Perfect tracking for maximum-phase nonlinear systems by iterative learning control

A method to track a desired trajectory by iterative learning control is proposed for uncertain maximum-phase nonlinear systems. The relation between the variations in the initial state, input and output is derived and it is shown that the inverse mapping from the desired output to the initial state and input is stable using the time reversal of unstable manifolds for a maximum-phase system as given by Doyle et al. Based on these facts, an input update law is proposed to find the initial state and the input for perfect tracking. Also, it is shown that perfect tracking can be made possible over a finite control horizon by using a non-causal input starting at any fixed state. Simulation results show that the proposed method works well.     

A generalised optimal linear quadratic tracker with universal applications – part 1: continuous-time systems

This paper presents a generalised optimal linear quadratic analog tracker (LQAT) with universal applications for the continuous-time (CT) systems. This includes: (1) a generalised optimal LQAT design for the system with the pre-specified trajectories of the output and the control input and additionally with both the input-to-output direct-feedthrough term and known/estimated system disturbances or extra input/output signals; (2) a new optimal filter-shaped proportional plus integral state-feedback LQAT design for non-square non-minimum phase CT systems to achieve a minimum phase-like tracking performance; (3) a new approach for computing the control zeros of the given non-square CT system; and (4) a one-learning-epoch input-constrained iterative learning LQAT design for the repetitive CT system.     

A generalised optimal linear quadratic tracker with universal applications. Part 2: discrete-time systems

Contrastive to Part 1, Part 2 presents a generalised optimal linear quadratic digital tracker (LQDT) with universal applications for the discrete-time (DT) systems. This includes (1) a generalised optimal LQDT design for the system with the pre-specified trajectories of the output and the control input and additionally with both the input-to-output direct-feedthrough term and known/estimated system disturbances or extra input/output signals; (2) a new optimal filter-shaped proportional plus integral state-feedback LQDT design for non-square non-minimum phase DT systems to achieve a minimum-phase-like tracking performance; (3) a new approach for computing the control zeros of the given non-square DT systems; and (4) a one-learning-epoch input-constrained iterative learning LQDT design for the repetitive DT systems.     

Asymptotic output tracking in a class of non-minimum phase nonlinear systems via learning-based inversion

Asymptotic output tracking of non-minimum phase (NMP) nonlinear systems has been a popular topic in control theory and applications. Many approaches have focused on finding solutions under minimal assumptions either in the target system or desired trajectories, as there is no general solution available. In this article, we propose a practical and simple solution for cases where the reference trajectory is periodic in time. Our approach employs a learning-based scheme to iteratively determine the desired feedforward input. Unlike previous learning-based frameworks, our method only requires the output tracking error to update the feedforward input iteratively and can be applicable to NMP systems. Our method retains the key advantages of the learning-based framework, including robustness to parameter uncertainties and periodic disturbances. We evaluate the effectiveness of our algorithm using simulation results with an inverted pendulum on a cart, a typical NMP nonlinear system.     

Composite adaptive locally weighted learning control for multi-constraint nonlinear systems

A composite adaptive locally weighted learning (LWL) control approach is proposed for a class of uncertain nonlinear systems with system constraints, including state constraints and asymmetric control saturation in this paper. The system constraints are tackled by considering the control input as an extended state variable and introducing barrier Lyapunov functions (BLFs) into the backstepping procedure. The system uncertainty is approximated by a composite adaptive LWL neural networks (NNs), in which a prediction error is constructed by using a series-parallel identification model, and NN weights are updated by both the tracking error and the prediction error. The update law with composite error feedback improves uncertainty approximation accuracy and trajectory tracking accuracy. The feasibility and effectiveness of the proposed approach have been demonstrated by formal proof and simulation results.     

