Xu Guang-zhao,Fu Qin,Du Li-li,Wu Jian-rong and Yu Peng-fei

(School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou,Jiangsu,215009)

Communicated by Li Yong

Abstract:This paper deals with the problem of iterative learning control for a class of linear continuous-time switched systems in the presence of a fixed initial shift.Here,the considered switched systems are operated during a finite time interval repetitively.According to the characteristics of the systems,a PD-type learning scheme is proposed for such switched systems with arbitrary switching rules,and the corresponding output limiting trajectories under the action of the PD-type learning scheme are given.Based on the contraction mapping method,it is shown that this scheme can guarantee the outputs of the systems converge uniformly to the output limiting trajectories of the systems over the whole time interval.Furthermore,the initial rectifying strategies are applied to the systems for eliminating the effect of the fixed initial shift.When the learning scheme is applied to the systems,the outputs of the systems can converge to the desired reference trajectories over a pre-specified interval.Finally,simulation examples illustrate the effectiveness of the proposed method.

Key words:iterative learning control,switched system,PD-type learning scheme,fixed initial shift,output limiting trajectory

1 Introduction

Since the complete algorithm of iterative learning control(ILC)was first proposed by Arimoto and Kawamura[1],it has become a hot issues of cybernetics(see[2]–[7]).The basic idea of ILC is to improve the control signal for the present operation cycle by feeding back the control error in the previous cycle.The classical formulation of ILC design problem is to find an update mechanism for the output trajectory of a new cycle based on the information from previous cycles so that the output trajectory converges asymptotically to the desired reference trajectory.Owing to its simplicity and effectiveness,ILC was found to be a good alternative in many areas and applications(e.g.,see[8]for detailed results).

In the process of ILC design,an interesting question is how to set the initial value of the iterative system properly at each iteration,such that the output trajectory of the iterative system can converge to the desired reference trajectory.In the previous works,a common assumption about this question is that the initial value at each iteration should be equal to the initial value of the desired reference trajectory(see[1]–[4]),or within its neighborhood(see[5]–[7]).In the case of perturbed initial conditions,boundedness of the tracking error is established and the error bound is shown to be proportional to the bound on initial condition errors(see[5]–[7]).Recently,the initial rectifying strategies are introduced in learning algorithm(see[11]–[13]).For a class of partially irregular multivariable plants,Porter and Mohamed[9]utilized initial impulse rectifying to eliminate the effect of the fixed shift,so that a complete reference trajectory tracking over the whole time interval was achieved.In the case of fixed initial shift,the output limiting trajectory for the first time under the actions of D-type and PD-type learning schemes was obtained in[10],the convergence performance under the action of a PID-type learning scheme was further considered in[11],and the convergence result was extended to nonlinear systems.Sun and Wang[12]addressed the initial shift problem of ILC for affine nonlinear systems with arbitrary relative degrees,and the uniform convergence of the output trajectory to a desired one jointed smoothly with a specified transient trajectory from the starting position was ensured in the presence of a fixed initial shift.Sun et al.[13]proposed a feedback-aided PD-type learning algorithm to solve the initial shift problem for linear time-invariant systems in the presence of a fixed initial shift.

Switched systems,each of which consists of a number of subsystems and a switching law,have attracted much attention in the field of control theory(see[14]).Recently,the ILC problem for linear continuous-time switched systems was put forward in[15],under the assumption condition that the initial value at each iteration is equal to the initial value of the desired reference trajectory,the convergence conclusions were obtained based on the D-type learning algorithms.The results obtained in[15]were further extended to nonlinear continuous-time switched systems in[16].

Stimulated by the works of[15],in this paper,we study the problem of iterative learning control algorithms for linear continuous-time switched systems in the presence of a fixed initial shift.A PD-type learning algorithm is put forward and the initial rectifying strategies are introduced to achieve the corresponding results.We propose two kinds of convergence conditions,one is discussed in[15]and the other is not,and the corresponding proofs are given.Our results shows that the output of the system converges uniformly to the corresponding output limiting trajectory over the whole time interval under the action of the PD-type learning scheme,and the output of the system converges uniformly to the corresponding desired reference trajectory over the pre-specified interval under the actions of the initial rectifying strategies.

