# Lebesgue-p Norm Convergence Analysis of PDα-Type Iterative Learning Control for Fractional-Order Nonlinear Systems

Authors: Lei Li
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The first-order and second-order PDα-type iterative learning control (ILC) schemes are considered for a class of Caputo-type fractional-order nonlinear systems. Due to the imperfection of the λ-norm, the Lebesgue-p (Lp) norm is adopted to overcome the disadvantage. First, a generalization of the Gronwall integral inequality with singularity is established. Next, according to the reached generalized Gronwall integral inequality and the generalized Young inequality, the monotonic convergence of the first-order PDα-type ILC is investigated, while the convergence of the second-order PDα-type ILC is analyzed. The resultant condition shows that both the learning gains and the system dynamics affect the convergence. Finally, numerical simulations are exploited to verify the results.


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## Body

## 1. Introduction

Iterative learning control (ILC) is an effective control developed for target trajectory tracking [1]. The key feature of the ILC is to improve the quality of control iteratively by using proportional, integral, and/or derivative tracking errors obtained from previous operation and finally to generate the control input that causes the desired output trajectory. Due to its satisfactory tracking performance by using less a prior knowledge, ILC has been widely applied to repetitive operations including robot manipulations and batch processes [2–4].Fractional calculus is a mathematical topic with more than three-hundred-year-old history, but its application to physics and engineering has attracted a lot of attention in the latest decades [5, 6]. They have been verified to be a powerful technique to model the memory and hereditary properties of many materials and processes [7–9]. Further, it has been acknowledged that a fractional-order controller performs well compared to an integer-order controller for a fractional-order system. This pushes the development of the fractional-order controllers [10, 11].Among different fractional-order controllers, fractional-order iterative learning control (FOILC) is becoming one of active research areas. In the 2010s, for anαth-order linear system, the authors investigated the αth-order derivative-type (Dα-type) ILC in time domain [12]. This investigation showed that the optimal ILC for an αth-order linear system is the ILC order being α. In the following years, many FOILC problems are presented for various fractional-order systems. Up to now, the FOILC area has attracted much attention, of which the convergence analysis is one of key issues. For more details, readers can refer to the works [13–19] and the references therein. Despite the nice results of existing investigations, there still remain some undesirable problems between the theoretical development and its practical application.The first one is that many nice results are derived under a questionable assumption that the desired control input exists [20–22]. However, from the engineering application perspective, the desired output trajectory should be predetermined by the target to be tracked, rather than constructed under the assumption that the desired control input existed. That is, the convergence analysis process relies on the information that seems to be known but is actually unknown to the desired control input.The second one is that the existing FOILC investigations analyzed the convergence by usingλ-norm-based analytical methods, and the convergence is guaranteed with the sufficiently large λ. As commented in [23, 24], the larger parameter λ may greatly inhibit the actual tracking error and ignore the influence of the state matrix and proportional learning gains to the condition. This conveys that the results of both the ILCs and FOILCs are mathematically good, but practically, it may result in the normal tracking error exceeding the practical tolerance even though the λ-norm tracking error is satisfactory. Thus, both the existing ILCs and FOILCs need to be refined. For this aspect, reference [25] has adopted Lp-norm to evaluate the tracking error and has derived the monotonic convergence of the conventional first-order PD-type ILC for a class of integer-order linear systems. The derivation tells that the convergence is not only dominated by the system input and output matrices and the derivative learning gain, but also by the system state matrix and the proportional learning gain. The result reflects the relationship between the system dynamics and the learning mechanism to the convergence. To be specific, as L2-norm measures the tracking error in the concept of energy, the convergence result in the sense of Lp-norm may boost the theoretical development near to practical execution. However, the result of the FOILC is a hanging issue. It is necessary to state that the integer derivative is a local operator, while the fractional derivative is a nonlocal operator which has many different properties; thus many theoretical approaches based on the integer-order control systems cannot have been directly applied to the fractional ones. Therefore, it is worthwhile to apply the Lp-norm for convergence analysis of existing FOILCs.In addition, as discussed in FOILCs [12, 15–17], the higher-order learning algorithms, which employ preceding control information of more than one iteration, have utility to lead a better performance in terms of both convergence rate and robustness, which is taken advantage of. As a matter of fact, with different choice of learning gains, the higher-order classical ILC algorithm can be perform slower and faster than or equivalent to the lower-order ones in terms of convergence rate [25]. However, such affirmation has not been seen valid for fractional-order iterative learning control systems.Motivated by the aforementioned hanging issue regarding the fractional-order systems and FOILC schemes, this paper develops the first-order and the second-order proportional-Caputo-fractional-order-derivative-type (PDα-type) ILCs for a class of Caputo-type fractional-order nonlinear dynamic systems and then applies the Lp-norm to investigate their convergence in an objective manner. The main contributions of this paper are that we establish a theoretical analysis framework on the monotonic convergence of the first-order PDα-type ILC for a Caputo-type fractional-order nonlinear system in the sense of Lp-norm. In the theoretical analysis, there is no need for the questionable assumption that the desired input exists and a novel Growall integral inequality with singularity is established for the strict convergence analysis. And then, the convergence is derived for the case when the second-order PDα-type is implemented on the systems and the convergent speed comparison of the second-order law with the first-order one is generalized to FOILCs.The rest of the paper is organized as follows. In Section2, the basic concepts, properties, and lemmas are described. In Section 3, the monotonic convergence of the first-order PDα-type ILC scheme and the convergence of the second-order PDα-type ILC are given. In Section 4, examples are presented to validate the theoretical results. Finally, some conclusions are drawn in Section 5.


