A new method to study ILC problem for time delay linear systems

R E S E A R C H

Open Access

A new method to study ILC problem for

time-delay linear systems

Zijian Luo

_{, Michal Feˇckan}

_{and JinRong Wang}

*_{Correspondence:} [email protected] 1_{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 apply a new method, a delayed matrix exponential, to study P-type and D-type learning laws for time-delay controlled systems to track the varying reference accurately by using a few iterations in a finite time interval. We present open-loop P- and D-type asymptotic convergence results in the sense of

λ

MSC: 34A37; 93C15; 93C40

Keywords: time-delay system; delayed exponential matrix; iterative learning control

1 Introduction

In the past decades, delay differential equations have been widely used in the fields of eco-nomics, physics, and engineering control. Existence, stability, and periodic solutions are studied extensively and there are many interesting and important results; see, for example, [–].

Since Uchiyama [] and Arimoto [] put forward the iterative learning control (ILC for short), a wide variety of iterative learning control problems and related issues are proposed and studied in recent decades. For example, ILC for varying reference trajectories [, ], ILC for fractional differential systems [, ], ILC for impulsive differential systems [, ], research on the robustness of ILC [, ], and so on.

It is a common phenomenon that time delays exist in many practical engineering issues. However, the prevalence of the phenomenon to the delay caused a lot of practical engi-neering problems. So the study of the control problem of time-delay system is paid more attention. Some effective methods for studying the iterative learning control for time-delay systems are provided by Sun [–].

After reviewing the previous works dealing with ILC problems for delay systems, we observe the following facts:

(i) A delay systemx˙(t) =Ax(t) +Bx(t–τ),t> is considered mostly as an integral system, whereA,Bare suitable matrices.

(ii) A uniform transition matrix associated withA,Bis not derived directly and the structure of solutionx(t)is not well characterized on every subintervals

[,τ], . . . , [nτ, (n+ )τ],n∈N.

(iii) An extended Gronwall inequality is used to derive convergence results instead of applying direct methods.

It is remarkable that Khusainov and Shuklin in [] initially introduced a delayed expo-nential matrix method to study the following linear differential equation with one delay term:

⎧ ⎨ ⎩

˙

x(t) =Ax(t) +Bx(t–τ), t≥,τ ≥,

x(t) =ϕ(t), –τ≤t≤, ()

whereAandBare matrices andϕis an arbitrary continuously differentiable vector func-tions. A representation of a solution of system () withAB=BAis given by using a so-called delay exponential matrix, which is defined as follows:

eBtτ =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

, t< –τ,

E, –τ ≤t< ,

E+Bt+B(t_–τ)+· · ·+Bk(t–(_kk_!–)τ)k, (k– )τ ≤t<kτ,k= , , . . . ,

()

forτ> , whereandEare then-dimensional zero and identity matrices, respectively. For more recent contributions on oscillating systems with pure delay, relative control-lability of system with pure delay, asymptotic stability of nonlinear multidelay differential equations, one can refer to [–] and the references therein.

Inspired by the references mentioned above, in this work, we discuss ILC for time-delay systems. More precisely, we study the following linear controlled systems with pure delay:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˙

xk(t) =Axk(t) +Bxk(t–τ) +uk(t), t∈[,T],

xk(t) =ϕ(t), –τ≤t≤,τ > ,

yk(t) =Cxk(t) +Duk(t),

()

whereTdenotes pre-fixed iteration domain length withT=Nτ andN∈N. Letϕ∈C τ := C_{([–τ, ],R}n_{),AandBbe twon×nmatrices such thatAB=BAandC,Dbe twom×n}

matrices,kdenotes thekth learning iteration, the variablesxk,uk∈Rnandyk∈Rmdenote

state, input, and output, respectively. By [], Corollary ., we derive that the statexk(·)

has the form

xk(t) =eA(t+τ)eBτtϕ(–τ) +

 –τ

eA(t–τ–s)eB(t–τ–s)

τ e

Aτ_{ϕ(s) –Aϕ(s)ds}

+

t



eA(t–τ–s)eB(t–τ–s)

τ eA

τ

uk(s)ds, ()

whereeBtτ is defined in () andB=e–AτB. By introducingeBt

τ for (), we state some possible advantages of our approach as follows.

