# Robustness Analysis of a Type of Iterative Algorithm for R-L Fractional Nonlinear Control Systems in the Sense ofLp Norm

Authors: Yanfang Li; Xianghu Liu; Guangjun Xu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The paper is concerned with the robustness analysis of a type of iterative algorithm for R-L fractional nonlinear control systems in the sense ofLp norm. Firstly, according to the Laplace transform and M-L function, the concept of mild solutions of the system is derived. Secondly, we give the sufficient conditions of robustness analysis of the PDα-type ILC algorithm with uncertain disturbances and then study the robust analysis of the second-order PDα-type ILC algorithm. At last, two fractional examples are given to demonstrate the results.


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## Body

## 1. Introduction

The aim of the paper is to analyze the robustness of a type of iterative algorithm in the sense ofLp norm of the following R-L fractional system:(1)DtαRLzt=Azt+But,t∈J=0,b,g1−α∗z0=z0,yt=Czt+Dut,where DtαRL denotes the R-L derivative of order α, 0<α<1, A,B,C∈Rn×n, ut is a control vector, and g1−α=t1−α/Γ1−α.Iterative learning control (ILC) was shown by Uchiyama in 1978 (in Japanese), and in recent years, more and more scholars have paid attention to the problems, among which are experts who study fractional calculus. The work of the fractional-order system in iterative learning control appeared in 2001. In the following decade, extensive attention has been paid to this field, great progress has been made [1–8], and many fractional nonlinear systems were investigated [9–17]. In recent years, the fractional ILC algorithm has played a great role in multiagent control information transmission, and for more information, one can see the references [13–16].In Li et al.’s study [17], the authors discussed a P-type ILC scheme for a class of fractional-order nonlinear systems with delay by using the λ-norm and Gronwall inequality and obtained the sufficient condition for the robust convergence of the tracking errors.In view of that theλ-norm often causes tracking errors that exceed the actual engineering range and cause inaccurate data, the authors Lan and Lin [18] used the Lp norm to discuss the convergence of iterative learning algorithms, and it objectively quantifies the essential characteristics of the tracking error and comprehensively reflects the behavior of the system. Zhang and Peng [19] used the generalized Young inequality of convolution and discussed the robustness of the PD-type fractional-order iteration and learning control algorithm in the sense of Lp norm, and the conditions of its robust convergence are obtained.The above references have analyzed the robustness of the algorithm of the Caputo-type fractional system, and we find the Caputo fractional derivative is often used to solve general diffusion problems. The R-L type fractional derivative has a wider application in viscoelastic problems because it does not require the function to be differentiable at the origin. As far as we all know, analyzing robustness with interference of the R-L type fractional system is an extremely interesting and challenging work.The rest of this paper is organized as follows. In Section2, according to the Laplace transform and M-L function, the concept of mild solutions of the system is derived. In Section 3, we give the sufficient conditions of robustness analysis of the PDα-type ILC algorithm with uncertain disturbances and then study the robust analysis of the second-order PDα-type ILC algorithm. In Section 4, two fractional examples are given to demonstrate the results.


## 2. Some Preliminaries for Fractional Systems

In this section, we show some definitions and preliminaries of theLp norm and Mittag–Leffler function. From [20–23], one can see the definitions of the R-L fractional integral and derivative.Definition 1.
The norm for then-dimensional vector Z=z1,z2,…,zn is defined as Z=max1≤i≤nzi, and the Lp norm is defined as Zp=∫0Tmaxzipdt1/p, where t∈0,T.Definition 2.
The definition of the two-parameter function of the Mittag–Leffler type is described by(2)Eα,βz=∑k=0∞zkΓαk+β,α>0,β>0,z∈C.
Ifβ=1, one has the Mittag–Leffler function of one parameter as follows:(3)Eαz=∑k=0∞zkΓαk+1.
Now, according to the results of the papers [17, 24–27], we will give the following lemma.Lemma 1.
(Lemma3, see [25]). The general solution of equation (1) is given by(4)zt=tα−1Eα,αA,tz0+∫0tt−sα−1EAt−sαBusds,where(5)Eα,βA,t=∑k=0∞Aktαk+β−1Γαk+β.Lemma 2.
(Definition 2.4, see [27]). The operators Eα,αt are exponentially bounded, and there is a constant C0=1/αA1−α/α, eαt=eA1/αt, M=eαb, and Eα,αA,t≤C0eαt≤C0M.Lemma 3.
(Ho¨lder inequality). Set p>0,q>0, and 1/p+1/q=1; if f∈LpΩ,g∈LqΩ, and f⋅g∈L1Ω, then f⋅gL1≤fLpgLq.


