Uniform design and analysis of iterative learning control for a class of impulsive first order distributed parameter systems

R E S E A R C H

Open Access

Uniform design and analysis of iterative

learning control for a class of impulsive

first-order distributed parameter systems

Xiulan Yu

_{and JinRong Wang}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Guizhou University, Guiyang, Guizhou 550025, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we provide uniform design and analysis framework for iterative learning control of a class of impulsive first-order distributed parameter systems in the time domain. In particular,P-type andD-type iterative learning controls with initial state learning are considered. We present convergence results for open-loop iterative learning schemes in the sense of theLp-norm and

λ

Keywords: iterative learning control; impulsive; first-order distributed parameter systems; initial state learning

1 Introduction

In , Arimotoet al.[] propose basic theories and algorithms on iterative learning con-trol (ILC) and point out that ILC is a useful practical concon-trol approach for systems which perform tasks repetitively over a finite time interval. The performance improvement can be made step by step, and the output trajectory can be realized to track the desired one, through updating the input signal by the error data. Through three decades of study and development, one has achieved significant progress in both theories and applications, and become one of the most active fields in intelligent control. For more details on the contri-butions for linear and nonlinear ordinary differential equations, the reader is referred to the monographs [–], and [–].

The issue on designing and analyzing an ILC for impulsive differential equations, dis-continuous systems [, ], distributed parameter systems or PDEs has not been fully investigated, and only a limited number of results [–] are available so far. In [], Liu

et al.exploreP-type iterative learning control law with initial state learning for impulsive ordinary differential equations to tracking the discontinuous output desired trajectory. In [], Xuet al.studyP-type andD-type ILC for linear first-order distributed parameter sys-tems in the sense of the sup-norm. In [], Huang and Xu apply aP-type steady-state ILC scheme to the boundary control of PDEs. In [], Huanget al.construct a uniform design and analysis framework for iterative learning control of linear inhomogeneous distributed parameter systems.

When dealing with impulsive distributed parameter systems, the ILC design and prop-erty analysis become far more challenging. The existing ILC design and analysis should be

improved. The main objective of this paper is extended [] to a study of the ILC for im-pulsive nonlinear first-order distributed parameter systems with initial error in the sense of theλ-norm (the symbol ofλ-norm is introduced by Arimotoet al.[];cf.[] and []) via semigroup theory.

This paper is a continuation of our recent related papers [, ]. The main contributions of the paper are summarized as follows.

(i) A uniform design and analysis framework is presented for ILC of a class of impulsive first-order distributed parameter systems in the time domain. Nevertheless, [] consider the ILC of impulsive ordinary differential equations in finite dimensional spaces.

(ii) Instead of considering ILC of linear first-order distributed parameter systems with-out initial error as in [], we consider a class of impulsive nonlinear first-order distributed parameter systems with initial error and more general discontinuous output tracking problem.

(iii) Instead of simplifying ILC updating law without initial state learning as in [], we consider ILC updating law with initial state learning.

(iv) Instead of choosing the sup-norm as in [], we use theLp-norm andλ-norm, re-spectively.

2 System description and problem statement

DenoteJ:= [,a] and letX,UandYbe three Hilbert spaces. We study ILC of the following impulsive nonlinear first-order distributed parameter systems:

˙

xk(t) =Axk(t) +f(t,xk(t)) +Buk(t), t∈J\D,k∈N,

xk(ti+) =xk(ti–) +gi(xk(ti)), i∈M,

()

and output equation

yk(t) =Cxk(t) +Duk(t), t∈J, ()

wherekdenotes the iterative times,xkis the state variable at thekth iteration,ukis the con-trol input at thekth iteration,ykis the system output at thekth iteration,D:={t,t, . . . ,tm}, M:={, , . . . ,m},xk: [,a]→X,uk: [,a]→U,yk: [,a]→Y. The linear unbounded operatorAis the infinitesimal generator of aC-semigroupT(t),t≥ inX,Bis a bounded linear operator fromUtoX,i.e.,B∈L(U,X),Cis a bounded linear operator fromXtoY,

i.e.,C∈L(X,Y), andDis a bounded linear operator fromUtoY,i.e.,D∈L(U,Y). The nonlinear termsf :J×X→Xandgi:X→Xwill be specified later. We have the im-pulsive time sequences{ti}i∈Msatisfying  =t<t<t<· · ·<tm<tm+=a. The jumps xk(ti–) :=limε→–x_k(t_i+ε) andx_k(t+

i) :=limε→+x_k(t_i+ε) represent the left and right limits

ofxk(t) att=ti, respectively.

