High-order iterative learning control for nonlinear systems

This paper presents a decentralized learning controller for large-scale nonlinear systems. A new learning method which is based on a performance index of sliding mode control is used to derive an iterative learning algorithm to obtain control actions. The effectiveness and the performance of the proposed approach are illustrated by the simulation results of a two-inverted pendulum system and a two-link manipulator. The attractive features also include a smaller residual error and robustness against nonlinear interactions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lots of iterative learning control (ILC) algorithms have been developed to improve tracking performance of systems that track same trajectory repetitively. In this paper, we use an open-closed-loop PD-type ILC scheme for the control of a class of nonlinear systems. Sufficient condition for guaranteeing the convergence of the learning scheme is given under a few reasonable assumptions, and then proved with linear operator theory. It is shown that the proportional coefficients of the ILC scheme is not a factor for guaranteeing the convergence, the condition is weaker than known results for similar ILC schemes.

To deal with iterative learning control of nonlinear systems with non‐repeatable control tasks, an adaptive PID‐type iterative learning controller is presented in this paper. The control structure is constructed based on a time‐varying boundary layer so that the initial state error problem can be solved and possible undesirable chattering behavior can be removed. The PID‐type iterative learning controller is designed by using the fact that a PID controller can be used as an approximator for an optimal controller in a compact set. An error equation is derived to represent the relation between the control input and system tracking error. Since the optimal PID gains for the best approximation are in general unavailable, the control parameters are tuned between successive iterations to ensure internal stability and error convergence. Under this adaptive PID‐type iterative learning controller, the referenced trajectory is allowed to be iteration‐varying which means that it is not required to be the same at each iteration. It is shown that all the adjustable parameters and the internal signals remain bounded for all iterations. Furthermore, the norm of tracking error vector at each time instant will asymptotically converge to a tunable residual set.

Most of the existing iterative learning control algorithms proposed for time-delay systems are based on the condition that the time-delay is precisely available, and the initial state is reset to the desired one or a fixed value at the start of each operation, which makes great limitation on the practical application of corresponding results. In this paper, a new iterative learning control algorithm is studied for a class of nonlinear system with uncertain state delay and arbitrary initial error. This algorithm needs to know only the boundary estimation of the state delay, and the initial state is updated, while the convergence of the system is guaranteed. Without state disturbance and output measurement noise, the system output will strictly track the desired trajectory after successive iteration. Furthermore, in the presence of state disturbance and measurement noise, the tracking error will be bounded uniformly. The convergence is strictly proved mathematically, and sufficient conditions are obtained. A numerical example is shown to demonstrate the effectiveness of the proposed approach.

A conditional iterative learning control (ILC) is derived for uncertain non-linear systems to track repetitive trajectory under the alignment initial condition. The learning of the conditional ILC is performed, only if the tracking error is dominated by the input instead of the initial conditions. The major advantage of this method is that it can simultaneously handle non-parametric uncertainties and the alignment initial condition. A simulation example is presented to illustrate the performance and implementation of the conditional ILC.

Iterative learning control (ILC) is a simple and effective method for the control of systems that perform same task repetitively. In this paper, open-closed-loop D-type ILC problem is investigated for a class of nonlinear time-varying systems. Under a few reasonable assumptions, sufficient condition for guaranteeing the convergence of the learning scheme is given and then proved with linear operator theory. The condition is independent of the specific form of the state space model, also it is weaker than known results for D-type ILC scheme.

A new iterative learning algorithm for nonlinear systems based on updated Newton method is investigated. The exchange row updated method is employed to derive the approximation of derivatives of the output function. The inverse of approximation of derivatives of the output function can be obtained by simple recurrent formula. This new algorithm has advantage of simplified calculation of the iterative learning control law in every iterative learning process. Sufficient conditions for convergence of this new updated Newton algorithm are given and proved. Moreover, a secant-type iterative learning control of new updated Newton method is presented. This approach avoid calculating partial derivatives such that the calculate work is largely reduced.

