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*Wen*-Jun *Cao* *Jian*-Xin Xu A Learning Variable Structure Controller of a Flexible One-Link Manipulator

"... I Introduction Variable Structure Control ͑VSC͒ is well noted for its simplicity in implementation and outstanding robustness to system uncertainties when in sliding mode ͓1-8͔. In recent years, many schemes have been proposed for flexible manipulators using VSC ͓9-11͔. In ͓12,13͔ we designed a swi ..."

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I Introduction Variable Structure Control ͑VSC͒ is well noted for its simplicity in implementation and outstanding robustness to system uncertainties when in sliding mode ͓1-8͔. In recent years, many schemes have been proposed for flexible manipulators using VSC ͓9-11͔. In ͓12,13͔ we designed a switching surface ϭ0 according to a reference model which has all real stable poles, hence tip vibration is reduced and tip position regulation can be achieved when the system approaches steady-state. However, to realize the ideal sliding mode, VSC requires infinite switching frequency which in practice is not achievable due to the limited sampling rate in digital implementation. The finite switching frequency associated with large control gains may cause chattering which is undesirable because it may excite the inherent flexible modes and is harmful to actuator devices. Replacing the signum function of the switching control with a continuous function can eliminate the chattering motion, but inevitably incurs less regulation accuracy. Generating the equivalent control profile is the ultimate objective of VSC in sliding mode, which assures perfect tracking and complete disturbance rejection. Under the boundedness and Lipschitz conditions of system dynamics, ͓1͔ provides an approach to measure the equivalent control: passing the control signal through a first-order low-pass filter. It requires that system stay strictly on the switching surface and the time constant of the filter approach zero, i.e., it demands an infinite switching frequency and an infinite filter bandwidth. Obviously, if the control environment is less severe compared with the worst case, we can come up with more appropriate control approaches. In this paper we focus on the flexible manipulator system under the repeatable control environment which has been considered in Ref. ͓14͔, etc. By virtue of repeatability, the past control sequence does reflect the dynamic characteristics of the uncertain system, and reflects the influence from reference signal as well as system perturbations to the regulation errors. The proposed Learning Variable Structure Control ͑LVSC͒ scheme has a simple structure consisting of two components in additive form: a standard VSC based on the known upper bounds using a continuous smoothing function, and a learning mechanism which simply adds up past control sequence which aims at extracting useful control knowledge from past control sequence so as to approximate the equivalent control. Rigorous proof based on energy function and functional analysis shows that LVSC system achieves the following novel properties: ͑1͒ The sequence of i where i denotes the iteration number converges uniformly to zero everywhere; ͑2͒ the uniformly bounded learning control sequence converges to the desired control profile, i.e., the equivalent control almost everywhere and the stringent requirements for obtaining equivalent control in ͓1͔ through filtering control signal are relaxed. In this paper we further address important issues arising from practical considerations and implement learning control in frequency domain by means of Fourier series expansion. Fourier series-based learning mechanism selectively updates coefficients of the learned frequency components and those coefficients are calculated by taking integration of control sequence over the entire control interval. Since the basis of Fourier space is orthogonal, the integral plays the role of averaging operation on the control sequence and alleviates the influence from unmodeled highfrequency flexible modes, noises and other nonrepeatable factors. In the subsequent discussions, the following basic notations will be used in all the section: ͉z͉ denotes the absolute value of a function z; ʈvʈϭ(v II Dynamic Formulation and Problem Statement. Refer to A Dynamic Model. The derivation of the dynamic equations of a one-link flexible manipulator given below follows then as in ͓15͔ where

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*Wen*-Jun *Cao* *Jian*-Xin Xu A Learning Variable Structure Controller of a Flexible One-Link Manipulator

"... I Introduction Variable Structure Control ͑VSC͒ is well noted for its simplicity in implementation and outstanding robustness to system uncertainties when in sliding mode ͓1-8͔. In recent years, many schemes have been proposed for flexible manipulators using VSC ͓9-11͔. In ͓12,13͔ we designed a swi ..."

Abstract
- Add to MetaCart

I Introduction Variable Structure Control ͑VSC͒ is well noted for its simplicity in implementation and outstanding robustness to system uncertainties when in sliding mode ͓1-8͔. In recent years, many schemes have been proposed for flexible manipulators using VSC ͓9-11͔. In ͓12,13͔ we designed a switching surface ϭ0 according to a reference model which has all real stable poles, hence tip vibration is reduced and tip position regulation can be achieved when the system approaches steady-state. However, to realize the ideal sliding mode, VSC requires infinite switching frequency which in practice is not achievable due to the limited sampling rate in digital implementation. The finite switching frequency associated with large control gains may cause chattering which is undesirable because it may excite the inherent flexible modes and is harmful to actuator devices. Replacing the signum function of the switching control with a continuous function can eliminate the chattering motion, but inevitably incurs less regulation accuracy. Generating the equivalent control profile is the ultimate objective of VSC in sliding mode, which assures perfect tracking and complete disturbance rejection. Under the boundedness and Lipschitz conditions of system dynamics, ͓1͔ provides an approach to measure the equivalent control: passing the control signal through a first-order low-pass filter. It requires that system stay strictly on the switching surface and the time constant of the filter approach zero, i.e., it demands an infinite switching frequency and an infinite filter bandwidth. Obviously, if the control environment is less severe compared with the worst case, we can come up with more appropriate control approaches. In this paper we focus on the flexible manipulator system under the repeatable control environment which has been considered in Ref. ͓14͔, etc. By virtue of repeatability, the past control sequence does reflect the dynamic characteristics of the uncertain system, and reflects the influence from reference signal as well as system perturbations to the regulation errors. The proposed Learning Variable Structure Control ͑LVSC͒ scheme has a simple structure consisting of two components in additive form: a standard VSC based on the known upper bounds using a continuous smoothing function, and a learning mechanism which simply adds up past control sequence which aims at extracting useful control knowledge from past control sequence so as to approximate the equivalent control. Rigorous proof based on energy function and functional analysis shows that LVSC system achieves the following novel properties: ͑1͒ The sequence of i where i denotes the iteration number converges uniformly to zero everywhere; ͑2͒ the uniformly bounded learning control sequence converges to the desired control profile, i.e., the equivalent control almost everywhere and the stringent requirements for obtaining equivalent control in ͓1͔ through filtering control signal are relaxed. In this paper we further address important issues arising from practical considerations and implement learning control in frequency domain by means of Fourier series expansion. Fourier series-based learning mechanism selectively updates coefficients of the learned frequency components and those coefficients are calculated by taking integration of control sequence over the entire control interval. Since the basis of Fourier space is orthogonal, the integral plays the role of averaging operation on the control sequence and alleviates the influence from unmodeled highfrequency flexible modes, noises and other nonrepeatable factors. In the subsequent discussions, the following basic notations will be used in all the section: ͉z͉ denotes the absolute value of a function z; ʈvʈϭ(v II Dynamic Formulation and Problem Statement. Refer to A Dynamic Model. The derivation of the dynamic equations of a one-link flexible manipulator given below follows then as in ͓15͔ where

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