MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2021/66615436661543Research ArticleRobustness Analysis of a Type of Iterative Algorithm for R-L Fractional Nonlinear Control Systems in the Sense of Lp NormLiYanfanghttps://orcid.org/0000-0003-0199-517XLiuXianghuXuGuangjunElsadanyAbdelalimDepartment of MathematicsZunyi Normal CollegeZunyi 563006GuizhouChinazync.edu.cn2021161202120212810202041220203112202016120212021Copyright © 2021 Yanfang Li et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper is concerned with the robustness analysis of a type of iterative algorithm for R-L fractional nonlinear control systems in the sense of Lp 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.

National Natural Science Foundation of China11661084Department of Education of Guizhou Province046Qian Hiao He KY093Qian Ke He Ping Tai Ren Cai 5784-08
1. Introduction

The aim of the paper is to analyze the robustness of a type of iterative algorithm in the sense of Lp norm of the following R-L fractional system:(1)DtαRLzt=Azt+But,tJ=0,b,g1αz0=z0,yt=Czt+Dut,where DtαRL denotes the R-L derivative of order α, 0<α<1, A,B,CRn×n, ut is a control vector, and g1α=t1α/Γ1α.

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 , and many fractional nonlinear systems were investigated . 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 .

In Li et al.’s study , 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 the λ-norm often causes tracking errors that exceed the actual engineering range and cause inaccurate data, the authors Lan and Lin  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  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 Section 2, 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 the Lp norm and Mittag–Leffler function. From , one can see the definitions of the R-L fractional integral and derivative.

Definition 1.

The norm for the n-dimensional vector Z=z1,z2,,zn is defined as Z=max1inzi, and the Lp norm is defined as Zp=0Tmaxzipdt1/p, where t0,T.

Definition 2.

The definition of the two-parameter function of the Mittag–Leffler type is described by(2)Eα,βz=k=0zkΓαk+β,α>0,β>0,zC.

If β=1, one has the Mittag–Leffler function of one parameter as follows:(3)Eαz=k=0zkΓαk+1.

Now, according to the results of the papers [17, 2427], we will give the following lemma.

Lemma 1.

(Lemma 3, see ). The general solution of equation (1) is given by(4)zt=tα1Eα,αA,tz0+0ttsα1EAtsαBusds,where(5)Eα,βA,t=k=0Aktαk+β1Γαk+β.

Lemma 2.

(Definition 2.4, see ). The operators Eα,αt are exponentially bounded, and there is a constant C0=1/αA1α/α, eαt=eA1/αt, M=eαb, and Eα,αA,tC0eαtC0M.

Lemma 3.

(Ho¨lder inequality). Set p>0,q>0, and 1/p+1/q=1; if fLpΩ,gLqΩ, and fgL1Ω, then fgL1fLpgLq.

3. Robustness Analysis of the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M43"><mml:mi>P</mml:mi><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>-Type ILC Algorithm with Uncertain Disturbances

In this section, we consider the following fractional equation:(6)DtαRLzkt=Azkt+Bukt+ωt,tJ=0,b,ykt=Czkt+Dukt+νt,where k=0,1,2,3,, and ωt and νt 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 t0,b, γ1 and γ2 are the parameters which will be determined, ydt is the given function, ek=ydtykt, 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.

Block diagram of the open- and closed-loop PDα-type ILC algorithm.

We denote that(9)κ1=I+γ2Dbα1/pγ2CC0MBqα1+1q,κ2=Iγ1Dbα1/pγ1CC0MBqα1+1q,κ3=bα1/pγ1C+γ2CC0MBωLpqα1+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, and κ1>κ2, and then, the sufficient condition for being uniformly bounded on J is limkukLpκ3/κ1κ2.

Proof.

Define(10)Δzkt=zdtzkt,Δukt=udtukt.

For tJ, one has Δzkαt=DtαRLΔzkt=AΔzkt+BΔukt and ek+1αt=CAΔzk+1t+BΔuk+1t.

