Creating a Sliding Mode in a Motion Control System by Adopting a Dynamic Defuzzification Strategy in an Adaptive Neuro Fuzzy Inference System

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1 Creating a Sliding Mode in a Motion Control System by Adopting a Dynamic Defzzification Strategy in an Adaptive Nero Fzzy Inference System M. Onder Efe Bogazici University, Electrical and Electronic Engineering Department, Bebek, 885 Istanbl, Trkey efemond@bon.ed.tr Okyay Kaynak Bogazici University, Electrical and Electronic Engineering Department, Bebek, 885 Istanbl, Trkey kaynak@bon.ed.tr Bogdan M. Wilamowski University of Wyoming Department of Electrical Engineering Laramie, WY 87, U.S.A. wilam@ieee.org Abstract: Adaptive nero-fzzy inference systems exhibit both the nmeric power of neral networks and verbal power of fzzy inference systems. Particlarly in control applications, the prime difficlty in selecting the parameters of sch systems stems from the navailability of the desired control inpts. In this paper, a novel approach is presented for the establishment of a sliding mode in the plant nder control. The approach presented adjsts solely the parameters of the defzzifier. At the adjstment stage, a dynamic adaptation law is proposed and it is proved that the particlarly chosen form of the adaptation strategy creates a sliding mode in the plant behavior while the parameters of the controller are also in a sliding mode. In the simlations, the dynamic model of observed that the method discssed is highly robst against the distrbances like varying payload conditions and noisy observations. I. INTRODUCTION Systems having strctral ncertainties or a known complicated strctre are difficlt to control. Modeling of the ncertainties or handling the deterministic complexity are typical problems freqently encontered in the field of systems and control engineering. From this point of view, the designer is generally in prsit of best tilization of what is known abot the system in hand. Sometimes the knowledge abot the system is best expressed in words. If the response of the system can be expressed in certain regions of the inpt space, the designer can come p with some IF-THEN statements, which constitte the basis of the fzzy inference systems. Fzzy Inference Systems are the most poplar constitent of the comptational intelligence area becase of their ability to represent hman expertise in the form of IF antecedent THEN conseqent statements. In this domain, the system behavior is modeled throgh the se of lingistic descriptions. Althogh the earliest work by Prof. Zadeh on fzzy systems was not paid as mch attention as it deserved in early 96s, since then the methodology has become a well-developed framework. The typical architectres of fzzy inference systems are those introdced by Wang [-], Takagi and Sgeno [3], and Jang [4]. In [], a fzzy system having Gassian membership fnctions, prodct inference rle and weighted average defzzifier is discssed. Takagi and Sgeno change the defzzification procedre where dynamic systems are sed for this prpose. The potential advantage of the method is that, nder certain constraints, the stability of the system can be stdied. Jang et al [4] propose an Adaptive Nero Fzzy Inference System (ANFIS), in which polynomials are sed in the defzzification stage. This strctre is commonly seen in the related literatre [5-6] and is sed in this paper too. The choice concerning the order of the polynomials and the variables to be sed in the defzzifier are left to the designer. Neral networks, on the other hand, are well known with their ability in generalizing the data and tolerating the falts. Therefore it is reasonable to expect that the ANFIS strctre, which is a sitable combination of neral networks and fzzy inference systems, can exhibit the above mentioned featres and can sccessflly be sed in control engineering practice. In control engineering practice, stability and robstness are of crcial importance. Becase of this, the implementation-oriented control engineering expert has always been in prsit of a design, which provide accracy as well as insensitivity to environmental distrbances and strctral ncertainties. At this point, it mst be emphasized that these ambigities can never be modeled accrately. When the designer tries to minimize the ambigities by the se of a detailed model, then the design becomes so tedios that its cost increases dramatically. A sitable way of tackling with ncertainties withot the se of complicated models is to introdce Variable Strctre Systems (VSS) theory based components into the system strctre. Variable Strctre Control (VSC) has sccessflly been applied to a wide variety of systems having ncertainties in the representative system models. The philosophy of the control strategy is simple, being based on two goals. First, the system is forced towards a //$. IEEE 894