(i) The structure of solutionx(t)is characterized on every subintervals.

Letydbe a desired trajectory and setek=yd–yk, which denotes output error andδuk=

uk+–uk.

For the system (), we consider the open-loop P-type ILC updating law

uk+(t) =uk(t) +Poek(t). ()

For the system () withD=, we consider the open-loop D-type ILC updating law

uk+(t) =uk(t) +Doe˙k(t), ()

wherePoandDo∈Rn×mare learning gain matrices.

The main objective of this paper is to use delayed exponential matrix to generate the control inputuksuch that the time-delay system outputykis tracking the iteratively

vary-ing reference trajectoriesydas accurately as possible whenk→ ∞uniformly on [,T] in

the sense of theλ-norm by adopting P-type ILC and D-type ILC.

Here we point out that our method is different from the method given in the previous reference, however, we obtain the same convergence results. Our method relies on a direct formula solution, so it is constructive.

The rest of this paper is organized as follows. In Section , we give some notations, concepts, and lemmas. In Sections  and , we give convergence results of P-type ILC and D-type for system (). Examples are given in Section  to demonstrate the applicability of our main results.

2 Preliminaries

LetJ⊂Rbe a finite interval andL(Rn_{) be the space of bounded linear operators inR}n_.

Denote byC(J,Rn_{) the Banach space of vector-value continuous functions fromJ→R}n

endowed with the∞-norm x =max_t∈J|x(t)|for a norm | · |onRn_{. We also consider}

on C(J,Rn_{) a λ-norm x}

λ=sup_t∈J{e–λt|x(t)|}, λ> . We introduce a space C(J,Rn) = {x∈C(J,Rn_{) :x˙ ∈C(J,R}n_{)}. For a matrix A:R}n_→Rn_{, we consider its matrix norm}

A =max_|x|=|Ax|generated by| · |.

Lemma .(see [], .., Chapter ) Let A∈L(Rn_{).For a given> ,there is a norm}

| · |onRn_{such that}

A ≤ρ(A) +,

whereρ(A)denotes the spectral radius of the matrix A.

Next, we give an alternative formula to compute the solution of linear system with pure delay, which is a direct corollary of [], Corollary ..

Lemma . Let f :J→Rn_{be a continuous function.The solution x∈C}_(J,Rn_)of

⎧ ⎨ ⎩

˙

x(t) =Ax(t) +Bx(t–τ) +f(t), t> ,τ≥,

has the form

Proof According to the formula () in [], we know the solution of system () has the form

Next, we submit the formula of delayed matrix exponential () to () to prove the result. We divide our proof into two steps.

Step . We prove that

Step . We check that

3 Convergence analysis of P-type

In this section, we give the first convergence result of P-type.

Theorem . Let yd(t),t∈[,T]be a desired trajectory for system().Ifρ(E–DPo) < ,

Proof Without loss of generality, we consider (j– )τ ≤t<jτ,j= , , . . . ,N. Linking ()

According to () and Lemma ., we have

ek+(t) = (E–DPo)ek(t) –C

For anyλ> A , we apply integration by parts via mathematical induction to derive

= – 

and, taking theλ-norm, we arrive at

ek+ λ≤

Remark . We use a delayed matrix exponential method to obtain convergence of the

uniform convergence only under conditionρ(E–DPo) <  in the end of the above proof.

Moreover, fixing∈(,–ρ(E–DPo)

 ), the smallest suitableλ> A is given by the equation

C Po

which is rather awkward to solve. On the other hand, () has a unique solution for any > . Moreover, it may have anyλ> A as a solution by varying C Po .