## 2. Preliminaries

In this section, we briefly give some basic definitions and properties related to fractional calculus [5, 6].Definition 1.
For an arbitrary integrable functionft, the definition of the fractional integrals of order α>0 is defined as(1)Dt-αft0=1Γα∫0tt-τα-1fτdτ,t∈0,∞,where Γ· is the Gamma function and Γα=∫0∞xα-1e-xdx.Definition 2.
For a given numberα>0, the α-order Caputo-type derivative of the function ft is defined as(2)Dtαft0C=1Γn-α∫0tfnτt-τα-n+1dτ,n-1<α<n,t∈0,∞,where n is an integer and fnt=dn/dtnft.Property 3.
If0<α<1, then Dt-α0Dtαft0C=ft-f0.Definition 4.
The Mittag-Leffler function is defined as(3)Eα,βz=∑k=0∞zkΓkα+β,α>0,β>0,z∈Cn×n,where Eαz=Eα,1z and E1,1z=ez.Definition 5 (see [26]).
For a given scalar functionf:0,T→R, its Lp-norm is defined as (4)f·p=∫0Tftpdt1/p,1≤p≤∞.For a time-varying vector function f:0,T→Rm,ft=f1t,…,fmtT, its Lp-norm is defined as(5)f·p=∫0Tmax1≤i≤mfitpdt1/p,1≤p≤∞.Lemma 6 (see [27]).
If the functionft∈Cn0,T, then the initial value problem, (6)Dtαxt0C=fxt,t,0<α<1,x0=x0,is equivalent to the following nonlinear Volterra integral equation: (7)xt=x0+1Γα∫0tt-τα-1fxτ,τdτ,and its solutions are continuous.For brevity, we setDtα0=Dtα0C in the following section.Lemma 7 (generalized Young inequality of convolution integral [26]).
For Lebesgue integrable scalar functionsg,h:0,T∈R, the generalized Young inequality of their convolution integral is (8)g∗h·r≤g·qh·p,where 1≤p,q,r≤∞ satisfy 1/r=1/p+1/q-1. Particularly, when r=p and thus q=1, then the inequality of convolution integral is(9)g∗h·p≤g·1h·p.Lemma 8 (see [2]).
Letan be a positive real sequence defined as (10)an≤ρ1an-1+ρ2an-2.If ρ1,ρ2 are nonnegative numbers satisfying(11)ρ=ρ1+ρ2<1,then the following holds:(12)limn→∞⁡an=0.Let us establish an extended Gronwall integral inequality with singularity, which is important to the convergence analysis in the next section. The proof is based on an iteration argument.Lemma 9.
Supposea,b≥0 (constant) and α>0,ct,xt and yt are nonnegative and locally integrable on 0,T0T0≤+∞. If (13)xt≤ct+∫0tt-sα-1ays+bxsds,t∈0,T0.Then(14)xt≤ct+∫0tΓα·Φα,αbΓαt-s·bcs+aysds,where Φα,αbΓαt-s=t-sα-1Eα,αbΓαt-sα.Proof.
See Appendix.


