Varying Trial Lengths Research Articles

Iterative learning based convergence analysis for nonlinear impulsive differential inclusion systems with randomly varying trial lengths

SummaryThis paper studies the finite‐time tracking problem for nonlinear impulsive differential inclusion systems with randomly varying trial lengths. First, we convert the set‐valued mapping in the differential inclusion systems to single‐valued mapping by a Steiner‐type selector. For the tracking problem of random discontinuous output trajectories, this paper defines a piecewise continuous variable by zero‐order holder to correct the tracking error of segmented continuity. Then, we introduce the average operator with forgetting factor to design three novel learning schemes, and establish convergence results by using the mathematical analysis tools such as impulsive Gronwall inequality and ‐norm. Finally, a numerical example verifies the validity of the theoretical results, and we compare the tracking performance of the output trajectories for different forgetting factors.

Event-Based Switching Iterative Learning Model Predictive Control for Batch Processes With Randomly Varying Trial Lengths.

Iterative learning model predictive control (ILMPC) has been recognized as an excellent batch process control strategy for progressively improving tracking performance along trials. However, as a typical learning-based control method, ILMPC generally requires the strict identity of trial lengths to implement 2-D receding horizon optimization. The randomly varying trial lengths extensively existing in practice can result in the insufficiency of learning prior information, and even the suspension of control update. Regarding this issue, this article embeds a novel prediction-based modification mechanism into ILMPC, to adjust the process data of each trial into the same length by compensating the data of absent running periods with the predictive sequences at the end point. Under this modification scheme, it is proved that the convergence of the classical ILMPC is guaranteed by an inequality condition relative with the probability distribution of trial lengths. Considering the practical batch process with complex nonlinearity, a 2-D neural-network predictive model with parameter adaptability along trials is established to generate highly matched compensation data for the prediction-based modification. To best utilize the real process information of multiple past trials while guaranteeing the learning priority of the latest trials, an event-based switching learning structure is proposed in ILMPC to determine different learning orders according to the probability event with respect to the trial length variation direction. The convergence of the nonlinear event-based switching ILMPC system is analyzed theoretically under two situations divided by the switching condition. The simulations on a numerical example and the injection molding process verify the superiority of the proposed control methods.

Quantised data-driven iterative learning bipartite consensus control for unknown heterogeneous linear MASs with varying trial lengths

This paper aims to realise the robust output bipartite consensus for unknown heterogeneous linear time-varying multiagent systems (MASs) subject to varying trial lengths, unknown measurement disturbances and data quantisation. To this end, inspired by the idea of quantised control, a quantised data-driven adaptive iterative learning bipartite consensus (AILBC) method is proposed. Specifically, to address the problem of varying trial lengths, a distributed auxiliary output prediction system is constructed based on the agents' input-output (I/O) dynamic relationship. An adaptive update protocol is developed to estimate the measurement disturbances and unknown parameters of I/O dynamic relationship. Subsequently, a quantised distributed data-driven iterative learning control (ILC) approach based on the quantised output information is proposed for MASs to achieve robust bipartite consensus tracking, with an attempt to relax the need of explicit model information. The bipartite consensus tracking errors are ultimately bounded through rigorous analysis, and this result is further extended to switching topologies. Finally, numerical simulations are conducted to verify the validity of the AILBC method.

Iterative Learning Control of Constrained Systems With Varying Trial Lengths Under Alignment Condition.

This brief is concerned with iterative learning control (ILC) of constrained multi-input multi-output (MIMO) nonlinear systems under the state alignment condition with varying trial lengths. A modified reference trajectory is constructed to meet the alignment condition by adjusting the reference trajectory to be spatially closed. Resorting to the barrier composite energy function (BCEF) approach, an adaptive ILC scheme is built to guarantee the bounded convergence of the resultant closed-loop system. Illustrative examples are presented to verify the validity of the proposed iteration scheme.