In this paper,the following notational conventions are adopted:for n-dimensional Euclidean space Rn,∥x∥ denotes Euclidean norm of a vector x=[x1,x2,···,xn]T.For a matrix A,∥A∥denotes its induced norm.For a function h:[0,T]→ Rnand a real number λ >0,∥h(·)∥sdenotes the supreme norm defined by

∥h(·)∥λdenotes the λ-norm defined by

From[2],∥h(·)∥sand ∥h(·)∥λare equivalent for a finite constant λ.Thus,the convergence results can be proved using either of them.

2 Problem Description

Consider the following linear continuous-time switched system(see[15]):

where x(t)∈Rn,u(t)∈Rr,y(t)∈Rqrepresent the state,control input and output of the system respectively,t denotes time variable. σ(t)is a switching law defined by σ(t):(1,2,···) → M={1,2,···,m},this means that the matrices(Aσ(t),Bσ(t),Cσ(t))are allowed to take values,at an arbitrary time,in the finite set

Without loss of generality,the arbitrary switching law σ(t)can be assumed as follows(see[15]–[16]):

Then the system(2.1)can be represented as:

where i∈ {1,2,···,m}.

It is assumed that system(2.2)is repeatable over t∈[0,T].Rewrite the system(2.2)at each iteration as:

where the subscript k is employed to mark the iteration index.System(2.3)is assumed to satisfy the following assumptions:

Assumption 1For each iteration index k,the initial value is always set to the fixed value x0,i.e.,xk(0)=x0.

Assumption 2The matrix CiBiis full row rank(when q≤r)or full column rank(when r≤ q),i∈ {1,2,···,m}.

Lemma 2.1[17]Suppose that{ak},{bk}(k=1,2,···)are two non-negative real sequences satisfying

3 PD-type Learning Algorithms

For a given trajectory yd(t),construct the PD-type learning scheme for the system(2.3)as follows(see[10]):

where Γ ∈ Rr×qand L ∈ Rq×qare the learning gain matrices,and all the eigenvalues of the matrix L have positive real part.While ek(t)=yd(t)?yk(t)is the tracking error at kth iteration.

It is assumed that there exists a(t)such that

For system(2.3),the dimension of the input is less than or equal to the dimension of the output,i.e.,r≤q,we have:

Theorem 3.1Suppose that Assumptions 1–2 and(3.2)are satisfied.If there exists a gain matrix Γ ∈ Rr×qsuch that

then,under the action of the learning scheme(3.1),the output sequence yk(t)converges uniformly to the corresponding output limiting trajectory(t)over the whole time interval

Differentiating both sides of the above expression,we obtain

From(3.5)and(3.6),we have Denote ?u?k(t)=u?d(t)? uk(t),?x?k(t)=x?d(t)? xk(t).From(2.3),(3.2)and(3.7),we get

Taking t∈[0,t1]and combining with(3.3),it yields

where c1= ∥ΓC1A1+LC1∥.

It follows from(2.3)and(3.2)that

Integrating both sides of(3.10)over[0,t],0≤t≤t1,and combining with(3.4)and Assumption 1,we obtain

Taking Euclidean norm on both sides of the above expression,we have

Taking λ such that

Substituting(3.11)into(3.9)results

Since 0 ≤ ρ <1,from(3.3),it is possible to choose λ sufficiently large so that ρ1<1.Then,(3.12)is a contraction inIt follows from(3.11)and(3.12)that

Furthermore,we have

Taking t∈[t1,t2]in(3.8)and combining with(3.3),it yields

where c3= ∥ΓC2A2+ΓLC2∥.

Integrating both sides of(3.10)over[t1,t],t1≤t≤t2,we obtain

Taking Euclidean norm on both sides of the above expression,which yields

Taking λ so that

Substituting(3.16)into(3.15)results

Since 0 ≤ ρ <1,by(3.3),it is possible to choose λ sufficiently large so that ρ2<1.Then,by(3.14)and Lemma 2.1,we have

together with(3.16)implies

Furthermore,we have

For t∈[t2,t3],by repeating the same procedure as that after(3.14),we can obtain

Furthermore,we have

From(3.13),(3.17)–(3.19),we know

Therefore

This completes the proof.