## 3. PDα-Type ILCs and Convergence Analysis for Fractional-Order Nonlinear Systems

Consider the following nonlinearα-order (0<α<1) systems:(15)Dtαxkt0C=fxkt,t+Bukt,ykt=Cxkt,t∈0,Txk0=0,where k refers to the operation number and 0,T is an operation time interval while α∈0,1. xkt∈Rn,ukt∈R and ykt∈R denoten-dimensional state vector, scalar control input, and output, respectively. B and C are matrices with appropriate dimensions. The function fxkt,t satisfies the global Lipschitz condition:(16)Cfxk+1t,t-fxkt,t≤L0Cxk+1t-xkt=L0yk+1t-ykt,where L0 is positive Lipschitz constant.In this section, the sufficient conditions are derived for convergence of the first-order and second-order PDα-type ILC algorithms for fractional-order nonlinear systems. Now, we give our main results.


## 4. Monotonic Convergence Analysis for First-Order PDα-Type ILC

To control the systems stated in (15), the first-order PDα-type ILC is given as follows:I1
(17)u1tis given arbitrarily,uk+1t=ukt+Lp1ekt+Ld1Dtαekt0,t∈0,T,k=2,3,4,….Here, Lp1 and Ld1 are the first-order proportional and fractional-order derivative learning gains, respectively. The expression ekt=ydt-ykt denotes the tracking error between the desired trajectory ydt and the system output ykt of the system (15) driven by ukt at the kth iteration.Theorem 10.
For the first-order PDα-type iterative learning control rue (I1) is applied to system (15), if the system matrices B, C, the order α and the Lipchitz constant L0 together with learning gains Lp1, Ld1satisfy the following condition:(18)ρ1=1-CBLd1+CBLp1+L0+L01-CBLd1Φα,αL0··1<1.Here, · stands for the absolute value and ·1 stands for the L1-norm of the function defined on the operation time interval 0,T.
Then, the output error is strictly monotonic convergence inLp-norm; that is, (1)
ek+1·p<ek·p;(2)
limk→∞⁡ek+1·p=0.Proof.
For the dynamic system (15) and the PDα-type ILC scheme (I1), from Lemma 6, we have (19)xk+1t=xk+10+1Γα∫0tt-τα-1fxk+1τ,τ+Buk+1τdτ,and, then, we get(20)ek+1t=ydt-yk+1t=ydt-ykt-yk+1t-ykt=ekt-Cxk+1t-xkt=ekt-Cxk+10+1Γα∫0tt-τα-1fxk+1τ,τ+Buk+1τdτ+Cxk0+1Γα∫0tt-τα-1fxkτ,τ+Bukτdτ=ekt-C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-CB·1Γα∫0tt-τα-1uk+1τ-ukτdτ=ekt-C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-CBLp1·1Γα∫0tt-τα-1ekτdτ-CBLd1·1Γα∫0tt-τα-1Dταekτdτ0.Applying Property 3 to the last term on the right side of (20), we have(21)ek+1t=1-CBLd1ekt-C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-CBLp1·1Γα∫0tt-τα-1ekτdτ.Taking absolution on both sides of (21) yields(22)ek+1t≤1-CBLd1ekt+1Γα∫0tt-τα-1Cfxk+1τ,τ-fxkτ,τdτ+CBLp1·1Γα∫0tt-τα-1ekτdτ.Applying Lipschitz condition to the second term on the right side of (22), we get(23)1Γα∫0tt-τα-1Cfxk+1τ,τ-fxkτ,τdτ≤L0·1Γα∫0tt-τα-1yk+1τ-ydτdτ+L0·1Γα∫0tt-τα-1ydτ-ykτdτ=L0·1Γα∫0tt-τα-1ek+1τdτ+L0·1Γα∫0tt-τα-1ekτdτ.Taking (23) into (22) obtains(24)ek+1t≤1-CBLd1ekt+L0·1Γα∫0tt-τα-1ek+1τdτ+CBLp1+L0·1Γα∫0tt-τα-1ekτdτ.Using Lemma 9 to inequality (24), we have(25)ek+1t≤1-CBLd1ekt+∫0tCBLp1+L0+L01-CBLd1Φα,αL0t-τekτdτ.Taking the Lp-norm on both sides of (25) and adopting the generalized Young inequality of convolution integral, we get(26)ek+1·p≤ρ1ek·p.
This completes the proof of Theorem10.Remark 11.
We can see, from the above derivation, that the convergence condition is quantified directly from theLp-norm, not by using the sufficiently large λ, and analyzed in terms of the tracking error rather than the control input error. Besides, the monotonic property of convergence can ensure the first-order PDα-type ILC rule to be practically implementable. Further, from condition (18), we can observe that the convergence is affected not only by the derivative learning gain and the system dynamics, but also by the proportional learning gain. That is, the result reflects the features of system dynamics and the mechanism of the algorithm to the convergence. Actually, the impact of the state dynamics and proportional learning gain, which are neglected in the existing FOILC investigations, exists but is significantly suppressed by the sufficiently large parameter λ. It should be noted that, as mentioned in [23], a large value of λ may have a huge tracking error, which is not allowable in practice.Remark 12.
Note that the convergence analysis for fractional-order linear time-invariant system has been investigated in my early work [28], in which the derivation is made by means of the state transition matrix in an equality form, and the convergence condition is (27)ρ=1-CBLd1+CΦ~α,α·BLp1+ABLd11,where A is the system matrix and Φ~α,αt=tα-1Eα,αAtα is the state transition matrix. However, in this paper, the investigated fractional-order system is nonlinear with its nonlinearity unknown; the proof of convergence is derived by means of inequality Lemma 9. Thus, the convergence condition ρ1 cannot degenerate to the condition ρ when the fractional-order nonlinear system is reduced to the corresponding linear case.


