Adaptive RCMAC sliding mode control for uncertain nonlinear systems
Neural Comput & Applic (2006) 15: 253–267 DOI 10.1007/s00521-006-0027-0
O R I GI N A L A R T IC L E
Chih-Min Lin Æ Chiu-Hsiung Chen
Adaptive RCMAC sliding mode control for uncertain nonlinear systems
Received: 24 May 2005 / Accepted: 4 January 2006 / Published online: 1 February 2006 Springer-Verlag London Limited 2006
Abstract An adaptive recurrent cerebellar-model-articulation-controller (RCMAC) sliding-mode control (SMC) system is developed for the uncertain nonlinear systems. This adaptive RCMAC sliding-model control (ARCSMC) system is composed of two systems. One is an adaptive RCMAC system utilized as the main controller, in which an RCMAC is designed to identify the system models. Another is a robust controller utilized to achieve system’s robust characteristics, in which an uncertainty bound estimator is developed to estimate the uncertainty bound so that the chattering phenomenon of control effort can be eliminated. The on-line adaptive laws of the ARCSMC system are derived in the sense of Lyapunov so that the system stability can be guaranteed. Finally, a comparison between SMC and ARCSMC for a chaotic system and a car-following system are presented to illustrate the effectiveness of the proposed ARCSMC system. Simulation results demonstrate that the proposed control scheme can achieve favorable control performances for the chaotic system and car-following systems without the knowledge of system dynamic functions. Keywords Adaptive control Æ Recurrent cerebellar-model-articulation-controller (RCMAC) Æ Sliding-mode control (SMC) Æ Chaotic system Æ Car-following system
1 Introduction It is well known that the sliding-mode control (SMC) is a powerful robust scheme for controlling the nonlinear systems with uncertainties [1, 2]. The most outstanding features of SMC are insensitive to system parameter C.-M. Lin (&) Æ C.-H. Chen Department of Electrical Engineering, Yuan-Ze University, No. 135, Yuandong Rd., 32003 Jhongli City, Taoyuan County, Taiwan, R.O.C. E-mail: [email protected] Tel.: +886-3-4638800 Fax: +886-3-4639355
variations, external disturbance rejection, and fast dynamic response [1]. Other remarkable advantages of this control approach are the simplicity of its implementation and the order reduction of the closed-loop systems. However, in practical applications, the SMC suffers two main disadvantages [2]. One is that it requires the system models that may be difficult to obtain in some cases. Another is that because the magnitude of uncertainty bound is unknown, the large bound is often required to achieve robust characteristics. This will lead the control input chattering. Recently, neural networks (NNs) have been applied for system identifications and controls [3–7]. The most useful property of NNs is their ability to uniformly approximate arbitrary input–output linear or nonlinear mappings on closed subsets. Based on this property, some researchers of NN-based sliding mode controllers have been developed which combines the advantages of the SMC with robust characteristics and the NNs with on-line learning ability; so that the stability, convergence, and robustness of the system can be improved [8–10]. For instance, Wai and Lin [8] have described a NN-based sliding mode control to estimate the bound of uncertainties. Nevertheless, the NNs are used as the compensating controller and sometimes the partial system parameters must be known. In [9], Tsai et al. presented a neuro-sliding mode control that utilized two parallel NNs to realize equivalent control and corrective control; thus, the system performance can be improved and the chattering can be eliminated. In [10], Da introduced an identification-based SMC and the bound of uncertainties is also not required. However, the above approaches suffer the computational complexity. According to the structure, the NNs can be mainly classified as feedforward neural networks (FNNs [3, 4, 9, 10]) and recurrent neural networks (RNNs [5–7]). As known, the FNN is a static mapping. Without the aid of tapped delays, the FNNs are unable to represent a dynamic mapping. Although much research has used the FNNs with tapped delays to deal with dynamic problems, the FNNs require a large number of neurons to represent dynamic
