[ieee 2014 american control conference - acc 2014 - portland, or, usa (2014.6.4-2014.6.6)] 2014...

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H based Disturbance Attenuation for Iterative Learning Control Krzysztof Galkowski, Pawel Dabkowski, Eric Rogers Abstract—Previous research has shown that repetitive pro- cesses, a class of 2D systems, can be used to design linear model based iterative learning control laws for convergence and transient performance, with supporting experimental bench- marking. In many applications attenuation of disturbances acting on the plant signals will also be required. The new results in this paper are control law design algorithms for this problem with disturbance attenuation measured by an Hnorm. I. I NTRODUCTION Many systems compete the same finite duration task over and over again, where once each is complete the system resets to the starting location and the next one begins. A generic example is a gantry robot executing a pick and place task where the sequence of operations is: i) collect an object, or payload, from a given location, ii) transfer it over a finite duration, iii) place the playload on a moving conveyor, iv) return to the starting location and v) repeat i)-iv) as many times as required or for a finite number and then stop for maintenance. In the literature, each execution is commonly known as a trial and its duration the trial length. Iterative learning control [1], or ILC for short, has been developed for such systems where the distinguishing feature is the use of information from previous trials to update the control signal applied on the next trial. In particular, once the system has completed each trial, the complete information generated is available for use in computing the control signal to be applied on the next trial, with the aim of sequentially improving performance from trial-to-trial. A major application area is industrial robotics, but many others have also arisen in the engineering domain and the survey papers [2] and [3] are possible starting points for the literature. More recently, there has been a technology transfer to next generation healthcare where ILC, with sup- porting clinical trials, has been used to adjust the level of electrical stimulation applied to the relevant muscles of a stroke patient undergoing robotic-assisted upper limb rehabilitation for everyday tasks, such as reaching out to a cup over a table top or reaching out and then extending the arm [4], [5], [6]. This work is partially supported by National Science Centre in Poland, grant No. 2011/01/B/ST7/00475 K. Galkowski is with the Institute of Control and Computation Engineer- ing, University of Zielona Gora, ul. Podgorna 50, 65-246 Zielona Gora, Poland. Pawel Dabkowski is with the Institute of Physiscs, Faculty of Physics, As- tronomy and Informatics, Nicholas Copernicus University, ul. Grudziadzka 5, 87-100 Torun, Poland E. Rogers is with Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK, E- mail: [email protected] Let α< denote the trial length and in the discrete time domain y ref (p), 0 p α - 1, the scalar or vector valued reference representing desired output behavior. Then if y k (p), 0 p α - 1,k 1, denotes the scalar or vector valued output on trial k, the error on this trial is e k (p)= y ref (p) - y k (p). Moreover, the construction of a sequence of input functions that improves performance from one trial to the next is equivalent to the following convergence conditions on the input and error lim k→∞ ||e k || =0, lim k→∞ ||u k - u || =0, (1) where || · || is a signal norm in a suitably chosen function space with a norm-based topology. These convergence con- ditions are for trial-to-trial performance and if the dynamics along the trial are discrete a commonly used setting for design is based on a form of lifting that enables the dynamic plant model in R to be treated, for single-input single-output (SISO) systems with a natural extension to the multivariable case, as a static system in R N , where N denotes the number of samples along the trial. If the along the trial transient dynamics are unsatisfactory, the only option in lifting design is to introduce a feedback control law and then complete the ILC design for the resulting controlled system. This step will be required if the plant is unstable when trial-to-trial error convergence is still possible due to the finite trial length and hence a two stage design procedure. An alternative that allows simultaneous consideration of trial-to-trial error convergence and transient response along the trials is to use the 2D systems setting. The use of 2D systems theory in ILC design is based on treating trial-to-trial updating in k as one direction of information propagation and the evolution of the dynamics over 0 p α as the other. The first use of 2D discrete linear systems theory in ILC design is credited to [7] where the Roesser state-space model was used. Repetitive processes [8] are another class of 2D systems characterized by a series of sweeps, termed passes (or trials), 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 that 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 that the output sequence of pass profiles generated can contain oscillations that increase in amplitude in the pass-to-pass direction. The finite pass length makes repetitive processes a more natural setting for ILC design and such designs have also been experimentally benchmarked [9], [10]. 2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA 978-1-4799-3274-0/$31.00 ©2014 AACC 4231

