Asymptotic state tracking in a class of nonlinear systems via learning-based inversion

Describes a learning approach to asymptotic state tracking in a class of nonlinear systems. The tracking problem considered concerns the case when the tracking-error dynamics are described by a set of time-varying nonlinear differential equations, which are periodic in time with a known period. Our iterative update scheme is based on the specific property that the learning system tends to oscillate in steady state. In fact, our approach extends in a very natural manner the idea of the well-known iterative learning control for the case of finite-time tracking problems to the case of infinite-time asymptotic tracking problems. The best advantage of the proposed leaning approach is that it is computationally simple and does not require one to solve any complicated equations based on full system dynamics. We explore the conditions under which a periodic nonlinear system exhibits a steady-state oscillation. Our work also can be viewed to provide a learning-based solution to the input-state inversion problem. The generality and practicality of our work is demonstrated through rigorous performance analysis and simulation using a robot manipulator.     

A homotopy method for exact output tracking of some non‐minimum phase nonlinear control systems

This paper presents a method for non‐causal exact dynamic inversion for a class of non‐minimum phase nonlinear systems, which seems to be an alternative to those existing in the literature. This method is based on a homotopy procedure that allows to find a ‘small’ periodic solution of a desired equation by a continuous deformation of a known periodic solution of a simpler auxiliary system. This method allows to face the exact output tracking problem for some non‐minimum phase systems that are well known in the literature, such as the inverted pendulum, the motorcycle and the CTOL aircraft. Copyright © 2008 John Wiley & Sons, Ltd.     

Hermite Polynomials for Iterative Output Replanning for Flat Systems Affected by Additive Noise

The paper considers the output tracking problem for nonlinear systems whose performance output is also a flat output of the system itself. A desired output signal is sought on the actual performance output by using a feedforward inverse input that is periodically updated with discrete‐time feedback of the sampled state of the system. The proposed method is based on an iterative output replanning that uses the desired output trajectory and the sampled state to replan an output trajectory whose inverse input helps in reducing the tracking error. This iterative replanning exploits the Hermite interpolating polynomials to achieve an overall arbitrarily smooth input and a tracking error that can be made arbitrarily small if the state sampling period is sufficiently small and mild assumptions are considered. Some simulation results are presented for the cases of a unicycle and a one‐trailer system affected by additive noise.     

一类不确定非线性系统的指数鲁棒输出跟踪

研究一类不确定非线性系统的鲁棒输出跟踪控制问题。应用输入／输出反馈线性化法和李亚普诺夫方法,提出一种基于不确定项上界的连续型鲁棒输出跟踪控制器设计方法。该控制器不仅可确保闭环系统的状态一致最终有界,使系统输出按指数规律跟踪期望输出,而且计算简单,更易实现。仿真结果证明了该方法的可行性与有效性。     

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.     

非线性滞后离散系统的学习控制算法

讨论了滞后非线性离散系统的学习控制问题,由于所给的学习算法及学习控制过程 中,没有涉及和用到相应于理想输出yd的理想输入ud及对应于系统的理想状态xd,故对被 控对象的动力学信息要求得很少,只是一种定性上的Lipschitz条件.所给出的控制算法不仅 收敛,而且也保证了对期望目标在通常意义下的跟踪(而不是像目前有些结果那样,只是跟踪 到期望目标的某一个邻域范围内).而且这些算法还以目前一些通常的算法为特例.     

Data‐driven high‐order terminal iterative learning control with a faster convergence speed

In this paper, a novel high‐order optimal terminal iterative learning control (high‐order OTILC) is proposed via a data‐driven approach for nonlinear discrete‐time systems with unknown orders in the input and output. The objective is to track the desired values at the endpoint of the operation cycle. The terminal tracking errors over more than one previous iterations are used to enhance the high‐order OTILC's performance with faster convergence. From rigor of the analysis, the monotonic convergence of the terminal tracking error is proved along the iteration direction. More importantly, the condition for a high‐order OTILC to outperform the low‐order ones is first established by this work. The learning gain is not fixed but iteratively updated by using the input and output (I/O) data, which enhances the flexibility of the proposed controller for modifications and expansions. The proposed method is data‐driven in which no explicit models are used except for the input and output data. The applications to a highly nonlinear continuous stirred tank reactor and a highly nonlinear fed‐batch fermentater demonstrate the effectiveness of the proposed high‐order OTILC design.     

