Linear Non Linear Iterative Learning Control

8/8/2019 Linear Non Linear Iterative Learning Control

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JIAN-XIN XU and YING TAN

Linear and Nonlinear Iterative

Learning Control

March 3, 2003

Springer

Berlin Heidelberg NewYork

Hong Kong London

Milan Paris Tokyo


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To our parents and

Iris Hong Chen and Elizabeth Huifan Xu

Jian-Xin Xu

Yang and my coming baby

Ying Tan


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Preface

Most existing control methods, whether conventional or advanced, robust orintelligent, linear or nonlinear, target at achieving the asymptotic convergenceproperty in tracking a given trajectory. On the other hand, most practical

control tasks, whether in process control or mechatronics, MEMS or spaceoriented, civil or military oriented, will have to be completed in a finite timeinterval. The scale of a finite interval can range from milliseconds to years.Tracking in finite horizon means the performance in transient process be-comes more important. Often, perfect tracking performance is required fromthe very beginning. Obviously, asymptotic convergence along the time axis isinadequate, as it only guarantees the performance at the steady state whenthe time horizon goes to infinity. What is more, when the control task is re-peated, the system will exhibit the same behavior. In practice there are manyprocesses repeating the same task in a finite interval, ranging from a weldingrobot in a VLSI production line, to a batch reactor in pharmaceutical indus-try. Most existing control methods, devised in the time domain, are not ableto fully capture and utilize the information available through the underlyingnature of the system repeatability.

Iterative Learning Control (ILC) differs from most existing control meth-ods in the sense that, it exploits every possibility to incorporate past controlinformation: the past tracking error signals and in particular the past con-trol input signals, into the construction of the present control action. This isrealized through memory based learning. First the long term memory com-ponents are used to store past control information, then the stored controlinformation is fused in a certain manner to form the feedforward part of thecurrent control action. In certain sense, ILC complements the existing controlmethods.

Since the birth of iterative learning control in early 1980s, the historyof ILC can be divided in to two phases. From early 1980s to early 1990s

was a linearly increasing period of ILC, in terms of reports and publicationsin theory and applications. From early 1990s, however, the research activi-ties in ILC undergo a nonlinear (exponential) increase. One such evidence is,


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VIII Preface

most premier control conferences have dedicated sessions related to iterativelearning control, in addition to the increasing publications, special issues, andreports on the variety of applications. In order to update readers with the lat-est advances in this active area, this book provide a comprehensive coverage

in most aspects of ILC, including linear and nonlinear ILC, lower order andhigher order ILC, contraction mapping based and Lyapunov based ILC, out-put tracking ILC and state tracking ILC, model based and black-box basedILC design, robust optimal design of ILC, quantified ILC performance analy-sis, ILC for systems with global and local Lipschitz continuous nonlinearities,ILC for systems with parametric and non-parametric uncertainties, ILC withnonlinear optimality, etc.

The book can be used as a reference or textbook for a course at graduatelevel. It is also suitable for self-study, as most topics addressed in the bookare self-contained in theoretical analysis, and accompanied by detailed exam-ples to help readers, such as control engineers and graduate students, bettercapture the essence and the global picture of each ILC scheme. To furtherfacilitate those who have interests but know little about ILC, two rudimen-

tary sections are provided in Chapter 1 and Chapter 7 respectively. The firstrudimentary section is written in such a way that it can be easily understoodeven by first year undergraduate students majoring in science and engineering.There are ten chapters in this monograph. Chapter 1 introduces the concept,rudiments and history of ILC. Chapters 2 - 6 reveal the intrinsic nature ofcontraction mapping based ILC. Chapters 7 - 9 extend the ILC to systemswith more general nonlinearities. In Chapters 7 - 8 the energy function ap-proaches, such as the Lyapunov technology, have been applied to repeatedlearning control problems. This serves as a bridge to connect the ILC fieldwith the majority of nonlinear control fields, such as nonlinear optimality,adaptive control, robust control, etc. Also, in Chapter 9 the black-box ap-proach using Wavelet network is integrated with ILC, which serves as anotherbridge to link the ILC field with the majority of intelligent control fields, such

as neural network, fuzzy logics, etc. Finally, Chapter 10 concludes the bookand points out several future research directions.