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Page 1: [IEEE 2014 American Control Conference - ACC 2014 - Portland, OR, USA (2014.6.4-2014.6.6)] 2014 American Control Conference - ℋ℞ based disturbance attenuation for iterative learning

H∞ based Disturbance Attenuation for Iterative Learning Control

Krzysztof Galkowski, Pawel Dabkowski, Eric Rogers

Abstract— Previous research has shown that repetitive pro-cesses, a class of 2D systems, can be used to design linearmodel based iterative learning control laws for convergence andtransient performance, with supporting experimental bench-marking. In many applications attenuation of disturbancesacting on the plant signals will also be required. The new resultsin this paper are control law design algorithms for this problemwith disturbance attenuation measured by an H∞ norm.

I. INTRODUCTION

Many systems compete the same finite duration task overand over again, where once each is complete the systemresets to the starting location and the next one begins. Ageneric example is a gantry robot executing a pick and placetask where the sequence of operations is: i) collect an object,or payload, from a given location, ii) transfer it over a finiteduration, iii) place the playload on a moving conveyor, iv)return to the starting location and v) repeat i)-iv) as manytimes as required or for a finite number and then stop formaintenance. In the literature, each execution is commonlyknown as a trial and its duration the trial length.

Iterative learning control [1], or ILC for short, has beendeveloped for such systems where the distinguishing featureis the use of information from previous trials to updatethe control signal applied on the next trial. In particular,once the system has completed each trial, the completeinformation generated is available for use in computing thecontrol signal to be applied on the next trial, with the aimof sequentially improving performance from trial-to-trial.

A major application area is industrial robotics, but manyothers have also arisen in the engineering domain and thesurvey papers [2] and [3] are possible starting points forthe literature. More recently, there has been a technologytransfer to next generation healthcare where ILC, with sup-porting clinical trials, has been used to adjust the levelof electrical stimulation applied to the relevant musclesof a stroke patient undergoing robotic-assisted upper limbrehabilitation for everyday tasks, such as reaching out toa cup over a table top or reaching out and then extendingthe arm [4], [5], [6].

This work is partially supported by National Science Centre in Poland,grant No. 2011/01/B/ST7/00475

K. Galkowski is with the Institute of Control and Computation Engineer-ing, University of Zielona Gora, ul. Podgorna 50, 65-246 Zielona Gora,Poland.

Pawel Dabkowski is with the Institute of Physiscs, Faculty of Physics, As-tronomy and Informatics, Nicholas Copernicus University, ul. Grudziadzka5, 87-100 Torun, Poland

E. Rogers is with Electronics and Computer Science,University of Southampton, Southampton SO17 1BJ, UK, E-mail: [email protected]

Let α < ∞ denote the trial length and in the discretetime domain yref (p), 0 ≤ p ≤ α − 1, the scalar or vectorvalued reference representing desired output behavior. Thenif yk(p), 0 ≤ p ≤ α− 1, k ≥ 1, denotes the scalar or vectorvalued output on trial k, the error on this trial is ek(p) =yref (p)−yk(p). Moreover, the construction of a sequence ofinput functions that improves performance from one trial tothe next is equivalent to the following convergence conditionson the input and error

limk→∞

||ek|| = 0, limk→∞

||uk − u∞|| = 0, (1)

where || · || is a signal norm in a suitably chosen functionspace with a norm-based topology. These convergence con-ditions are for trial-to-trial performance and if the dynamicsalong the trial are discrete a commonly used setting fordesign is based on a form of lifting that enables the dynamicplant model in R to be treated, for single-input single-output(SISO) systems with a natural extension to the multivariablecase, as a static system in RN , where N denotes the numberof samples along the trial.

