Assessment of gradient-based iterative learning controllers using a multivariable test facility with varying interaction

A multiple input, multiple output (MIMO) experimental test facility has been developed for the evaluation, benchmarking and comparison of iterative learning control (ILC) strategies. The system addresses the distinct lack of experimental studies for the multivariable case and enables controller performance and robustness to be rigorously investigated over a broad range of operating conditions. The electromechanical facility is multi-configurable with up to 3 inputs and permits both exogenous disturbance injection and a variable level of coupling to be applied between input and output pairs. To confirm its suitability for evaluation and comparison of ILC, theoretical results are derived for two popular forms of gradient-type ILC algorithm, linking interaction with fundamental performance limitations. The test facility is then used to establish how well theoretical predictions match experimental results. The analysis is then extended to provide solutions to address this performance degradation, and these are again confirmed using the test facility.     

Internal-model-based design of repetitive and iterative learning controllers for linear multivariable systems

Repetitive and iterative learning control are two modern control strategies for tracking systems in which the signals are periodic in nature. This paper discusses repetitive and iterative learning control from an internal model principle point of view. This allows the formulation of existence conditions for multivariable implementations of repetitive and learning control. It is shown that repetitive control can be realized by an implementation of a robust servomechanism controller that uses the appropriate internal model for periodic distrubances. The design of such controllers is discussed. Next it is shown that iterative learning control can be implemented in the format of a disturbance observer/compensator. It is shown that the resulting control structure is dual to the repetitive controller, and that both constitute an implementation of the internal model principle. Consequently, the analysis and design of repetitive and iterative learning control can be generalized to the powerful analysis and design procedure of the internal model framework, allowing to trade-off the convergence speed for periodic-disturbance cancellation versus other control objectives, such as stochastic disturbance suppression.     

Credit assigned CMAC and its application to online learning robust controllers

In this paper, a novel learning scheme is proposed to speed up the learning process in cerebellar model articulation controllers (CMAC). In the conventional CMAC learning scheme, the correct numbers of errors are equally distributed into all addressed hypercubes, regardless of the credibility of the hypercubes. The proposed learning approach uses the inverse of learned times of the addressed hypercubes as the credibility (confidence) of the learned values, resulting in learning speed becoming very fast. To further demonstrate online learning capability of the proposed credit assigned CMAC learning scheme, this paper also presents a learning robust controller that can actually learn online. Based on robust controllers presented in the literature, the proposed online learning robust controller uses previous control input, current output acceleration, and current desired output as the state to define the nominal effective moment of the system from the CMAC table. An initial trial mechanism for the early learning stage is also proposed. With our proposed credit-assigned CMAC, the robust learning controller can accurately trace various trajectories online.     

Digital iterative learning control of linear multivariable plants

It is shown that digital iterative learning controllers can be designed for linear multivariable plants using only the step-response matrices of such plants. This demonstration is effected by proving a fundamental theorem which establishes precise sufficient conditions under which iterative learning control is achieved by such digital controllers. These general results are illustrated by the presentation of numerical results for the digital iterative learning control of a third-order linear multivariable plant with two inputs and two outputs.     

An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions.     

Digital iterative learning control of compensated linear multivariable plants

It is shown that digital iterative learning controllers can be designed for irregular linear multivariable plants by introducing appropriate digital compensators. This demonstration is effected by proving a fundamental theorem which establishes precise sufficient conditions under which iterative learning control is achieved by such digital controllers and compensators in the case of first-order irregular plants. These general results are illustrated by the presentation of numerical results for the digital iterative learning control of a third-order partially irregular plant with two inputs and two outputs. The extension of the results to higher-order irregular plants is discussed.     

System identification and on-line robust control of a multivariable system

System identification and on-line robust control have been developed for a multi-variable system with dead times. For system identification, a modified Astrom and Hagglund' s autotuning method is applied to obtain the transfer function matrix. An accurate transfer function matrix can be obtained using the proposed method. However, if the system has noises, an accurate transfer function matrix may not be obtained even if a relay with hysteresis is used. Modelling error is unavoidable. An on-line robust control based on a stability index is proposed to improve the performance of the control system     

The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods

The problem of designing realistic multivariable controllers to solve the servomechanism problem is considered in this paper. Specifically, it is desired to find a controller for a plant to solve the robust servomechanism problem, so that closed-loop stability and asymptotic regulation occur, and also so that other desirable properties of the controlled system, such as fast response, low-interaction, integrity, tolerance to plant variations, etc., occur. The method of design is based on using state space methods via a two-stage process: 1) using theory, determine the existence of a solution and control structure required to solve the problem, and 2) using nonlinear programming methods, determine the unknown controller parameters so as to minimize a performance index for the system subject to certain constraint requirements. Numerous examples, varying from a single-input/single-output to a four-input/four-output system, are given to illustrate the design method, and the results obtained are compared with the results obtained by using other alternate design methods. In all cases, the controllers obtained have been highly competitive with controllers obtained by alternate design methods.     