DenotePC(J,X) :={x:J→X:x, continuous att∈J\D, andxis continuous from the left and has right hand limits att∈D}endowed with theλ-normxλ=supt∈Je–λtx(t)X for someλ> . DefineLp_{(J,X) :={x:J→Xis strongly measurable :}a

x(s)

p

Xds<∞}, endowed with the normxLp= (a

x(t)

p Xdt)



p_,p∈_(,∞_{). Obviously, (PC(J,X),} ·

λ)

and (Lp_{(J,X), ·}

By aPC-mild solution of () with initial valuex() =x∈X, we mean the functionxk∈

PC(J,X) can be rewritten as the following expression []:

xk(t) =T(t)x+

t



T(t–s)fs,xk(s)

+Buk(s)

ds

+

<tj<t

T(t–tj)gj

xk(tj)

, t∈J. ()

By adopting the same methods as in [], Theorem ., one can obtain the existence and uniqueness of a mild solution of () withx() =xwhenf andgisatisfy the standard Lipschitz conditions.

Submitting () into (), we have

yk(t) =CT(t)x+

t



CT(t–s)fs,xk(s)

+Buk(s)

ds

+

<tj<t

CT(t–tj)gj

xk(tj)

+Duk(t), t∈J.

Letyd(·) be the desired trajectory. Denoteuk:=uk+(t) –uk(t),xk:=xk+(t) –xk(t), andek(t) :=yd(t) –yk(t) wherekrepresents the iteration index. Consider the open-loop

P-type ILC updating law with initial state learning:

xk+() =xk() +Lek(),

uk+(t) =uk(t) +γek(t),

()

and the open-loopD-type ILC updating law with initial state learning:

xk+() =xk() +Lek(),

uk+(t) =uk(t) +γe˙k(t),

()

whereL,L∈L(Y,X) andγ,γ∈L(Y,U) are unknown parameters to be determined.

Concerning the system (), we will designP-type andD-type iterative learning schemes to generate the control inputuk(·) such that the system piecewise continuous outputyk(·) tracks the discontinuous desired output trajectoryyd(·) as accurately as possible ask→ ∞ fort∈Jin the sense of suitable norms. We shall give two convergence results for open-loop iterative learning schemes in the sense of theLp-norm andλ-norm, respectively, in the next sections.

3 Convergence results forP-type ILC updating law

We need the following assumptions:

(H) A:D(A)⊆X→Xis the generator of aC-semigroupT(t),t≥onX. DenoteM:= sup_t∈JT(t)L(X,X).

(H) f :J×X→X is strongly measurable for the first variable and continuous for the second variable. Moreover, there exists aLf > such that

(H) There exists aLg> such that

gi(u) –gi(v)_X≤Lgu–vX, u,v∈X,t∈J,i∈M.

(H) LetIY be an identity operator inYandIY–Dγ–CL∈L(Y,Y)satisfy

ρ:=IY–Dγ–CLL(Y,Y)< . ()

Now we are ready to give the first convergence result in the sense of theLp_-norm.

Theorem . Assume that(H)-(H)hold.If

IY–DγL(Y,Y)q √

a+CL(X,Y)MBL(U,X)γL(Y,U) q √

a_{( +ML}

g)meMLfa< , ()

then for arbitrary initial input u, ()guarantees that yktends to yd∈Lp(J,Y)as k→ ∞

in the sense of the Lp_{-norm where <p,q<∞and}

p+



q= .

Proof Linking () and (), we have

ek+(t) =yd(t) –yk+(t) = (IY–Dγ)ek(t) –Cxk(t). ()

In what follows, we proveek+Lp→ ask→ ∞.

Step . We prove thatek+()Y→ ask→ ∞. In fact, fort= , by using () we have

ek+() = (I–Dγ)ek() –CLek(). ()

Substituting () into () and taking theY-norm, we have

ek+()_Y=IY–Dγ–CLL(Y,Y)ek()_Y:=ρek()_Y,

which implies that

_ek+()_Y≤ρke()_Y. ()

Linking () and (), we conclude that

lim

k→∞

ek+()_Y= . ()

Step . For anyt∈(ti,ti+],i= , , . . . ,m, we have

xk(t)_X

=xk+(t) –xk(t)_X

≤Mxk()_X+MLf

t



xk(s)_X+BL(U,X)uk(s)_U

+MLg

<tj<t

xk(tj)_X

≤MLL(Y,X)ek()_Y+MLf

t



xk(s)_Xds

+MBL(U,X)γL(Y,U)

t



ek(s)_Yds+MLg

<tj<t

xk(tj)_X.

Using the impulsive Gronwall inequality (see []) and the Hölder inequality, we have

xk(t)_X

≤

MLL(Y,X)ek()_Y+MBL(U,X)γL(Y,U)

t



ek(s)_Yds

( +MLg)meMLfa

≤MLL(Y,X)ek()_Y+MBL(U,X)γL(Y,U)q √

aekLp( +ML_g)meMLfa_{, ()}

where _p+_q=  andp,q> .

Taking theY-norm for () and substituting () into it, we have

ek+(t)_Y

≤ IY–DγL(Y,Y)ek(t)_Y+CL(X,Y)

MLL(Y,X)ek()_Y

+MBL(U,X)γL(Y,U) q √

aekLp( +ML_g)meMLfa

≤ IY–DγL(Y,Y)ek(t)_Y+CL(X,Y)MLL(Y,X)( +MLg)meMLfaek()_Y

+CL(X,Y)MBL(U,X)γL(Y,U)q √

aekLp( +ML_g)meMLfa.