If the along the trial transient dynamics are unsatisfactory,the only option in lifting design is to introduce a feedbackcontrol law and then complete the ILC design for theresulting controlled system. This step will be required if theplant is unstable when trial-to-trial error convergence is stillpossible due to the finite trial length and hence a two stagedesign procedure. An alternative that allows simultaneousconsideration of trial-to-trial error convergence and transientresponse along the trials is to use the 2D systems setting.The use of 2D systems theory in ILC design is basedon treating trial-to-trial updating in k as one direction ofinformation propagation and the evolution of the dynamicsover 0 ≤ p ≤ α as the other.

The first use of 2D discrete linear systems theory in ILCdesign is credited to [7] where the Roesser state-space modelwas used. Repetitive processes [8] are another class of 2Dsystems characterized by a series of sweeps, termed passes(or trials), through a set of dynamics defined over a fixedfinite duration known as the pass length. On each pass anoutput, termed the pass profile, is produced that acts asa forcing function on, and hence contributes to, the dynamicsof the next pass profile. This, in turn, leads to the uniquecontrol problem that the output sequence of pass profilesgenerated can contain oscillations that increase in amplitudein the pass-to-pass direction. The finite pass length makesrepetitive processes a more natural setting for ILC design andsuch designs have also been experimentally benchmarked [9],[10].

2014 American Control Conference (ACC)June 4-6, 2014. Portland, Oregon, USA

978-1-4799-3274-0/$31.00 ©2014 AACC 4231

Page 2: [IEEE 2014 American Control Conference - ACC 2014 - Portland, OR, USA (2014.6.4-2014.6.6)] 2014 American Control Conference - ℋ℞ based disturbance attenuation for iterative learning

This paper uses the repetitive process setting to developalgorithms for ILC design when the plant is subject todisturbances with attenuation measured by an H∞ norm.Throughout the paper, M > 0 and M < 0 are usedto denote symmetric positive definite and negative definitematrices respectively. Also the null and identity matrices withcompatible dimensions are denoted by 0 and I respectively.Finally, ∗ denotes the transpose of a block entry in asymmetric matrix and r(·) denotes the spectral radius ofsquare matrix, i.e., if the h × h matrix H has eigenvaluesλi, 1 ≤ i ≤ h, then r(H) = max1≤i≤h |λi|.

II. BACKGROUND

The plant to be controlled is assumed to be adequatelymodeled by linear time-invariant dynamics that, after sam-pling, can be represented by a controllable and observ-able discrete linear state-space model defined by the tripleA,B,C. In an ILC setting the state-space description ofthe uncontrolled dynamics is written as

xk(p+ 1) = Axk(p) +Buk(p), 0 ≤ p ≤ α− 1,

yk(p) = Cxk(p), (2)

where the integer subscript k ≥ 0 denotes the trial numberand on trial k xk(p) ∈ Rn is the state vector, yk(p) ∈ Rm isthe output vector, uk(p) ∈ Rr is the vector of control inputsand the trial duration α < ∞. If the signal to be tracked isdenoted by yref (p) then ek(p) = yref (p)−yk(p) is the erroron trial k, and the most basic requirement is to satisfy theconditions given in (1).

Trial-to-trial error convergence does not require that (2) isstable, i.e., r(A) < 1, since, for example, it is easily shownthat an update law of the form uk+1(p) = uk(p)+Lek(p+1),where L is an r×m matrix to be designed, gives this propertyprovided r(I − CBL) < 1. The reason for this is the finitetrial duration over which even an unstable linear system canonly produce a bounded output. For cases where r(A) ≥ 1,this allows for the production of large errors for small valuesof k and/or large values of α.

Even if r(A) < 1 there could be unacceptably largeoscillations in the dynamics produced along the early trialsfor many practical applications, such as a gantry robot whosetask is to collect an object from a location, place it on amoving conveyor, and then return for the next one and so on.If, for example, the object has an open top and is filled withliquid, and/or is fragile in nature, then unwanted vibrationsduring the transfer time could have very detrimental effects.In such cases there is also a need to control the dynamicsalong the trials, i.e., in p.