Multidimensional Taylor network adaptive control for MIMO time‐varying uncertain nonlinear systems with noises

In this paper, an adaptive control approach based on the multidimensional Taylor network (MTN) is proposed here for the real‐time tracking control of multiple‐input–multiple‐output (MIMO) time‐varying uncertain nonlinear systems with noises. Two MTNs are used to formulate the optimum control and adaptive filtering approaches. The feed‐forward MTN controller (MTNC) is developed to realize the precise tracking control. The closed‐loop errors between the filtered outputs and expected values are directly chosen as the MTNC's inputs. A valid initial value selection scheme for the weights of the MTNC, which can ensure the initial stability of adaptive process, is introduced. The proposed MTNC can update its weights online according to errors caused by system's uncertain factors, based on stable learning rate. The resilient backpropagation algorithm and the adaptive variable step size algorithm via linear reinforcement are utilized to update the MTNC's weights. The MTN filter (MTNF) is developed to eliminate measurement noises and other stochastic factors. The proposed adaptive MTN filtering system possesses the distinctive properties of the Lyapunov theory–based adaptive filtering system and MTN. Lyapunov function of the filtering errors between the measured values and MTNF's outputs is defined. By properly choosing the weights update law in the Lyapunov sense, the MTNF's outputs can asymptotically converge to the desired signals. The design is independent of the stochastic properties of the input disturbances. Simulation of the MTN‐based control is conducted to test the effectiveness of the presented results.     

Output-feedback Dynamic Surface Control for a Class of Nonlinear Non-minimum Phase Systems

In this paper, an output-feedback tracking controller is proposed for a class of nonlinear non-minimum phase systems. To keep the unstable internal dynamics bounded, the method of output redefinition is applied to let the stability of the internal dynamics depend on that of redefined output, thus we only need to consider the new external dynamics rather than internal dynamics in the process of designing control law. To overcome the explosion of complexity problem in traditional backstepping design, the dynamic surface control (DSC) method is firstly used to deal with the problem of tracking control for the nonlinear non-minimum phase systems. The proposed outputfeedback DSC controller not only forces the system output to asymptotically track the desired trajectory, but also drives the unstable internal dynamics to follow its corresponding bounded and causal ideal internal dynamics, which is solved via stable system center method. Simulation results illustrate the validity of the proposed output-feedback DSC controller.      

Flexible Links Manipulators: from Modelling to Control

This paper relates recent results obtained in the field of modelling and control of flexible link manipulators and proposes an investigation of the problem raised by this type of systems (at least in the planar case). First, adopting the modal floating frame approach and the Newton–Euler formalism, we propose an extension of the models for control to the case of fast dynamics and finite deformations. This dynamic model is based on a nonlinear generalisation of the standard Euler–Bernoulli kinematics. Then, based on the models recalled we treat the end-effector tracking problem for the one-link case as well as for the planar multi-link case. For the one-link system, we propose two methods, the first one is based on causal stable inversion of linear non-minimum phase model via output trajectory planning. The other one is an algebraic scheme, based on the parametrization of linear differential operators. For the planar multi-link case the control law proposed is based on causal stable inversion over a bounded time domain of nonlinear non-minimum phase systems. Numerical tests are presented together with experimental results, displaying the well behaved of these approaches.     

Iterative learning control for linear discrete time nonminimum phase systems

This paper investigates iterative learning control for linear discrete time nonminimum phase systems. First, iterative learning control with advanced output data is considered for maximum phase systems. Next, the results are extended to nonminimum phase systems. The stability of the inverse mapping from the desired output to the input is proven based on the results for maximum phase systems. The input should be updated with the output which is more advanced than the input by the sum of the relative degree of the system and the number of nonminimum phase zeros. An example is given to indicate the importance of proper advances of output in the input update law.     

Robust output tracking control of a class of non-minimum phase systems and application to VTOL aircraft

In this article, we study the output tracking control of a class of MIMO nonlinear non-minimum phase systems in the presence of input disturbances. In order to attenuate the effects of disturbances, the method of uncertainty and disturbance estimator (UDE) is extended to the controller design for non-minimum phase systems. Due to the fact that the accumulated disturbances is composed of internal states and external disturbances, a different stability analysis is given, and the overall closed-loop system is proved to be semi-globally stable. The proposed state-feedback controller not only forces system outputs to asymptotically track desired trajectories, but also drives the unstable internal dynamics to follow bounded and causal ideal internal dynamics (IID) solved via stable system centre (SSC) method. Simulation results demonstrate that the proposed controller achieves excellent tracking and disturbance rejection performance via the example of VTOL aircraft which has been the benchmark of nonlinear non-minimum phase systems.     