254
responses in the time domain. Moreover, the weight updates of the FNNs do not utilize the internal network information and the function approximation is sensitive to the training data. For the RNNs, of particular interest is their ability to deal with time varying input or output through their own natural temporal operation [6]. Thus, the RNN is a dynamic mapping and demonstrates good control performance in the presence of unmodeled dynamics. However, no matter for the FNNs or RNNs, the learning is slow since all the weights are updated during each learning cycle. Therefore, the effectiveness of the NN is limited in problems requiring on-line learning. Recently, the cerebellar model articulation controller (CMAC) is classified as a non-fully connected perceptron-like associative memory network with overlapping receptive fields. Compared with the neural networks, the CMACs have been adopted widely for the closed-loop control of complex dynamical systems because of its fast learning property, good generalization capability, and simple computation. The application of CMAC is not only limited to control problems but also a model-free function approximator. The CMAC has already been validated that it can approximate a nonlinear function over a domain of interest to any desired accuracy [11]. The advantages of using CMAC over NNs in many practical applications have been presented in recent literatures [12, 13]. However, the conventional CMAC uses constant binary or triangular receptive-field basis functions. The disadvantage is that the derivative information is not preserved. For acquiring the derivative information of input and output variables, Chiang and Lin [14] developed a CMAC network with differentiable Gaussian receptive-field basis function, and provided the convergence analyses of this network. This makes CMAC a suitable candidate for a wide class of nonlinear system control [15, 16]. However, the major drawback of these CMACs is that they belong to static network structure. To resolve the static CMAC problem and to overcome the disadvantage of SMC, an adaptive recurrent cerebellar-model-articulation-computer sliding-mode control (ARCSMC) system is proposed in this study. In this system, a recurrent CMAC (RCMAC) is proposed, which includes the delayed self-recurrent units in the association memory space so that it presents a dynamic CMAC. The developed control system consists of two parts. One is an adaptive RCMAC system utilized as the main controller, in which an RCMAC is used to identify the system models. Another is a robust controller utilized to achieve system’s robust characteristics, in which an uncertainty bound estimator is developed to estimate the uncertainty bound. The on-line adaptive laws of the ARCSMC system are derived in the sense of Lyapunov, so that the system stability can be guaranteed. The proposed ARCSMC system can resolve the problems of requirement of system models and uncertainty bound in the SMC. Finally, a comparison between SMC and ARCSMC for a chaotic system and a car-following system are presented to illustrate the effectiveness of the proposed ARCSMC system. The major contributions of
this study are as follows: (1) to solve the problems of requirement of system models and uncertainty bound in the SMC, (2) to eliminate the chattering phenomenon in the control effort, and (3) the successful applications of the ARCSMC system for accurate control of a chaotic system and a car-following system. This paper is organized as follows: Problem formulation is described in Sect. 2. The design of the sliding mode control is briefly reviewed in Sect. 3. The design procedures of the proposed ARCSMC scheme are constructed in Sect. 4. In Sect. 5, a comparison between SMC and the proposed ARCSMC for a chaotic system and a car-following system are presented. Conclusions are drawn in Sect. 6.