Iteration learning control for 2D linear discrete systems with randomly varying trail lengths

This paper considers the problem of randomly varying trial lengths of a class of two-dimensional (2D) linear systems represented by Roesser model, and two new iterative learning control (ILC) schemes are proposed. One is a conventional P-type control rule with a modified tracking error; the other is an ILC law with an iteration-average operator. Both learning control schemes are developed by a 2D stochastic variable to describe varying trial lengths. Under the Bernoulli distribution assumption and the initial state conditions, the convergence analysis is performed rigorously in the probability sense. Finally, illustrative examples have been provided to verify the theoretical results.

Iterative learning control based on dynamic time warping for a class of discrete‐time nonlinear systems with varying trial lengths and terminus constraint

AbstractThis article proposes a dynamic time warping (DTW)‐based iterative learning control (ILC) scheme for discrete‐time nonlinear systems to tackle the path learning problem with varying trial lengths and terminus constraint. By incorporating the improved DTW algorithm, the varying trial lengths are aligned as a desired length. Meanwhile, this algorithm can find the most similar waypoints between the output and the desired paths, which can be used to design an updating law and facilitate the convergence of path learning. Different from the existing ILC methods based on the probability distribution function for learning trajectory in the time domain, the method in this article can be applied to learn the spatial path corresponding to the desired trajectory. Furthermore, the learning property in the presence of variable initial states is discussed under the proposed method. Several illustrative examples manifest the validity of the proposed DTW‐based ILC algorithm.

Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths

Abstract In this paper, we introduce a new kind of conformable stochastic impulsive differential systems (CSIDS) involving discrete distribution of Bernoulli. For random discontinuous trajectories, we modify the tracking error of piecewise continuous variables by a zero-order holder. First, the improved P-type and PD α -type learning laws of the random iterative learning control (ILC) scheme are designed through global and local averaging operators. Next, we establish sufficient conditions for convergence of the tracking error in the expectation sense and prove the main results by using the impulsive Gronwall inequality and mathematical analysis tools. Finally, the theoretical results are verified by two numerical examples, and the tracking performance is compared for different conformable order of α.

An iterative learning approach to formation control of discrete‐time multi‐agent systems With varying trial lengths

AbstractIn this article, an iterative learning approach is proposed for the formation control of discrete‐time multi‐agent systems, where the trial length of each learning iteration is randomly varying. In particular, a modified state error related to the prescribed formation is defined by taking into account the nonuniform actual trial length that could be different from the desired one. Then, a P‐type iterative learning protocol is established for switching networks of agents subject to nonuniform trial lengths, and the convergence analyses are given for the fixed and the iteration‐varying initial conditions respectively. It shows that through iterative learning, the given formation will be maintained among multiple agents in the entire time interval of one trial. In the end, simulations are done to demonstrate the correctness of the obtained theoretical results.

Iterative learning control for repetitive tasks with randomly varying trial lengths using successive projection

SummaryThis article proposes an effective iterative learning control (ILC) approach based on successive projection scheme for repetitive systems with randomly varying trial lengths. A modified ILC problem is formulated to extend the classical ILC task description to incorporate a randomly varying trial length, while its design objective considers the mathematical expectation of its tracking error to evaluate the task performance. To solve this problem, this article employs the successive projection framework to give an iterative input signal update law by defining the corresponding convex sets based on the design requirements. This update law further yields an ILC algorithm, whose convergence properties are proved to be held under mild conditions. In addition, the input signal constraint can be embedded into the design without violating the convergence properties to obtain an alternative algorithm. The performance of the proposed algorithms is verified using a numerical model to show the effectiveness at occasions with and without input constraints.

Iterative learning control for impulsive multi-agent systems with varying trial lengths

In this paper, we introduce iterative learning control (ILC) schemes with varying trial lengths (VTL) to control impulsive multi-agent systems (I-MAS). We use domain alignment operator to characterize each tracking error to ensure that the error can completely update the control function during each iteration. Then we analyze the system’s uniform convergence to the target leader. Further, we use two local average operators to optimize the control function such that it can make full use of the iteration error. Finally, numerical examples are provided to verify the theoretical results.