2 desired dynamics, second, the system is maintained on that differential geometry. In the literatre, the former dynamics is named the reaching mode, while the latter is called the sliding mode. The control strategy borrows its name from the latter dynamic behavior, and is called Sliding Mode Control (SMC). Earliest notion of SMC strategy was constrcted on a second order system in the late 96s by Emelyanov [7]. The work stiplated that a special line cold be defined on the phase plane, sch that any initial state vector cold be driven towards the plane and then be maintained on it, while forcing the error dynamics towards the origin. Since then, the theory has greatly been improved and the sliding line has taken the form of a mltidimensional srface, called the sliding srface arond which a switching control action takes place. Nmeros contribtions to VSS theory have been made dring the last decade, some of them are as follows: Hng et al [8] has reviewed the control strategy for linear and nonlinear systems. In [8], the switching schemes ptting the differential eqations into canonical forms and generating simple SMC based controls are considered in detail. Gao et al [9], apply the SMC scheme to robotic maniplators and discss the qality of the scheme. The latest stdies consider this robstness property by eqipping the system with comptationally intelligent methods. In [] and [], fzzy inference systems are proposed for SMC scheme. A standard fzzy system is stdied and the relevant robstness analyses are carried ot. Particlarly, the work presented in [] emphasizes that the robstness and stability properties of intelligent control strategies can be stdied throgh the se SMC theory. It is shown in the paper in this way that the approach is robst i. e. it can compensate the deficiencies cased by poor modeling of plant dynamics and external distrbances. In [-3], it is demonstrated that the theory of VSS can well be sed for the prpose of learning. The method discssed is based on the stabilization of gradient based training strategies with the aim of redcing the parametric change effort. The method discssed in this paper is first proposed by Sira-Ramirez and Colina-Morles for learning in Adaptive Linear Elements (ADALINE) [4]. The paper gives the example of an inverse dynamics identification of a Kapitsa pendlm with a single ADALINE. Y et al discss the same algorithm for ADALINE with the improvement on ncertainty bond adaptation [5]. The strategy adopted is based on the adaptive adjstment of ncertainty bonds. The bond adaptation strategy in [5] works well, as long as there is no noisy observations. Otherwise, once the mean vale of the error remains close to the origin, the mechanism starts integrating the absolte vale of the noise signal, which reglarly gives increments to the ncertainty bond parameter. This cases instability in the long rn. Therefore, in this paper a constant bond is adopted. The approach presented in this paper ses the backgrond discssed in [4-5]. The primary difference is that the method is modified to establish a sliding mode in the phase plane of the plant nder control, while keeping the controller parameters in an eqivalent sliding motion. This paper is organized as follows: Second section describes the ANFIS architectre, the following section presents the derivation of the learning algorithm. In the forth section how the SMC design is performed is discssed. The next section describes the plant sed as the test bed. In the sixth section, the simlation reslts are presented. Conclsions constitte the last part of the paper. 