on the fuzzy syste to be controlled as well as the desired trajectory. We have derived a sile and ractical necessary and sufficient condition, in relation to stability of noniniu hase systes, for analytically deterining the controller stability. We have given three concrete nuerical exales to deonstrate racticality and utility of our new results. The results obtained in this aer cover a very general class of TS fuzzy dynaic systes, which are actually, as we have shown, nonlinear and tie-varying ARX systes. Our results are not only unique and hence valuable to fuzzy systes but also useful to the conventional studies of outut tracking control of nonlinear tievarying systes. REFERENCES Fig. 5. Outut of the unstable TS fuzzy dynaic syste controlled by an outut tracking controller in Exale 3, which is designed in Exale 2 using our feedback linearization technique. Sign reresents the desired outut trajectory whereas sign + reresents the syste outut. The figure shows that erfect tracking is achieved. Note that the final fixed osition of the desired trajectory, r f ; is 0.7, instead of 0.4 shown in Fig. 3 for Exale 2. Fig. 6. Outut of the outut tracking controller in Exale 3. Because of the change of the final osition of the desired trajectory fro 0.4 in Exale 2 to 0.7 in Exale 3, the controller becoes unstable, as redicted by using (5). VI. CONCLUSION We have roved that a general class of TS fuzzy dynaic systes is nonlinear tie-varying ARX systes and have established a sile necessary and sufficient condition for analytically deterining local stability of the fuzzy systes. The condition can also be used to check quality of a TS fuzzy odel against the hysical syste odeled and invalidate the odel if the conditions warrant. Based on the revealed structure of the fuzzy systes, we have develoed a feedback linearization ethod for systeatically designing a controller to control any given TS fuzzy syste of the general class, stable or not, so that erfect outut tracking is obtained. Our design ethod always roduces stable controllers for a large ortion of the general TS fuzzy systes that are coonly encountered [those with = 0 in (3)], regardless of desired trajectories as long as they are bounded. For the reaining TS fuzzy systes [those with in (3)], we have roved [] G. Chen, T. T. Pha, and J. J. Weiss, Fuzzy odeling of control systes, IEEE Trans. Aeros. Electron. Syst., vol. 3, , 995. [2] B. Friedland, Advanced Control Syste Design. Englewood Cliffs, NJ: Prentice-Hall, 996. [3] D. P. Filev and R. R. Yager, A generalized defuzzification ethod via BAD distributions, Int. J. Intell. Syst., vol. 6, , 99. [4] J. W. Grizzle and P. V. Kokotovic, Feedback linearization of saleddata systes, IEEE Trans. Autoat. Contr., vol. 33, , 988. [5] D. A. Linkens and H. O. Nyongesa, Learning systes in intelligent control: An araisal of fuzzy, neural and genetic algorith control alications, Proc. Inst. Elect. Eng., vol. 43, , 996. [6] K. Liu and F. L. Lewis, Adative tuning of fuzzy logic identifier for unknown nonlinear systes, Int. J. Adat. Contr. Signal Process., vol. 8, , 994. [7] L. Ljung, Syste identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 987. [8] J. Sjoberg, Q. Zhang, and L. Ljung, Nonlinear black-box odeling in syste identification: A unified overview, Autoatica, vol. 3, , 995. [9] J.-J. E. Slotine and W. Li, Alied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 99. [0] T. Takagi and M. Sugeno, Fuzzy identification of systes and its alications to odeling and control, IEEE Trans. Syst., Man Cybern., vol. SMC-5,. 6 32, 985. [] K. Tanaka and M. Sano, A robust stabilization roble of fuzzy controller systes and its alications to backing u control of a trucktrailer, IEEE Trans. Fuzzy Syst., vol. 2,. 9 34, 994. [2] K. Tanaka, T. Ikeda, and H. O. Wang, Robust stabilization of a class of uncertain nonlinear systes via fuzzy control: Quadratic stabilizability, H control theory, and linear atrix inequalities, IEEE Trans. Fuzzy Syst., vol. 4,. 3, 996. [3] P. M. Van Den Hof and R. J. P. Schraa, Identification and control Closed-loo issues, Autoatica, vol. 3, , 995. [4] H. O. Wang, K. Tanaka, and M. F. Griffin, An aroach to fuzzy control of nonlinear systes: Stability and design issue, IEEE Trans. Fuzzy Syst., vol. 4,. 4 23, 996. [5] L. Wang and R. Langari, Building Sugeno-tye odels using fuzzy discretization and orthogonal araeter estiation techniques, IEEE Trans. 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9 298 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO., FEBRUARY 999 [23], Sufficient conditions on general fuzzy systes as function aroxiators, Autoatica, vol. 30, , 994. [24], The Takagi Sugeno fuzzy controllers using the silified control rules are nonlinear variable gain controllers, Autoatica, vol. 34, , 998. [25], Constructing nonlinear variable gain controllers via the Takagi Sugeno fuzzy control, IEEE Trans. Fuzzy Syst., vol. 6, , 998. A Color Texture Based Visual Monitoring Syste For Autoated Surveillance George Paschos and Kion P. Valavanis Abstract This aer describes a visual onitoring syste that erfors scene segentation based on color and texture inforation. Color inforation is cobined with texture and corresonding segentation algoriths are develoed to detect and easure changes (loss/gain) in a given scene or environent over a eriod of tie. The xyy color sace is used to reresent the color inforation. The two chroaticity coordinates (x; y) are cobined into one, thus, roviding the chroinance (sectral) art of the iage, while Y describes the luinance (intensity) inforation. The roosed color texture segentation syste rocesses luinance