Remark 3.1The convergence condition(3.3)was discussed in[15]and Theorem 3.1 extends the results in[15]to the systems with fixed initial shifts.Especially,when x0=xd(0),the conclusion of Theorem 3.1 is the same as that in[15].

For the system(2.3),since the dimension of the output is less than or equal to the dimension of the input,i.e.,q≤r,we have:

Theorem 3.2Suppose that Assumptions 1–2 and(3.2)are satisfied.If there exists a gain matrix Γ ∈ Rr×qsuch that

then,under the action of the learning scheme(3.1),the output sequence yk(t)converges uniformly to the corresponding output limiting trajectory(t)over the whole time interval[0,T],under the action of the learning scheme(3.1),i.e.,

Proof. Denote δxk(t)=xk+1(t)? xk(t),δuk(t)=uk+1(t)? uk(t).From(2.3),(3.1)and(3.7),we have

It is easy to yield that

From(3.21)we obtain

Taking t∈[0,t1]and combining with(3.20),it yields

It follows from(3.4)and Assumption 1 that

Then we have

Substituting(3.24)into(3.23)results

Integrating both sides of(3.21)over[0,t],0≤t≤t1,and combining with Assumption 1,we obtain

As in the proof of Theorem 3.1,we derive

Substituting(3.26)and(3.24)into(3.25)results

Therefore,we have

From(3.26)–(3.28)we obtain

Furthermore,we have

Taking t∈[t1,t2]in(3.22)and combining with(3.20),it yields

Substituting(3.32)into(3.30)yields

Integrating both sides of(3.21)over[t1,t],t1≤t≤t2,we can obtain

As in the proof of Theorem 3.1,we derive

Substituting(3.34)and(3.32)into(3.33)results

where

By(3.29),(3.30)and Lemma 2.1,we have

From(3.31)we obtain

together with(3.29)and(3.36)implies

From(3.30),(3.34),(3.35)and(3.37),we have

Furthermore,we have

By repeating the same procedure as that after(3.30)again and again,we can obtain

From(3.28),(3.37)and(3.40),we know

This completes the proof.

Remark 3.2Generally speaking,PD-type learning algorithms for continuous systems can guarantee that the output limiting trajectory of the system asymptotically converges to the desired reference trajectory as time increases(see[10],[11]and[13]).It follows from

that the PD-type learning algorithm proposed in this paper also has the same properties.

4 Initial Rectifying Strategies

In order to achieve a complete reference trajectory tracking over a pre-specified interval,the initial rectifying strategies(see[11]–[13])are introduced to the learning scheme in this section.And the initial rectifying term is added to the PD-type learning scheme to eliminate the effect of the fixed initial shift beyond a small initial time interval.

Construct the PD-type iterative learning scheme with initial rectifying term for the system(2.3)as follows:

where r(t)represents the initial rectifying term.

If e?(t)is required approximately equal to zero over a finite interval[0,h],that is,

then we may choose the initial rectifying term r(t)=e?Ltδh(t)(yd(0)? C1x0),where δh(t)is a function satisfyingwhen t<0 or t>h.The function δh(t)to meet the requirements may be chosen as follows:

Remark 4.1From(4.1),we know that,if r(t)is a continuous function,then e?(t)is a differentiable function.So the objective of taking δh(t)such as(4.2)is to guarantee the differentiability of e?(t).

For the system(2.3),the dimension of the output is less than or equal to the dimension of the input,i.e.,r≤q,we have

Theorem 4.1Suppose that Assumptions 1–2 and(3.2)are satisfied.If there exists a gain matrix Γ ∈ Rr×qsuch that

then,under the action of the learning scheme(3.41),the output sequence yk(t)converges uniformly to the corresponding output limiting trajectory y?d(t)over the whole time interval

Proof.Note that y?d(0)=C1x0.It follows from the system(3.1)that y?d(0)=C1x?d(0).Thus,it is reasonable to set the initial value:x?d(0)=x0.By(3.5),

We differentiate both sides of the above expression to obtain

According to the definitions of e?(t)and r(t),it is easy to see that

From(4.3)–(4.5),we have

The rest is on exactly the same lines as that after(3.7)in the proof of Theorem 3.1.