The many designs based on lifting, see the relevant refer-ences in [2], [3] for a selection of these, would proceed incases where the plant is unstable or the transient performancealong the trials is unsatisfactory, by first designing a feedbackcontroller to stabilize the plant and/or obtain acceptable alongthe trial dynamics. This would be followed by ILC designfor the controlled dynamics resulting from the previous step.To detail the setting for the lifting designs, consider, for

simplicity, the SISO case with N samples along the trial.Also introduce for each k

Uk =[uk(0) uk(1) . . . uk(N − 1)

]T,

Yk =[yk(1) yk(2) . . . yk(N)

]T,

often referred to in the literature as the input and outputsuper-vectors, respectively. Then the plant dynamics aredescribed in lifted form by the static equation in k

Yk = HUk, (3)

where H is a lower triangular matrix whose non-zero entriesare constructed from the system Markov parameters. In termsof the trial-to-trial error, the lifted representation is of theform Ek+1 = HEk, where Ek is the corresponding errorsupervector. This discrete updating equation in k is used asa basis ILC design. This method is prone to computationalissues if the matrices involved are of high dimensions and isnot applicable to design with differential dynamics.

An alternative is to use a 2D systems setting, where, asdiscussed in the previous section, ILC fits naturally into theclass of repetitive processes [8]. The unique characteristic ofa repetitive process is a series of sweeps, or trials, througha set of dynamics defined over a fixed finite duration knownas the trial length. On each trial an output, termed the trialprofile, is produced which acts as a forcing function on,and hence contributes to, the dynamics of the next trial.The unique control problem is that the output sequence oftrial profiles generated can contain oscillations that increasein amplitude in the trial-to-trial direction. (In the repetitiveprocess literature the word pass is used instead of trial.)

Repetitive processes cannot be controlled using standardsystems theory and algorithms. Instead, the starting point isthe rigorous stability theory [8] based on an abstract model ofthe dynamics in a Banach space setting that includes a verylarge class of processes with linear dynamics and a constanttrial length as special cases.

Consider discrete linear repetitive processes described bythe following state-space model over 0 ≤ p ≤ α− 1, k ≥ 1,

xk(p+ 1) = Axk(p) +Buk(p) +B0yk−1(p),yk(p) = Cxk(p) +Duk(p) +D0yk−1(p),

(4)

where on trial k, xk(p) ∈ Rn is the state vector, yk(p) ∈ Rm

is the trial profile vector, uk(p) ∈ Rr is the control inputvector, and α is the finite trial length. To complete theprocess description, it is necessary to specify the initial, orboundary, conditions, i.e. the state initial vector on each trial,xk+1(0), k ≥ 0, and the initial trial profile y0(p), 0 ≤ p ≤ α.

Given the boundary conditions and the control inputvector, the state and pass profile vectors on trial k = 1 canbe computed from (4) over 1 ≤ p ≤ α. Then the processresets to p = 0 and given the input vector (u2(p)) the stateand pass profile vectors can be computed for k = 2 and soon. This updating structure is exactly that of ILC. In the nextsection, the use of a repetitive process setting to analyze andILC schemes and, in particular, the stability theory of theseprocesses is employed to develop algorithms for control law

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design for trial-to-trial error convergence and along the trialperformance with disturbance attenuation.

This paper starts from the assumption that the disturbancesare additive on both the state and trial profile dynamics,giving the following state-space model for analysis over0 ≤ p ≤ α− 1, k ≥ 0,

xk+1(p+ 1) = Axk+1(p) + Buk+1(p) + B0yk(p)

+ B1ωk+1(p),

yk+1(p) = Cxk+1(p) + Duk+1(p) + D0yk(p)

+ D1ωk+1(p), (5)

where, on trial k, ωk(p) ∈ Rs is a disturbance vector on boththe state and trial profile vectors and is assumed to belong tothe function space `s2. The remainder of the notation is thatfor the disturbance-free model and the vector ω is assumedto have finite energy as measured by the `s2 norm defined for2D discrete signals as follows.