H∞ tracking-based sliding mode control foruncertain nonlinear systems via an adaptive fuzzy-neural approach

A novel adaptive fuzzy-neural sliding-mode controller with H(infinity) tracking performance for uncertain nonlinear systems is proposed to attenuate the effects caused by unmodeled dynamics, disturbances and approximate errors. Because of the advantages of fuzzy-neural systems, which can uniformly approximate nonlinear continuous functions to arbitrary accuracy, adaptive fuzzy-neural control theory is then employed to derive the update laws for approximating the uncertain nonlinear functions of the dynamical system. Furthermore, the H(infinity) tracking design technique and the sliding-mode control method are incorporated into the adaptive fuzzy-neural control scheme so that the derived controller is robust with respect to unmodeled dynamics, disturbances and approximate errors. Compared with conventional methods, the proposed approach not only assures closed-loop stability, but also guarantees an H(infinity) tracking performance for the overall system based on a much relaxed assumption without prior knowledge on the upper bound of the lumped uncertainties. Simulation results have demonstrated that the effect of the lumped uncertainties on tracking error is efficiently attenuated, and chattering of the control input is significantly reduced by using the proposed approach.     

Conditional learning control for alignment initial condition

This article tries to handle the alignment initial condition for contraction mapping based iterative learning control, such that the system can operate continuously without any initial condition reset. This goal is achieved for a class of nonlinear systems through the proposed conditional learning control, which has several advantages over the alternative method, adaptive learning control. The conditional learning control guarantees that sufficient knowledge can be learned to update the input and achieve perfect output tracking, despite the non-identical initial conditions. The sufficient conditions of either monotonic or strictly monotonic convergence of the input sequence, and the choice of learning gains are given. The performance of the proposed method is illustrated by simulated examples.     

Improved adaptive fuzzy backstepping control of a magnetic levitation system based on Symbiotic Organism Search

Magnetic levitation systems have become very important in many applications. Due to their instability and high nonlinearity, such systems pose a challenge to many researchers attempting to design high-performance and robust tracking control. This paper proposes an improved adaptive fuzzy backstepping control for systems with uncertain input nonlinear function (uncertain parameters and structure), and applies it to a magnetic levitation system, which is a typical representative of such systems. An adaptive fuzzy system is used to approximate unknown, partially known or uncertain input nonlinear functions of a magnetic levitation system. An adaptation law is obtained based on Ljapunov analysis in order to guarantee closed-loop stability and good tracking performance. Initial adaptive and control parameters have been initialized with Symbiotic Organism Search optimization algorithm, due to strong non-linearity and instability of the magnetic levitation system. The theoretical background of the proposed control method is verified with a simulation study and implementation on a laboratory experimental application.     

Robust output tracking of nonlinear systems with mismatched uncertainties

The problem of robust output tracking for a class of uncertain nonlinear systems which do not satisfy the conventional matching condition is considered. The main assumption on the uncertainty is that the triangularity condition is satisfied. Based on backstepping method and input/output linearization approach, we propose a class of non-adaptive state feedback controllers which can guarantee exponential stability of the tracking error for the uncertain nonlinear systems first. Next, adaptive control laws are developed so that no prior knowledge of the bounds on the uncertainties is required. By updating these upper bounds, we design a class of adaptive robust controllers. It is shown that under the proposed adaptive robust control the tracking error of the controlled system converges to zero as time approaches infinity.     

基于最优状态转移的非因果稳定逆

本文研宄非最小相位系统的精确跟踪问题.理想情况下,非最小相位系统针对参考轨迹的精确跟踪可以通过非因果稳定逆方法实现,但控制输入需从负无穷处开始作用.而在实际情况下应用非因果稳定逆算法时,控制输入通过延拓提前作用的时间是有限的,只能得到近似的跟踪效果.本文提出了一种基于最优状态转移的非因果稳定逆算法,能够在实际情况下实现非最小相位系统对参考轨迹的精确跟踪,放松了稳定逆方法对系统的初始状态和延拓时间的限制,而且在相同跟踪效果的条件下,比近似稳定逆方法的延拓时间更短.对比仿真结果验证了所提方法的性能.     

