Neural Sliding Mode Control for Magnetic Levitation ... - IEEE Xplore
2010 IEEE International Conference on Control Applications Part of 2010 IEEE Multi-Conference on Systems and Control Yokohama, Japan, September 8-10, 2010
Neural sliding mode control for magnetic levitation systems Xiaoou Li, Wen Yu
Abstract— Neural sliding mode control (NSMC) may decrease chattering of the sliding mode control (SMC) and improve control accuracy of the neural control (NC). There are some problems with the common parallel structure, such as the chattering is big at start stage. In order to overcome the above problem, we propose a new serial structure for NSMC, it is called two-stage neural sliding control. A dead-zone NC is used to make the tracking error bounded, then super-twisting second-order SMC is applied to guarantee finite time convergence. This new controllers has less chattering during its discrete realization, and ensures finite time convergence. Real-time experiments for a magnetic levitation system are presented to compare this new NSMC with regular controllers, such as PID, NC, SMC, and normal NSMC.
I. I NTRODUCTION Various control schemes have been developed in the literature for nonlinear systems [15], [13]. The traditional proportional and derivative (PD) controller is very simple and does not require any knowledge of mechanical systems. However it gives very large actuation and cannot guarantee zero tracking error with the existence of disturbances [4]. While model-based nonlinear control can remove this error, it is usually restricted to the case that the model is exactly known. Adaptive control can compensate unknown dynamics if the structure is known [17], [13]. The robust version of adaptive control may achieve a good performance with the system uncertainties and external disturbances [15], [16]. Non-adaptive control may also get high quality performance with the parameters or structure uncertainties. Robust feedback control [2] and optimal control [9] may guarantee closed-loop stable if the disturbances are bounded. All of these works assume an exact or partial knowledge of the nonlinear dynamics. Obviously, this is a requirement that generally cannot be met in practice. Normal sliding mode control (SMC) can eliminate large system uncertainties by a variable structure compensator with chattering problems [10]. Boundary layer SMC can assure no chattering happens when tracking error is less than but the tracking error converges to it is not asymptotically stable [17]. A new generation of SMC based on second-order sliding-mode has been recently developed by [12]. In [8], super-twisting and robust exact Xiaoou Li is with the Departamento de Computación, CINVESTAVIPN, Av.IPN 2508, México D.F., 07360, México. Wen Yu is with the Departamento de Control Automatico, CINVESTAV-IPN, Av. IPN 2508, 07360, México D.F., México.
978-1-4244-5363-4/10/$26.00 ©2010 IEEE
615
differentiator techniques were used to guarantee finite time convergence. But they require to know the nominal part of the system and an upper bound for the acceleration. The tracking error of SMC converges to zero if its gain is bigger than the upper bound of the unknown nonlinear function. When we do not have complete information, a modelfree nonlinear controller is needed. Neural networks can be considered as an alternative model-free controller because they offer potential benefit for nonlinear modeling. A good deal of researches demonstrate that neural networks indeed fulfill the promise of providing model-free learning controllers [7]. A neuro-adaptive controller by using a neural networks plus a servo feedback control was proposed in [5]. A hybrid neuro control for robot tracking was discussed in [1], where static neural networks are used to learn mass matrix, centrifugal and Coriolis force. A modified continuous-time version of backpropagation algorithm was given in [6] which does not need off-line learning and plant Jacobian. Because they used function approximation theory, the algorithms are sensitive to the training data and local minima. Due to neural modeling error, neural control (NC) cannot assure that tracking errors are asymptotically stable. Normal combination of NC and SMC applies them parallel, where SMC is used to compensate tracking error of the neural control [7]. The gain of SMC should be bigger than the upper bound of the neural approximation error. The chattering is smaller than pure SMC, because the upper bound of neural modeling error is smaller than the upper bound of whole nonlinear function. The final tracking error is asymptotically stable [6]. These neural sliding mode control (NSMC) apply NC and SMC at same time, and cause two problems. 1) When control error is big, the chattering of SMC become large. 2) When control error is small, the affect of NC loses, because this error is from unmodeled of NC, it cannot be decreased further by neural networks training. Also this type of NSMC cannot assure finite time convergence [20]. In order to overcome the above problems of NSMC, in this paper, NC and SMC are connected serially, it is called two-stage neural control. The neural network is used to approximate nonlinear systems. A dead-zone training algorithm is applied for the NC. After the tracking error enters the dead-zone, a super-twisting second-order SMC is used to guarantee finite time convergence of the NC. This type of control can ensure finite time convergence of NC and less chattering of SMC. This new NSMC
is successfully applied to a magnetic levitation system. Results from experimental tests carried out to validate the controller are presented.