While preparing the book, the authors benefited greatly from stimulat-ing discussions and judicious suggestions by ILC experts worldwide. Discus-sions with kevin Moore, Zeungnam Bien, Suguru Arimoto, Richard Longman,Zhihua Qu, David Owens, Yangquan Chen, Toshiharu Sugie, Danwei Wang,Tae-Yong Kuc, Chiang-Ju Chien, and many others, helped us clarify variousaspects of the iterative learning control problems, which in turn motivatedus to explore the underlying nature and properties of ILC, thereby lead tothis book. The authors would like to express their special appreciation tothe LNCIS series editor, Dr Thomas Ditzinger, for his strong support andprofessionalism.


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Preface IX

Singapore, Jian-Xin XuFebuary, 2003 Ying Tan


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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is Iterative Learning Control . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Simplest ILC: an example . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 ILC for Non-affine Process . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 ILC for Dynamic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 D-Type ILC for Dynamic Process . . . . . . . . . . . . . . . . . . . 111.1.5 Can We Relax the Identical Initialization Condition? . . 131.1.6 Why ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2 History of ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Book Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Robust Optimal Design for the First Order Linear-type

ILC Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Convergence Properties in Iteration Domain . . . . . . . . . . . . . . . . 24

2.4 Robust Optimal Design for Convergence Speed . . . . . . . . . . . . . 272.5 Robust Optimal Design for Global Uniform Bound . . . . . . . . . . 302.6 Monotonic Convergence Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Analysis of Higher Order Linear-type ILC Schemes . . . . . . . . 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Convergence Speed Analysis of the Second Order ILC . . . . . . . . 443.4 m-th Order ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


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XII Contents

4 Linear ILC Design for MIMO Dynamic Systems . . . . . . . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 The Linear-type ILC Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.5 Robust Optimal Design for MIMO Dynamic Systems . . . . . . . . 604.6 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Nonlinear-type ILC Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Convergence Analysis for Linear-type ILC Scheme . . . . . . . . . . . 735.4 The Newton-type ILC Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5 The Secant-type ILC Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6 Nonlinear ILC Design for MIMO Dynamic Systems . . . . . . . 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3 The Newton-type ILC Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 886.4 The Secant-type ILC Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Composite Energy Function Based Learning Control . . . . . . 977.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 From Contraction Map to Energy Function Approach . . . . . . . . 98

7.2.1 ILC Bottleneck GLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.2 What Can We Learn From Adaptive Control . . . . . . . . . 1017.2.3 ILC with Composite Energy Function . . . . . . . . . . . . . . . . 1037.3 General Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4 Learning Control Configuration and Convergence Analysis . . . . 1097.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 Quasi-Optimal Iterative Learning Control . . . . . . . . . . . . . . . . . 1178.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.3 Nonlinear Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.4 Synthesized Quasi-Optimal Learning Control

Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


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Contents XIII

9 Learning Wavelet Control Using Constructive Wavelet

Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.2 Fundamentals of Wavelet Networks . . . . . . . . . . . . . . . . . . . . . . . . 132

9.3 LWC Design for Affine Nonlinear Uncertain Systems . . . . . . . . . 1349.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.3.2 Design and Analysis of LWC . . . . . . . . . . . . . . . . . . . . . . . . 137

9.4 LWC for Non-affine Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . 1449.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449.4.2 LWC Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10 Conclusions and Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . 15710.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15710.2 Recommendation for Future Research . . . . . . . . . . . . . . . . . . . . . . 158