## 3. Robustness Analysis of thePDα-Type ILC Algorithm with Uncertain Disturbances

In this section, we consider the following fractional equation:(6)DtαRLzkt=Azkt+Bukt+ωt,t∈J=0,b,ykt=Czkt+Dukt+νt,where k=0,1,2,3,…, and ωtandνt are uncertain disturbances.For system (6), we apply the following open- and closed-loop PDα-type ILC algorithm:(7)uk+1t=ukt+γ1ekt+γ2ek+1αt,where t∈0,b, γ1 and γ2 are the parameters which will be determined, ydt is the given function, ek=ydt−ykt, and ekαt=DtαRLek. For convenience, one can see Figure 1. The initial state of each iterative learning is as follows:(8)zk+10=zk0+Bγ1ekt.Figure 1
Block diagram of the open- and closed-loopPDα-type ILC algorithm.We denote that(9)κ1=I+γ2D−bα−1/pγ2CC0MBqα−1+1q,κ2=I−γ1D−bα−1/pγ1CC0MBqα−1+1q,κ3=bα−1/pγ1C+γ2CC0MBωLpqα−1+1q+γ1+γ2νLp.Theorem 1.
Assume that each iteration state meets algorithm (7) and the initial state is zk0=zd0; then, there exists m>0 such that κ1>0,κ2>0,κ3<m,andκ1>κ2, and then, the sufficient condition for being uniformly bounded on J is limk⟶∞ukLp≤κ3/κ1−κ2.Proof.
Define(10)Δzkt=zdt−zkt,Δukt=udt−ukt.
Fort∈J, one has Δzkαt=DtαRLΔzkt=AΔzkt+BΔukt and ek+1αt=CAΔzk+1t+BΔuk+1t.
According to system (6), we have(11)zk+1t=tα−1Eα,αA,tz0+∫0tt−sα−1Eα,αAt−sαBuk+1s+ωtds,zdt=tα−1Eα,αA,tz0+∫0tt−sα−1Eα,αAt−sαBudsds,and thus, using the ILC algorithms (7) and (8), we derive(12)Δzk+1t=∫0tt−sα−1Eα,αAt−sαBΔuk+1s+ωtds,Δzkt=∫0tt−sα−1Eα,αAt−sαBΔuks+ωtds,so(13)Δuk+1t=Δukt−γ1ydt−ykt−γ2ydt−yk+1t,=Δukt−γ1CΔzkt+DΔukt−νt−γ2CΔzk+1t+DΔuk+1t−νt,=Δukt−γ1C∫0tt−sα−1Eα,αAt−sαBΔuks+ωtds+γ1DΔukt−γ1νt,−γ2C∫0tt−sα−1Eα,αAt−sαBΔuk+1s+ωtds+γ2DΔuk+1t−γ2νt.
Hence,(14)Δuk+1tI+γ2D=ΔuktI−γ1D−γ1C∫0tt−sα−1Eα,αAt−sαBΔuks+ωtds−γ1νt−γ2C∫0tt−sα−1Eα,αAt−sαBΔuk+1s+ωtds−γ2νt.
By taking theLp norm, we obtain(15)Δuk+1LpI+γ2D≤ΔukLpI+γ2D+bα−1/pγ1CC0MBΔukLp+ΔωLpqα−1+1q+γ1νLp+bα−1/pγ2CC0MBΔuk+1Lp+ΔωLpqα−1+1q+γ2νLp,denoting(16)κ1=I+γ2D−bα−1/pγ2CC0MBqα−1+1q,κ2=I−γ1D−bα−1/pγ1CC0MBqα−1+1q,κ3=bα−1/pγ1C+γ2CC0MBωLpqα−1+1q+γ1+γ2νLp.
Consequently,κ1Δuk+1Lp≤κ2ΔukLp+κ3. So, there exists a positive m, such that κ3<m and κ1>0,κ2>0,andκ1>κ2, and then limk⟶∞ukLp≤κ3/κ1−κ2, which implies ekt is uniformly bounded on J.