= 0 then the following matrix Riccati equation + + + = 0
has a positive solution . So it is reasonable to introduce the second assumption. Let note the identification error as,
II. N EURO IDENTIFIER FOR NONLINEAR SYSTEMS A controlled nonlinear system can be expressed as: ·
= ( )
∈ < ∈
Neural sliding mode control for magnetic levitation systems Xiaoou Li, Wen Yu
Abstract— Neural sliding mode control (NSMC) may decrease chattering of the sliding mode control (SMC) and improve control accuracy of the neural control (NC). There are some problems with the common parallel structure, such as the chattering is big at start stage. In order to overcome the above problem, we propose a new serial structure for NSMC, it is called two-stage neural sliding control. A dead-zone NC is used to make the tracking error bounded, then super-twisting second-order SMC is applied to guarantee finite time convergence. This new controllers has less chattering during its discrete realization, and ensures finite time convergence. Real-time experiments for a magnetic levitation system are presented to compare this new NSMC with regular controllers, such as PID, NC, SMC, and normal NSMC.
I. I NTRODUCTION Various control schemes have been developed in the literature for nonlinear systems [15], [13]. The traditional proportional and derivative (PD) controller is very simple and does not require any knowledge of mechanical systems. However it gives very large actuation and cannot guarantee zero tracking error with the existence of disturbances [4]. While model-based nonlinear control can remove this error, it is usually restricted to the case that the model is exactly known. Adaptive control can compensate unknown dynamics if the structure is known [17], [13]. The robust version of adaptive control may achieve a good performance with the system uncertainties and external disturbances [15], [16]. Non-adaptive control may also get high quality performance with the parameters or structure uncertainties. Robust feedback control [2] and optimal control [9] may guarantee closed-loop stable if the disturbances are bounded. All of these works assume an exact or partial knowledge of the nonlinear dynamics. Obviously, this is a requirement that generally cannot be met in practice. Normal sliding mode control (SMC) can eliminate large system uncertainties by a variable structure compensator with chattering problems [10]. Boundary layer SMC can assure no chattering happens when tracking error is less than but the tracking error converges to it is not asymptotically stable [17]. A new generation of SMC based on second-order sliding-mode has been recently developed by [12]. In [8], super-twisting and robust exact Xiaoou Li is with the Departamento de Computación, CINVESTAVIPN, Av.IPN 2508, México D.F., 07360, México. Wen Yu is with the Departamento de Control Automatico, CINVESTAV-IPN, Av. IPN 2508, 07360, México D.F., México.
978-1-4244-5363-4/10/$26.00 ©2010 IEEE
615
differentiator techniques were used to guarantee finite time convergence. But they require to know the nominal part of the system and an upper bound for the acceleration. The tracking error of SMC converges to zero if its gain is bigger than the upper bound of the unknown nonlinear function. When we do not have complete information, a modelfree nonlinear controller is needed. Neural networks can be considered as an alternative model-free controller because they offer potential benefit for nonlinear modeling. A good deal of researches demonstrate that neural networks indeed fulfill the promise of providing model-free learning controllers [7]. A neuro-adaptive controller by using a neural networks plus a servo feedback control was proposed in [5]. A hybrid neuro control for robot tracking was discussed in [1], where static neural networks are used to learn mass matrix, centrifugal and Coriolis force. A modified continuous-time version of backpropagation algorithm was given in [6] which does not need off-line learning and plant Jacobian. Because they used function approximation theory, the algorithms are sensitive to the training data and local minima. Due to neural modeling error, neural control (NC) cannot assure that tracking errors are asymptotically stable. Normal combination of NC and SMC applies them parallel, where SMC is used to compensate tracking error of the neural control [7]. The gain of SMC should be bigger than the upper bound of the neural approximation error. The chattering is smaller than pure SMC, because the upper bound of neural modeling error is smaller than the upper bound of whole nonlinear function. The final tracking error is asymptotically stable [6]. These neural sliding mode control (NSMC) apply NC and SMC at same time, and cause two problems. 1) When control error is big, the chattering of SMC become large. 2) When control error is small, the affect of NC loses, because this error is from unmodeled of NC, it cannot be decreased further by neural networks training. Also this type of NSMC cannot assure finite time convergence [20]. In order to overcome the above problems of NSMC, in this paper, NC and SMC are connected serially, it is called two-stage neural control. The neural network is used to approximate nonlinear systems. A dead-zone training algorithm is applied for the NC. After the tracking error enters the dead-zone, a super-twisting second-order SMC is used to guarantee finite time convergence of the NC. This type of control can ensure finite time convergence of NC and less chattering of SMC. This new NSMC
is successfully applied to a magnetic levitation system. Results from experimental tests carried out to validate the controller are presented.
= 0 then the following matrix Riccati equation + + + = 0
has a positive solution . So it is reasonable to introduce the second assumption. Let note the identification error as,
II. N EURO IDENTIFIER FOR NONLINEAR SYSTEMS A controlled nonlinear system can be expressed as: ·
= ( )
∈ < ∈
