AN ALTERNATIVE DECOUPLED SINGLE-INPUT FUZZY SLIDING MODE CONTROL WITH APPLICATIONS

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1 AN ALTERNATIVE DECOUPLED SINGLE-INPUT FUZZY SLIDING MODE CONTROL WITH APPLICATIONS Fang-Ming Y, Hng-Yan Chng* and Chen-Ning Hang Department of Electrical Engineering National Central University, Chngli, Taiwan 0, R.O.C. sliding srface to obtain a fy sliding mode controller described in the previos works. The advantage of sch controllers is that the nmber of rles reqired is redced from mn to m in Hwang [], [7] and Bartolini [8] or nm in Kng [9]. In these stdies, we can find that the FSMC is one of the redcing fy rles methods. Frthermore, in general, since the FSMC combines fy control and sliding mode control principles, the closed-loop system has better performance, with robstness and stability, than that sing only one control theory. Moreover, another objective of designing fy controllers is to apply them to higher order systems. The large majority of fy controllers are limited to systems with predominantly second-order dynamics. As proven in [0], [], and [], conventional fy controllers which se the system error and its derivative in the fy rle base are a type of PID control. The action of sch fy controllers are eqivalent to that of fll-state feedback controllers for second-order systems and, hence, these systems can always be stabilied. However, for a forthorder system, sch as the cart-pole system, sing a conventional fy controller will reslt in a large nmber of rles (m. For these systems, the intitive sense is that some rles may not be activated if a stabiliing rle base is determined. In most stdies, the fy controller of second-order systems is designed on a phase plane bilt by error e and change of error e& that are prodced from the states and. For eample, in a cart-pole system only the pole sbsystem is considered ignoring the cart sbsystem and it is ths impossible to achieve a good control arond the set point (distance=0. In this stdy, a decopled single-inpt fy sliding mode controller design is proposed. This controller garantees some properties, sch as the robst performance and stability properties. In addition, a class of forth-order nonlinear systems is investigated. Lo and Ko [] proposed a method called decopled fy slidingmode control to cope with the above isse. Thogh Chen, Y, and Chng proposed another kind of method called Decopled Fy Controller Design with Single-Inpt Fy Logic, in which convergence is faster than that of []. It can t show ot the effectiveness and feasibility of the practical system. Herein, a new approach, with the better evoltion performance than [5] and [] as well as ABSTRACT A decopled single-inpt fy sliding mode control design is proposed. By sing this approach, the order of the rle base is redced greatly. In addition, compared with previos reslts, five fy rles are given to control a class of forth-order nonlinear systems. Using this method, the system can achieve asymptotic stability and the evoltion of system will be better as well as converge faster than that of previos reports. One simlation stdies of a cart-pole system is presented to demonstrate the effectiveness of the method. Frthermore, by sing this approach, eperimental reslts are given to assre the robstness and stability of system. KEY WORDS Fy logic control, decopled, seesaw system. INTRODUCTION It is well known that fy logic controllers (FLC s have been proven to be a powerfl tool in controlling nonlinear systems since the work of Mamdani [] was proposed in 97. The fy control algorithm can be regarded as model-free control algorithms in contrast to a conventional feedback control algorithm. However, some isses still eist in the control of comple systems via fy logic controllers, for eample, No general stability analysis tools can be applied to FLC s. The large amont of fy rles for a high-order system make the analysis comple. The design parameters of membership fnctions affect the performance of the fy system and sitable membership fnctions may be obtained throgh a considerable time-consming and a trial-and-error procedre. To overcome these formidable tasks, many researchers spend mch effort to investigate the above three problems. Nevertheless, the main problem with these approaches is that the nmber of rles sed in the rle base remains qite large. In recent years, the concept of the fy sliding mode control (FSMC [], [], [] has been reported. To decrease the nmber of rles in the rle base, several athors have sggested sing a composite state, called a * Corresponding athor