Semi‐Global Output Feedback Tracking to Reference System with Input for a Benchmark Nonlinear System

In this paper, we solve the output feedback semi‐global tracking problem of the nonlinear benchmark RTAC system, in which the reference signals are output trajectories of a simple control systems subject to some input. The introduction of the input enlarges the class of the reference signals that can be tracked. And the proposed solution covers the previous results of output feedback global tracking, namely the solution is still global when the input is zero. Finally, a simulation illustrates the effectiveness of our approach.     

Adaptive neural network tracking control for uncertain nonlinear systems with input delay and saturation

In this article, the adaptive tracking control problem is considered for a class of uncertain nonlinear systems with input delay and saturation. To compensate for the effect of the input delay and saturation, a compensation system is designed. Radial basis function neural networks are directly utilized to approximate the unknown nonlinear functions. With the aid of the backstepping method, novel adaptive neural network tracking controllers are developed, which can guarantee all the signals in the closed‐loop system are semiglobally uniformly ultimately bounded, and the system output can track the desired signal with a small tracking error. In the end, a simulation example is given to illustrate the effectiveness of the proposed methods.     

A learning approach to tracking in mechanical systems with friction

This paper describes a novel learning control scheme for tracking periodic trajectories in mechanical systems with friction. It is based on the fact that the solution of the closed-loop system tends to be periodic in steady state. When the closed-loop system reaches the steady state, the proposed learning control scheme updates the control input. By doing this iteratively, the proposed learning control scheme eventually can drive the tracking error to zero. Neither the information of the system mass nor the parametric model for friction is required for successful tracking. In particular the proposed learning control scheme can be implemented at cheap cost on a commercially available microprocessor. Furthermore, its generality is well supported through rigorous convergence analysis     

Global asymptotic tracking for nonminimum‐phase nonlinear systems in output feedback form

The problem of global asymptotic tracking by output feedback is studied for a class of nonminimum‐phase nonlinear systems in output feedback form. It is proved that the problem is solvable by an n‐dimensional output feedback controller under the two conditions: (a) the nonminimum‐phase nonlinear system can be rendered minimum‐phase by a virtual output; and (b) the internal dynamics of the nonlinear system driven by a desired signal and its derivatives has a bounded solution trajectory. With the help of a new coordinate transformation, a constructive method is presented for the design of a dynamic output tracking controller. An example is given to validate the proposed output feedback tracking control scheme.     

Adaptive Learning Control of Nonlinear Systems by Output Error Feedback

This paper addresses the problem of designing an output error feedback control for single-input, single-output nonlinear systems with uncertain, smooth, output-dependent nonlinearities whose local Lipschitz constants are known. The considered systems are required to be observable, minimum phase with known relative degree and known high frequency gain sign: linear systems are included. The reference output signal is assumed to be smooth and periodic with known period. By developing in Fourier series expansion a suitable periodic input reference signal, an output error feedback adaptive learning control is designed which ldquolearnsrdquo the input reference signal by identifying its Fourier coefficients: bounded closed loop signals and exponential tracking of both input and output reference signals are obtained when the Fourier series expansion is finite, while arbitrary small tracking errors are exponentially achieved otherwise. The resulting control is not model based, is independent of the system order and depends only on the relative degree, the reference signal period and the high frequency gain sign.     

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.     

Multi‐parametric extremum seeking‐based iterative feedback gains tuning for nonlinear control

We study in this paper the problem of iterative feedback gains auto‐tuning for a class of nonlinear systems. For the class of input–output linearizable nonlinear systems with bounded additive uncertainties, we first design a nominal input–output linearization‐based robust controller that ensures global uniform boundedness of the output tracking error dynamics. Then, we complement the robust controller with a model‐free multi‐parametric extremum seeking control to iteratively auto‐tune the feedback gains. We analyze the stability of the whole controller, that is, the robust nonlinear controller combined with the multi‐parametric extremum seeking model‐free learning algorithm. We use numerical tests to demonstrate the performance of this method on a mechatronics example. Copyright © 2016 John Wiley & Sons, Ltd.     