For the above inequality, one can take theLp_{-norm to derive that}

ek+Lp

≤ CL(X,Y)MLL(Y,X)( +MLg)meMLfa q

√

aek()_Y

+IY–DγL(Y,Y)q √

a+CL(X,Y)MBL(U,X)γL(Y,U) q √

a_{( +ML}

g)meMLfa

× ekLp.

Finally, one can use () and () to derive that

lim

k→∞ek+L

p= .

The proof is completed.

Remark . The condition () in Theorem . seems to be a bit strong since we choose theLp_{-norm. However, one can choose another suitable norm, theλ-norm, to weaken this} condition.

Theorem . Assume that(H)-(H)hold and

ρ:=IY–DγL(Y,Y)< . ()

For arbitrary initial input u, ()guarantees that yktends to yd∈PC(J,Y)as k→ ∞in

the sense of theλ-norm for a sufficiently largeλ> .

Proof By our assumptions and Theorem ., we know () holds. Next, we only need to proveek+λ→ ask→ ∞.

Note that

e–λt

t



ek(s)_Yds≤ 

λekλ.

Then () turns to

xk(t)_X≤

MLL(Y,X)ek()_Y+

eλt_MB

L(U,X)γL(Y,U)

λ ekλ

×( +MLg)meMLfa. ()

Substituting () into () again, we have

ek+(t)_Y ≤ IY–DγL(Y,Y)ek(t)_Y+CL(X,Y)

MLL(Y,X)ek()_Y

+e

λt_MB

L(U,X)γL(Y,U)

λ ekλ

( +MLg)meMLfa.

For the above inequality, one can take theλ-norm to derive that

ek+λ≤ CL(X,Y)MLL(Y,X)( +MLg)meMLfa)ek()_Y

+

IY–DγL(Y,Y)+

MBL(U,X)γL(Y,U)( +MLg)meMLfa λ

ekλ.

Then for someλ>  large enough and linking (), we obtain

ek+λ≤ρekλ,

which gives

lim

k→∞ek+λ= .

The proof is completed.

Remark . One can use the same method in Theorem . to weaken assumption () in Theorem . if one replaces the standardLp_{-norm with an exponentially weighted term}

4 Convergence results forD-type ILC updating law

–Cd dt

<tj<t

T(t–tj)xk+(tj)

–CT()xk+(t) –C

t

 d

dtT(t–s)xk+(s)ds

=IY–CT()Bγ

˙ ek(t) –C

t



AT(t–s)Bγe˙k(s)ds

–CT()xk+(t) –C

t



AT(t–s)xk+(s)ds

–C <tj<t

AT(t–tj)xk+(tj).

This yields

e_˙k+(t)_Y

≤IY–CT()Bγ_L(Y,Y)e˙k(t)_Y

+CL(X,Y) max

t∈[,a]

AT(t)_L(X,X)BL(U,X)γL(Y,U)

t



_˙ek(s)_Yds

+MCL(X,Y)xk+(t)_X+CL(X,Y) max

t∈[,a]

_AT(t)

L(X,X)

t



xk+(s)ds

+CL(X,Y) max

t∈[a–tm,a–t]

AT(t)_L(X,X)

<tj<t

xk+(tj).

Taking theλ-norm, we get

˙ek+λ≤δ˙ekλ

+CL(X,Y)maxt∈[,a]AT(t)L(X,X)BL(U,X)γL(Y,U)

λ ˙ekλ:=I

+MCL(X,Y)xk+λ

+CL(X,Y)maxt∈[,a]AT(t)L(X,X)

λ xk+λ:=I

+ m

j=

CL(X,Y)maxt∈[a–tm,a–t]AT(t)L(X,X)

eλ(t–tj) xk+λ:=I.

Obviously, the termsIi,i= , , , tend to zero if we chooseλ>  large enough. Concerning () and changingktok+ , we take theλ-norm,

xk+λ≤MLL(Y,X)( +MLg)meMLfaek+()_Y:=I

+MBL(U,X)γL(Y,U)( +MLg) m_eMLfa

λ ek+λ:=I.

Using () and takingλ>  large enough, we see thatIi,i= , , tend to zero. This yields

xk+λ→ ask→ ∞.

Next, noting thatδ< , we obtain

lim

Step . We compute the tracking error at each iteration,

Taking theλ-norm, we get

ek+λ≤ek+()_Y+



λ˙ek+λ. ()

Linking () and () and () we derive the desired results.

5 Example

In this section, we consider the following impulsive partial differential equation with Neu-mann boundary conditions:

The authors declare that they have no competing interests.

Authors’ contributions

Author details

1_{College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, P.R. China.} 2_{Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China.}

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too. This work is supported by the National Natural Science Foundation of China (11201091) and Outstanding Scientific and Technological Innovation Talent Award of Education Department of Guizhou Province ([2014]240).

Received: 26 March 2015 Accepted: 10 August 2015

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