2 with practical system evoltion, is proposed. That is another kind of decopled and single-inpt fy sliding mode controller is sed to replace the fy controller of [5], [6] and []. This approach possesses a sole fy inpt variable of order m as given in Table and Fig., rather than the controllers which are of order m on the phase plane constitted by e and e& or higher order for single-inpt systems, where m is the nmber of fification levels. Frthermore, the Lyapnov fnction is employed to get the fy control otpt, and then to assre the system asymptotic stability. Compared with [5] and [], five fy control rles are sed as shown in Table. If one needs a finer control, one can easily add or modify the rles by means of the Table and Fig.. Using this approach, the system can achieve asymptotic stability and it will converge faster than that of [] and the system evoltion performance is better than that of [5]. Meanwhile, the eperimental reslts are given to show the effectiveness and feasibility of the approach. The rest of the paper is divided into five sections. In Section, the systems are described. In Section, the single-inpt fy sliding mode control is presented. In Section, the detail design of the decopled fy logic controller is proposed. In Section 5, the proposed controller is sed to control a cart-pole system. In Section 6, the eperimental reslts are given to show the system robstness and stability by sing the approach. Finally, we conclde with Section 7.. SYSTEM DESCRIPTION Consider a second-order nonlinear system as the following state-space model in a canonical form: ( = = f ( + b( + d( ( y( = where T = [, ] is the state vector, ( f and b ( are nonlinear fnctions, is the control inpt, and d ( is eternal distrbance. The distrbance is assmed to be bonded as d D(. For this kind of the second order system, many kinds of control methods, sch as, fy control, PID control, hierarchical fy control and sliding mode control etc are tilied. A control law can be easily designed to make the second order system ( arrive at or control goal. However, for nonlinear models sch as a cart-pole system, the system dynamic representation is generally not in a canonical form eactly. Rather, it has a form shown below: ( = = f( + b ( + d( ( ( = = f ( + b ( + d where f, T = [,,, ] is the state vector, ( b (, b ( are nonlinear fnctions, d ( t, f ( and, are the control inpts, and d are eternal distrbances. The distrbances are assmed to be bonded as d D, d D. From (, one can design and respectively, however, this approach is only tilied to control a sbsystem in (. For eample, if the model is a cart-pole system, we only control either the pole or the cart of a system sch as (. Hence, the idea of decopling is employed to design a controller otpt to govern the whole system.. SINGLE-INPUT FUZZY SLIDING MODE CONTROL In this section, the idea of single-inpt fy sliding mode control (SIFSMC approach will be demonstrated. For the second order system (, a switching line, fnction of s, is chosen as s = c + ( By taking the time derivative of both sides of (, we can obtain s & = c + = c + f ( + b( + d ( Then, mltiplying both sides of the above eqation by s gives s s& = sc + sf ( + sb( + sd (5 Here, we assme that b ( > 0. In (, it is seen that s& increases as increases and vice versa. Eqations (5 and ( provides the information that if s > 0, the decreasing will make s& decrease as to lead s& s decrease, and that if s < 0, the increasing will make make s& increase as to lead s& s decrease. Now, we