Decoupled fuzzy controller design with single-input fuzzy logic

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1 Fuzzy Sets and Systems 9 (00) Decoupled fuzzy controller design with single-input fuzzy logic Shi-Yuan Chen, Fang-Ming Yu, Hung-Yuan Chung Department of Electrical Engineering, National Central University, Chungli, Taiwan 30, ROC Received 7 April 000; received in revised form June 00; accepted 9 June 00 Abstract A decoupled fuzzy controller design with single-input fuzzy logic is proposed. We will utilize the approach of the single-input fuzzy logic to explore the fourth-order nonlinear systems. In addition, ve fuzzy rules are given to control a class offourth-order nonlinear systems. Using this approach, the system can achieve asymptotic stability and the response ofsystem will converge faster than that ofprevious reports. Two simulation studies ofa cart pole system and a ball beam system are presented to demonstrate the eectiveness ofthe method. c 00 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy control; Sliding mode control; Signed distance. Introduction Fuzzy logic controllers (FLCs) have been proven to be a powerful tool since the work of Mamdani [] was proposed in 974. The fuzzy control algorithm can be regarded as model-free control algorithms in contrast to a conventional feedback control algorithm. However, some issues still exist in the control of complex systems using fuzzy logic controllers, for example, () No general stability analysis tools can be applied to FLCs. () The large amount offuzzy rules for a high-order system make the analysis complex. (3) The design parameters ofmembership functions aect the performance of the fuzzy system and suitable membership functions may be obtained via a Corresponding author. Tel.: ; fax: address: hychung@ee.ncu.edu.tw (H.-Y. Chung). considerable time-consuming and a trial-and-error procedure. Hence, many researchers spend much eort to investigate the above three problems to overcome these formidable tasks. In recent years, the concept ofthe fuzzy sliding mode control (FSMC) [6,8,5] has been reported. To decrease the number ofrules in the rule base, several authors have suggested using a composite state, called a sliding surface to obtain a fuzzy sliding mode controller described in the previous works. The advantage ofsuch controllers is that the number ofrules required is reduced from m n to m in Hwang [6], [7] and Bartolini [] or nm in Kung [9]. So, the FSMC is one ofthe reducing fuzzy rules methods. In general, since the FSMC combines fuzzy control and sliding mode control principles, the closed-loop system has better performance than that using only one control theory. Moreover, another problem ofdesigning fuzzy controllers is applying them to higher order systems /0/$ - see front matter c 00 Elsevier Science B.V. All rights reserved. PII: S (0)0030-0

2 336 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) Table Rule table for a signed distance fuzzy logic control d s NB NM ZE PM PB u PB PM ZE NM NB The large majority offuzzy controllers are limited to systems with predominantly second-order dynamics. As proven in [,0,4], conventional fuzzy controllers which use the system error and its derivative in the fuzzy rule base are a type of PID control. The action ofsuch fuzzy controllers are equivalent to that of full-state feedback controllers for second-order systems and, hence, these systems can always be stabilized. However, for a fourth-order system, such as the ball-and-beam system, the system may not be stabilized by using a PID controller and, therefore, using a conventional fuzzy controller will result in a large number ofrules (m 4 ). For these systems, the intuitive sense is that some rules may not be activated ifa stabilizing rule base is determined. In most studies, the fuzzy controller of second-order systems is designed on a phase plane built by error e and change oferror ė that are produced from the states x and ẋ. For example, in a cart pole system only the pole subsystem is considered ignoring the cart subsystem and it is thus impossible to achieve a good control around the set point (distance = 0). In this study, a decoupled fuzzy controller design with single-input fuzzy logic is proposed. This controller guarantees some properties, such as the robust performance and stability properties. In addition, a class offourth-order nonlinear systems is investigated. Lo and Kuo [] proposed a method called decoupled fuzzy slidingmode control to cope with the above issue. Another kind of fuzzy approach [3] is used to replace the fuzzy controller of[]. It is the signed distance that proposes a sole fuzzy input variable of order m as given in Table and Fig. 3, rather than the controllers which are oforder m on the phase plane constituted by e and ė or higher order for single-input systems, where m is the number offuzzication levels. Furthermore, the Lyapunov function is employed to get the signed distance, and then get the control input u, as well as by means ofthe method ofdecoupled fuzzy sliding mode controller to get better performance. Compared with [,3], ve fuzzy control rules are used as shown in Table. Ifone needs a ner control, one can easily add or modify the rules by means of the Table and Fig. 3. Using this approach, the system can achieve asymptotic stability and it will converge faster than that of[]. The rest ofthe paper is divided into ve sections. In Section, the systems are described. In Section 3, the signed distance fuzzy logic control is presented. In Section 4, the detail design ofthe decoupled fuzzy logic controller is proposed. In Section 5, the proposed controller is used to control a cart-pole system and a ball-beam system. Finally, we conclude with Section 6.. System description Consider a second-order nonlinear system, which can be represented by the following state space model in a canonical form ẋ (t) =x (t); ẋ (t) =f(x)+b(x)u + d(t); y(t) =x (t); () where x =[x ;x ] T is the state vector, f(x) and b(x) are nonlinear functions, u is the control input, and d(t) is external disturbance. The disturbance is assumed to be bounded as d(t) 6D(t). For this kind ofthe second-order system, we can use many kinds ofcontrol methods, such as, fuzzy control, PID control, sliding mode control, etc. A control law u can be easily designed to make the second-order system () arrive at our control goal. However, for such nonlinear models as a cart pole system, the system dynamic representation is generally not in a canonical form exactly. Rather, it has a form shown below: ẋ (t) =x (t); ẋ (t) =f (x)+b (x)u + d (t); ẋ 3 (t) =x 4 (t); ẋ 4 (t) =f (x)+b (x)u + d (t); () where x =[x ;x ;x 3 ;x 4 ] T is the state vector, f (x), f (x) and b (x), b (x) are nonlinear functions, u, u are the control inputs, and d (t), d (t) are external disturbances. The disturbances are assumed to be bounded as d (t) 6D (t), d (t) 6D (t). From (), one can design u and u, respectively, however,