### 4.1. Convergence Analysis for Second-Order PDα-Type ILC

In this section, we go on considering the second-order PDα-type ILC algorithm, which is constructed by employing the control inputs and their output errors of the latest previously adjacent operations in a weighting average form as follows:I2
(28)u1tis given arbitrarily,u2t=u1t+Lp1e1t+Ld1Dtαe1t0,uk+1t=r1ukt+Lp1ekt+Ld1Dtαekt0+r2uk-1t+Lp2ek-1t+Ld2Dtαek-1t0,k=2,3,4,….Here, Lp2and Ld2 denote the second-order proportional and fractional-order derivative learning gains, respectively. The weighting coefficients r1 and r2 satisfy 0≤r1<1, 0≤r2≤1, and r1+r2=1.Theorem 13.
For the second-order PDα-type iterative learning control rule (I2) is applied to system (15), if the system matrices B, C, the order α, and the Lipchitz constant L0 together with the learning gains Lp1, Lp2, Ld1, Ld2satisfy the following conditions:(1)
ρ1=1-CBLd1+CBLp1+L0+L01-CBLd1Φα,αL0··1<1.(2)
ρ2=1-CBLd2+CBLp2+L0+L01-CBLd2Φα,αL0··1<1.
Then, the learning scheme(I2) is convergent, that is, limk→∞⁡ek+1·p=0.Proof.
From the dynamic system (15) and the PDα-type ILC scheme (I2), we have(29)ek+1t=ydt-yk+1t=r1ydt-ykt+r2ydt-yk-1t-yk+1t-r1ykt-r2yk-1t=r1ekt+r2ek-1t-Cxk+1t-r1xkt-r2xk-1t=r1ekt+r2ek-1t-Cxk+10+1Γα∫0tt-τα-1fxk+1τ,τ+Buk+1τdτ+r1Cxk0+1Γα∫0tt-τα-1fxkτ,τ+Bukτdτ+r2Cxk-10+1Γα∫0tt-τα-1fxk-1τ,τ+Buk-1τdτ=r1ekt+r2ek-1t-r1C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-r2C·1Γα∫0tt-τα-1fxk+1τ,τ-fxk-1τ,τdτ-CB·1Γα∫0tt-τα-1uk+1τ-r1ukτ-r2uk-1τdτ=r1ekt+r2ek-1t-r1C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-r2C·1Γα∫0tt-τα-1fxk+1τ,τ-fxk-1τ,τdτ-CB·1Γα∫0tt-τα-1r1Lp1ekτ+r2Lp2ek-1τdτ-CB·1Γα∫0tt-τα-1r1Ld1Dταekτ0+r2Ld2Dταek-1τ0dτ.Applying Property 3 to the last term on the right side of (29) yields(30)CB·1Γα∫0tt-τα-1r1Ld1Dταekτ0+r2Ld2Dταek-1τ0dτ=r1CBLd1ekt+r2CBLd2ek-1t.Submitting (30) into (29) and taking absolute values on both sides of (29) obtain(31)ek+1t≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1·1Γα∫0tt-τα-1Cfxk+1τ,τ-fxkτ,τdτ+r2·1Γα∫0tt-τα-1Cfxk+1τ,τ-fxk-1τ,τdτ+r1CBLp1·1Γα∫0tt-τα-1ekτdτ+r2CBLp2·1Γα∫0tt-τα-1ek-1τdτ≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1L0·1Γα∫0tt-τα-1yk+1τ-ydτdτ+r1L0·1Γα∫0tt-τα-1ydτ-ykτdτ+r2L0·1Γα∫0tt-τα-1yk+1τ-ydτdτ+r2L0·1Γα∫0tt-τα-1ydτ-yk-1τdτ+r1CBLp1·1Γα∫0tt-τα-1ekτdτ+r2CBLp2·1Γα∫0tt-τα-1ek-1τdτ=r11-CBLd1ekt+r21-CBLd2ek-1t+L0·1Γα∫0tt-τα-1ek+1τdτ+r1L0+CBLp1·1Γα∫0tt-τα-1ekτdτ+r2L0+CBLp2·1Γα∫0tt-τα-1ek-1τdτ.Using Lemma 9 to equality (31) derives that(32)ek+1t≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1∫0tCBLp1+L0+L01-CBLd1Φα,αL0t-τekτdτ+r2∫0tCBLp2+L0+L01-CBLd2Φα,αL0t-τek-1τdτ.Adopting the Lp-norm on both sides of (32) implies that(33)ek+1·p≤r1ρ1ek·p+r2ρ2ek-1·p,and, then, according to Lemma 8 and assumptions (1) and (2), it is therefore finally evident that limk→∞⁡ek+1·p=0.Remark 14.
It can be observed that whenr2=0, the second-order PDα-type ILC updating law (I2) degenerates to the first-order PDα-type ILC updating law (I1). Then the convergence becomes(34)ρ1=1-CBLd1+CBLp1+L0+L01-CBLd1Φα,αL0··1<1.Corollary 15.
Assume thatlimk→∞⁡ek+1·p/ek·p exists. Analogous to the discussion regarding the convergent speed in [25], we can make assertions as follows:(1)
Ifρ2<ρ12<1, then the second-order algorithm (I2) is Qp-faster than the first-order (I1).(2)
Ifρ12=ρ2<1, then the scheme (I2) is Qp-equivalent to the scheme (I1).(3)
Ifρ12<ρ2<1, then the second-order strategy (I2) is Qp-slower than the first-order (I1).Remark 16.
From the view point of speed, comparing(1) and (3) of Corollary 15, we can conclude that if we choose a suitable learning gains, the second-order updating law (I2) may not be a preferred candidate for the systems. However, if we pursue more freedom in choosing the learning gains and better robustness to noise, the second-order updating law (I2) is a useful alternative.