## 4. Robust Analysis of the Second-OrderPDα-Type ILC Algorithm

In this section, we consider the following second-orderPDα-type ILC algorithm:(17)u2t=u1t+γ1e1t+γ2e1αt,uk+1t=r1ukt+γ1ekt+γ2ekαt+r2uk−1t+γ3ek−1t+γ4ek−1αt,k=2,3,…,where r1+r2=1.The initial state of the system is as follows:(18)zk+10=zk0+BL1ekt+BL2ekαt.For convenience, one can see Figure2.Figure 2
Block diagram of the second-orderPDα-type ILC algorithm.Assume that the initial state of each iterative learning meets (18), where L1andL2 are the parameters which will be determined.Note(19)K1=r1+r1γ1D+r1γ2CB+r1γ1C+r1γ2CAbα−1/pC0MBqα−1+1q,K2=r2+r2γ3D+r2γ4CB+r2γ3C+r2γ4CAbα−1/pC0MBqα−1+1q,K3=r1γ1C+r1γ2CA+r2γ3C+r2γ4CAbα−1/pC0Mqα−1+1qωLp.Theorem 2.
Suppose system (6) satisfies the second-order PDα-type ILC algorithm and the initial state of each iteration satisfies (18), then there exists positive p such that K1+K2<1,andK3⟶0. Since k⟶∞, Δuk+1Lp is uniformly bounded, which guarantees that limk⟶∞ekλ=0andt∈J.Proof.
According to Lemma1, we yield(20)zk+1t=tα−1Eα,αA,tz0+∫0tt−sα−1Eα,αAt−sαBuk+1s+ωtds,Δzk+1t=∫0tt−sα−1Eα,αAt−sαBΔuk+1s+ωtds,Δzkt=∫0tt−sα−1Eα,αAt−sαBΔuks+ωtds,and then,(21)Δuk+1t=r1Δukt+γ1CΔzkt+γ1DΔukt+γ2CAΔzkt+BΔukt+r2Δuk−1t+γ3CΔzk−1t+γ3DΔuk−1t+γ4CAΔzk−1t+BΔuk−1t,=r1+r1γ1D+r1γ2CBΔukt+r2+r2γ3D+r2γ4CBΔuk−1t+r1γ1C+r1γ2CA∫0tt−sα−1Eα,αAt−sαBΔuks+ωtds+r2γ3C+r2γ4CA∫0tt−sα−1Eα,αAt−sαBΔuk−1s+ωtds.
By taking theLp norm, it yields(22)Δuk+1Lp≤r1+r1γ1D+r1γ2CBΔuktLp+r1γ1C+r1γ2CAbα−1/pC0MBqα−1+1qΔuktLp+r2+r2γ3D+r2γ4CBΔuk−1tLp+r2γ3C+r2γ4CAbα−1/pC0MBqα−1+1qΔuk−1tLp+r1γ1C+r1γ2CAbα−1/pC0Mqα−1+1qωLp+r2γ3C+r2γ4CAbα−1/pC0Mqα−1+1qωLp.
For brevity, note that(23)K1=r1+r1γ1D+r1γ2CB+r1γ1C+r1γ2CAbα−1/pC0MBqα−1+1q,K2=r2+r2γ3D+r2γ4CB+r2γ3C+r2γ4CAbα−1/pC0MBqα−1+1q,K3=r1γ1C+r1γ2CA+r2γ3C+r2γ4CAbα−1/pC0Mqα−1+1qωLp,and one can deduce Δuk+1Lp≤K1ΔukLp+K2Δuk−1Lp+K3.
There exists a constantp>0, which satisfies K1+K2<1andK3⟶0. Since k⟶∞, Δuk+1Lp is uniformly bounded. The proof is completed.


## 5. Simulations

In this section, we will give two simulation examples to demonstrate the validity of the algorithms.