2 Problem formulation Consider the nth-order nonlinear dynamic systems expressed in the following form: 8 x_ 1 ¼ x2 > > > > > > < x_ 2 ¼ x3 > > > x_ n ¼ f ðx1 ; x2 ; ; xn Þ þ gðx1 ; x2 ; ; xn ÞuðtÞ þ dðtÞ > > > : y ¼ x1 ð1Þ or, equivalently the form ( xðnÞ ¼ f ðxÞ þ gðxÞuðtÞ þ dðtÞ y¼x
ð2Þ
where f(Æ) and g(Æ) are unknown but bounded real continuous functions, u(t) 2< and y 2< are control input and system output, respectively, d(t) 2< is an unknown external disturbance, and x ¼ ½x1 ; x2 ; . . . ; xn T ¼ ½x; x_ ; . . . ; xðn1Þ T 2
O R I GI N A L A R T IC L E
Chih-Min Lin Æ Chiu-Hsiung Chen
Adaptive RCMAC sliding mode control for uncertain nonlinear systems
Received: 24 May 2005 / Accepted: 4 January 2006 / Published online: 1 February 2006 Springer-Verlag London Limited 2006
Abstract An adaptive recurrent cerebellar-model-articulation-controller (RCMAC) sliding-mode control (SMC) system is developed for the uncertain nonlinear systems. This adaptive RCMAC sliding-model control (ARCSMC) system is composed of two systems. One is an adaptive RCMAC system utilized as the main controller, in which an RCMAC is designed to identify the system models. Another is a robust controller utilized to achieve system’s robust characteristics, in which an uncertainty bound estimator is developed to estimate the uncertainty bound so that the chattering phenomenon of control effort can be eliminated. The on-line adaptive laws of the ARCSMC system are derived in the sense of Lyapunov so that the system stability can be guaranteed. Finally, a comparison between SMC and ARCSMC for a chaotic system and a car-following system are presented to illustrate the effectiveness of the proposed ARCSMC system. Simulation results demonstrate that the proposed control scheme can achieve favorable control performances for the chaotic system and car-following systems without the knowledge of system dynamic functions. Keywords Adaptive control Æ Recurrent cerebellar-model-articulation-controller (RCMAC) Æ Sliding-mode control (SMC) Æ Chaotic system Æ Car-following system
1 Introduction It is well known that the sliding-mode control (SMC) is a powerful robust scheme for controlling the nonlinear systems with uncertainties [1, 2]. The most outstanding features of SMC are insensitive to system parameter C.-M. Lin (&) Æ C.-H. Chen Department of Electrical Engineering, Yuan-Ze University, No. 135, Yuandong Rd., 32003 Jhongli City, Taoyuan County, Taiwan, R.O.C. E-mail: [email protected] Tel.: +886-3-4638800 Fax: +886-3-4639355
variations, external disturbance rejection, and fast dynamic response [1]. Other remarkable advantages of this control approach are the simplicity of its implementation and the order reduction of the closed-loop systems. However, in practical applications, the SMC suffers two main disadvantages [2]. One is that it requires the system models that may be difficult to obtain in some cases. Another is that because the magnitude of uncertainty bound is unknown, the large bound is often required to achieve robust characteristics. This will lead the control input chattering. Recently, neural networks (NNs) have been applied for system identifications and controls [3–7]. The most useful property of NNs is their ability to uniformly approximate arbitrary input–output linear or nonlinear mappings on closed subsets. Based on this property, some researchers of NN-based sliding mode controllers have been developed which combines the advantages of the SMC with robust characteristics and the NNs with on-line learning ability; so that the stability, convergence, and robustness of the system can be improved [8–10]. For instance, Wai and Lin [8] have described a NN-based sliding mode control to estimate the bound of uncertainties. Nevertheless, the NNs are used as the compensating controller and sometimes the partial system parameters must be known. In [9], Tsai et al. presented a neuro-sliding mode control that utilized two parallel NNs to realize equivalent control and corrective control; thus, the system performance can be improved and the chattering can be eliminated. In [10], Da introduced an identification-based SMC and the bound of uncertainties is also not required. However, the above approaches suffer the computational complexity. According to the structure, the NNs can be mainly classified as feedforward neural networks (FNNs [3, 4, 9, 10]) and recurrent neural networks (RNNs [5–7]). As known, the FNN is a static mapping. Without the aid of tapped delays, the FNNs are unable to represent a dynamic mapping. Although much research has used the FNNs with tapped delays to deal with dynamic problems, the FNNs require a large number of neurons to represent dynamic