Robust adaptive learning control for nonlinear systems with extended matching unstructured uncertainties

The exponential output tracking problem for a class of single‐input, single‐output uncertain nonlinear systems, including systems with extended matching unstructured uncertainties and without a well‐defined global relative degree, is addressed. Conditions on the uncertain system dynamics are derived, which allow us to design a state‐feedback learning control achieving semi‐global exponential output tracking of sufficiently smooth and periodic reference signals of known period, while guaranteeing ??₂ and ??_∞ transient performances during the learning phase. The application of the proposed learning approach to the position tracking control problem for uncertain permanent magnet step motors with non‐sinusoidal flux distribution and uncertain position‐dependent load torque allows us to provide a solution to a yet unsolved problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct adaptive fuzzy backstepping control of uncertain nonlinear systems in the presence of input saturation

In this paper, a novel direct adaptive fuzzy control approach is presented for uncertain nonlinear systems in the presence of input saturation. Fuzzy logic systems are directly used to tackle unknown nonlinear functions, and the adaptive fuzzy tracking controller is constructed by using the backstepping recursive design techniques. To overcome the problem of input saturation, a new auxiliary design system and Nussbaum gain functions are incorporated into the control scheme, respectively. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB), and the tracking error converges to a small neighborhood of the origin. A simulation example is included to illustrate the effectiveness of the proposed approach. Two key advantages of the scheme are that (i) the direct adaptive fuzzy control method is proposed for uncertain nonlinear system with input saturation by using Nussbaum function technique and (ii) The number of the online adaptive learning parameters is reduced.     

Model‐Reference Adaptive Moment Control of Uncertain Nonlinear Stochastic Systems

In this paper, a new model‐reference adaptive moment control method is proposed to control the first and second moments of an uncertain nonlinear system with additive external stochastic excitation. This method has established a closed‐loop control system that calculates an adaptive stochastic nonlinear input by introducing a Lyapunov function and adaptive update law. The proposed adaptive structure is innovative in trying to minimize two errors simultaneously: the moments tracking error and the error between the nonlinear system output and reference model. Furthermore, the proposed method can control the expected and covariance matrices of the states without needing to solve the complicated Fokker‐Planck‐Kolmogorov differential equation or using the approximate methods. Simulation has been performed on two practical examples, which show a good performance for the designed controller.     

Output feedback adaptive DSC for nonlinear systems with guaranteed L-infinity tracking performance

In this paper, the problem of output tracking for a class of uncertain nonlinear systems is considered. First, neural networks are employed to cope with uncertain nonlinear functions, based on which state estimation is constructed. Then, an output feedback control system is designed by using dynamic surface control (DSC). To guarantee the L-infinity tracking performance, an initialization technique is presented. The main feature of the scheme is that explosion of complex- ity problem in backstepping control is avoided, and there is no need to update the unknown parameters including control gains as well as neural networks weights, the adaptive law with one update parameter is necessary only at the first design step. It is proved that all signals of the closed-loop system are semiglobally uniformly ultimately bounded and the L-infinity performance of system tracking error can be guaranteed. Simulation results demonstrate the effectiveness of the proposed scheme.     

带有干扰观测器的线控转向系统复合自适应神经网络控制

考虑车辆线控转向(SbW)系统存在不确定动态特性以及外界干扰影响.本文提出一种带有干扰观测器的复合自适应神经网络实现SbW系统的精确建模与稳定控制.首先,利用神经网络在线逼近系统不确定动态,避免控制器设计中使用到系统模型的先验知识.然后,结合系统的跟踪误差与建模误差提出一种新的复合自适应学习率来更新神经网络的权值,从而加快跟踪误差的收敛速度.最后通过设计干扰观测器补偿系统受到摩擦力矩、回正力矩与神经网络逼近误差的影响,提高了系统的抗干扰能力.李雅普诺夫稳定性理论证明了闭环系统的跟踪误差信号一致最终有界.数值仿真与硬件在环实验结果验证了该控制方法的有效性和优越性.