Definition 1: [11] The `s2 norm of a vector γk(p) ∈ Rs

defined over [0,∞], [0,∞] is

||γ||k2 =

√√√√ ∞∑p=0

∞∑k=0

ωTk (p)ωk(p),

where γk(p) is said to be a member of `s2[0,∞], [0,∞],or `s2 for short, if ||ω||2 <∞.

Definition 2: A discrete linear repetitive process describedby (5) is said to have H∞ disturbance attenuation, or H∞norm bound, γ if it is stable along the trial and

sup06=ω∈`sp2

||y||2||ω||2

< γ. (6)

The following result gives an LMI characterization ofdisturbance attenuation in the sense of Definition 2.

Theorem 1: [12] A discrete linear repetitive process de-scribed by (5) is stable along the trial and has H∞ distur-bance attenuation γ > 0 if there exist matrices P1 > 0 andP2 > 0 such that the following LMI with P = diag(P1, P2)is feasible

−P PΦ PΩ 0∗ −P 0 CT

2

∗ ∗ −γ2I 0∗ ∗ ∗ −I

< 0, (7)

where

Φ =

[A B0

C D0

], Ω =

[B1

D1

], C2 =

[0 I

].

In the next section the problem of control law design toensure stability along the trial and a prescribed degree ofH∞ disturbance attenuation is solved for ILC. Moreover, theresults given in the remainder of this paper extend naturallyto the case when the additive disturbance to the state and passprofile vectors are arise from different vectors that belong tothe `2 space.

III. ILC ANALYSIS AND DESIGN

The model for ILC design is

xk(p+ 1) = Axk(p) +Buk(p) +B1ωxk(p),

yk(p) = Cxk(p) +D1ωyk(p),

(8)

where ωx ∈ Rs1 and ωy ∈ Rs2 are disturbance vectors thatcannot be measured. In ILC, the novel feature is the use ofprevious trial information to compute the current trial inputwhere at any instant p information generated on previoustrials, such as at instant p + 1, can be used. Hence ILCallows the use of information that is non-causal in standardsystems theory. An ILC law for application on trial k + 1often has the form

uk(p) = uk−1(p) + ∆uk(p), (9)

where ∆uk(p) is the correction applied to the control vectorimplemented on the previous trial.

Introduce the vectors

ηk+1(p+ 1) := xk+1(p)− xk(p),ζxk+1(p) := ωx

k+1(p)− ωxk(p),

ζyk+1(p) := ωyk+1(p)− ωy

k(p),(10)

with the assumptions that ωxk+1(p) 6= ωx

k(p) andωyk+1(p) 6= ωy

k(p). Also introduce, for analysis purposes,

∆uk+1(p) = K1ηk+1(p+ 1) +K2ek(p+ 1). (11)

Then

ηk+1(p+ 1) = (A+BK1)ηk+1(p) +BK2ek(p)

+B1ζxk+1(p− 1),

ek+1(p) = −C(A+BK1)ηk+1(p) + (I − CBK2)ek(p)

− CB1ζxk+1(p− 1)−D1ζ

yk+1(p),

(12)

which is a linear discrete repetitive process state-space modelof the form (5), i.e.,

ηk+1(p+ 1) = Aηk+1(p) + B0ek(p) + B1ζk+1(p),

ek+1(p) = Cηk+1(p) + D0ek(p) + D1ζk+1(p),(13)

where

A = A+BK1, B0 = BK2, B1 =[B1 0

],

C = −C(A+BK1), D0 = (I − CBK2),

D1 =[−CB1 −D1

], ζk+1(p),=

[ζxk+1(p− 1)ζyk+1(p)

].

(14)

In the repetitive process setting, η is the current trial (orpass) state vector and the output (or pass profile) vector isthe current trial error. Hence linear repetitive process stabilitytheory can be applied to the analysis and regulation of trial-to-trial error convergence and the transient dynamics alongthe trials.

Remark 1: r(D0) = r(I − CBK2) < 1 is precisely thecondition obtained for trial-to-trial error convergence usingthe 2D Roesser state-space model [7]. This condition willguarantee convergence in k but places no restriction on the

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dynamics produced along the trials, e.g., on the eigenvaluesof the state matrix A.