3 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) Without loss ofgenerality, Eq. (4) can be rewritten as follows: d = ẋ + c x : (5) The signed distance d s is dened for an arbitrary point B(x; ẋ) as follows: d s = sgn(s) ẋ + c x = ẋ + c x s = ; (6) where Fig.. Derivation ofa signed distance. this approach is only utilized to control a subsystem in (). For example, ifthe model is a cart pole system, we only control either the pole or the cart ofa system such as (). Hence, the idea ofdecoupling is employed to design a control u to govern the whole system. 3. Signed distance fuzzy logic control In this section, the idea of[3] named the signed distance is used, and the feasibility of the present approach will be demonstrated. The switching line is de- ned by s: ẋ + c x =0: (3) First, we introduce a new variable called the signed distance. Let A(x; ẋ) be the intersection point ofthe switching line and the line perpendicular to the switching line from an operating point B(x ; ẋ ), as illustrated in Fig.. Next, d is evaluated. The distance between A(x; ẋ) and B(x ; ẋ ) can be given by the following expression: d =[(x x ) +(ẋ ẋ ) ] = = ẋ + c x : (4) f { f or s 0; sgn(s) = or s 0: (7) For the second-order system (), a switching line is chosen as s = c x + x : (8) By taking the time derivative ofboth sides of(8), we can obtain ṡ = c ẋ +ẋ = c x + f(x)+b(x)u + d: (9) Then, multiplying both sides ofthe above equation by s gives sṡ = sc x + sf(x)+sb(x)u + sd: (0) Here, we assume that b(x) 0. In (9), it is seen that ṡ increases as u increases and vice versa. Eq. (0) provides the information that if s 0, the decreasing u will make sṡ decrease and that if s 0, the increasing u will make sṡ decrease. Now, we choose a Lyapunov function V = d s : () Then V = d s ḋ s = sṡ : () Hence, it is seen that if s 0, then d s 0, decreasing u will make sṡ decrease so that V 0 and that if s 0,

