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.     

Iterative learning control design with high‐order internal model for discrete‐time nonlinear systems

In this paper, a high‐order internal model (HOIM)‐based iterative learning control (ILC) scheme is proposed for discrete‐time nonlinear systems to tackle the tracking problem under iteration‐varying desired trajectories. By incorporating the HOIM that is utilized to describe the variation of desired trajectories in the iteration domain into the ILC design, it is shown that the system output can converge to the desired trajectory along the iteration axis within arbitrarily small error. Furthermore, the learning property in the presence of state disturbances and output noise is discussed under HOIM‐based ILC with an integrator in the iteration axis. Two simulation examples are given to demonstrate the effectiveness of the proposed control method. Copyright © 2016 John Wiley & Sons, Ltd.     

Computationally‐Light Non‐Lifted Data‐Driven Norm‐Optimal Iterative Learning Control

Computational complexity and model dependence are two significant limitations on lifted norm optimal iterative learning control (NOILC). To overcome these two issues and retain monotonic convergence in iteration, this paper proposes a computationally‐efficient non‐lifted NOILC strategy for nonlinear discrete‐time systems via a data‐driven approach. First, an iteration‐dependent linear representation of the controlled nonlinear process is introduced by using a dynamical linearization method in the iteration direction. The non‐lifted NOILC is then proposed by utilizing the input and output measurements only, instead of relying on an explicit model of the plant. The computational complexity is reduced by avoiding matrix operation in the learning law. This greatly facilitates its practical application potential. The proposed control law executes in real‐time and utilizes more control information at previous time instants within the same iteration, which can help improve the control performance. The effectiveness of the non‐lifted data‐driven NOILC is demonstrated by rigorous analysis along with a simulation on a batch chemical reaction process.     

High‐order internal model‐based iterative learning control design for nonlinear distributed parameter systems

This article deals with the problem of iterative learning control algorithm for a class of nonlinear parabolic distributed parameter systems (DPSs) with iteration‐varying desired trajectories. Here, the variation of the desired trajectories in the iteration domain is described by a high‐order internal model. According to the characteristics of the systems, the high‐order internal model‐based P‐type learning algorithm is constructed for such nonlinear DPSs, and furthermore, the corresponding convergence theorem of the presented algorithm is established. It is shown that the output trajectory can converge to the desired trajectory in the sense of (L²,λ) ‐norm along the iteration axis within arbitrarily small error. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.     

Compensation‐based data‐driven ILC with input and output package dropouts

This paper explores the problem of random data loss at both input and output sides and proposes a compensation‐based data‐driven iterative learning control (cDDILC) to refrain from deteriorating of the control performance due to the data loss. A linear data model is first established to describe the input‐output dynamics of a repetitive control system in the iteration domain. The linear data model, which only virtually exists in the computer without any physical backgrounds, is employed as a predictive model to estimate and compensate the lost output data. Meanwhile, the lost input data is replaced by the corresponding input of the same time instant in the latest previous iterations. Then, a cDDILC is proposed by introducing two Bernoulli random variables to describe the stochastic data loss at both input and output sides. The proposed cDDILC method is data driven and independent of a precise plant model. Although the design and analysis of the cDDILC start from a MIMO linear repetitive system, one can easily extend the results to a MIMO nonlinear nonaffine one. Theoretical analysis and simulations confirm the efficiency of the proposed cDDILC method.     

Adaptive Terminal ILC for Iteration‐varying Target Points

Terminal iterative learning control (TILC) has been developed to reduce the error between system output and a fixed desired point at the terminal end of operation interval over iterations. In this work, the desired terminal point is not fixed but allowed to change run‐to‐run among a set of fixed points and a new adaptive terminal iterative learning control scheme is developed to achieve learning objective over iterations. The control signal is updated from the measured terminal value at the end of a run, instead of the whole output trajectory. Although the reference terminal point is iteration‐varying, the new adaptive TILC guarantees that the tracking error converges to zero iteratively. Both rigorous mathematical analysis and simulation results confirm the applicability and effectiveness of the proposed approach.     

Iterative Learning Control for Linear Discrete‐Time Systems with Unknown High‐Order Internal Models: A Time‐Frequency Analysis Approach

This work focuses on the iterative learning control (ILC) for linear discrete‐time systems with unknown initial state and disturbances. First, multiple high‐order internal models (HOIMs) are introduced for the reference, initial state, and disturbances. Both the initial state and disturbance consist of two components, one strictly satisfies HOIM and the other is random bounded. Then, an ILC scheme is constructed according to an augmented HOIM that is the aggregation of all HOIMs. For all known HOIMs, an ILC design criterion is introduced to achieve satisfactory tracking performance based on the 2‐D theory. Next, the case with unknown HOIMs is discussed, where a time‐frequency‐analysis (TFA)‐based ILC algorithm is proposed. In this situation, it is shown that the tracking error inherits the unknown augmented HOIM that is an aggregation of all unknown HOIMs. Then, a TFA‐based method, e.g., the short‐time Fourier transformation (STFT), is employed to identify the unknown augmented HOIM, where the STFT could ignore the effect of the random bounded initial state and disturbances. A new ILC law is designed for the identified unknown augmented HOIM, which has the ability to reject the unknown the initial state and disturbances that strictly satisfy HOIMs. Finally, a gantry robot system with iteration‐invariant or slowly‐varying frequencies is given to illustrate the efficiency of the proposed TFA‐based ILC algorithm.     