## 4.1. Convergence Analysis for Second-Order PDα-Type ILC

In this section, we go on considering the second-order PDα-type ILC algorithm, which is constructed by employing the control inputs and their output errors of the latest previously adjacent operations in a weighting average form as follows:I2
(28)u1tis given arbitrarily,u2t=u1t+Lp1e1t+Ld1Dtαe1t0,uk+1t=r1ukt+Lp1ekt+Ld1Dtαekt0+r2uk-1t+Lp2ek-1t+Ld2Dtαek-1t0,k=2,3,4,….Here, Lp2and Ld2 denote the second-order proportional and fractional-order derivative learning gains, respectively. The weighting coefficients r1 and r2 satisfy 0≤r1<1, 0≤r2≤1, and r1+r2=1.Theorem 13.
For the second-order PDα-type iterative learning control rule (I2) is applied to system (15), if the system matrices B, C, the order α, and the Lipchitz constant L0 together with the learning gains Lp1, Lp2, Ld1, Ld2satisfy the following conditions:(1)
ρ1=1-CBLd1+CBLp1+L0+L01-CBLd1Φα,αL0··1<1.(2)
ρ2=1-CBLd2+CBLp2+L0+L01-CBLd2Φα,αL0··1<1.
Then, the learning scheme(I2) is convergent, that is, limk→∞⁡ek+1·p=0.Proof.
From the dynamic system (15) and the PDα-type ILC scheme (I2), we have(29)ek+1t=ydt-yk+1t=r1ydt-ykt+r2ydt-yk-1t-yk+1t-r1ykt-r2yk-1t=r1ekt+r2ek-1t-Cxk+1t-r1xkt-r2xk-1t=r1ekt+r2ek-1t-Cxk+10+1Γα∫0tt-τα-1fxk+1τ,τ+Buk+1τdτ+r1Cxk0+1Γα∫0tt-τα-1fxkτ,τ+Bukτdτ+r2Cxk-10+1Γα∫0tt-τα-1fxk-1τ,τ+Buk-1τdτ=r1ekt+r2ek-1t-r1C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-r2C·1Γα∫0tt-τα-1fxk+1τ,τ-fxk-1τ,τdτ-CB·1Γα∫0tt-τα-1uk+1τ-r1ukτ-r2uk-1τdτ=r1ekt+r2ek-1t-r1C·1Γα∫0tt-τα-1fxk+1τ,τ-fxkτ,τdτ-r2C·1Γα∫0tt-τα-1fxk+1τ,τ-fxk-1τ,τdτ-CB·1Γα∫0tt-τα-1r1Lp1ekτ+r2Lp2ek-1τdτ-CB·1Γα∫0tt-τα-1r1Ld1Dταekτ0+r2Ld2Dταek-1τ0dτ.Applying Property 3 to the last term on the right side of (29) yields(30)CB·1Γα∫0tt-τα-1r1Ld1Dταekτ0+r2Ld2Dταek-1τ0dτ=r1CBLd1ekt+r2CBLd2ek-1t.Submitting (30) into (29) and taking absolute values on both sides of (29) obtain(31)ek+1t≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1·1Γα∫0tt-τα-1Cfxk+1τ,τ-fxkτ,τdτ+r2·1Γα∫0tt-τα-1Cfxk+1τ,τ-fxk-1τ,τdτ+r1CBLp1·1Γα∫0tt-τα-1ekτdτ+r2CBLp2·1Γα∫0tt-τα-1ek-1τdτ≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1L0·1Γα∫0tt-τα-1yk+1τ-ydτdτ+r1L0·1Γα∫0tt-τα-1ydτ-ykτdτ+r2L0·1Γα∫0tt-τα-1yk+1τ-ydτdτ+r2L0·1Γα∫0tt-τα-1ydτ-yk-1τdτ+r1CBLp1·1Γα∫0tt-τα-1ekτdτ+r2CBLp2·1Γα∫0tt-τα-1ek-1τdτ=r11-CBLd1ekt+r21-CBLd2ek-1t+L0·1Γα∫0tt-τα-1ek+1τdτ+r1L0+CBLp1·1Γα∫0tt-τα-1ekτdτ+r2L0+CBLp2·1Γα∫0tt-τα-1ek-1τdτ.Using Lemma 9 to equality (31) derives that(32)ek+1t≤r11-CBLd1ekt+r21-CBLd2ek-1t+r1∫0tCBLp1+L0+L01-CBLd1Φα,αL0t-τekτdτ+r2∫0tCBLp2+L0+L01-CBLd2Φα,αL0t-τek-1τdτ.Adopting the Lp-norm on both sides of (32) implies that(33)ek+1·p≤r1ρ1ek·p+r2ρ2ek-1·p,and, then, according to Lemma 8 and assumptions (1) and (2), it is therefore finally evident that limk→∞⁡ek+1·p=0.Remark 14.
It can be observed that whenr2=0, the second-order PDα-type ILC updating law (I2) degenerates to the first-order PDα-type ILC updating law (I1). Then the convergence becomes(34)ρ1=1-CBLd1+CBLp1+L0+L01-CBLd1Φα,αL0··1<1.Corollary 15.
Assume thatlimk→∞⁡ek+1·p/ek·p exists. Analogous to the discussion regarding the convergent speed in [25], we can make assertions as follows:(1)
Ifρ2<ρ12<1, then the second-order algorithm (I2) is Qp-faster than the first-order (I1).(2)
Ifρ12=ρ2<1, then the scheme (I2) is Qp-equivalent to the scheme (I1).(3)
Ifρ12<ρ2<1, then the second-order strategy (I2) is Qp-slower than the first-order (I1).Remark 16.
From the view point of speed, comparing(1) and (3) of Corollary 15, we can conclude that if we choose a suitable learning gains, the second-order updating law (I2) may not be a preferred candidate for the systems. However, if we pursue more freedom in choosing the learning gains and better robustness to noise, the second-order updating law (I2) is a useful alternative.