### 5.1.PDα-Type ILC with Initial State Error

Consider the following one-dimensional systems as follows:(24)Dt0.6RLxt=xk2t+0.1ut+ωkt,t∈J=1,2,x0=2,yt=xt+0.3ukt+νt,with the iterative learning control and initial state error(25)uk+1t=ukt+0.5ekt+0.5ek+1αt,xk+10=xk0+0.1ekt,whereAx⋅=x⋅2. Now, we can choose α=0.6, B=0.1, C=1, p=2, γ1=γ2=0.5, ωt=10−3sin0.001t,andνt=10−5t. For the system, we use the PDα-type ILC algorithm and set the initial control u0⋅=0, ydt=5sinet2, and t∈0,2. One can calculate M≈3>0,κ1=0.47,κ2=0.17,andκ3<0.01=m, and then, all conditions of Theorem 1 are satisfied.The state trajectories of system (24) with initial conditions are given in Figure 3 and Table 1, and with the increase of the number of iterations, it can track the desired trajectory gradually. Consistent with the theoretical analysis in the previous section, the algorithm has a faster convergence speed. At the end of the fourth iteration, the algorithm has converged. From Figures 3 and 4, the curve is basically completely fitted, showing that the system algorithm is well robust.Figure 3
Simulation results of outputyk.Table 1
.Numerical simulation of the output of the system in Section5.1 and the desired trajectory.
kykydtkkykydtkkykydtk12.08182.053918−4.9051−4.9774355.01834.994521.25801.245419−0.9237−0.964536−4.0133−4.091530.32940.3088204.76844.791037−4.6198−4.70374−0.7118−0.7414210.81930.791238−3.3214−3.39565−1.8284−1.867722−4.7823−4.8564393.22063.19836−2.9501−2.9989234.10164.085740−4.7914−4.87917−3.9613−4.019024−2.2443−2.2993414.93184.92168−4.6935−4.7576252.18032.1603422.11762.08409−4.9269−4.993626−3.9964−4.066943−4.8627−4.953510−4.4206−4.4837274.35424.3339444.59434.579011−2.9888−3.0410283.84353.7982454.25924.240312−0.6484−0.6824292.67052.651546−2.3747−2.4479132.17382.1620304.94694.904847−4.3241−4.4139144.44564.475531−1.9316−1.989148−2.2423−2.3162154.75334.7927324.48464.4505493.33873.3090162.08442.0698333.10673.0883502.89452.860317−2.5569−2.6095342.10992.0826Figure 4
The tracking error of the systems.


### 5.2.PDα-Type ILC with Random Disturbance

Consider a two-dimensional ILC system; we setα=0.7, ωt=10−10sinπt/1000,νt=10−3, Ax1x2=2x12x22,B=0111,C=1101,andD=0111 and construct the second-order PDα-type ILC algorithm as follows:(26)uk+1t=0.1ukt+0.1ekt+0.1ekαt+0.1uk−1t+0.2ek−1t+0.2ek−1αt,k=2,3,…We also select other parameters and initial values of the algorithm as follows:u0⋅=0, ydt=y1dty2dt=5sint4t3,t∈0,1.9, r1=1,r2=0.5,γ1=1,andγ2=0.5. It is easy to show that M≈3>0,K1=0.264,K2=0.428,andK3⟶0, and all conditions of Theorem 2 are satisfied. In the simulation, ∗∗∗ denotes the desired trajectory of state 1, ⋄⋄⋄ denotes the desired trajectory of state 2, and solid lines (—–) in different colors denote the output of the system. In Figure 5, we use k1 to represent the iteration of state 1 and use k2 to represent the iteration of state 2, and the tracking error is shown in Figure 6, which implies the number of iterations and tracking error.Figure 5
Simulation results of outputyk.Figure 6
The tracking error of the system.From Figure6 and Table 2, one can find that the tracking error tends to zero quickly, so the output of the system can track the desired trajectory almost perfectly.Table 2
Numerical simulation of the output of the system in Section5.2 and the desired trajectory.
kykydtkkykydtk10010022.38082.397120.01810.004034.17824.207330.05340.032044.95384.987440.13490.108054.51824.546450.28600.256062.97842.992360.52990.500070.73780.705670.88980.86408−1.7256−1.753981.38871.37209−3.7370−3.784092.09002.048010−4.8293−4.8876102.89582.916011−4.7343−4.7946113.95004.000012−3.4746−3.5277125.30635.324013−1.3580−1.3970136.88466.9120141.09811.0755148.74888.7880153.32593.28491510.922710.9760164.70194.68991613.430013.5000174.95734.94671716.294616.3840184.00603.99241819.540319.6520192.09362.06051923.190923.328020−0.3191−0.37572027.270227.4360


