## Iterative Learning Control for Discrete Linear Systems with Zero Markov Parameters using Repetitive Process Stability Theory

*February 14, 2012 at 7:14 am* *
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**Abstract**—This paper considers iterative learning control for the practically relevant case of deterministic discrete linear plants where the first Markov parameter is zero. A 2D systems approach that uses a strong form of stability for linear repetitive processes is used to develop a one step control law design for both trial-to-trial error convergence and along the trial performance. The resulting design computations are completed using linear matrix inequalities, and results from applying the control law to one axis of a gantry robot are also given by way of experimental verification.

**I. INTRODUCTION
**

Iterative learning control (ILC) is a technique for controlling systems operating in a repetitive, or trial-to-trial, mode with the requirement that a reference trajectory defined over a finite interval 0 ≤ t≤α, where α denotes the trial length or duration, is followed to a high precision. Examples include robotic manipulators that are required to repeat a given task to high precision, chemical batch processes or, more generally, the class of tracking systems.

Since the original work [1], the general area of ILC has been the subject of intense research effort. An initial source for the literature here are the survey papers [2], [3]. One approach to the analysis and design of ILC schemes is to use 2D systems theory where one direction of information propagation is from trial-to-trial and the other is along the trial, and it is in this setting that the results reported in this paper are developed.

In ILC, the control law must ensure convergence of the trial-to-trial error, where the error on any trial is the difference between the reference signal and the output and is defined over a finite duration. Hence it is possible for trial-to trial error convergence to occur even if the along the trial dynamics are unstable. For discrete linear systems, a common way to approach ILC design is to use lifting and thereby write the dynamics in terms of a standard difference equation, and for unstable examples the route is to first design a stabilizing feedback control law and then apply lifting to the resulting controlled process dynamics. The result is a two stage design process.

An alternative to lifting is to use a 2D systems approach, where previous work [4], [5] has shown that ILC schemes can be designed for a class of discrete linear systems by, as one option, extending techniques developed for 2D linear systems using the framework of linear repetitive processes. Also these designs have been experimentally verified on a gantry robot executing a pick and place operation that is typical of many industrial applications to which ILC is well suited. The basis of the results in [4], [5] is stability along the pass, or trial, for linear repetitive processes that, in the case of the along the trial dynamics, demands bounded-input-bounded-output (BIBO) stability uniformly with respect to the trial duration. Hence it is possible to simultaneously design for trial-to-trial error convergence and along the trial response. This allows the design of a single control law and adds to the options to the designer which in some cases may be more attractive than pre-stabilization followed by ILC design.

Some ILC algorithms for linear dynamics require that the first Markov parameter of the plant state-space model is not the zero matrix, that is, for the state-space triple {A,B,C}, CB≠0. A specific example is, in the single-input-single-output (SISO) case for simplicity, P-type ILC of the form

where the integer k ≥ 0 denotes the trial number, uk(p) is the trial input, ek(p) is the difference on trial k between the reference signal and the output yk(p); and Г

is a gain to be selected. Trial-to-trial error convergence holds in this case provided

Hence if CB = 0; or in an implementation rounding errors enforce this condition this simple structure algorithm cannot be used. There has been work on a 2D systems approach to design where the first Markov parameter is non-zero, coupled with experimental verification on a gantry robot [6], and this paper extends these results to the case when CB = 0; Supporting experimental results from application to the same gantry robot are given and the natural extension to cases when the first h > 1 Markov parameters are zero noted.

In this paper, the null and identity matrices with the required dimensions are denoted by 0 and I; respectively. Also Г> 0 and Г< 0 are used to denote symmetric matrices that are positive definite and negative definite, respectively. The symbol * is used to denote entries in symmetric matrices, and r(•) is used to denote the spectral radius of a given matrix, that is, if H is a q x q matrix then r(H) = max_{1}≤ i≤q│λi│< 1; where λi ,1 ≤ i ≤ q; is an eigenvalue of H.

**II. BACKGROUND AND INITIAL ANALYSIS
**

The systems considered in this paper are assumed tobe adequately represented by discrete linear time-invariant systems described by the state-space triple {A,B,C} In an ILC setting for linear time-invariant dynamics, the statespace model is written as

where on trial k, xk(p) є R^{n }is the state vector, yk(p) єR^{m }is the output vector, uk(p) єR^{r }is the vector of control inputs, and the trial length α < ∞. If the signal to be tracked is denoted by yref (p) then ek(p) = yref (p) –yk(p) is the

error on trial k. The most basic requirement is to force the error to converge in k.

As discussed in the introduction to this paper, ILC can be treated in a 2D systems setting where information propagation in one direction is from trial-to-trial and the other along the trial. In the case of discrete linear dynamics [7], the Roesser state-space model [8] has been used to design a control law to ensure trial-to-trial error convergence, but applications will arise where it is also necessary to control the along the trial dynamics. For example, consider a gantry robot whose task is to collect an object from a location, place it on a moving conveyor, and then return for the next one and so on. If, for example, the object has an open top and is filled with liquid, and/or is fragile, unwanted vibrations during the transfer time could have very detrimental effects.

For discrete dynamics, one approach in cases such as that outlined above is to design a feedback control scheme to control the process output and then proceed to ILC design for trial-to-trial error convergence by, for example, a lifted standard linear systems state-space model of the controlled dynamics. In implementation terms, a two loop control scheme results and this paper considers the alternative where a 2D systems setting is used to design a single controller for both tasks. Linear repetitive processes are a distinct class of 2D linear systems where the duration of information propagation in one of the two directions is finite, and next the relevant background on these processes is given.

The unique characteristic of a repetitive process [9] is a series of sweeps, termed passes, through a set of dynamics defined over a fixed finite duration known as the pass length. On each pass an output, termed the pass profile, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This, in turn, leads to the unique control problem where the output sequence of pass profiles generated can contain oscillations that increase in amplitude in the pass-to-pass direction.

To introduce a formal definition, let α< +∞ denote the pass length. Then in a repetitive process the pass profile yk(p) generated on pass k acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile yk+1(p); p = 0,1,.,…..,

α-1; k≥0.

Attempts to control these processes using standard (or 1D) systems theory and algorithms fail (except in a few very restrictive special cases) precisely because such an approach ignores their inherent 2D systems structure, that is, information propagation occurs from pass-to-pass (k direction) and along a given pass (t direction) and also the initial conditions are reset before the start of each new pass. To remove these deficiencies, a rigorous stability theory has been developed [9] based on an abstract model of the dynamics in a Banach space setting that includes a very large class of processes with linear dynamics and a constant pass length as special cases, including those described by (3) below.

The abstract model for linear constant pass length repetitive processes is

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