Convergence analysis for iterative learning control of impulsive linear discrete delay systems

This paper addresses convergence of iterative learning control for impulsive linear discrete delay systems with randomly varying trial lengths when coefficient matrices of systems are permutable. With the aid of the explicit representation of solutions expressed in discrete matrix delayed exponential, we provide two sufficient conditions of convergence to guarantee that tracking errors uniformly converge to zero in the sense of expectation for the above impulsive controlled systems by designing two proper update learning laws with the modified tracking errors. Finally, two illustrative examples are given to verify the theoretical results.

Adaptive iterative learning control for 2D nonlinear systems with nonrepetitive uncertainties

AbstractThis paper explores the iterative learning control (ILC) problem for two‐dimensional (2D) multi‐input multi‐output nonlinear parametric systems by taking all nonrepetitive uncertainties of stochastic initial shifts, different tracking tasks and nonuniform trial lengths into consideration. A 2D stochastic variable is defined with Bernoulli stochastic distribution for the first time to handle iteration‐varying trial lengths. The desired output is incorporated into the learning control law as a feedback to compensate the iterative changes of the tracking tasks. An iterative 2D parameter updating law is established using a new defined virtue tracking error to well address the systems uncertainties and varying trial lengths. Consequently, a novel 2D adaptive ILC (2D‐AILC) is presented by incorporating the control law and parameter updating law. The convergence is proved by introducing both the Lyapunov stability principle and the 2D key technique lemma into the repetitive 2D systems even though the dynamic evolution along with two‐dimensional directions makes it more difficult in the mathematic analysis. The simulation study tests the theoretical results: the presented 2D‐AILC scheme can still accomplish a tracking task exactly even though there exist random initial states, nonrepetitive reference trajectories, and iteration‐varying trial lengths.

Data-driven nonlinear ILC with varying trial lengths

This work considers randomly varying trial lengths of a nonlinear and non-affine repetitive system and proposes a data-driven nonlinear iterative learning control (DDNILC) method via dynamic linearization. A nonlinear learning control law and a nonlinear parameter estimation law are developed by introducing a stochastic variable to describe the varying trial lengths. The learning gain is both nonlinear and iteration-time-varying, and is iteratively estimated through the parameter updating law so that the system uncertainties can be addressed more effectively. Moreover, the results are extended to the MIMO case. The proposed methods as well as the introduced dynamic linearization are data-driven without the need of an explicit model. Under the Bernoulli distribution assumption, the convergence analysis is performed rigorously in the probability sense. Illustrative examples are provided to verify the derived theoretical results.

Iterative learning formation control for continuous-time multi-agent systems with randomly varying trial lengths

This paper investigates the iterative learning control for continuous-time multi-agent formation systems to realize the desired formation, where the trial lengths could be randomly varying at each iteration. To be specific, an ILC (iterative learning control) protocol with an iteratively moving average operator is established for multiple nonlinear agents with switching topologies, where a modified formation tracking error is defined and the control information from several previous trials is used to deal with the varying trail lengths. The convergence conditions are derived by using a redefied λ-norm with mathematical expectation for both the zero and varying initial shift cases, which ensure that the formation performance can still be maintained during the whole motion process when the actual trial length is greater or less than the desired one. In the end, simulation results illustrate the effectiveness of the proposed method.

A Two-Dimensional Approach to Iterative Learning Control with Randomly Varying Trial Lengths

In this paper, iterative learning control (ILC) is considered to solve the tracking problem of time-varying linear stochastic systems with randomly varying trial lengths. Using the two-dimensional Kalman filtering technique, the authors can establish a recursive framework for designing the learning gain matrix along both time and iteration axes by optimizing the trace of input error covariance matrix. It is strictly proved that the input error converges to zero asymptotically in mean square sense and thus the tracking error covariance converges. The extensions to that prior distribution of nonuniform trial lengths is unknown are also investigated with an asymptotical estimation method. Numerical simulations are provided to verify the effectiveness of the proposed framework.