For system(2.3)that the dimension of the output is less than or equal to the dimension of the input,i.e.,q≤r,we have

Theorem 4.2Suppose that Assumptions 1–2 and(3.2)are satisfied.If there exists a gain matrix Γ ∈ Rr×qsuch that

then,under the action of the learning scheme(3.41),the output sequence yk(t)converges uniformly to the corresponding output limiting trajectory y?d(t)over the whole time interval[0,T],under the action of the learning scheme(3.44),i.e.,yd(t)? e?(t),

Proof. Replace(3.7)with(4.6)and(3.1)with(3.41),then the proof can be completed by using the same procedure as that in Theorem 3.2.

If we take a positive number h sufficiently small,then the above two theorems show that the initial rectifying strategies can achieve a complete reference trajectory tracking beyond a small initial time interval[0,h].

Remark 4.2Assumption 2 can guarantee the existence of the gain matrices Γ in Theorems 3.1,3.2,4.1 and 4.2.

5 Simulation Examples

Example 5.1Systems that the dimension of the input is less than or equal to the dimension of the output.Consider the following linear continuous-time switched system,which contains two subsystems:

where σ(t)is an arbitrary switching sequence with σ(t)={1,2},and

the subscript k is employed to mark the iteration index.Set the initial value at each iteration with the fi xed valuethen

(1)PD-type learning algorithm.For the given desired reference trajectory:

we have

We produce a random sequence σ(t),t=0,0.2,0.4,0.6,0.8,1 with the values 1 and 2,as shown in Fig.5.1.

Fig.5.1 The random switching rule of σ(t)

If σ(t)=1,system(5.1)is(A1,B1,C1),otherwise,if σ(t)=2,(5.1)is(A2,B2,C2).Take the initial control u0(t)=0.Under the action of the learning scheme(3.1),and by using the mathematical software Matlab,it is easy to see that∥e?k∥stends to zero as k → ∞,as shown in Fig.5.2.

Fig.5.2 Tracking errors of yk

(2)Initial rectifying strategies.Take h=0.1,and construct

Take the desired reference trajectory as:

A random sequence σ(t)is shown in Fig.5.3.

Fig.5.3 The random switching rule of σ(t)

Take the initial control u0(t)=0.Under the action of the learning scheme(4.1),and by using the mathematical software Matlab,it is easy to see that ∥e?k∥stends to zero as k→∞,as shown in Fig.5.4.

Fig.5.4 Tracking errors of yk

Example 5.2Systems that the dimension of the output is less than or equal to the dimension of the input.Consider the following linear continuous-time switched system,

which contains two subsystems:

where σ(t)is an arbitrary switching sequence with σ(t)={1,2},and

the subscript k is employed to mark the iteration index.Set the initial value at each iteration to the fixed valuethen

(1)PD-type learning algorithm.For the given desired reference trajectory:

A random sequence σ(t)is shown in Fig.5.5.

Fig.5.5 The random switching rule of σ(t)

Fig.5.6 Tracking errors of yk

(2)Initial rectifying strategies.Take h=0.1,and construct

Take the desired reference trajectory as:yd(t)=3e2t?3e?tf(t),where

then we have

A random sequence σ(t)is shown in Fig.5.7.

Fig.5.7 The random switching rule of σ(t)

and by using the mathematical software Matlab,it is easy to see that ∥e?k∥stends to zero as k→∞,as shown in Fig.5.8.

Fig.5.8 Tracking errors of yk

6 Conclusion

This paper studies the PD-type iterative learning control problem for linear continuous time switched systems in the presence of a fixed initial shift.Two kinds of the systematic structure are considered in this paper,one is that the dimension of the input is less than or equal to the dimension of the output,and the other is that the dimension of the output is less than or equal to the dimension of the input.By the PD-type learning scheme,the convergence theorems of the output tracking errors are established based on the contraction mapping method.Our results show that,under the action of the PD-type learning scheme,the output of the system converges uniformly to the corresponding output limiting trajectory over the whole time interval,while under the actions of the initial rectifying strategies,the output of the system converges uniformly to the corresponding desired reference trajectory beyond a small initial time interval under the actions of the initial rectifying strategies.The simulation results are consistent with theoretical analysis.How to apply the methods in this paper to nonlinear switched systems,it remains further research.