Constructing modal robust controllers using a transfer function of a closed system

A method of constructing modal robust controllers in a one-loop control system that provides an optimal relation between the control quality and robustness is proposed. The method can be applied to systems of any order. The mathematical apparatus is extremely simple and can be reduced to dividing polynomials.     

An LMI approach for robust iterative learning control with quadratic performance criterion

This paper presents the design of iterative learning control based on quadratic performance criterion (Q-ILC) for linear systems subject to additive uncertainty. The robust Q-ILC design can be cast as a min–max problem. We propose a novel approach which employs an upper bound of the worst-case performance, then formulates a non-convex quadratic minimization problem to get the update of iterative control inputs. Applying Lagrange duality, the Lagrange dual function of the non-convex quadratic problem is equivalent to a convex optimization over linear matrix inequalities (LMIs). An LMI algorithm with convergence properties is then given for the robust Q-ILC design. Finally, we provide a numerical example to illustrate the effectiveness of the proposed method.     

Learning rates in the digital iterative learning control of linear multivariable plants

The learning rates achievable in the digital iterative learning control of linear multi-variable plants are investigated. It is shown that the irregularity and stability characteristics of the plants under control impose severe constraints on the achievable learning rates. These results are not only significant in their own right but also strongly motivate the introduction of compensators to increase the learning rates achievable in irregular plants. These general results are illustrated by the presentation of numerical results for the iterative learning digital control of an uncompensated fourth-order completely irregular plant with two inputs and two outputs     

Synthesis of fixed-structure robust controllers using a constrained particle swarm optimizer with cyclic neighborhood topology

The purpose of this paper is to develop design scheme based on an easy-to-use meta-heuristic approach with superior reliability and validity for fixed-structure robust controllers, satisfying both multiple control specifications and system stability conditions. For this purpose, a particle swarm optimizer is first developed, which reduces the probability of premature convergence to local optima in the PSO (particle swarm optimization) by exploiting the particle’s local social learning based on the idea of cyclic-network topology. Next, it is shown how to obtain a fixed-structure robust controller with constraints on multiple H_∞ specifications and system stability based on the developed PSO technique incorporated with a simple constraint handling method. Finally, typical numerical examples are studied to show the applicability of the proposed methodology to the synthesis of fixed-structure robust controllers. These examples clearly verify that the developed design scheme gives a novel and powerful impetus with remarkable reliability to fixed-structure robust controller syntheses.     

Modelling and motion control of a mechatronic system using BGM with intelligent controllers

In this research paper, a mechatronics system such as a pan tilt platform (PTP) has been considered for motion control under intelligent controllers. A proportional-derivative (PD) controller is considered for comparison of results obtained from fuzzy and hybrid controllers. The trajectory following performance of the mechatronics system is found against these controllers. The results of simulations show that hybrid fuzzy controller reduce the tracking error effectively in lesser settling time. The intelligent controllers require knowledge base of error and derivative of error to compensate the PTP dynamics. The intelligent controllers have similar trends as the PD controllers and compensated both electrical and mechanical dynamics. The PD controller requires position measurement. The intelligent controllers have knowledge base consisting of position and velocity data. Thus intelligent controllers have position measurement along with knowledge base for position control system. The best results were achieved with hybrid fuzzy controllers. They meet the desired specifications.     