Adaptive learning tracking for robot manipulators with varying trial lengths

This paper proposes adaptive iterative learning control schemes for robot manipulator systems with iteration-varying lengths. To prove the asymptotical convergence of the joint position tracking error along the iteration axis, this paper develops a new composite energy function based on the newly introduced auxiliary variables for the analysis. Moreover, the traditional assumption of identical initialization condition is relaxed to be arbitrarily varying and then an initial rectifying mechanism is introduced to tackle initial shift problem of robotic systems. Illustrative simulations on a two degree-of-freedom robot manipulator are provided to verify the theoretical results.

Iterative learning control for noninstantaneous impulsive fractional‐order systems with varying trial lengths

SummaryIn this paper, we propose five iterative learning control (ILC) schemes for noninstantaneous impulsive fractional‐order systems with randomly varying trial lengths. We introduce a domain alignment operator to establish a rigorous convergence analysis for nonlinear fractional‐order systems. This operator guarantees that the input, state, output, and tracking error are constrained in a function space that is designed in advance. Moreover, with the help of this operator, we extend the conventional ILC scheme from discrete systems to continuous and algebraic systems with incorporating redundant tracking information. In addition, nonlinear ILC schemes are also presented with a geometric analysis concept. All proposed schemes are shown to be convergent to the desired tracking trajectory. Two illustrative examples are provided to verify the theoretical results.

Iterative learning control for linear discrete delay systems via discrete matrix delayed exponential function approach

ABSTRACTThis paper proposes iterative learning control (ILC) for linear discrete delay systems with randomly varying trial lengths without knowing prior information on the probability distribution of random iteration length. Based on matrix delayed exponential function approach, an explicit solution to the linear discrete delay controlled systems is used to generate a sequence of outputs that approximate the desired reference by adopting two ILC update laws in the presence of randomly iteration-varying lengths. A new and direct mathematical technique is explored to deal with ILC for linear discrete delay systems. Two illustrative examples are provided to verify the theoretical results.

Adaptive learning tracking for uncertain systems with partial structure information and varying trial lengths

This paper considers the adaptive iterative learning control (ILC) for continuous-time parametric nonlinear systems with partial structure information under iteration-varying trial length environments. In particular, two types of partial structure information are taken into account. The first type is that the parametric system uncertainty can be separated as a combination of time-invariant and time-varying part. The second type is that the parametric system uncertainty mainly contains time-invariant part, whereas the designed algorithm is expected to deal with certain unknown time-varying uncertainties. A mixing-type adaptive learning scheme and a hybrid-type differential-difference learning scheme are proposed for the two types of partial structure information cases, respectively. The convergence analysis under iteration-varying trial length environments is strictly derived based on a novel composite energy function. Illustrative simulations are provided to verify the effectiveness of the proposed schemes.

Iterative learning scheme‐based fault estimation design for nonlinear systems with varying trial lengths and specified constraints

SummaryThis technical note deals with a fault estimation (FE) problem for nonlinear systems with varying trial lengths subjected to specified constraints. An iterative learning observer is developed to achieve FE and state reconstruction simultaneously, which thus to consider the state error and fault estimating information from previous iteration to improve the FE performance in the current iteration. Compared with conventional observer‐based FE methods, the proposed method only requires the bounds of parameter matrices rather than the precise model of the system. To deal with the missing and redundancy problems caused by varying trial lengths, a truncation operator is presented to design the FE law. Different from existing iterative learning methods, the presented method aims at the nonlinear systems with specified constraints, such as filling systems. Furthermore, the λ‐norm method and mathematical induction are employed to obtain the solutions of iterative learning matrices and observer gain matrix. Finally, illustrative examples are introduced to demonstrate the validity and the effectiveness of the proposed FE approach.