4 338 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) then d s 0, increasing u will make sṡ decrease so that V 0. So we can ensure that the system is asymptotically stable. From the above relation, we can conclude that u d s : (3) Hence, the fuzzy rule table can be established on a onedimensional space of d s as shown in Table instead of a two-dimensional space of x and ẋ. The control action can be determined by d s only. Hence, we can easily add or modify rules for ne control. For implementation, a triangular type membership function is chosen for the aforementioned fuzzy variables, as shown in Fig.. Remark. The rule table can be established on a onedimensional space like Table. It is seen that the total number ofrules is greatly reduced compared with conventional FLCs. In other words, a traditional fuzzy system has four states to be controlled and the range ofeach state variable is divided into ve fuzzy sets. The number ofrules forming the knowledge database is 5 4 = 65. With the present method, only one variable (d s ) needs to be fuzzied and only ve rules are necessary to establish the knowledge database. The signed distance fuzzy logic control (SDFLC) is shown in Fig. 3 where d s and u are the input and output ofthe signed distance fuzzy logic control, respectively. The input ofthe proposed fuzzy controller is d s, which is a fuzzied variable of d s. The output ofthe fuzzy controller is U, which is the fuzzied variable of u. All the universes ofdiscourse ofd s and U are arranged from to. Thus, the range ofnonfuzzy variables d s and u must be scaled to t the universe ofdiscourse offuzzied variable d s and U with scaling factors K and K, respectively, namely, d s = K d s (t); (4) u(t) =K U: (5) 4. Design of decoupled fuzzy logic controller In this section, the idea ofthe signed distance of fuzzy logic control is used in Section 3. In Eq. (), we rst dene one switching line as s = c (x z)+x (6) and another switching line as s = c x 3 + x 4 : (7) The control objective is to drive the system state to the original equilibrium point. The switching line variables s and s are reduced to zeros gradually at the same time by an intermediate variable z. In Eq. (6), z is a value transferred from s,ithas a value proportional to s and has the range proper to x. Eq. (6) denotes that the control objective of u is changed from x =0, x =0 to x = z, x =0. Because the controller u = u is used to govern the whole system, the bound of x can be guaranteed by letting z 6 Z u ; 0 Z u ; (8) where Z u is the upper bound of abs(z). Eq. (8) implies that the maximum absolute value of x will be limited. Summarizing what we have mentioned above, z can be dened as z = sat(s = z )Z u ; 0 Z u ; (9) where z is the boundary layer of s to smooth z, z transfers s to the proper range of x, and the denition of sat( ) function is { sgn( ) if ; sat( ) = if : (0) Notice that z is a decaying oscillation signal because Z u is a factor less than one. Remark. Consider Eq. (6). If s = 0, then x = z, x = 0. Since z is a value transferred from s, when s 0, then z 0 and x 0. From Eq. (7), if the condition s 0, the control objective can be achieved. Remark 3. Although SDFLC may be extended to the order of n (n =; ; 3;:::) system, it cannot be applied in higher-order system as in []. In a similar way, It can carry over to the third-order system. The two-level decoupled SDFLC is proposed and is shown in Fig. 4.

5 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) Fig.. Fuzzy variable oftriangular type. Fig. 3. The block diagram ofthe SDFLC. Fig. 4. The block ofthe decoupled SDFLC. 5. Computer simulations In this section, we shall demonstrate that the decoupled SDFLC is applicable to both the cart pole system and the ball beam system [] to verify the theoretical development. 5.. Inverted pendulum The structure ofan inverted pendulum is illustrated in Fig. 5 and its dynamic is described below: ẋ = x ; ẋ = m tg sin x m p L sin x cos x x + cos x u L( 4 3 m +d; t m p cos x ) ẋ 3 = x 4 ; ẋ 4 = 4 3 m plx sin x + m p g sin x cos x 4 3 m t m p cos x + 4 3( 4 3 m u + d; () t m p cos x ) where x = is the angle ofthe pole with respect to the vertical axis; x = the angle velocity ofthe pole with

6 340 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) Fig. 5. Structure ofan inverted pendulum. Fig. 6. Angle evolution ofthe pole. ( ), z ( ). respect to the vertical axis; x 3 = x the position ofthe cart; x 4 =ẋ the velocity ofthe cart; and m t = m c + m p. In what follows, we dene the following variables: s = c ( z)+ = c (x z)+x ; () s = c x +ẋ = c x 3 + x 4 (3) and z = sat(s = z )Z u ; 0 Z u : (4) In the simulation, the following specications are used: m p =0:05 kg; m c =kg; L =0:5m; g =9:8m=s ; c =5; c =0:5; z =5; Z u =0:945; d 6 0:0873; K =; K =40 and initial values are = 60 ; =0; x =0; ẋ =0: Figs. 6 8 shows the simulation result. It is found that the pole and the cart can be stabilized to the equilibrium point. Fig. 7. Position evolution ofthe cart. 5.. Ball beam system Consider a ball beam system as depicted in Fig. 9. The mathematical expression ofthis system can be written as ẋ = x ; ẋ = u + d; ẋ 3 = x 4 ; ẋ 4 = B(x 3 x g sin x ); (5) where x = is the angle ofthe beam with respect to the vertical axis; x = the angle velocity ofthe beam with respect to the vertical axis; x 3 = r the position ofthe ball; x 4 =ṙ the velocity ofthe ball; B = MR =