Iterative learning control for uncertain nonlinear networked control systems with random packet dropout

This paper proposes a novel networked iterative learning control (NILC) scheme with adjustment factor for a class of discrete‐time uncertain nonlinear systems with stochastic input and output packet dropout modeled as 0‐1 Bernoulli‐type random variable. Firstly, the equivalence relation between the realizability of controlled system and the input‐output coupling parameter (IOCP) is established. Secondly, in order to overcome the main obstacle arising from the unknown IOCP, an identification technique is developed for it. Thirdly, it is strictly proved that, under certain conditions, the tracking errors driven by the developed NILC scheme are convergent to zero along iteration direction in the sense of expectation. Finally, an example is given to demonstrate the effectiveness of the proposed NILC scheme and the merits of adjustment factor.     

SECOND‐ORDER NONSINGULAR TERMINAL SLIDING MODE DECOMPOSED CONTROL OF UNCERTAIN MULTIVARIABLE SYSTEMS

This paper proposes a second‐order nonsingular terminal sliding mode decomposed control method for multivariable linear systems with internal parameter uncertainties and external disturbances. First, the systems are converted into the block controllable form, consisting of an input‐output subsystem and a stable internal dynamic subsystem. A special second‐order non‐singular terminal sliding mode is proposed for the input‐output subsystem. The control law is designed to drive the states of the input‐output subsystem to converge to the equilibrium point asymptotically. Then the states of the stable zero‐dynamics of the system converge to the equilibrium point asymptotically. The method proposed in the paper has advantages for higher‐dimensional multivariable systems, in the sense that it simplifies the design and makes it possible to realize a robust decomposed control. Meanwhile, because of the adoption of the second‐order sliding mode, the control signal is continuous. Simulation results are presented to validate the design.     

Iterative learning control for output‐constrained nonlinear systems with input quantization and actuator faults

In this work, we propose a novel iterative learning control algorithm to deal with a class of nonlinear systems with system output constraint requirements and quantization effects on the system control input. Actuator faults have also been considered, which include multiplicative, additive, and stuck actuator faults. To the best of our knowledge, this is the first reported work in the iterative learning control literature to deal with quantization effects for the control input of nonlinear systems under the effects of actuator faults and system output constraints. Under the proposed scheme, using backstepping design and composite energy function approaches in the analysis, we show that uniform convergence of the state tracking errors can be guaranteed over the iteration domain, and the constraint requirement on the system output will not be violated at all time. In the end, a simulation study on a single‐link robot model is presented to demonstrate the effectiveness of the proposed scheme.     

On iterative learning control design for tracking iteration-varying trajectories with high-order internal model

In this paper, iterative learning control (ILC) design is studied for an iteration-varying tracking problem in which reference trajectories are generated by high-order internal models (HOIM). An HOIM formulated as a polynomial operator between consecutive iterations describes the changes of desired trajectories in the iteration domain and makes the iterative learning problem become iteration varying. The classical ILC for tracking iteration-invariant reference trajectories, on the other hand, is a special case of HOIM where the polynomial renders to a unity coefficient or a special first-order internal model. By inserting the HOIM into P-type ILC, the tracking performance along the iteration axis is investigated for a class of continuous-time nonlinear systems. Time-weighted norm method is utilized to guarantee validity of proposed algorithm in a sense of data-driven control.     

Gradient‐based back‐propagation dynamical iterative learning scheme for the neuro‐fuzzy inference system

In this paper, a gradient‐based back propagation dynamical iterative learning algorithm is proposed for structure optimization and parameter tuning of the neuro‐fuzzy system. Premise and consequent parameters of the neuro‐fuzzy model are initialized randomly and then tuned by the proposed iterative algorithm. The learning algorithm is based on the first order partial derivative of the output with respect to the structure parameters. The first order derivative of the model output with respect to the structure parameters determines the sensitivity of the model to structure parameters. The sensitivity values are then used to set the tuning factors and parameters updating step sizes. Therefore, an adaptive dynamical iterative scheme is achieved which adapts the learning procedure to the current state of the performance during the optimization process. Larger tuning step sizes make the convergence speed higher and vice versa. In this regard, this parameter is treated according to the calculated sensitivity of the model to the parameter. The proposed learning algorithm is compared with the least square back propagation method, genetic algorithm and chaotic genetic algorithm in the neuro‐fuzzy model structure optimization. Smaller mean square error and shorter learning time are sought in this paper, and the performance of the proposed learning algorithm is versified regarding these criteria.     