4 Convergence analysis of D-type

In this section, we discuss the ILC convergence of D-type.

Theorem . Let yd(t),t∈[,T],be a desired trajectory for system()with D=.Ifρ(E–

Similar to the proof of Theorem ., one can apply Lemma . to derive that

+ CA

In analogy to the computation in ()-(), inequality () becomes

_e˙k+(t)e–λt≤ρ(E–CDo) +e˙k(t)e–λt+ (W+W) Do ˙ek λ, ()

we needCto be surjective.

5 Simulation examples

In this section, four numerical examples are presented to demonstrate the validity of the designed method. In order to simulate the tracking errors of trajectories, we adopt for sim-plicityL_{-norm in our simulations. ThisL}_{-norm can be used, since by the fact that e}

Figure 1 The system output and the tracking error for (20).

Table 1 The tracking error of each iteration for (20)

k 1 2 3 4 5 6 7 8 9 10

Error 18.41 4.31 2.68 1.91 1.33 0.95 0.68 0.48 0.34 0.24

eλT _e

k λfor a suitableλ> A , we obtain ek → ask→ ∞,i.e.,limk→∞supt∈J|ek(t)|=

, which yieldsek→ inL(J,Rn).

Example . Consider

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˙

xk(t) =xk(t) +xk(t– .) +uk(t), x(t)∈R,t≥,

xk(t) =t, –.≤t≤,

yk(t) =xk(t) + .uk(t),

()

and P-type ILC

uk+(t) =uk(t) +ek(t).

The original reference trajectory is

yd(t) =

⎧ ⎨ ⎩

sin(πt), t∈[, .],

sin(πt) + cos(πt) + , t∈[., ].

Set t∈[, ],τ = .,ϕ(t) =t. Forn=m= , A= ,B= ,C= , andD= .. It is not difficult to find thatB=e–AτB=e–.and

et_.=

⎧ ⎨ ⎩

 +e–.t, t∈[, .],

 +e–._t+e– (t–.)

! , t∈[., ].

Next, we setPo= . Obviously,ρ( –DPo) = . < . Thus, all conditions of Theorem .

are satisfied, soyk(t) uniformly converges toyd(t), fort∈[, ].

The upper figure of Figure  shows the outputykof equation () of the th iterations

and the reference trajectoryyd. The lower figure of Figure  shows theL-norm of the

Figure 2 The system output and the tracking error for (21).

Table 2 The tracking error of each iteration for (21)

k 1 2 3 4 5 6 7 8 9 10

Error 18.41 10.80 7.16 4.99 3.60 2.65 1.98 1.49 1.34 0.86

k 11 12 13 14 15 16 17 18 19 20

Error 0.66 0.51 0.39 0.30 0.23 0.18 0.14 0.11 0.08 0.06

Example . Consider

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˙

xk(t) =xk(t) +xk(t– .) +uk(t), x(t)∈R,t≥,

xk(t) =t, –.≤t≤,

yk(t) =xk(t),

()

and D-type ILC

uk+(t) =uk(t) + .˙ek(t).

The original reference trajectory is the same as for Example .. Set t∈[, ],τ = ., ϕ(t) =t. Forn=m= ,A= ,B= ,C=  andD= . It is not difficult to find thatBand et_.are the same as for Example .. Next, we setDo= .. Obviously,ρ( –CDo) = . < .

Thus, all conditions of Theorem . are satisfied.

The upper figure of Figure  shows the equation () outputykof the th iterations and

the reference trajectoryyd. The lower figure of Figure  shows theL-norm of the tracking

error (see also Table ) in each iteration.

Example . Consider

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˙

xk(t) =xk(t) +xk(t– .) +uk(t), xk(t),uk(t)∈R,t≥,

xk(t) = (et,et)T, –.≤t≤,

yk(t) = (, )xk(t) + (, )uk(t),

()

and P-type ILC

uk+(t) =uk(t) +

. –.