The main result of this paper is the following.Theorem 2: A discrete linear repetitive process described

by (13) is stable along the trial and has H∞ disturbanceattenuation γ > 0 if there exist matrices W1 > 0, W2 > 0,N1 and N2 such that the following LMI is feasible

−W1 0 Υ1 Υ2 B1 0 0∗ −W2 −CΥ1 W2 − CΥ2 −CB1 −D1 0∗ ∗ −W1 0 0 0 0∗ ∗ ∗ −W2 0 0 W2

∗ ∗ ∗ ∗ −γ2I 0 0∗ ∗ ∗ ∗ ∗ −γ2I 0∗ ∗ ∗ ∗ ∗ ∗ −I

< 0,

(15)

where Υ1 = AW1 + BN1, Υ2 = BN2. If this LMI isfeasible, stabilizing control law matrices are given by

K1 = N1W−11 , K2 = N2W

−12 . (16)

Proof: This consists of the following steps: a) applythe result of Theorem 1 with

Φ =

[A B0

C D0

],Ω =

[B1

D1

], C2 =

[0 I

],

b) set P1 = W−11 , P2 = W−12 and left- and right-multiplyby diag(W1,W2,W1,W2, I, I, I) and c) set N1 = K1W1

and N2 = K2W2.

Remark 2: Theorem 2 gives a sufficient condition forsolvability of H∞ control law design problem. A desiredcontrol law (16) can be determined by solving the followingconvex optimization problem:

min σ subject to (15) (where σ = γ2) (17)The control law resulting from application of Theorem 2 canbe expressed in a form that explains its action in physicalor applications focused terms. In particular, simple algebraicmanipulations enable the control law (9) to be written as

uk(p) = uk−1(p) +K1(xk(p)− xk−1(p))

+K2(yref (p+ 1)− yk−1(p+ 1)). (18)

The second term in this last equation is phase-advance onthe previous trial error, which is used in many alreadyreported ILC implementations. Moreover, the data requiredfor this term is causal since yk−1(p) for all p is availableon completion of trial k − 1 and the ability to use suchinformation is one of the distinguishing features of ILC. Inpractical terms, it introduces anticipation of future errors inthe along the trial direction into the control law computation.If such a term is not present the ILC law can be replaced bya standard feedback control action and the benefits of ILCare lost. The first term in (18) is activated by the differencebetween the state vectors on the current and previous trialsand if all entries in these vectors are not available formeasurement an observer is one option.

0 50 100 150 200 250−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Sample Number

Fig. 1. The reference trajectory yref (p) for the considered axis of thegantry robot.

IV. CASE STUDY

Prior to any form of physical application, it is necessaryto examine control law performance in simulation and thissection gives a case study on the application of the new resultof this paper to one axis of a multi-axis gantry robot. Thisrobot emulates pick and place operation and has previouslybeen used for testing and comparing the performance ofother ILC algorithms, see, for example, [13]. Each axis ofthis robot has been modeled based on frequency responsetests where, since the axes are orthogonal, it is assumed thatthere is minimal interaction between them. An approximatecontinuous-time transfer-function representation for each ofthe three axes has been obtained through frequency responsetests and applying straight line approximations to the Bodegain plots. In this paper only the final result for one axis isused in the form of the following 7th order transfer-function(those for the other two axes can be found in [13])

G(s) =13077183.4436(s+ 113.4)

s(s2 + 61.57s+ 1.125× 104)

(s2 + 30.28s+ 2.13× 104)

(s2 + 227.9s+ 5.647× 104)(s2 + 466.1s+ 6.142× 105),

(19)

and the disturbance is simulated noise added to the statevector and the trial output.

The component of the 3D trajectory for this axis is shownin Fig. 1. and the results given in this section were obtainedfor the discrete representation of (19) obtained using zero-order hold discretization with a sampling period of 0.01 sec.The matrices B1 and D1 used are

B1 =[

1 1 1 1 1 1 1]T, D1 = 1

and disturbances on the state, ωx, and output ωy dynamicswere generated using the Matlab function ((rand(1, α −1)− .5) ∗ 2) ∗ 0.0358/50 with range plus and minus 2% ofthe maximum yref values. These choices are for illustrativepurposes only.