254
responses in the time domain. Moreover, the weight updates of the FNNs do not utilize the internal network information and the function approximation is sensitive to the training data. For the RNNs, of particular interest is their ability to deal with time varying input or output through their own natural temporal operation [6]. Thus, the RNN is a dynamic mapping and demonstrates good control performance in the presence of unmodeled dynamics. However, no matter for the FNNs or RNNs, the learning is slow since all the weights are updated during each learning cycle. Therefore, the effectiveness of the NN is limited in problems requiring on-line learning. Recently, the cerebellar model articulation controller (CMAC) is classified as a non-fully connected perceptron-like associative memory network with overlapping receptive fields. Compared with the neural networks, the CMACs have been adopted widely for the closed-loop control of complex dynamical systems because of its fast learning property, good generalization capability, and simple computation. The application of CMAC is not only limited to control problems but also a model-free function approximator. The CMAC has already been validated that it can approximate a nonlinear function over a domain of interest to any desired accuracy [11]. The advantages of using CMAC over NNs in many practical applications have been presented in recent literatures [12, 13]. However, the conventional CMAC uses constant binary or triangular receptive-field basis functions. The disadvantage is that the derivative information is not preserved. For acquiring the derivative information of input and output variables, Chiang and Lin [14] developed a CMAC network with differentiable Gaussian receptive-field basis function, and provided the convergence analyses of this network. This makes CMAC a suitable candidate for a wide class of nonlinear system control [15, 16]. However, the major drawback of these CMACs is that they belong to static network structure. To resolve the static CMAC problem and to overcome the disadvantage of SMC, an adaptive recurrent cerebellar-model-articulation-computer sliding-mode control (ARCSMC) system is proposed in this study. In this system, a recurrent CMAC (RCMAC) is proposed, which includes the delayed self-recurrent units in the association memory space so that it presents a dynamic CMAC. The developed control system consists of two parts. One is an adaptive RCMAC system utilized as the main controller, in which an RCMAC is used to identify the system models. Another is a robust controller utilized to achieve system’s robust characteristics, in which an uncertainty bound estimator is developed to estimate the uncertainty bound. The on-line adaptive laws of the ARCSMC system are derived in the sense of Lyapunov, so that the system stability can be guaranteed. The proposed ARCSMC system can resolve the problems of requirement of system models and uncertainty bound in the SMC. Finally, a comparison between SMC and ARCSMC for a chaotic system and a car-following system are presented to illustrate the effectiveness of the proposed ARCSMC system. The major contributions of
this study are as follows: (1) to solve the problems of requirement of system models and uncertainty bound in the SMC, (2) to eliminate the chattering phenomenon in the control effort, and (3) the successful applications of the ARCSMC system for accurate control of a chaotic system and a car-following system. This paper is organized as follows: Problem formulation is described in Sect. 2. The design of the sliding mode control is briefly reviewed in Sect. 3. The design procedures of the proposed ARCSMC scheme are constructed in Sect. 4. In Sect. 5, a comparison between SMC and the proposed ARCSMC for a chaotic system and a car-following system are presented. Conclusions are drawn in Sect. 6.
2 Problem formulation Consider the nth-order nonlinear dynamic systems expressed in the following form: 8 x_ 1 ¼ x2 > > > > > > < x_ 2 ¼ x3 > > > x_ n ¼ f ðx1 ; x2 ; ; xn Þ þ gðx1 ; x2 ; ; xn ÞuðtÞ þ dðtÞ > > > : y ¼ x1 ð1Þ or, equivalently the form ( xðnÞ ¼ f ðxÞ þ gðxÞuðtÞ þ dðtÞ y¼x
ð2Þ
where f(Æ) and g(Æ) are unknown but bounded real continuous functions, u(t) 2< and y 2< are control input and system output, respectively, d(t) 2< is an unknown external disturbance, and x ¼ ½x1 ; x2 ; . . . ; xn T ¼ ½x; x_ ; . . . ; xðn1Þ T 2