Figure 3 The system output and the tracking error for (22).

Table 3 The tracking error of each iteration for (22)

k 1 2 3 4 5 6 7 8 9 10

Error 10.09 5.41 3.14 1.62 0.76 0.33 0.14 0.06 0.02 0.01

The original reference trajectory is

yd(t) =

⎧ ⎨ ⎩

sin(πt), t∈[, .],

sin(πt) + sin(πt) + , t∈[., ].

Sett∈[, ],τ= .,ϕ(t) = (et_,et₎T_{. Next,n= ,m= ,A=B=E,Eis the identity matrix,}

C= (, ) andD= (, ). It is not difficult to find that

B=e–AτB=

.   .

,

and

et_.=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

E, t∈[–., ],

E+Bt, t∈[, .],

E+Bt+B (t–.)

! , t∈[., ].

Next, we setPo= (., –.)T. Obviously,ρ(E–DPo) = . < . Thus, all conditions of

Theorem . are satisfied. Thenyk(t) uniformly converges toyd(t), fort∈[, ].

The upper figure of Figure  shows the equation () outputykof the th iterations and

the reference trajectoryyd. The lower figure of Figure  shows theL-norm of the tracking

error (see also Table ) in each iteration.

Example . For system

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˙

xk(t) =xk(t) +xk(t– .) +uk(t), xk(t),uk(t)∈R,t≥,

xk(t) = (t,t)T, –.≤t≤,

yk(t) = (, )xk(t),

Figure 4 The system output and the tracking error for (23).

Table 4 The tracking error of each iteration for (23)

k 1 2 3 4 5 6 7 8 9 10

Error 10.09 6.17 5.18 4.62 4.02 3.40 2.81 2.30 1.87 1.50

k 11 12 13 14 15 16 17 18 19 20

Error 1.21 0.96 0.77 0.61 0.49 0.39 0.31 0.24 0.19 0.15

we take the D-type ILC

uk+(t) =uk(t) +

 –.

˙ ek(t).

The original reference trajectory is the same as for Example .. Obviously, all conditions of Theorem . are satisfied. Thenyk(t) uniformly converges toyd(t), fort∈[, ].

The upper figure of Figure  shows the outputykof equation () of the th iterations

and the reference trajectory yd. The lower figure of Figure  shows theL-norm of the

tracking error (see also Table ) in each iteration.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have made equal contributions. All authors have read and approved the final manuscript.

Author details

1_{Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China.}2_{Department of Mathematical} Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, Bratislava, 842 48, Slovakia.

Acknowledgements

The authors are grateful to the referees and Associate Professor Dr. Qian Chen for their careful reading of the manuscript and valuable comments. The authors also thank the editor for help. The first and third authors acknowledge National Natural Science Foundation of China (11661016; 11201091), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), United Foundation of Guizhou Province ([2015]7640) and Outstanding Scientific and Technological Innovation Talent Award of Education Department of Guizhou Province ([2014]240). The second author acknowledges the Slovak Grant Agency VEGA No. 1/0078/17 and the Slovak Research and Development Agency under the contract No. APVV-14-0378.

Received: 4 August 2016 Accepted: 6 January 2017

References

1. Balachandran, B, Kalmár-Nagy, T, Gilsinn, DE: Delay Differential Equations. Springer, Berlin (2009)

2. He, J: Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul.2, 235-236 (1997)

4. Wen, Y, Zhou, XF, Zhang, Z, Liu, S: Lyapunov method for nonlinear fractional differential systems with delay. Nonlinear Dyn.82, 1015-1025 (2015)

5. Abbas, S, Benchohra, M, Rivero, M, Trujillo, JJ: Existence and stability results for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations. Appl. Math. Comput.247, 319-328 (2014) 6. Zhang, GL, Song, MH, Liu, MZ: Exponential stability of the exact solutions and the numerical solutions for a class of

linear impulsive delay differential equations. J. Comput. Appl. Math.285, 32-44 (2015)