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161


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1

Introduction

According to Merrian-Websters Collegiate Dictionary, the term learning isdefined as

the act or experience of one that learns knowledge or skill acquired by instruction or study modification of a behavioral tendency by experience (as exposure to con-

ditioning)

In a word, learning generally implies a gaining or transfer of knowledge. Inthis book, the primary goal is centered on iterative learning control. The termiterative indicates a kind of action that requires the dynamic process berepeatable, i.e., the dynamic system is deterministic and the tracking controltasks are repeatable over a finite tracking interval. This kind of control prob-lems is frequently encountered in many industrial processes, such as wafermanufacturing process, batch reactor process, IC welding process, and vari-ous assembly lines or production lines, etc. The motivation of iterative learn-ing control comes from a deeper recognition, that knowledge can be learnedfrom experience. In other words, when a control task is performed repeatedly,we gain extra information from a new source: past control input and trackingerror profiles, which can be viewed as a kind of experience. This kind of ex-perience serves as a new source of knowledge related to the dynamic processmodel, and accordingly reduces the need for the process model knowledge.The new knowledge learned from the experience provides the possibility ofimproving the tracking control performance.

1.1 What is Iterative Learning Control

Let us start from a new class of control tasks: perfect tracking in a finite

time interval under a repeatable control environment. The perfect trackingtask implies that the target trajectory must be strictly followed from the verybeginning of the execution. The repeatable control environment implies an


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2 1 Introduction

identical target trajectory and the same initialization condition for all re-peated control trials. Many existing control methods are not able to fulfillsuch a task, because they only warrant an asymptotic convergence, and beingmore essential, they are unable to learn from previous control trials, whether

succeeded or failed. Without learning, a control system can only produce thesame performance without improvement, even if the task repeats consecu-tively. ILC was proposed to best meet this kind of control tasks. The idea ofILC is straightforward: use the control information of the preceding trial toimprove the control performance of the present trial. This is realized throughmemory based learning.

Fig. 1.1 shows one such schematic diagram,

Fig. 1.1. Memory Based Learning

where the subscript i denotes the i-th control trial. Assume that the controller

is memoryless. It can be seen, in addition to the standard feedback loop,a set of memory components are used to record the control signal of thepreceding trial, ui(t), which is incorporated into the present control, ui+1(t),in a pointwise manner. The sole purpose is to embed an internal model intothe feed-through loop. Let us see how this can be achieved. Assume that thetarget trajectory, yd(t), is repeated over a fixed time interval, and the plant isdeterministic with exactly the same initialization condition. Suppose that theperfect output tracking is achieved at the i-th trial, i.e. yd(t)yi(t) = 0, whereyi(t) is the system output at the i-th trial. The feedback loop is equivalentlybroken up. ui(t) who did the perfect job will be preserved in the memory forthe next trial. In the sequel ui+1(t) = ui(t), which warrants a perfect trackingwith a pure feedforward.

A typical ILC, shown in Fig. 1.2, is somehow still different from Fig. 1.1.


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1.1 What is Iterative Learning Control 3

Fig. 1.2. A typical ILC

The interesting idea is to further remove the time domain feedback fromthe current control loop. Inappropriately closing the loop may lead to insta-

bility. To design an appropriate closed-loop controller, much of the processknowledge is required. Now suppose the process dynamics cannot escape toinfinity during the tracking period that is always a finite interval in ILC tasks.We need not even be bothered to design a stabilizing controller in the timedomain, as far as the control system converges gradually when the learningprocess repeats. In this way an ILC can be designed with the minimum systemknowledge.

In the field of ILC, such repeated learning trials are frequently describedby words like cycles, runs, iterations, repetitions, passes, etc. Since the ma-

jority of ILC related work done hitherto is under the framework of contractionmapping characterized by an iterative process, it is thus more appropriate touse the words iteration(s), iteration axis, iteration domain, etc. to describesuch an iterative learning process, as we shall follow in the rest of this book.

