ITERATIVE LEARNING CONTROL - GONE WILD S. R. Koscielniak TRIUMF Vancouver B.C. Canada Abstract

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ITERATIVE LEARNING CONTROL - GONE WILD

S. R. Koscielniak*, TRIUMF, Vancouver, B.C., Canada

Abstract

Before AI and neural nets, the excitement was about it-

erative learning control (ILC): the idea to train robots to

perform repetitive tasks, or train a system to reject quasi-

periodic disturbances. The excitement waned after the dis-

covery of “learning transients” in systems which satisfy the

ILC asymptotic convergence (AC) stability criteria. The

transients may be of long duration, persisting long after ei-

genvalues imply they should have decayed, and span or-

ders of magnitude. They occur both for causal and non-

causal learning.

The field recovered with the introduction of tests for

“monotonic convergence of the vector norm”, but no deep

and truly satisfying explanation was offered. Here we ex-

plore solutions of the ILC equations that couple the itera-

tion index to the within-trial sample index. This sheds light

on the causal learning – for which the AC test gives a re-

peated eigenvalue.

Moreover, since 2016, this author has demonstrated that

a new class of solutions, which are soliton-like, satisfy the

recurrence equations of ILC and offer additional insight to

long-term behaviour. A soliton is a wave-like object that

emerges in a dispersive medium that travels with little or

no change of shape at an identifiable speed. This paper is

the first public presentation of the soliton solutions, which

may occur for both causal (i.e. look back) and noncausal

(i.e. look ahead) learning functions that have diagonal band

structure for their matrix representation.

INTRODUCTION

The TRIUMF-Elinac[1,2] is a 30 MeV, 10 mA c.w. ca-

pable 1.3 GHz SRF electron linear accelerator. The linacne

has a 300 keV electron gun in an SF6-filled tank, a 10 MeV

injector cryomodule with one 9-cell cavity, and a 20 MeV

accelerator cryomodule (ACM) with two cavities, and a

target station; all linked by beam transports. The linac is

extensible to 50 MeV by the installation of a second ACM.

The E-linac is nominally a c.w. accelerator, but must op-

erate at much lower power for commissioning and beam

development. So e-linac is pulsed with a variety of repeti-

tion rate and pulse length, leading to significant transient

beam-loading of the SRF cavities. Feed-forward compen-

sation (FFC) of beam-loading transients is essential. Ini-

tially, the RF group maintained an argument that the beam

would pass before the RF transients grew, and that FFC

was not needed; later they conceded the need for FFC.

A different FFC[t] function is needed for every combi-

nation of repetition rate and pulse length. In 2013, LLRF

group proposed [3] to generate the many FFC[t] time func-

tions by Iterative Learning [4]. Based on numerical simu-

lation, they chose a 4-term noncausal Iterative Learning

* shane@triumf.ca

Control (ILC): Q=I and L=I+⅓(↑+↑↑+ ↑↑↑). Here Q is a

filter and ↑ is a lift applied to the FF vector. But numerical

cases do not guarantee stability or convergence, because

they cannot span the entire space of initial conditions. So

present author, embarked on a complete analysis, culminat-

ing in the discovery in 2016 of wave solutions and predic-

tion of unstable learning.

The E-linac LLRF system is a digital system, and well

suited to implementing ILC. The RF waveform is demod-

ulated and sampled. The control signals are implemented

digitally. All samples can be recorded and held as super-

vectors.

ITERATIVE LEARNING CONTROL

System (or plant P) with its own internal (stable) re-

sponse. Place a ILC wrapper around P and iterate wrapped

system from one trial to next. During the trial, the plant is

a free system with a driven input. At the end of an individ-

ual trial, the input is updated based on results from the ILC

wrapper. “Learning” is a matrix-map iteration of input to

output vector. Nested within the matrix is the internal re-

sponse of the plant during a trial, which is different each

trial. In time domain the iterations are non-linear. Hope-

fully the plant and its inputs & outputs and the map all “set-

tle down” so that the wrapped system converges.

ILC occurs in a 2-dimensional space: within a trial, in-

dex k; and from one iteration to the next, index j or n.

Symbols & Equations

Vectors (internal index k): u = input, y = output, d = dis-

turbance (repetitive), yd = desired output (repetitive),

ej=(yd-y)j = error. Operators: P = “the plant”, i.e. the sys-

tem, Q = filter, L = learning function.

During the trial: yj=Puuj + Pdd. From one trial to the

next, “u” learns from the previous trial (or trials). uj+1=

Q[uj+Lej] where ej=(yd-y)j . Elimination of e leads to iter-

ative maps for u and y. These maps are function of the in-

ternal gains (K) and time constants (τ) of P, and the itera-

tion gains (ν) of L; or the set {K,τ;ν} for short.

Converges to what?

Mapping: operator M such that xn+1=M:xn where xn may

be scalar or vector. If M=M[n], mapping is non-linear.

Mappings have fixed points which may be stable or unsta-

ble, and are not necessarily zeros of x. Within the basin of

attraction, a map will (eventually) converge on a stable

fixed point. Convergence is not a synonym for “stable”.

“Stability” answers the question: “what happens when x is

perturbed from its fixed point?”. The perturbations are in-

finitesimal, the system is locally linear, and the response is

a decaying oscillation if “stable”.

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ITERATIVELEARNINGCONTROL-GONEWILDS.R.Koscielniak*,TRIUMF,Vancouver,B.C.,CanadaAbstractBeforeAIandneuralnets,theexcitementwasaboutit-erativelearningcontrol(ILC):theideatotrainrobotstoperformrepetitivetasks,ortrainasystemtorejectquasi-periodicdisturbances.Theexcitementwanedafterthedis-coveryof“learnin...

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ITERATIVE LEARNING CONTROL - GONE WILD S. R. Koscielniak TRIUMF Vancouver B.C. Canada Abstract

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