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Solving the convex optimization problem (17) and com-puting the control law matrices using (16) gives the controllaw matrices

K1 =[Ka

1 Kb1

],

Ka1 =

[−1.077 −29.01 −2.063 −4.996

],

Kb1 =

[10.99 15.17 −47.87

],

K2 = 2026,

with associated minimum H∞ disturbance attenuationγ = 10.

As representative results for this design, Fig. 2 shows themean squared error over 50 trials (where this quantity iscomputed along each trial) Figures 3, 4, 5 show the output,input and error on trial k = 50 respectively. Figures 6, 7 and8 show the progression against trial number k of the input,error and output respectively.

0 5 10 15 20 25 30 35 40 45 5010

−7

10−6

10−5

10−4

10−3

Trial number

X−

axis

mse

(m

2 )

Mean Squared Error

Fig. 2. Mean squared error (mse) against trial number (k).

0 20 40 60 80 100 120 140 160 180 200−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Along the trial direction

Out

put f

or th

e tr

ial n

umbe

r k

= 5

0

Output signalReference signal

Fig. 3. The output on trial number 50.

These results illustrate the potential of this design andhence experimental verification is feasible. The controllaw (18) does require access to the state vectors producedon successive trials and if all entries are not available formeasurement one option is to use an observer to estimatethe entries that cannot be directly measured. Another optionis to seek to design a control law that only uses outputinformation. This problem in the disturbance-free case is

0 20 40 60 80 100 120 140 160 180 200−8

−6

−4

−2

0

2

4

6

8

10

12

Along the trial direction

Inpu

t for

the

tria

l num

ber

k =

50

Fig. 4. The input on trial number 50.

0 20 40 60 80 100 120 140 160 180 200−3

−2

−1

0

1

2

3x 10

−3

Along the trial direction

Err

or fo

r th

e tr

ial n

umbe

r k

= 5

0

Fig. 5. The error on trial number 50.

010

2030

4050

60

0

50

100

150

200−15

−10

−5

0

5

10

15

Trial number

Input progression

Along the pass direction

Inpu

t val

ue

Fig. 6. Input progression against trial number k.

considered in [10] where the resulting control law can bewritten as

uk(p) = uk−1(p) +K3(yk(p)− yk−1(p))

+ K4(yk(p− 1)− yk−1(p− 1))

+ K5(yref (p+ 1)− yk−1(p+ 1)).

The design results in the previous section extend naturallyto this case.

V. CONCLUSIONS AND FURTHER WORK

This paper has considered the design of ILC laws fordiscrete linear systems when there is also a requirement

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010

2030

4050

60

0

50

100

150

200

250−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Trial number

Error progression

Along the pass direction

Err

or v

alue

Fig. 7. Error progression.

010

2030

4050

60

0

50

100

150

200

250−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Trial number

Output progression

Along the pass direction

Out

put v

alue

Fig. 8. Output progression.

for disturbance attenuation. The analysis uses the repetitiveprocess setting and results in LMI based designs, where thefinal form of the control law contains a phase-lead termand other terms activated by only plant output. A casestudy, based on the model of one axis of a gantry robotused to benchmark ILC laws with the disturbance modeledas simulated noise, confirms the potential of the design.Experimental verification of these designs would require ameans of introducing disturbances. This is feasible with arecently commissioned two-input two-output system [14],which can be configured such that one channel can be usedto introduce disturbances and will form immediate furtherwork.

Further research on the algorithm development side shouldinclude design in the presence of model uncertainty. Moti-vated by the phase-lead ILC term present in the control law,extension to make further use of previous trial terms shouldbe undertaken and also higher-order ILC where informationfrom a finite, greater than one, number of trials is used thecomputation of the current trial input.