## 5. Simulation Illustrations

In this simulation, we consider the following fractional-order nonlinear system:(35)Dt0.50x1tDt0.50x2t=0.2x1t0.1sin⁡x1t+0.2x2t+00.5ut,yt=01x1tx2t,x10x20=00.The operation time period is0,1, the desired trajectory is ydt=12t21-t, and the initial control is u1t=0.For the first-orderPD0.5-type ILC scheme (I1), the first-order proportional and fractional-order derivative learning gains are chosen as Lp1=0.2 and Ld1=1.5, respectively. It can be seen L0=0.2 and Φα,αL0··1=1.3630; then it is easy to verify that ρ1=0.7271<1, which means that the monotonic convergence condition (18) is satisfied. The outputs by the scheme (I1) at the 2nd, 3rd, and 4th operations are shown in Figure 1, respectively. The monotonic tracking error in the sense of L2-norm is shown in Figure 2.Figure 1
System outputs of(I1).Figure 2
Tracking errors in the sense ofL2-norm.For the second-orderPD0.5-type ILC (I2), we consider two cases as follows.Case 1.
In schemes(I1) and (I2), the first-order learning gains are set as Lp1=0.2and Ld1=1.5, respectively. In scheme (I2), the weighting coefficients are assigned as r1=0.5 and r2=0.5, and the second-order learning gains are selected as Lp2=0.1 and Ld2=1.2, respectively. It is computed that ρ1=0.7271<1 and ρ2=0.8498<1, which is included in the case that ρ12<ρ2. It shows that the output error of (I1) convergence is faster than that of (I2). The corresponding outputs by scheme (I2) at the 3rd, 5th, and 7th operations are displayed in Figure 3. It shows that the output follows the desired trajectory well as the iteration increases. The comparison of the tracking error in the sense of L2-norm made by the updating laws (I1) and (I2) is shown in Figure 4. It shows that the tracking errors of both the first-order and second-order laws are convergent and the first-order law (I1) is convergence faster than second-order law (I2).Figure 3
System outputs of(I2).Figure 4
Comparison of tracking errors.Case 2.
In schemes(I1) and (I2), the first-order learning gains are chosen as Lp1=0.1 and Ld1=1, respectively. In (I2), the weighting coefficients are assigned as r1=0.3 and r2=0.7, and the second-order learning gains are set as Lp2=0.1 and Ld2=1.7, respectively. It is computed that ρ1=0.9771<1 and ρ2=0.5316<1, which belongs to the case that ρ2<ρ12. The outputs by scheme (I2) at the 2nd, 4th, and 6th iterations are exhibited in Figure 5. Figure 6 shows that the tracking error of the updating law (I1) convergence is slower than that of (I2) in the sense of L2-norm.Figure 5
System outputs of(I2).Figure 6
Comparison of tracking errors.


## 6. Conclusion

In this paper, for a class of fractional-order nonlinear systems, the first-order and second-order PDα-type ILC strategies are developed and the sufficiency for convergence is analyzed by means of evaluating the tracking error in the sense of Lp-norm. For analysis, it is found that the sufficient conditions of convergence not only depend on all of system dynamics, but also rely on all of learning gains. Moreover, the convergence speed comparison of the second-order law with the first-order one has been affirmed. We have clarified that, for a fractional-order nonlinear system, the results of the second-order law, that is, its convergence being faster than, equivalent to, or slower than first-order scheme, are validated. All theoretical results were conducted by simulations.



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*Source: 5157267-2018-03-01.xml*