一种非线性多环递归自适应跟踪控制

针对控制输入有界的变参数广义高阶非线性系统跟踪控制问题,给出一种多环递归跟踪的鲁棒控制方法.通过分层引入虚拟跟踪器,将高阶系统分解为多个独立子系统;内环虚拟跟踪器使内环输出指数收敛于外环虚拟输入,最内环设计自适应控制器补偿参数摄动和外部干扰,并保证输出指数收敛于外环虚拟输入;多环递归跟踪实现系统输出精确跟踪期望输入,理论证明闭环系统的全局渐近收敛性.数值仿真验证了多环跟踪控制器的可行性和合理性.     

Constrained data-driven optimal iterative learning control

A constrained optimal ILC for a class of nonlinear and non-affine systems, without requiring any explicit model information except for the input and output data, is proposed in this work. In order to address the nonlinearities, an iterative dynamic linearization method without omitting any information of the original plant is introduced in the iteration direction. The derived linearized data model is equivalent to the original nonlinear system and reflects the real-time dynamics of the controlled plant, rather than a static approximate model. By transferring all the constraints on the system output, control input, and the change rate of input signals into a linear matrix inequality, a novel constrained data-driven optimal ILC is developed by minimizing a predesigned objective function. The optimal learning gain is unfixed and updated iteratively according to the input and output measurements, which enhances the flexibility regarding modifications and expansions of the controlled plant. The results are further extended to the point-to-point control tasks where the exact tracking performance is required only at certain points and a constrained data-driven optimal point-to-point ILC is proposed by only utilizing the error measurements at the specified points only.     

带有输入饱和及输出受限非仿射系统固定时间跟踪控制

针对具有量化输入饱和及输出受限的非线性非仿射系统,提出固定时间自适应神经网络跟踪控制方法.引入中值定理解决系统具有非仿射结构的问题;基于反步法,使用Barrier Lyapunov函数约束系统输出,并利用RBF神经网络逼近未知函数;根据固定时间控制理论设计输入信号,该输入信号由滞后量化器量化,以降低控制信号的通信速率,并保证该系统在满足量化输入饱和及输出受限的条件下,系统可以在固定时间内跟踪上期望信号,且该系统收敛时间与初始状态无关.最后通过Matlab仿真软件验证所设计控制器的有效性.     

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.     

Robust‐decentralized tracking control for a class of uncertain MIMO nonlinear systems with time‐varying delays

A new control design method based on signal compensation is proposed for a class of uncertain multi‐input multi‐output (MIMO) nonlinear systems in block‐triangular form with nonlinear uncertainties, unknown virtual control coefficients, strongly coupled interconnections, time‐varying delays, and external disturbances. By this method, the controller design is performed in a backstepping manner. At each step of backstepping procedure, a nominal virtual controller is first designed to get desired output tracking for the nominal disturbance‐free subsystem, and then a robust virtual compensator is designed to restrain the effect of the uncertainties, delays involved in the subsystem, and the couplings among the subsystems. The designed controller is linear and time‐invariant, so the explosion of complexity in the control law is avoid. It is proved that robust stability and robust practical tracking property of the closed‐loop system can be ensured, and the tracking errors can be made as small as desired. Copyright © 2013 John Wiley & Sons, Ltd.     

一类非线性系统的自组织模糊CMAC神经网络自适应重构跟踪控制

提出了一种非线性系统的自组织模糊CMAC(SOFCMAC)神经网络自适应重构跟踪控制方法,首先通过构造增广系统,设计出线性渐近跟踪控制器,然后采用SOFCMAC神经网络在线重构系统的非线性特性,以消除非线性特性引起的系统误差,可保证非线性系统闭环稳定并使系统输出跟踪期望输出.仿真算例证明了SOFCMAC神经网络自适应重构跟踪控制系统的稳定性.