Sampled‐data iterative learning control for continuous‐time nonlinear systems with iteration‐varying lengths

In this work, sampled‐data iterative learning control (ILC) method is extended to a class of continuous‐time nonlinear systems with iteration‐varying trial lengths. In order to propose a unified ILC algorithm, the tracking errors will be redefined when the trial length is shorter or longer than the desired one. Based on the modified tracking errors, 2 sampled‐data ILC schemes are proposed to handle the randomly varying trial lengths. Sufficient conditions are derived rigorously to guarantee the convergence of the nonlinear system at each sampling instant. To verify the effectiveness of the proposed ILC laws, simulations for a nonlinear system are performed. The simulation results show that if the sampling period is set to be small enough, the convergence of the learning algorithms can be achieved as the iteration number increases.     

Data‐based iterative learning mechanism for unknown input‐output coupling parameters/matrices

In this paper, we explore how to get the information of input‐output coupling parameters (IOCPs) for a class of uncertain discrete‐time systems by using iterative learning technique. Firstly, by taking advantage of repetitiveness of control system and informative input and output data, we design an iterative learning scheme for unknown IOCPs. It is shown that we can get the exact values of IOCPs one by one through running the repetitive system T+1 times if the control system is with identical initial state and noise free. Secondly, we give the iterative learning scheme for unknown IOCPs in the presence of measurement noise, system noise, or initial state drift and analyze the influence factors on the performance of developed iterative learning scheme. Meanwhile, we introduce the maximum allowable control deviation into the iterative learning mechanism to minimize the negative impact of noise on the performance of learning scheme and to enhance the robust of iterative learning scheme. Thirdly, for a class of multiple‐input–multiple‐output systems, we also develop iterative learning mechanism for unknown input‐output coupling matrices. Finally, an illustrative example is given to demonstrate the effectiveness of proposed iterative learning scheme.     

Formation control for a class of nonlinear multiagent systems using model‐free adaptive iterative learning

In this paper, the problem of formation control is considered for a class of unknown nonaffine nonlinear multiagent systems under a repeatable operation environment. To achieve the formation objective, the unknown nonlinear agent's dynamic is first transformed into a compact form dynamic linearization model along the iteration axis. Then, a distributed model‐free adaptive iterative learning control scheme is designed to ensure that all agents can keep their desired deviations from the reference trajectory over the whole time interval. The main results are given for the multiagent systems with fixed communication topologies and the extension to the switching topologies case is also discussed. The feature of this design is that formation control can be solved only depending on the input/output data of each agent. An example is given to demonstrate the effectiveness of the proposed method.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

High‐precision formation control of nonlinear multi‐agent systems with switching topologies: A learning approach

Arbitrary high precision is considered one of the most desirable control objectives in the relative formation for many networked industrial applications, such as flying spacecrafts and mobile robots. The main purpose of this paper is to present design guidelines of applying the iterative schemes to develop distributed formation algorithms in order to achieve this control objective. If certain conditions are met, then the control input signals can be learned by the developed algorithms to accomplish the desired formations with arbitrary high precision. The systems under consideration are a class of multi‐agent systems under directed networks with switching topologies. The agents have discrete‐time affine nonlinear dynamics, but their state functions do not need to be identical. It is shown that the learning processes resulting from the relative output formation of multi‐agent systems can converge exponentially fast with the increase of the iteration number. In particular, this work induces a distributed algorithm that can simultaneously achieve the desired relative output formation between agents and regulate the movement of multi‐agent formations as desired along the time axis. The illustrative numerical simulations are finally performed to demonstrate the effectiveness and performance of the proposed distributed formation algorithms. Copyright © 2014 John Wiley & Sons, Ltd.     

Data‐Driven Adaptive Critic Approach for Nonlinear Optimal Control via Least Squares Support Vector Machine

This paper develops an online adaptive critic algorithm based on policy iteration for partially unknown nonlinear optimal control with infinite horizon cost function. In the proposed method, only a critic network is established, which eliminates the action network, to simplify its architecture. The online least squares support vector machine (LS‐SVM) is utilized to approximate the gradient of the associated cost function in the critic network by updating the input‐output data. Additionally, a data buffer memory is added to alleviate computational load. Finally, the feasibility of the online learning algorithm is demonstrated in simulation on two example systems.     

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.     

Two‐stage auxiliary model gradient‐based iterative algorithm for the input nonlinear controlled autoregressive system with variable‐gain nonlinearity

This article focuses on the parameter estimation problem of the input nonlinear system where an input variable‐gain nonlinear block is followed by a linear controlled autoregressive subsystem. The variable‐gain nonlinearity is described analytical by using an appropriate switching function. According to the gradient search technique and the auxiliary model identification idea, an auxiliary model‐based stochastic gradient algorithm with a forgetting factor is presented. For the sake of improving the parameter estimation accuracy, an auxiliary model gradient‐based iterative algorithm is proposed by utilizing the iterative identification theory. To further optimize the performance of the algorithm, we decompose the identification model of the system into two submodels and derive a two‐stage auxiliary model gradient‐based iterative (2S‐AM‐GI) algorithm by using the hierarchical identification principle. The simulation results confirm the effectiveness of the proposed algorithms and show that the 2S‐AM‐GI algorithm has higher identification efficiency compared with the other two algorithms.