choose a Lyapnov fnction: V = s (6 Then V & = ss& (7 Hence, it is seen that if s > 0, decreasing will make s& decrease as to case s& s to decrease so that V & < 0 and that if s < 0, increasing will make s& increase as to case s& s to decrease so that V & < 0. So we can ensre that the system is asymptotically stable. From the above relation, we can conclde that: s (8 Hence, the fy rle table can be established on a onedimensional space of s as shown in Table instead of a two-dimensional space of and. The control action can be determined by s only. Meanwhile, we can easily add or modify rles for fine control. For implementation, a trianglar type membership fnction is chosen for the aforementioned fy variables, as shown in Fig.. 8

3 Table. Rle table for the SIFSMC. s NB NM ZE PM PB PB PM ZE NM NB N B NM ZE PM PB - Fig.. Fy variable of trianglar type Remark : The rle table can be established on a onedimensional space like Table. It is seen that the total nmber of rles is greatly redced compared with conventional FLCs, which depends on a two-dimensional space of and for second order system. In other words, a traditional fy system has for states to be controlled and the range of each state variable is divided into five fy sets. The nmber of rles forming the knowledge database is 5 = 65. With the present method, only one variable s needs to be fified and only five rles are necessary to establish the knowledge database. The single-inpt fy sliding mode control (SIFSMC is shown in Fig. where s and are the inpt and otpt of the single-inpt fy sliding mode control, respectively. The inpt of the proposed fy controller is ~ s, which is a fified variable of s. The otpt of the fy controller is U, which is the fified variable of. All the niverses of discorse of ~ s and U are arranged from to. Ths, the range of nonfy variables s and mst be scaled to fit the niverse of discorse of fified variable ~ s and U with scaling factors K and K, respectively, namely, define one switching line as s = c ( + ( and another switching line as s = c + ( The control objective is to drive the system states to the original eqilibrim point. The switching line variables s and s are redced to eros gradally at the same time by an intermediate variable. In eqation (, is a vale transferred from s, it has a vale proportional to s and has the range proper to. Eqation ( denotes that the control objective of is changed from = 0, = 0 to =, = 0. Becase the controller = is sed to govern the whole system, the bond of can be garanteed by letting Z, 0 < Z < ( where Z is the pper bond of abs (. Eqation ( implies that the maimm absolte vale of will be limited. Smmariing what we have mentioned above, can be defined as tanh ( s / Φ Z, 0 < Z < ( = where Φ is the bondary layer of s to smooth, Φ transfers s to the proper range of. Notice that is a decaying oscillation signal becase Z is a factor less than one. Remark. Consider eqation (. If s = 0, then =, = 0. Since is a vale transferred from s, when s 0, then 0 and 0. From eqation (, if the condition s 0, the control objective can be achieved. The two-level decopled SIFSMC is proposed and is shown in Fig.. ~ s = K s( (9 t K U (0 = s SMC Z s( K ~ s Fy Control U Rles K ( _ s SMC SIFSMC Plant Fig.. The block diagram of the SIFSMC.. DESIGN OF DECOUPLED FUZZY LOGIC CONTROLLER In this section, the idea of the single-inpt fy sliding mode control is sed in Section. In Eq. (, we first Fig.. The block diagram of the decopled SIFSMC. 5. COMPUTER SIMULATIONS In this section, we shall demonstrate that the decopled SIFSMC is applicable to the cart-pole system [5, ] and the practical seesaw system [] to verify the theoretical 9