5. ADAPTIVE NEURO FUZZY INFERENCE SYSTEMS (ANFIS) Adaptive Nero-Fzzy Inference Systems are realized by an appropriate combination of neral and fzzy systems. This hybrid combination enables to tilize both the verbal and the nmeric power of intelligent systems. As is known from the theory of fzzy systems, different fzzification and defzzification strategies with different rle base strctres can reslt in varios soltions to a given task. This paper considers the ANFIS strctre with first order Sgeno model containing nine rles. Bell shaped membership fnctions with prodct inference rle are sed at the fzzification level. Fzzifier otpts the firing strengths for each rle. The vector of the firing strengths is normalized and the reslting vector is defzzified by tilizing the first order Sgeno model. The strctre for two inpts and one otpt is illstrated in Fig. and a sample rle is described below for m- inpt one otpt ANFIS. IF is U i, AND is U i, AND AND m is U i,m THEN f i = Y i, + + Y i,m m +Y i,m+ In the IF part of this representation, lowercase variables denote the inpts, ppercase variables stand for the fzzy sets corresponding to the domain of each lingistic label. The ANFIS otpt is clearly a linear fnction of the adjstable defzzifier parameters denoted by Y i,j. The membership fnctions sed are described by () in which, c ij denotes the center of i th rle s j th membership fnction. a ij and b ij determine the shape of the fnction. The parameters of the membership fnctions are selected sch that the region of interest is covered appropriately. The selection is depicted in Fig.. The overall realization performed by the system considered is given in (), where linear fnctions of inpt variables are sed as defzzifier with algebraic prodct aggregation method. ( ) µ ij j = b ij j cij, + a ij R m f i µ ij ( j ) i = j = R F = = R m fi wni µ ( ) i= ij j i = j = () () 895

3 where, m µ ij j = wni = R m µ k = j= ( ) kj j ( ) j (3) In (3), the vector of firing strengths is normalized and the reslting vector is represented by w n. With the definition given in (3), and the realization described by (), the controller otpt can be expressed more compactly as follows. where, T F = wn Y (4) [ w w ] T w n = n... nr (5) y ym y, m+ = y ym y, m+ Y (6) yr yrm yr, m+ = [ ] T m (7) 3. DERIVATION OF THE LEARNING ALGORITHM In this section, it is assmed that the physical constraints reqire the following ineqalities to hold tre. Y < BY ; < B ; < B & w ; n < Bw& n (8) If one defines the adjstable parameter matrix as given in (6), defining the error at the otpt of the controller as in (9), the Lyapnov fnction in () cold be a sitable fnction for describing the learning performance. The time derivative of the fnction is as given by (). where, ec = F F d (9) V = e c () V = e c e c () e c = F F d () In order to evalate the expression in (), we have the following relation: F T T T = w Y n + wn Y + wn Y (3) Theorem: If the dynamic defzzification strategy is adopted as described in (4), the learning strategy becomes stable in the sense of Lyapnov. T wn = ksign( ec) T T wnwn (4) Y Proof: If (4) is sbstitted into (3) with the aid of (9), (), (4) and (3), the error dynamics in (5) is obtained. Using the bonds of the ncertainties mentioned at the beginning of the section leads to (6). T T ( ec) + w ny + wny F d e c = ksign (5) T T V = kec + ( w ny + wny F d) ec (6) < ( BY( Bw B & + Bw B ) BF k) e n n & + & d c In order to have a negative time derivative for the Lyapnov fnction in (), the parameter k mst satisfy the following ineqality with the fact that B wn =. ( Bw& B ) n + B & + B & Fd k > BY (7) 4. SLIDING MODE CONTROL DESIGN The analysis presented in the previos section aims to minimize the Lyapnov fnction in (), which is an instantaneos cost measre sed in most nero-fzzy control strategies. It is apparent that the se of the presented analysis in control applications entails the desired vales of the controller otpts. Therefore, for the applications in which the desired signals are navailable, the method cannot be sed withot any modification. In this part, parallel to the philosophy of variable strctre controller design procedre, a switching fnction is defined and described by (8). The symbol e seen in (8) is the discrepancy between the reference state vale and observed state vale. s = e + λe (8) If one replaces e c of (4) with s of (8), it is straightforward to prove that the Lyapnov fnction in (9) is minimized in time and its time derivative is enforced to have negative vales de to the adjstment strategy in (4). s V = (9) For this case, the selection of k vales mst be reasonably large for compensating the bonds introdced with the new selection. The details of the analysis are not inclded de to the space limit. 5. PLANT MODEL In the simlations the dynamic model of a two degrees of freedom direct drive robotic maniplator, which is illstrated in Fig. 3, is sed as the test bed. Since the dynamics of sch a mechatronic system is modeled by nonlinear and copled differential eqations, precise otpt tracking becomes a difficlt objective de to the strong interdependency between the variables involved. Besides, the ambigities on the friction related dynamics in the plant model make the design mch more complicated. Therefore the methodology adopted mst be intelligent in some sense. 896