The concept of performance improvement under a repeated operationprocess have long been observed, analyzed and applied [93] [129]. The ar-ticles by [7, 38, 9, 12, 64, 65] have formed the initial framework of ILC, underwhich subsequent developments have taken place over the years. Since then,though undergoing rapid progress, the main framework of ILC has been deter-mined and the majority of ILC schemes developed hitherto are still within thisframework, which is characterized by two key features: the linear pointwiseupdating law and iterative convergence, and is subject to two fundamentalconditions: the global Lipschitz continuity (GLC) condition and the identi-cal initialization condition (i.i.c). The contraction mapping methodology, amethod commonly used in function approximation and numerical analysis,has accordingly been brought up in iterative learning control design. In orderto capture the concept of ILC from a more quantified point of view, in therest of this section we shall conduct a rudimentary course briefing on ILC.


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4 1 Introduction

1.1.1 The Simplest ILC: an example

To illustrate the underlying concepts and properties of ILC, let us start withthe simplest ILC problem: for a given process

y(t) = g(t)u(t)

where g(t) = 0 is defined over a period [0, T], find the control input, u(t),

such that the target trajectory

yd(t) t [0, T]

can be perfectly tracked. Without loss of generality we assume that yd(t) andg(t) are bounded functions.

If g(t) is known a priori, this problem becomes a little trivial, as we cansimply calculate the desired control signal directly by inverting the process,which is exactly an open-loop approach

ud(t) = yd(t)g(t)

t [0, T].

However, we know that any open-loop control schemes are sensitive to theplant modeling inaccuracy. In our case, if the exact values of g(t) are notavailable, the above simple open-loop scheme does not work. Let us assumethat g(t), though unknown, spans within 0 < 1 g(t) 2 < , where 1and 2 are known lower and upper bounds. Can we find the desired controlprofile ud(t)? One may think of a two-stage approach. From the input-outputrelationship and the availability of the measurement of y(t) and u(t), firstidentify the time-varying gain g(t) point-wisely for t [0, T]. Then the desiredcontrol signal can be computed according to the inverse relationship, ud(t) =yd(t)/g(t) t [0, T], provided that the function g(t) is captured perfectly

over [0, T].Can we merge the two stage control into one stage, that is, can we directly

acquire the desired control signal without any parametric or function identifi-cation? This will make the control system more efficient, and avoid extra errorincurred by any intermediate computation, e.g. a large numerical error mayoccur if g(t) takes a very small value at some instant t and is inverted. If thecontrol task runs once only and ends, we are not able to directly achieve thedesired control signal. When the same control task is repeated many times, wecan acquire the control signal iteratively by the following simplest iterativelearning control scheme

ui+1(t) = ui(t) + qyi(t) t [0, T] (1.1)

where the subscript i Z+ is the iteration index, Z+ = 0, 1, is the set ofnon-negative integers. u0(t) can be either generated by any control method orsimply set to be zero. In fact all we need for u0(t) is to guarantee a bounded


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output y0(t). q is a constant learning gain, and yi(t)= yd(t) yi(t) is

the output tracking error sequence. Let us see how does this scheme workiteratively, and investigate conditions which ensure the system learnability.

First of all, we are going to execute the control action many times as i

evolves, with the ultimate objective of finding out the desired control signal,ud(t), with respect to yd(t). Once yd(t) is given, ud(t) should be fixed. Thisimplies that the system, in this particular case the function g(t), defined over[0, T], must be identical for any iterations. We know that a deterministicsystem will produce the same response when the same input repeats. Wethus define a repeatable control environment: a deterministic system with thecontrol task repeated over a fixed time interval. The repeatability is the veryfirst necessary condition for any deterministic learning controller to effectivelyperform.

