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Auto Tuning Neuron to Sliding Mode Control Application of an Auto-Tuning Neuron to Sliding Mode Control Wei-Der Chang, Rey-Chue Hwang, and Jer-Guang Hsieh IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART C: APPLICATIONS AND REVIEWS, VOL. 32, NO. 4, NOVEMBER 2002 4992c106

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Abstract INTRODUCTION AUTO-TUNING NEURON CONTROL STRUCTURE AND TUNING ALGORITHM ILLUSTRATIVE EXAMPLES CONCLUSION REFERENCES Directory

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This paper presents a control strategy that incorporates an auto- tuning neuron into the sliding mode control (SMC) in order to eliminate the high control activity and chattering due to the SMC. The main difference between the auto-tuning neuron and the general one is that a modified hyperbolic tangent function with adjustable parameters is mployed. In this proposed control structure, an auto-tuning neuron is then used as the neural controller without any connection weights.. The control law will be switched from the sliding control to the neural control, when the state trajectory of system enters in some boundary layer. In this way, the chattering phenomenon will not occur. The results of numerical simulations are provided to show the control performance of our proposed method. Index TermsAuto-tuning neuron, Lyapunov approach, sliding mode control (SMC), switching control. Abstract

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A useful and powerful control scheme to deal with the existence of the model uncertainty, or imprecision, is the sliding mode control (SMC). As we know, the model uncertainty or imprecision may arise from insufficient information about the system or from the purposeful simplification of mathematical model representation of plant, e.g., order reduction. The control law of SMC, however, is an intense switching action similar to that of bang-bang control, when the state trajectory of system reaches around the sliding surface. This leads to the appearance of chattering across the sliding surface and may excite the high- frequency unmodeled dynamics of the system, undesirable in most real applications. A simple method for solving the discontinuous control law and chattering action is to introduce a boundary layer. This method, however, does not ensure the convergence of the state trajectory of system to the sliding surface, and probably results in the existence of the steady-state error. In addition, analysis of a system dynamics within the boundary layer is very complicated. INTRODUCTION

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For solving the drawbacks, a number of studies have been published. In, a control strategy was proposed based upon an on-line estimator constructed by a recurrent neural network to eliminate the chattering. In, the controller consists of the traditional SMC and Gaussian neural network. At the beginning, the SMC is used to force the state trajectory of system toward the sliding surface. Then the control law is switched from the SMC to Gaussian neural network control if the state trajectory of system reaches the boundary layer. A fuzzy-neural network similar to that in was employed to replace the SMCwhen the state trajectory of the system is within the boundary layer. In this paper, a simple neural controller will be proposed to solve the drawbacks. The neural controller is constructed by using only one auto-tuning neuron with three adjustable parameters with no weight connection. The difference between the auto- tuning neuron and thegeneral neuron is that a newmodified hyperbolic tangent function a[1+ exp(-bx)]^-1[1-exp( - bx)] is used as its activation function where the two parameters a and b are adjustable.

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Hence, the output range and shape of this function are free and will not be restricted in a certain interval. The applicability and flexibility of the proposed neural controller are promising. Comparison with the traditional multilayer neural network controller, the complicated architectures and the heavy computation for updating parameters can be simplified and reduced greatly. In fact, it is not necessary to require so many adjustable parameters for a direct adaptive neural controller. In a real on-line control process, a full connection neural controller will also affect the reaction time of the overall control system, and undoubtedly increase the difficulty for hardware implementation. The main purpose in this study is to combine a proposed auto- tuning neuron with the SMC for a class of simple nonlinear systems. The single auto-tuning neuron used as the direct adaptive neural controller will be activated and replace the SMC when the state trajectory of system goes into the boundary layer in order to eliminate the chattering resulting from the SMC. The detailed control structures and adaptation laws of this method will be described in the next section.