A new sliding mode–based iterative learning control algorithm is proposed and its stability is proved. An important characteristic of the algorithm is that the entire control signal is comprised of the sliding mode part and the iterative learning part, while the current signal of the iterative part is computed by its previous value and the sliding mode error. As a result, repeatable disturbances are cancelled out by the iterative learning part, while non–repeatable uncertainties are dealt with by the sliding mode part. The effectiveness is illustrated by simulations.

An adaptive iterative learning controller (ILC) is proposed for a class of nonlinear systems with hysteresis described by Bouc-Wen model. First, the property of Bouc-Wen mode is discussed and the upper bound is derived .Then an iterative learning controller is proposed through the Lyapunov-like function. This controller can eliminate the oscillation and overshoot caused by hysteresis effectively. Finally, simulation results are provided to illustrate the effectiveness of the proposed controller.

In this paper, a PD-type Sampled-Data Iterative Learning Control algorithm is proposed for a class of nonlinear systems with time delays and uncertain disturbances, including random input disturbance and random output measurement noise. By introducing Taloy’ s norm, a rigorous proof is given for the convergence at each sampling point. A sufficient condition is derived to ensure that the real output can track the desired trajectory within a small error if the sampling period is small enough. At last, simulation result illustrates the validity of the proposed control law.

In the past decade, there has been considerable research interest in learning control methods based on reinforcement learning (RL) and approximate dynamic programming (ADP). As an important class of function approximation techniques, kernel methods have been recently applied to improve the generalization ability of RL and ADP methods but most previous works were only based on simulation. This paper focuses on experimental studies of real-time online learning control for nonlinear systems using kernel-based ADP methods. Specifically, the kernel-based dual heuristic programming (KDHP) method is applied and tested on real-time control systems. Two kernel-based online learning control schemes are presented for uncertain nonlinear systems by using simulation data and online sampling data, respectively. Learning control experiments were performed on a single-link inverted pendulum system as well as a double-link inverted pendulum system. From the experimental results, it is shown that both online learning control schemes, either using simulation data or using real sampling data, are effective for approximating near-optimal control policies of nonlinear dynamical systems with model uncertainties. In addition, it is demonstrated that KDHP can achieve better performance than conventional DHP, which uses multilayer perceptron neural networks.

In this paper, a hybrid deep learning neural network controller (HDLNNC) for nonlinear systems is proposed. The proposed controller structure consists of a multi-layer feed-forward neural network, which can be trained based on the hybrid deep learning. The Lyapunov stability criterion is used to develop an adaptive learning rate due to the learning rate of the updating parameters plays a worthy role in achieving the stability of a system. To show the robustness of the proposed controller and its performance, several tests such as disturbance signals and parameter variations are carried on a numerical example. In this concern, the practical implementation of the proposed HDLNNC is executed on a real system. The results indicate that the proposed controller is able to improve the system performance compared with other existing controllers.

SummaryThis paper proposes robust iterative learning control schemes for continuous‐time nonlinear systems with various nonparametric uncertainties under nonuniform trial length circumstances. The nonuniform trial length is described by a random variable, which causes a random data missing problem while designing and analyzing algorithms for the precise tracking problem. Three common types of nonparametric uncertainties are taken into account: norm‐bounded uncertainty, variation‐norm‐bounded uncertainty, and norm‐bounded uncertainty with unknown coefficients. A novel composite energy function is introduced with the help of a newly defined virtual tracking error for the asymptotical convergence of the proposed schemes. Extensions to multiple‐input–multiple‐output cases are also elaborated. Illustrative simulations are provided to verify the theoretical results.

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.

The output tracking control problem via state feedback is addressed for a class of single input-single output nonlinear systems which are affected by extended matching unstructured uncertainties. Under the assumption that the output reference signal is sufficiently smooth and periodic with known period, a robust adaptive learning control is designed, which learns the unstructured unknown periodic disturbance signals due to system uncertainties by identifying the Fourier coefficients of any truncated approximation while guaranteeing L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> transient performances. For any initial condition of the system in an arbitrary given compact set, by properly setting the control parameters: i) the output tracking error exponentially converges to a residual set which may be arbitrarily reduced by increasing the number of terms in the truncated Fourier series expansions; ii) when the unknown periodic disturbances can be represented by finite Fourier series expansions, the output tracking error exponentially converges to zero.