Synthesis of globally optimal controllers for robust performance tounstructured uncertainty

This paper solves the problem of synthesis of controllers achieving globally optimal robust performance against unstructured time-varying and/or nonlinear uncertainty. The performance measure considered is the infinity-to-infinity induced norm of a system's transfer function. The solution utilizes sensitivity analysis of linear programming and the theory of parametric programming     

Plasma multivariable robust control system design and simulation for a thermonuclear tokamak-reactor

The paper reports results on the design and analysis of the multivariable feedback H_infin; robust system for plasma current, position and shape control in the fusion energy advanced tokamak (FEAT) developed in the International Thermonuclear Experimental Reactor (ITER) project. The system contains the fast loop with the SISO plasma vertical speed robust controller and the slow loop with the MIMO plasma current and shape robust controller. The goal is to study the resources of the system robustness to achieve a higher degree of the FEAT operation reliability. Two H_infin; block diagonal controllers {K _SISO, K _MIMO} were designed by a mixed sensitivity approach in the framework of the disturbance rejection configuration. These controllers were compared with block diagonal decoupling, PI and LQG controllers at the set of FEAT key scenario points according to the multiple-criterion: nominal performance at minor disruptions, robust stability and robust performance. The H_infin; controllers showed larger multivariable stability margin and better nominal performance.     

A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties

In this paper, we present a new robust iterative learning control (ILC) design for a class of linear systems in the presence of time-varying parametric uncertainties and additive input/output disturbances. The system model is described by the Markov matrix as an affine function of parametric uncertainties. The robust ILC design is formulated as a min–max problem using a quadratic performance criterion subject to constraints of the control input update. Then, we propose a novel methodology to find a suboptimal solution of the min–max optimization problem. First, we derive an upper bound of the worst-case performance. As a result, the min–max problem is relaxed to become a minimization problem in the form of a quadratic program. Next, the robust ILC design is cast into a convex optimization over linear matrix inequalities (LMIs) which can be easily solved using off-the-shelf optimization solvers. The convergences of the control input and the error are proved. Finally, the robust ILC algorithm is applied to a physical model of a flexible link. The simulation results reveal the effectiveness of the proposed algorithm.     

Universal learning network and its application to robust control

Universal learning networks (ULNs) and robust control system design are discussed, ULNs provide a generalized framework to model and control complex systems. They consist of a number of interconnected nodes where the nodes may have any continuously differentiable nonlinear functions in them and each pair of nodes can be connected by multiple branches with arbitrary time delays. Therefore, physical systems which can be described by differential or difference equations and also their controllers can be modeled in a unified way. So, ULNs constitute a superset of neural networks or fuzzy neural networks. In order to optimize the systems, a generalized learning algorithm is derived for the ULNs, in which both the first order derivatives (gradients) and the higher order derivatives are incorporated. The derivatives are calculated by using forward or backward propagation schemes. These algorithms for calculating the derivatives are extended versions of back propagation through time (BPTT) and real time recurrent learning (RTRL) by Williams in the sense that generalized nonlinear functions and higher order derivatives are dealt with. As an application of ULNs, the higher order derivative, one of the distinguished features of ULNs, is applied to realizing a robust control system in this paper. In addition, it is shown that the higher order derivatives are effective tools to realize sophisticated control of nonlinear systems. Other features of ULNs such as multiple branches with arbitrary time delays and using a priori information will be discussed in other papers.     

Stability in a non-autonomous iterative system: An application to oligopoly

This paper reconsiders the relation between oligopoly and perfect competition, more specifically the problem of emergent instability when the number of competitors increases, as pointed out by several authors. A process of mixed short and long run dynamics is set up. In the short run the competitors are subject to capacity limits due to fixed capital stocks, in the long run they may renew these stocks and so in the moments of reinvestment have access to a constant returns technology. The evolution of the system depends on the number of competitors, the interval between their entry on the market, and the durability of capital. The main result is a theorem showing that if capital has a durability of more periods than the spacing of reinvestment times among the firms, multiplied with their total number, then the system always contracts to the Cournot equilibrium state.     

Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties

This paper proposes the design of the integrated output feedback and iterative learning control (ILC) for batch processes with uncertain perturbations and interval time-varying delays, where the main idea is to transform the design into a robust delay-range-dependent H_∞ control of a 2D system described by a state-space model with varying delays. A sufficient criterion for delay-dependent H_∞ noise attenuation is derived through linear matrix inequality (LMI) by introducing a comprehensive 2D difference Lyapunov–Krasovskii functional candidate and adding a differential inequality to the difference in the Lyapunov function for the 2D system. Based on the criterion obtained, the delay-range-dependent output feedback controller combined with ILC is then developed. The developed system ensures that the closed-loop system for all admissible uncertainties is asymptotically stable and has a prescribed H_∞ performance level in terms of the LMI constraint. The controller is obtained by solving an LMI optimization problem with simple calculations and less constraint conditions. Moreover, the conditions can also be directly extended from delay-range-dependent to general delay-dependent stability. Applications in injection velocity control demonstrate the effectiveness and feasibility of the proposed method.