and chroinance searately. Luinance is rocessed in three stages: filtering, soothing, and boundary detection. Chroinance is rocessed in two stages: histogra ulti-thresholding, and region growing. Two or ore iages ay be cobined at the end in order to detect scene changes, using logical ixel oerators. As a case study, the ethodology is used to deterine wetlands loss/gain. For coarison uroses, results in both the xyy and HIS color saces are resented. I. INTRODUCTION Texture has been widely acceted as a feature of riary iortance in iage rocessing and couter vision since it rovides unique inforation about the hysical characteristics of surfaces, objects, and scenes [], [2]. An iage ay reresent a secific textural attern, while in other cases, an iage ay be coosed of two or ore textural atterns. In the first case, the roble encountered is that of classification, since a single texture has to be recognized. In the second case, one has to searate the different textures fro each other within a single iage, thus, erforing an iage segentation task. There has been considerable research in the area of texture analysis (i.e., descrition, segentation, classification) [3] [5]. However, ost of the work has focused on ethods using gray-level iages, where only the luinance (intensity) coonent of the iage signal is utilized. Only liited work has been reorted in the literature related to the use of color in texture analysis [6], [7]. In order to incororate the chroatic inforation into texture analysis, assuing that the RGB color sace is used, the following choices exist. ) Each color band (i.e., R; G; B) is rocessed searately. 2) Inforation across different bands (e.g., cross-correlations RG, RB, GB) is extracted. 3) Both individual color band and cross-band inforation is used. Manuscrit received Seteber 9, 996; revised May 0, 998. G. Paschos is with the Couter Science Division, Florida Meorial College, Miai, FL USA. K. P. Valavanis is with the Robotics and Autoation Laboratory, A-CIM Center, University of Southwestern Louisiana, Lafayette, LA USA. Publisher Ite Identifier S (99) ) A coosite easure to describe the chroatic inforation is used. Methods based on one of the first three choices have been recently reorted [6], [7]. The fourth alternative is exlored in this research using the xyy color sace [8]. The roosed Color Texture Analysis Syste is shown in Fig.. The ain goal of the syste is to searate a given iage into two arts, naely, a Region of Interest (ROI), and the rest of the iage (i.e., the background). A ROI is tyically an area of the iage that reresents soething eaningful in the corresonding real-world scene. For exale, an aerial iage ay cature a iece of land surrounded by water. The land, in this case, is the ROI, and the surrounding water is the background. The syste erfors analysis on luinance and chroinance in arallel, and, at the final stage, results are cobined to detect changes (i.e., loss/gain) in a secific area of the iage (ROI). Processing starts by transforing a given iage fro RGB to xyy (Fig. ). This roduces the luinance coonent (Y ) directly, whereas the two chroaticity values (x; y) are cobined to rovide for a single-valued chroinance. Textural inforation, such as sizes and orientations of basic iage features (e.g., edges, blobs), is contained in the luinance coonent. Thus, a set of filters tuned to different sizes and orientations is alied on luinance and roduces a corresonding set of filtered iages. Soothing of the filtered iages follows, thus, eliinating surious/negligible regions. The soothed iages are cobined into a single iage, based on a neighborhood ixel siilarity easure, and boundaries of otential ROI s are extracted using a ercetron-tye rocessing echanis. The result of luinance rocessing is, thus, a Boundary Iage. Croinance rocessing roceeds in two stages. First, the chroinance histogra is couted and ultile thresholds are identified. Secondly, these thresholds are used to segent the chroinance iage into a corresonding nuber of regions (i.e., otential ROI s). Thus, the result of chroinance rocessing is a Region Iage. Using a region exansion algorith, the Boundary and Region Iages are cobined to locate the desired Region of Interest (e.g., wetland area). The result is a ROI Iage showing the identified ROI. The final stage involves the coarison of two or ore ROI iages to locate ossible scene changes. Tyically, two or ore iages of the sae real-world scene are taken at different ties. Each of these iages will result in a corresonding ROI Iage, after going through the various segentation stages (i.e., luinance and chroinance rocessing). Change detection and easureent is erfored by coaring two such ROI Iages using logical ixel oerators. The end result of this research is threefold: ) incororation of texture and color attributes for scene analysis; 2) develoent of coutationally efficient and easily ileentable algoriths for the analysis of color textures; 3) develoent of aroriate neural network architectures for iage segentation and classification. One of the ain alications of the roosed syste is in the onitoring of wetlands. Such environents exerience changes over tie (i.e., artial loss/gain of wetland area). The develoent of autonoous surveillance systes caable of collecting data over a eriod of tie and analyzing the using a variety of visual roerties in order to identify such changes is, thus, iortant. The ethodology resented in this aer rovides the analysis coonent of such an autonoous syste. It incororates color and texture visual attributes into a unified fraework and utilizes the to detect and easure loss/gain /99$ IEEE