## 5.1.PDα-Type ILC with Initial State Error

Consider the following one-dimensional systems as follows:(24)Dt0.6RLxt=xk2t+0.1ut+ωkt,t∈J=1,2,x0=2,yt=xt+0.3ukt+νt,with the iterative learning control and initial state error(25)uk+1t=ukt+0.5ekt+0.5ek+1αt,xk+10=xk0+0.1ekt,whereAx⋅=x⋅2. Now, we can choose α=0.6, B=0.1, C=1, p=2, γ1=γ2=0.5, ωt=10−3sin0.001t,andνt=10−5t. For the system, we use the PDα-type ILC algorithm and set the initial control u0⋅=0, ydt=5sinet2, and t∈0,2. One can calculate M≈3>0,κ1=0.47,κ2=0.17,andκ3<0.01=m, and then, all conditions of Theorem 1 are satisfied.The state trajectories of system (24) with initial conditions are given in Figure 3 and Table 1, and with the increase of the number of iterations, it can track the desired trajectory gradually. Consistent with the theoretical analysis in the previous section, the algorithm has a faster convergence speed. At the end of the fourth iteration, the algorithm has converged. From Figures 3 and 4, the curve is basically completely fitted, showing that the system algorithm is well robust.Figure 3
Simulation results of outputyk.Table 1
.Numerical simulation of the output of the system in Section5.1 and the desired trajectory.
kykydtkkykydtkkykydtk12.08182.053918−4.9051−4.9774355.01834.994521.25801.245419−0.9237−0.964536−4.0133−4.091530.32940.3088204.76844.791037−4.6198−4.70374−0.7118−0.7414210.81930.791238−3.3214−3.39565−1.8284−1.867722−4.7823−4.8564393.22063.19836−2.9501−2.9989234.10164.085740−4.7914−4.87917−3.9613−4.019024−2.2443−2.2993414.93184.92168−4.6935−4.7576252.18032.1603422.11762.08409−4.9269−4.993626−3.9964−4.066943−4.8627−4.953510−4.4206−4.4837274.35424.3339444.59434.579011−2.9888−3.0410283.84353.7982454.25924.240312−0.6484−0.6824292.67052.651546−2.3747−2.4479132.17382.1620304.94694.904847−4.3241−4.4139144.44564.475531−1.9316−1.989148−2.2423−2.3162154.75334.7927324.48464.4505493.33873.3090162.08442.0698333.10673.0883502.89452.860317−2.5569−2.6095342.10992.0826Figure 4
The tracking error of the systems.


## 5.2.PDα-Type ILC with Random Disturbance

Consider a two-dimensional ILC system; we setα=0.7, ωt=10−10sinπt/1000,νt=10−3, Ax1x2=2x12x22,B=0111,C=1101,andD=0111 and construct the second-order PDα-type ILC algorithm as follows:(26)uk+1t=0.1ukt+0.1ekt+0.1ekαt+0.1uk−1t+0.2ek−1t+0.2ek−1αt,k=2,3,…We also select other parameters and initial values of the algorithm as follows:u0⋅=0, ydt=y1dty2dt=5sint4t3,t∈0,1.9, r1=1,r2=0.5,γ1=1,andγ2=0.5. It is easy to show that M≈3>0,K1=0.264,K2=0.428,andK3⟶0, and all conditions of Theorem 2 are satisfied. In the simulation, ∗∗∗ denotes the desired trajectory of state 1, ⋄⋄⋄ denotes the desired trajectory of state 2, and solid lines (—–) in different colors denote the output of the system. In Figure 5, we use k1 to represent the iteration of state 1 and use k2 to represent the iteration of state 2, and the tracking error is shown in Figure 6, which implies the number of iterations and tracking error.Figure 5
Simulation results of outputyk.Figure 6
The tracking error of the system.From Figure6 and Table 2, one can find that the tracking error tends to zero quickly, so the output of the system can track the desired trajectory almost perfectly.Table 2
Numerical simulation of the output of the system in Section5.2 and the desired trajectory.
kykydtkkykydtk10010022.38082.397120.01810.004034.17824.207330.05340.032044.95384.987440.13490.108054.51824.546450.28600.256062.97842.992360.52990.500070.73780.705670.88980.86408−1.7256−1.753981.38871.37209−3.7370−3.784092.09002.048010−4.8293−4.8876102.89582.916011−4.7343−4.7946113.95004.000012−3.4746−3.5277125.30635.324013−1.3580−1.3970136.88466.9120141.09811.0755148.74888.7880153.32593.28491510.922710.9760164.70194.68991613.430013.5000174.95734.94671716.294616.3840184.00603.99241819.540319.6520192.09362.06051923.190923.328020−0.3191−0.37572027.270227.4360


## 6. Conclusion

In this paper, we show the concept of mild solutions of the R-L fractional system and considered two cases of thePDα-type ILC algorithm. The sufficient conditions of robustness analysis of the PDα-type ILC algorithm with uncertain disturbances were given by the corresponding theorems and proved. At last, two R-L fractional examples are given to demonstrate the results.



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