7 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) Fig. 8. Control output ofexample. Fig. 0. Angle evolution ofthe beam. ( ), z( ). Fig. 9. Structure ofa ball beam system. (J b + MR ); J b the moment ofinertia ofthe ball; M the mass ofthe ball; R the radius ofthe ball; g the acceleration ofgravity. The objective is to keep the ball close to the center of the beam and the beam close to the horizontal position. Dene s = c (x z)+x ; (6) s = c x 3 + x 4 (7) and z = sat(s = z )Z u ; 0 Z u : (8) In the simulation, the following parameters are used: B =0:743; J b = 0 6 ; M =0:05 kg; R =0:0 m; g=9:8m=s Fig.. Position evolution ofthe ball. c =5; c =0:5; z =5; Z u =0:945; d 6 0:08; K =; K =40; the initial values are x = =60 ; x = =0; x 3 = r =0; x 4 =ṙ =0: The simulation results are shown in Figs. 0. It is found that and r converge to zero, respectively. From Figs. 6 and 7 to 0 and, with the same c =5,

8 34 S.-Y. Chen et al. / Fuzzy Sets and Systems 9 (00) combined with each other. This may be an important and interesting topic in future. Acknowledgements The authors wish to thank the nancial support ofthe National Science Council ofthe Republic of Taiwan under Contract NSC89-3-E References Fig.. Control output ofexample. c =0:5 as in [5], it is seen that the convergence is faster than that of []. 6. Conclusions The present approach was used to replace the fuzzy controller oflo s approach. Compared with previous results, only ve fuzzy control rules are needed as given in Table. For a ner control, we can easily add or modify rules by way of the rule table for a signed distance fuzzy logic control and the SDFLC. Next, simulation results show that the pole and the cart can be stabilized to the equilibrium, and r ofthe ball beam system converge to zero, and the performance is better than that of[]. Although the SDFLC can be extended to an nth-order system, however, the decoupled method only satises the fourth-order system like the cart pole system. Hence, the same weak point ofthe approach is that when applied to systems higher than fourth order such as a double-inverted pendulum system (which needs a three-level control), ifthe cart is to be stabilized at the origin as in [], then the method described here will fail. If there exists a new SMC method, which can deal with a higher-order system larger than the fourth- and the higher-order system being not a canonical form, then the SDFLC may be [] G. Bartolini, A. Ferrara, Multivariable fuzzy sliding mode control by using a simplex ofcontrol vectors, in: S.G. Tzafestas, A.N. Venetsanopoulos (Eds.), Fuzzy Reasoning in Information, Decision, and Control Systems, Kluwer, Amsterdam, The Netherlands, 994, pp [] C.L. Chen, P.C. Chen, C.K. Chen, Analysis and design of fuzzy control system, Fuzzy Sets Systems 57 () (993) [3] Byung-Jae Choi, Seong-Woo Kwak, Byung Kook Kim, Design ofa single-input fuzzy logic controller and its properties, Fuzzy Sets and Systems 06 (999) [4] S. Galicher, L. Foulloy, Fuzzy controllers: synthesis equivalence, IEEE Trans. Fuzzy Systems 3 (995) [5] J.S. Glower, J. Munighan, Design fuzzy controllers from a variable structures standpoint, IEEE Trans. Fuzzy Systems 5 () (997) [6] Guang-Chyan Hwang, Shih-Chang Lin, A stability approach to fuzzy control design for nonlinear systems, Fuzzy Sets and Systems 48 (99) [7] Y.R. Hwang, M. Tomizuka, Fuzzy smoothing algorithms for variable structure systems, IEEE Trans. Fuzzy Systems (994) [8] Sung-Woo Kim, Ju-Jang Lee, Design ofa fuzzy controller with fuzzy sliding surface, Fuzzy Sets and Systems 7 (995) [9] C.C. Kung, S.C. Lin, Fuzzy controller design: a sliding mode approach, in: S.G. Tzafestas, A.N. Venetsanopoulos (Eds.), Fuzzy Reasoning in Information, Decision, and Control Systems, Kluwer, Amsterdam, The Netherlands, 994, pp [0] K. Liu, F.L. Lewis, Some issues about fuzzy control, in: proc. IEEE Conf. Decision Control, San Antonio, TX, Vol., December 993, pp [] Ji-Chang Lo, Ya-Hui Kuo, Decoupled fuzzy sliding-mode control, IEEE Trans. Fuzzy Systems 6 (3) (998) [] E.H. Mamdani, Applications of fuzzy algorithms for simple dynamic plants, Proc. IEE, Vol., 974, pp