4 The general form of robot dynamics is described by () where Μ(θ), V( θ, θ ), τ and f c stand for the state varying inertia matrix, vector of Coriolis terms, applied torqe inpts and friction terms respectively. The plant parameters are given in Table in standard nits. M ( θ) θ + V( θ θ ) = τ fc, () If the anglar positions and anglar velocities are described as the state variables of the system, for copled and first order differential eqations can define the model. In () and (), the terms seen in () are given explicitly. p + p3cos( θ) p + p3cos( θ) M( θ ) = () p + p3cos( θ) p θ ( ) ( θ + θ ) p3sin( θ) V θ, θ = θ () p3sin( θ) In above, p = M p, p = M p and p 3 = M p. Here M p denotes the payload mass. The details of the plant model are presented in [4]. Table. Maniplator Parameters Motor Rotor Inertia.67 I Arm Inertia.334 I Motor Rotor Inertia.75 I 3 Motor Stator Inertia.4 I 3C Arm Inertia.63 I 4 Motor Mass 73. M Arm Mass 9.78 M Motor Mass 4. M 3 Arm Mass 4.45 M 4 Arm Length.359 L Arm Length.4 L Arm Center of Gravity.36 L 3 Arm Center of Gravity. L 4 Axis Friction 5.3 f c Axis Friction. f c Torqe Limit 45. τ max Torqe Limit 39. τ max 6. SIMULATION RESULTS In the simlation stdies presented, the plant introdced in the fifth section is controlled by the proposed control scheme. The aim is to prodce some torqe signals that establish a sliding motion in the phase plane for each link. As the controller, the architectre discssed in the second section is adopted with nine rles (R=9) and two inpts (m=) for each link. The strctre of the control system is as illstrated in Fig. 4, in which the plant is in an ordinary feedback loop. Based on the tracking error vector, first the vale of s( e, e ) is evalated and this qantity is applied to the adjstment mechanism. In evalating the vale of the qantity s, the slope of the switching srface (λ) has been set to.5. In practical implementations a nmber of difficlties are encontered, which make it difficlt to achieve an accrate trajectory tracking. The simlation stdies carried ot address these difficlties. The first difficlty to be alleviated is the varying payload conditions, which introdces abrpt changes in the dynamics of the system nder control. As the reference trajectory illstrated in Fig. 5. implies, the motion starts with no payload. At time t= sec, a payload of kg is grasped and released at time t=5 sec. The same variation is repeated at time t=9 sec and t= sec. After the time t=5, the maniplator is kept motionless. The second difficlty is the existence of observation noise. The information sed by the controller is corrpted by a Gassian distribted random noise having zero mean and variance eqal to 33e-6. The peak magnitde of the noise signal is within ± with probability very close to nity. The third difficlty is the nonzero positional initial conditions. In order to demonstrate the reaching mode performance of the algorithm, the first link is moved to π/5 radians and the second link is moved to π/5 radians initially. Under these conditions, the state tracking error graph in Fig. 6 is obtained. The trend in position and velocity errors clearly stiplate that the algorithm is able to achieve precise tracking objective. The motion in the phase plane is illstrated in the top row of Fig. 7. The pper sbplots of the figre show that for both links, after a fast reaching mode, a sliding mode is enforced and is maintained by prodcing a sitable control signal. In the bottom row of the figre, it is shown for both links that the Lyapnov fnction in (9) is minimized. In order to show the minimization activity of the algorithm presented, the vertical axes for these sbplots are selected as logarithmic. It is seen that some small magnitde spikes occr in time and they are dampened ot qickly. We relate these spikes to the difficlties stated above. What shold be emphasized as a last point is the torqe signal prodced by the controller. As seen in Fig. 8, the controller otpts are directly applied to the maniplator withot exceeding the limits of the applicable control ranges. The control signal has a smooth characteristic, which does not violate the potential limits of the actators. Dring the simlations k, k and the sampling rate have been set to, and.5 msec respectively. Frthermore, in order to redce the chattering effect in the sliding mode, the fnction in (3) has been sed instead of the sign fnction in the dynamic defzzification strategy described in (4). 7. CONCLUSIONS ec sign( e c ) (3) ec +.5 In this paper, a novel method for establishing a sliding motion in the phase plane of a nonlinear plant is discssed. The method is based on the adoption of a dynamic defzzification strategy in an adaptive nero- 897