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In order to eliminate the high-frequency control and chattering around the sliding surface caused by the SMC, we now introduce an auto-tuning neuron to be the direct adaptive neural controller to replace the SMC, when the state trajectory of system goes into the boundary layer AUTO-TUNING NEURON

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where the activation function h( ) : < ! <is a modified hyperbolic tangent function; a is the saturated level; and b is the slope value. Note, that these two adjustable parameters, a and b, influence mainly the output range and the curve shape of this activation function, as shown in Fig. 2. In this case, since the output range of u2 can be automatically tuned according to certain adaptation mechanism, it is unnecessary to consider the scaling problem of the controller. For convenience, let = [; a; b]T 2 <3 represent the vector of adjustable parameters.We wish to adjust, such that, the control objective can be achieved.

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In this study, the control structure using an auto-tuning neuron with the SMC can be schematically shown in Fig. 3, where u1 is the sliding control defined in (6); u2 is the neural control defined in (10); and u = d(u1; u2) is a function of u1 and u2 defined by In (11), s(e) is a scalar function described in (5); > 0 is the boundary layer thickness; is a small positive value to form an intermediate region CONTROL STRUCTURE AND TUNING ALGORITHM

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is a function of error e used as a weighting factor of u1 and u2. From(11) and (12), we know that (e) 2 (0; 1]. The overall control procedure can be summarized as follows. First, if the initial state of system is outside the boundary layer, then it is subject to the sliding control u1 forcing the state trajectory toward the boundary layer. Second, if it goes near the boundary layer, i.e., entering in the intermediate region, let the control input u be a convex combination of u1 and u2 such that the control switching between u1 and u2 can be smooth and continuous. Finally, only the adaptive neural control u2 is activated when it enters in the boundary layer.

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To illustrate the use of the proposed method, the following two examples are provided. Note, that the sampling time is set to be 0.02 in these simulations. Our control objective is to regulate the system output x to the desired output xd = 0. For instance, with the initial state x(0) = 1:5, Fig. 5 shows the results by using the traditional SMC. Fig. 6 shows the results by using our proposed method. It can be easily seen from Fig. 5 that the high-frequency control and the chattering around x = 0have occurred. From Fig. 6, we conclude that the control input is modulated and its output state x is asymptotically controlled to the desired value xd = 0 by using the proposed method. ILLUSTRATIVE EXAMPLES

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In this paper, we have proposed a control strategy that consists of a general SMC and a neural control constructed by an auto- tuning neuron. In order to eliminate the high control activity and chattering due to the SMC, the control law here is smoothly switched from the sliding control to the neural control, when the state trajectory of system enters in some boundary layer. Thus, the chattering phenomenon around the sliding surface will never occur. For the adaptive neural control, we have presented a stable tuning mechanism based on the Lyapunov stability theory to guarantee the convergence of the system output. From the results of two numerical simulations, we conclude that the proposed method can perform successful control. It is interesting to consider the switching between SMC and PID control, whether the PID control is produced by some classical rules, e.g., Ziegler-Nichols tuning, or by rules based on auto- tuning neurons. The latter is still under our investigation. No fair comments can be made at this point. CONCLUSION

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[1] J. J. E. Slotine and W. P. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [2] Y. Fang, T.W. S. Chow, and X. D. Li, Use of a recurrent neural network in discrete sliding-mode control, in Proc. Inst. Electr. Eng., Control Theory Appl., vol. 146, Jan. 1999, pp. 84–90. [3] R. M. Sanner and J. J. E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Networks, vol. 3, pp. 837–863, Nov. 1992. [4] F. P. Da andW. Z. Song, Sliding mode adaptive control based on fuzzy neural networks, Control and Decision, vol. 13, no. 4, pp. 301–305, 1998. in Chinese. [5] C. T. Chen andW. D. Chang, A feedforward neural network with function shape autotuning, Neural Netw., vol. 9, no. 4, pp. 627–641, 1996. [6] W. D. Chang, R. C. Hwang, and J. G. Hsieh, Adaptive control of multivariable dynamic systems using independent self-tuning neurons, in Proc. 10th Int. Conf. Tools with Artificial Intelligence, Taipei, Taiwan, R.O.C, 1998, pp. 68–73. [7] L. X. Wang, A Course in Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1997. REFERENCES

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