4 development. Inverted Pendlm The strctre of an inverted pendlm is illstrated in Fig. and its dynamic is described below: = mt g sin mpl sin cos + cos = + d L ( mt mp cos = (5 mpl sin + mpg sin cos = mt mp cos + + d ( mt mp cos where = θ the angle of the pole with respect to the vertical ais; = & θ the angle velocity of the pole with respect to the vertical ais; = the position of the cart; = the velocity of the cart; m = m + m t c p method and Ref. [5]. It is fond that the pole and the cart can be stabilied to the eqilibrim point. Nevertheless, in these Figres, we can find that or proposed method have better performances than that of [5] and []. Under the same initial conditions, or methods have smaller angle variation of the pole, smaller position variation of the car and smaller control otpt variation. Fig. 5. Angle evoltion of the pole. θ Fig. 6. Position evoltion of the cart. Fig.. Strctre of an inverted pendlm system. In what follows, we define the following variables: s = c ( θ + & θ = c ( + (6 c + = c s = & + (7 and = tanh ( s / Φ Z, 0 < Z < (8 In the simlation, the following specifications are sed: m = 0.05kg, m = kg, L = 0.5m, g = 9.8m / s, c = 5, c p = 0.5, Φ c = 5, Z = 0.95, d 0.087, K =, K = 0 and initial vales are o θ = 60, & θ = 0, = 0, = 0 Fig. 5 7 shows the simlation reslt of or proposed Fig. 7. Control otpt of Simlation. 0

5 6. SEESAW SYSTEM EXPERIMENTAL RESULTS θ The angle of inclination of the wedge ϕ The angle that the cart makes with the vertical line of the wedge A. System Model Physical Concept According to the basic physical concept, if the nstable system of seesaw mechanism, the vertical line along the center of gravity of the inverted wedge is not passing throgh the flcrm perpendiclarly, the inverted wedge will reslt in a torqe and rotating ntil reaching the stable state. If we want to balance the inverted wedge, we have to pt an eternal force to prodce an appropriate opposite torqe. For this reason, we eqip the inverted wedge with a cart to balance the nstable system. The eqipment can move the cart to prodce the appropriate torqe against the internal force. Dynamical Eqation The balancing mechanism of the seesaw is shown in Fig. 8 []. From the principle of the balance of force and the balance of torqe, the dynamical eqation can be represented as + mg sin θ B = m &, ( Mg sin θ r + mg ( θ + φ ( + r + r µθ& = I & θ sin (9 where I is the inertia of the wedge which is given below. a b I = ρ abc + (0 The parameters of the dynamical eqation are denoted as in Table. r r M Fig. 8. The balancing mechanism of the inverted wedge. Table. The involved parameters of the dynamical eqation. m The mass of cart r The height of wedge M The mass of wedge r The height of center of mass of wedge I The inertia of wedge µ The damping coefficient Distance of the cart from the origin θ µ B φ B m of the angle The damping coefficient of the cart In what follows, we define = and = c + θ = c ( c + = c = θ, = & θ, =, s ( θ & + ( s = & + ( with = tanh ( s / Φ Z, 0 < Z <. ( In the eperiment, the following specifications are sed: c = 5, c = 0.5, Φ = 5, Z = 0.95, d 0.087, K =, K = 0. Hardware Description The seesaw consists of a DC Servo Motor, a potentiometer for measring the seesaw angle, a potentiometer for measring the cart position, an inverted wedge made of plastic board, a high performance data acqisition card, PCL-88H, and a -bit personal compter (Pentim- as a pc-based controller. The Fig. 9 is the practical hardware strctre of the seesaw system. Fig. 9. The practical hardware strctre of the seesaw system. Software Description In this stdy, we se the software, Borland C++ Bilder (BCB, to design all controllers and to apply the hardware. The reasons why we se this software are given below:. It offers the timer component. The timer component can offer the fi time. Then, we can set the trigger time in the millisecond by conting the CPU time and the sampling vale in the fied sampling time.. All designs are nder the windows, since the windows operation has become a trend in ftre application. Using the windows operation is helpfl for an engineer and for those who are not an engineer to operate.. We can revise the parameters of the controller, and control the hardware real time. We can se the image component or other component to draw the response of the controller in real time. Under the windows, it is easily seen whether the design is satisfactory or not.