The repetitive process setting is not the only way toaddress ILC design for disturbance attenuation and in thispaper some brief discussion of the relative merits whencompared to the lifting setting have been given. Further re-search, informed by simulation and experimental evaluation,is also required in this general area where the most likelyoutcome is different design settings whose relative meritswill also be applications dependant. One important point isthat the repetitive process setting extends to allow differential

dynamics along the trial, using the theory of differentiallinear repetitive processes, but there is no lifted model forsuch dynamics.

REFERENCES

[1] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operations ofrobots by learning,” Journal of Robotic Systems, vol. 1, no. 2, pp.123–140, 1984.

[2] D. A. Bristow, M. Tharayil, and A. Alleyne, “A survey of iterativelearning control,” IEEE Control Systems Magazine, vol. 26, no. 3, pp.96–114, 2006.

[3] Hyo-Sung Ahn, YangQuan Chen, and K. L. Moore, “Iterative learningcontrol: brief survey and categorization 1998 – 2004,” IEEE Trans-actions on Systems, Man and Cybernetics Part C, vol. 37, no. 6, pp.1099–1121, 2007.

[4] C. T. Freeman, A.-M. Hughes, J. H. Burridge, P. H. Chappell, P. L.Lewin, and E. Rogers, “Iterative learning control of FES applied tothe upper extremity for rehabilitation,” Control Engineering Practice,vol. 17, no. 3, pp. 368–381, 2009.

[5] A.-M. Hughes, C. T. Freeman, J. H. Burridge, P. H. Chappell,P. L. Lewin, and E. Rogers, “Feasibility of iterative learning controlmediated by functional electrical stimulation for reaching after stroke,”Neurorehabilitation and Neural Repair, vol. 23, no. 6, pp. 559 – 568,2009.

[6] C. T. Freeman, E. Rogers, A.-M. Hughes, J. H. Burridge, and K. L.Meadmore, “Iterative learning control in health care: Electrical stim-ulation and robotic-assisted upper-limb stroke rehabilitation,” IEEEControl Systems Magazine, vol. 32, no. 1, pp. 18 –43, 2012.

[7] J. E. Kurek and M. B. Zaremba, “Iterative learnig control synthesisbased on 2-D system theory,” IEEE Transactions on Automatic Con-trol, vol. 38, no. 1, pp. 121–125, 1993.

[8] E. Rogers, K. Gałkowski, and D. H. Owens, Control Systems Theoryand Applications for Linear Repetitive Processes, ser. Lecture Notes inControl and Information Sciences. Berlin, Germany: Springer-Verlag,2007, vol. 349.

[9] L. Hładowski, K. Gałkowski, Z. Cai, E. Rogers, C. T. Freeman, andP. L. Lewin, “Experimentally supported 2D systems based iterativelearning control law design for error convergence and performance,”Control Engineering Practice, vol. 18, no. 4, pp. 339–348, 2010.

[10] ——, “Output information based iterative learning control law designwith experimental verification,” ASME Journal of Dynamic Systems,Measurement and Control, vol. 134, no. 2, pp. 021 012/1–021 012/10,2012.

[11] C. Du and L. Xie, H∞ Control and Filtering of Two-dimensionalSystems, ser. Lecture Notes in Control and Information Sciences.Berlin, Germany: Springer-Verlag, 2002, vol. 278.

[12] W. Paszke, K. Gałkowski, E. Rogers, and D. H. Owens, “H∞ andguaranteed cost control of discrete linear repetitive processes,” LinearAlgebra and its Applications, vol. 412, no. 2–3, pp. 91–131, 2006.

[13] J. D. Ratcliffe, P. L. Lewin, E. Rogers, J. J. Hatonen, and D. H. Owens,“Norm-optimal iterative learning control applied to gantry robots forautomation applications,” IEEE Transactions on Robotics, vol. 22,no. 6, pp. 1303–1307, 2006.

[14] D. Van, C. T. Freeman, and P. L. Lewin, “Development of a multivari-able test facility for the evaluation of iterative learning controllers,” inProceedings of the American Control Conference, Montreal, Canada,2012, pp. 621–626.

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