Under the repeatable control environment, can the simplest ILC warrantsa convergence sequence of yi(t) to yd(t), or ui(t) to ud(t), as i ? There

are two ways we can prove the convergence, either yi(t) 0, or ui(t)=

ud(i) ui(t) 0, when i . For simplicity we will omit the time t for allvariables from 0 to T if not otherwise mentioned. First we demonstrate theconvergence of the output tracking sequence

yi+1 = yd yi+1 = yd gui+1 = yd g(ui + qyi)

= (yd gui) qgyi = (1 qg)yi.

Consequently

|yi+1| |1 qg| |yi|.

On the other hand, we know 0 < 1 g(t) 2 < , hence a conservativeselection of the learning gain is

q = 12 .

It is easy to verify

0 |1 qg| 2 1

2= < 1

and

|yi+1|

|yi| < 1, i Z+,

which shows

limi

|yi| lim

ii+1|y

0| 0

because y0(t) and yd(t) are finite in [0, T].


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6 1 Introduction

Now let us show, as an alternate way, that ui 0. Note

ui+1 = ud ui+1 = ud (ui + qyi)

= (ud ui) qyi = ui qyi.

On the other hand,

yi = yd yi = gud gui = gui.

By substituting yi we have

ui+1 = (1 qg)ui,

from which we can see that the convergence conditions are the same for thesequences yi and ui. In this particular problem, the convergence of ui toud implies the convergence ofyi to yd. For more complicated problems relatedto the dynamic process or MIMO cases, they may show some differences.

Following the above demonstration on the simplest ILC, a question may

arise: does the simplest ILC still work if the process is nonlinear in controlinput u (non-affine-in-input)? In the following we will address this problem.

1.1.2 ILC for Non-affine Process

A non-affine-in-input process can be described by

y(t) = g(u(t), t) t [0, T]

where g(u, t) is nonlinear in u, e.g. g = ueu. It is worth to point out that, evenifg is known a priori, the closed form ofg1 may not exist for most nonlinearfunctions. Thus we are not able to find the desired control profile by invertingthe process, consequently ud = g1(yd(t), t) is not achievable. Moreover, in

practice g could be only partially known. To capture the desired control signal,we need to look for a more powerful approach, which is again the simplestILC (1.1) associated with certain condition imposed on the function g.

Let us first derive the convergence of the input sequence, ui ud. Assumethat g is continuously differentiable to all the arguments, using the Mean ValueTheorem

yi = yd yi = g(ud, t) g(ui, t) = g(ud, t) g(ud ui, t)

= g(ud, t) [g(ud, t) gu(i, t)ui] = gu(i, t)ui

where gu= f

u, and i [ud |ui|, ud + |ui|]. Following the preceding

derivation, and substituting the above relationship

ui+1 = ui qyi = ui qgu(i, t)ui

= (1 qgu(i, t))ui,


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hence

|ui+1| |1 qgu(i, t)||ui|.

In order to let |1 qgu(i, t)| = < 1, the function g needs to meet thefollowing condition:

(C1) gu must have known lower and upper bounds, both are of the same signand strictly nonzero. Assume 1 the lower bound and 2 the upper bound,then either 0 < 1 2 or 0 > 2 1.

With this condition a learning gain q can be chosen to make strictly lessthan one. A conservative design is q = 12 (for simplicity we only considerpositive gu). Note the similarity between the present non-affine and precedinglinear cases. gu is the equivalent process gain, like g(t) in the linear case, thusit naturally leads to the same convergence condition within the same boundingcondition. However, in the non-affine case the process gain gu is depending onthe control input u. Thus it is also necessary to limit u, especially when guturns out to be a radially unbounded function of u, i.e.

lim|u|

|gu| .

In such circumstance we have to limit u to a compact set U. By virtue of thecontinuous differentiability of g, gu is bounded on U. For example considerg = ueu and u [0, um], then gu = eu +ueu, 1 = 1, and 2 = eum+umeum

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Linear Non Linear Iterative Learning Control