7. Uchiyama, M: Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. Soc. Instrum. Control Eng.14, 706-712 (1978)

8. Arimoto, S, Kawamura, S: Bettering operation of robots by learning. J. Robot. Syst.1, 123-140 (1984)

9. Xu, JX, Xu, J: On iterative learning for different tracking tasks in the presence of time-varying uncertainties. IEEE Trans. Syst. Man Cybern. B34, 589-597 (2004)

10. Ahn, HS, Moore, KL, Chen, YQ: Iterative Learning Control: Robustness and Monotonic Convergence in the Iteration Domain. Communications and Control Engineering Series. Springer, Berlin (2007)

11. Luo, Y, Chen, YQ: Fractional order controller for a class of fractional order systems. Automatica45, 2446-2450 (2009) 12. Li, Y, Chen, YQ, Ahn, HS: On thePDα_{-type iterative learning control for the fractional-order nonlinear systems. In: Proc.}

Amer. Control Conference, pp. 4320-4325 (2011)

13. Bien, ZZ, Xu, JX (eds.): Iterative Learning Control: Analysis, Design, Integration and Applications. Springer, Media (2012)

14. Li, ZG, Chang, YW, Soh, YC: Analysis and design of impulsive control systems. IEEE Trans. Autom. Control46, 894-897 (2001)

15. Sun, MX: Robust convergence analysis of iterative learning control systems. Control Theory Appl.15, 320-326 (1998) 16. Lee, HS, Bien, Z: Design issues on robustness and convergence of iterative learning controller. Intell. Autom. Soft

Comput.8, 95-106 (2002)

17. Sun, MX, Chen, YQ, Huang, BJ: High order iterative learning control system for nonlinear time-delay systems. Acta Autom. Sin.20, 360-365 (1994)

18. Sun, MX: Iterative learning control algorithms for uncertain time-delay systems (I). J. Xi’an Instit. Technol.17, 259-266 (1997)

19. Sun, MX: Iterative learning control algorithms for uncertain time-delay systems (II). J. Xi’an Instit. Technol.18, 1-8 (1998)

20. Khusainov, DY, Shuklin, GV: Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Žilina Math. Ser.17, 101-108 (2003)

21. Khusainov, DY, Shuklin, GV: Relative controllability in systems with pure delay. Int. J. Appl. Math.2, 210-221 (2005) 22. Medved’, M, Pospišil, M, Škripokvá, L: Stability and the nonexistence of blowing-up solutions of nonlinear delay

systems with linear parts defined by permutable matrices. Nonlinear Anal. TMA74, 3903-3911 (2011) 23. Medved’, M, Pospišil, M: Sufficient conditions for the asymptotic stability of nonlinear multidelay differential

equations with linear parts defined by pairwise permutable matrices. Nonlinear Anal. TMA75, 3348-3363 (2012) 24. Diblik, J, Feˇckan, M, Pospišil, M: Representation of a solution of the Cauchy problem for an oscillating system with two

delays and permutable matrices. Ukr. Math. J.65, 58-69 (2013)

25. Boichuk, A, Diblik, J, Khusainov, D, R˚užiˇcková, M: Fredholm’s boundary-value problems for differential systems with a single delay. Nonlinear Anal. TMA72, 2251-2258 (2010)

26. Boichuk, A, Diblik, J, Khusainov, D, R˚užiˇcková, M: Boundary value problems for delay differential systems. Adv. Differ. Equ.2010, Article ID 593834 (2010)

27. Boichuk, A, Diblik, J, Khusainov, D, R˚užiˇcková, M: Boundary-value problems for weakly nonlinear delay differential systems. Abstr. Appl. Anal.2011, Article ID 631412 (2011)