5 fzzy inference system sed as the controller. It is seen that the algorithm discssed is able to compensate deficiencies cased by the imperfect observations of the state variables, abrptly changing plant dynamics, initial condition errors and complex plant dynamics. From these points of view, the method proposed is highly promising for control prposes. 8. REFERENCES [] Wang, L., Adaptive Fzzy Systems and Control, Design and Stability Analysis, PTR Prentice Hall, 994. [] Wang, L., A Corse in Fzzy Systems and Control, PTR Prentice Hall, 997. [3] Takagi T., and M. Sgeno, Fzzy Identification of Systems and Its Applications to Modeling and Control, IEEE Trans. on Systems, Man, and Cybernetics, v.5, no., pp.6-3, Janary 985. [4] Jang, J.-S. R., Sn, C.-T., Miztani, E., Nero- Fzzy and Soft Compting, PTR Prentice Hall, 997. [5] Efe, M. O. and O. Kaynak, A Comparative Stdy of Soft Compting Methodologies in Identification of Robotic Maniplators, Robotics and Atonomos Systems, v.3, no:3, pp.-3,. [6] Efe, M. O., O. Kaynak and I. J. Rdas, A Novel Comptationally Intelligent Architectre for Identification and Control of Nonlinear Systems, IEEE Int. Conf. on Robotics and Atomation (ICRA 99), May -5, Detroit, Michigan, U.S.A., pp.73-77, 999. [7] Emelyanov, S. V., Variable Strctre Control Systems, Moscow, Naka, 967. [8] Hng, J. Y., W. Gao and J. C. Hng, Variable Strctre Control: A Srvey, IEEE Trans. on Indstrial Electronics, v.4, no., pp.-, Febrary 993. [9] Gao, W., J. C. Hng, Variable Strctre Control of Nonlinear Systems: A New Approach, IEEE Trans. on Indstrial Electronics, v.4, no., pp.45-55, Febrary 993. [] Y. Byngkook and W. Ham, Adaptive Fzzy Sliding Mode Control of Nonlinear Systems, IEEE Transactions on Fzzy Systems, v.6, (998) [] K. Erbatr, O. Kaynak, A. Sabanovic and I. Rdas, Fzzy Adaptive Sliding Mode Control of a Direct Drive Robot, Robotics and Atonomos Systems, v.9, (996) 5-7. [] Efe, M. O., and O. Kaynak, Stabilizing and Robstifying the Learning Mechanisms of Artificial Neral Networks in Control Engineering Applications, to appear in Int. Jornal of Intelligent Systems,. [3] Efe, M. O., O. Kaynak and B. M. Wilamowski, Stable Training of Comptationally Intelligent Systems By Using Variable Strctre Systems Techniqe, to appear in IEEE Transactions on Indstrial Electronics, April. [4] Sira-Ramirez, H. and E. Colina-Morles, A Sliding Mode Strategy for Adaptive Learning in Adalines, IEEE Trans. on Circits and Systems I: Fndamental Theory and Applications, v.4, no., pp.-, Dec. 995 [5] Y, X., M. Zhihong and S. M. M. Rahman, Adaptive Sliding Mode Approach for Learning in a Feedforward Neral Network, Neral Compting & Applications, v.7, pp.89-94, 998. [6] Direct Drive Maniplator R&D Package User Gide, Integrated Motions Incorporated, 74 Gillman Street, Berkeley, California 947, U.S.A., 999. ACKNOWLEDGMENTS This work is spported by Bogazici University Research Fnd (Project no: 99A and A3D) and by NSF (Grant No: 99633). µ ij(j) Α Α Β Π Π Β w w w Ν Ν w Figre. Architectre of ANFIS w f w f j Figre. Membership fnctions sed in the simlations Σ F 898

6 .8 BASE Angle Error. ELBOW Angle Error e. e BASE Velocity Error ELBOW Velocity Error e e -.4 Figre 3. Physical strctre of the maniplator Figre 6. State tracking errors θ d - + Σ e EVALUATE S w n ANFIS CONTROLLER e c τ = F UPDATE Y MATRIX ROBOT Y θ de/dt Link e x -4 3 V de/dt Link e 3 x -4 V Figre 4. Strctre of the Control System - Time (sec) - Time (sec) Figre 7. Motion in phase plane for each link (top row) and the behavior of the Lyapnov fnction in time (bottom row) BASE Angle ELBOW Angle 4 Evalated BASE Torqe Evalated ELBOW Torqe 5 θ d θ d τ e τ e BASE Velocity ELBOW Velocity Applied BASE Torqe Applied ELBOW Torqe θ d θ d τ a 5 τ a Figre 5. Reference state trajectories Figre 8. Evalated and applied torqe signals 899