6 B. Reslts Based on what we want to balance the angle of inverted wedge and the position of cart, we can give the proper commands for balancing control in different sitations, and ths constrct the controller of the system. The eperimental reslts for decopled SIFSMC are shown in Fig. 0, and with the eternal distrbance shown in Fig. In the reslts of eperiment, it is fond that once the states reach the eqilibrim point, the system is staying in robst condition even nder the circmstance of adding eternal distrbance. Figs. 0- show that θ and converge to ero with or withot eternal distrbance, respectively. In a word, the present method of decopled SIFSMC possesses the advantages of rle redction, robstness and chattering elimination, as well as withot the eact mathematical model. Fig. 0. The evoltion of the practical seesaw system for decopled SIFSMC: (a position, (b angle, and (c control otpt. (a Remark : For the practical seesaw system, de to the fact that the initial states is hard to set the same. Hence, the reasonable initial angle and the position taken are as shown in Figs 0-. (b (a (c Fig.. The position evoltion of the practical seesaw system with distrbance for decopled SIFSMC: (a position, (b angle, and (c control otpt. 7. CONCLUSIONS (b (c The present approach was sed to replace the fy controller of Lo s approach. Compared with previos reslts, only five fy control rles are needed as given in Table. For a finer control, we can easily add or modify rles by way of the rle table for a single-inpt fy logic control of variable s. Net, simlation reslt show that the pole and the cart can be stabilied to the eqilibrim, and the performance is better than that of [5] and []. Also, the eperimental reslts of seesaw system show the performances of robstness even nder the circmstance of reasonable eternal distrbance. Althogh the decopled SIFSMC can be etended to an n-th order system, the decopled method only satisfies the forth-order system

7 like the cart-pole system. Hence, the same weak point of the approach is that when applied to systems higher than forth order sch as a doble-inverted pendlm system (which needs a three-level-control, if the cart is to be stabilied at the origin as in [], then the method described here will fail. If there eists a new SMC method, which can deal with a higher order system larger than the forth-order and the higher order system being not a canonical form, then the decopled SIFSMC may be combined with each other. This may be an important and interesting topic in ftre. 8. ACKNOWLEDGEMENT The athors wish to thank the financial spport of the National Science Concil of the Repblic of Taiwan nder Contract NSC-9--E [0] C. L. Chen, P. C. Chen, and C. K. Chen, Analysis and design of fy control system, Fy Sets Syst. vol. 57, no., pp. 5-0, Jly 99. [] K. Li and F. L. Lewis, Some isses abot fy control, in proc. IEEE Conf. Decision Contr., San Antonio, TX, Dec. 99, vol., pp [] S. Galicher and L. Folloy, Fy controllers: Synthesis eqivalence, IEEE Trans. Fy Syst., vol., pp. 0-8, May 995. [] M. L. Lee, H. Y. Chng, Hierarchical Fy Control with Applications to Seesaw Systems, M. S. thesis, Department of Electrical Engineering, National Central University, 000. [] Ji-Chang Lo, Ya-Hi Ko, Decopled Fy Sliding-Mode Control, IEEE Transactions on Fy Systems, vol. 6. No.. Agst 998, pp REFERENCES [] E.H. Mamdani, Applications of fy algorithms for simple dynamic plants, Proc. IEE 97, pp [] Gang-Chyan Hwang, Shih-Chang Lin, A stability approach to fy control design for nonlinear systems, Fy Sets and Systems vol. 8 99, pp [] Sng-Woo Kim, J-Jang Lee, Design of a fy controller with fy sliding srface, Fy Sets and Systems vol.7 995, pp [] Jacob S. Glower, Jeffrey Mnighan, Design Fy Controllers from a Variable Strctres Standpoint, IEEE Transactions on Fy Systems, vol. 5. No.. Febrary 997, pp. 8-. [5] S. Y. Chen, F. M. Y and H. Y. Chng, Decopled Fy Controller Design with Single-Inpt Fy Logic, Fy Sets and Systems vol. 9, 00, pp. 5-. [6] Byng-Jae Choi, Seong-Woo Kwak, Byng Kook Kim, Design of a single-inpt fy logic controller and its properties, Fy Sets and Systems vol , pp [7] Y. R. Hwang and M. Tomika, Fy smoothing algorithms for variable strctre systems, IEEE Trans. Fy Syst., vol., pp. 77-8, Nov. 99. [8] G. Bartolini and A. Ferrara, Mltivariable fy sliding mode control by sing a simple of control vectors, in Fy reasoning in Information, Decision, and Control Systems, S. G. Tafestas and A. N. Venetsanopolos, Eds. Amsterdam, The Netherlands: Klwer, 99, pp [9] C. C. Kng and S. C. Lin, Fy controller design: A sliding mode approach, in Fy Reasoning in Information, Decision, and Control Systems, S. G. Tafestas and A. N. Venetsanopolos, Eds. Amsterdam, The Netherlands: Klwer, 99, pp