Blazej Cichy

Iterative learning control for spatio-temporal dynamics using Crank-Nicholson discretization

Iterative learning control is now well established for linear and nonlinear dynamics in terms of both the underlying theory and experimental application. This approach is specifically targeted at cases where the same operation is repeated over a finite duration with resetting between successive repetitions. Each repetition or pass is known as a trial and the key idea is to use information from previous trials to update the control input used on the current one with the aim of improving performance from trial-to-trial. In this paper, new results on ILC applied to systems that arise from discretization of bi-variate partial differential equations describing spatio-temporal systems or processes are developed. Theses are based on Crank-Nicholson discretization of the governing partial differential equation, resulting in an unconditionally numerically stable approximation of the dynamics. It is also shown that this setting allows the selection of a finite number of points for sensing and actuation. The resulting control laws can be computed using Linear Matrix Inequalities (LMIs). Finally, an illustrative example is given and areas for further research are discussed.

2D systems based robust iterative learning control using noncausal finite-time interval data

Abstract This paper uses a 2D system setting in the form of repetitive process stability theory to design an iterative learning control law that is robust against model uncertainty. In iterative learning control the same finite duration operation, known as a trial over the trial length, is performed over and over again with resetting to the starting location once each is complete, or a stoppage at the end of the current trial before the next one begins. The basic idea of this form of control is to use information from the previous trial, or a finite number thereof, to compute the control input for the next trial. At any instant on the current trial, data from the complete previous trial is available and hence noncausal information in the trial length indeterminate can be used. This paper also shows how the new 2D system based design algorithms provide a setting for the effective deployment of such information.

An approach to iterative learning control for spatio-temporal dynamics using nD discrete linear systems models

Iterative Learning Control (ILC) is now well established in terms of both the underlying theory and experimental application. This approach is specifically targeted at cases where the same operation is repeated over a finite duration with resetting between successive executions. Each execution is known as a trial and the key idea is to use information from previous trials to update the control input used on the current one with the aim of improving performance from trial-to-trial. In this paper, the subject area is the application of ILC to spatio-temporal systems described by a linear partial differential equation (PDE) using a discrete approximation of the dynamics, where there are a number of construction methods that could be applied. Here explicit discretization is used, resulting in a multidimensional, or nD, discrete linear system on which to base control law design, where n denotes the number of directions of information propagation and is equal to the total number of indeterminates in the PDE. The resulting control laws can be computed using Linear Matrix Inequalities (LMIs) and a numerical example is given. Finally, a natural extension to robust control is noted and areas for further research briefly discussed.

Stabilization of a class of uncertain "wave" discrete linear repetitive processes

Repetitive processes are a distinct class of two-dimensional (2D) systems (i.e. information propagation in two independent directions occurs) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or (most often) 2D systems theory. In this paper we continue the development of a systems theory for a recently proposed model of these processes necessary to represent terms which arise in some applications areas but are not included in the currently used models. The new results are on the design of control laws for stabilization in the case when there is uncertainty associated with the defining state-space model

Control law design for discrete linear repetitive processes with non-local updating structures

Repetitive processes are a class of 2D systems where information propagation in one direction is of finite duration. These processes make a series of sweeps, termed passes, through a set of dynamics and on completion of each pass resetting to the starting position occurs ready for the start of the next pass. The control problem is that the previous pass output, termed the pass profile, acts as a forcing function on the current pass and can result in oscillations that increase in amplitude from pass-to-pass. In the case of discrete dynamics, these processes have structural links with 2D systems described by the well known Roesser and Fornasini–Marchesini state-space models but some applications require updating structures that cannot be represented by these models. This requirement arises either in adequately modeling the dynamics or as a result of the control law structure and requires the development of a systems theory for eventual use in applications. In this paper such a theory is advanced through the development of new control law design algorithms.

Wave repetitive process approach to a class of ladder circuits

In this paper a 2D systems setting is used to develop new results on modeling and the stability analysis of electrical multi element ladder circuits from the repetitive process standpoint.

Discrete linear repetitive processes with smoothing

Repetitive processes are a distinct class of two-dimensional (2D) systems (i.e. information propagation in two independent directions occurs) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or (often) 2D systems theory. In this paper we continue the development of a control systems theory for discrete linear repetitive processes with inter-pass smoothing which is required to represent dynamics which arise in some applications areas but are not included in the models previously considered. The new results are on the stability analysis and the design of control laws for stabilization, including the case when there is uncertainty associated with the defining state-space model.

An unconditionally stable finite difference scheme systems described by second order partial differential equations

An unconditionally stable finite difference scheme for systems whose dynamics are described by a second-order partial differential equation is developed. The scheme is motivated by the well-known Crank-Nicolson discretization which was developed for first-order systems. The stability of the finite-difference scheme is analysed by von Neumanns method. Using the new scheme, a discrete in time and space model of a deformable mirror is derived as the basis for control law design. The convergence of this scheme for various values of the discretization parameters is checked by numerical simulations.

Stability of a class of 2D linear systems with smoothing

Repetitive processes are a distinct class of twodimensional (2D) systems (i.e. information propagation in two independent directions occurs) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or (often) 2D systems theory. In this paper we begin the development a systems theory for a model of these processes necessary to represent terms which arise in some applications areas but are not included in the currently used models.

An unconditionally stable approximation of a circular flexible plate described by a fourth order partial differential equation

An unconditionally stable finite difference scheme for systems whose dynamics are described by a second-order partial differential equation is developed with use of regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization which was developed for first-order systems. The stability of the finite-difference scheme is analyzed by von Neumanns method. Using the new scheme, a discrete in time and space model of a deformable mirror is derived as the basis for control law design. The convergence of this scheme for various values of the discretization parameters is checked by numerical simulations.