According to system (6), we have(11)zk+1t=tα1Eα,αA,tz0+0ttsα1Eα,αAtsαBuk+1s+ωtds,zdt=tα1Eα,αA,tz0+0ttsα1Eα,αAtsαBudsds,and thus, using the ILC algorithms (7) and (8), we derive(12)Δzk+1t=0ttsα1Eα,αAtsαBΔuk+1s+ωtds,Δzkt=0ttsα1Eα,αAtsαBΔuks+ωtds,so(13)Δuk+1t=Δuktγ1ydtyktγ2ydtyk+1t,=Δuktγ1CΔzkt+DΔuktνtγ2CΔzk+1t+DΔuk+1tνt,=Δuktγ1C0ttsα1Eα,αAtsαBΔuks+ωtds+γ1DΔuktγ1νt,γ2C0ttsα1Eα,αAtsαBΔuk+1s+ωtds+γ2DΔuk+1tγ2νt.

Hence,(14)Δuk+1tI+γ2D=ΔuktIγ1Dγ1C0ttsα1Eα,αAtsαBΔuks+ωtdsγ1νtγ2C0ttsα1Eα,αAtsαBΔuk+1s+ωtdsγ2νt.

By taking the Lp norm, we obtain(15)Δuk+1LpI+γ2DΔukLpI+γ2D+bα1/pγ1CC0MBΔukLp+ΔωLpqα1+1q+γ1νLp+bα1/pγ2CC0MBΔuk+1Lp+ΔωLpqα1+1q+γ2νLp,denoting(16)κ1=I+γ2Dbα1/pγ2CC0MBqα1+1q,κ2=Iγ1Dbα1/pγ1CC0MBqα1+1q,κ3=bα1/pγ1C+γ2CC0MBωLpqα1+1q+γ1+γ2νLp.

Consequently, κ1Δuk+1Lpκ2ΔukLp+κ3. So, there exists a positive m, such that κ3<m and κ1>0,κ2>0, and κ1>κ2, and then limkukLpκ3/κ1κ2, which implies ekt is uniformly bounded on J.

4. Robust Analysis of the Second-Order <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M81"><mml:mi>P</mml:mi><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>-Type ILC Algorithm

In this section, we consider the following second-order PDα-type ILC algorithm:(17)u2t=u1t+γ1e1t+γ2e1αt,uk+1t=r1ukt+γ1ekt+γ2ekαt+r2uk1t+γ3ek1t+γ4ek1αt,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 Figure 2.

Block diagram of the second-order PDα-type ILC algorithm.

Assume that the initial state of each iterative learning meets (18), where L1 and L2 are the parameters which will be determined.

Note(19)K1=r1+r1γ1D+r1γ2CB+r1γ1C+r1γ2CAbα1/pC0MBqα1+1q,K2=r2+r2γ3D+r2γ4CB+r2γ3C+r2γ4CAbα1/pC0MBqα1+1q,K3=r1γ1C+r1γ2CA+r2γ3C+r2γ4CAbα1/pC0Mqα1+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, and K30. Since k, Δuk+1Lp is uniformly bounded, which guarantees that limkekλ=0 and tJ.

Proof.

According to Lemma 1, we yield(20)zk+1t=tα1Eα,αA,tz0+0ttsα1Eα,αAtsαBuk+1s+ωtds,Δzk+1t=0ttsα1Eα,αAtsαBΔuk+1s+ωtds,Δzkt=0ttsα1Eα,αAtsαBΔuks+ωtds,and then,(21)Δuk+1t=r1Δukt+γ1CΔzkt+γ1DΔukt+γ2CAΔzkt+BΔukt+r2Δuk1t+γ3CΔzk1t+γ3DΔuk1t+γ4CAΔzk1t+BΔuk1t,=r1+r1γ1D+r1γ2CBΔukt+r2+r2γ3D+r2γ4CBΔuk1t+r1γ1C+r1γ2CA0ttsα1Eα,αAtsαBΔuks+ωtds+r2γ3C+r2γ4CA0ttsα1Eα,αAtsαBΔuk1s+ωtds.

By taking the Lp norm, it yields(22)Δuk+1Lpr1+r1γ1D+r1γ2CBΔuktLp+r1γ1C+r1γ2CAbα1/pC0MBqα1+1qΔuktLp+r2+r2γ3D+r2γ4CBΔuk1tLp+r2γ3C+r2γ4CAbα1/pC0MBqα1+1qΔuk1tLp+r1γ1C+r1γ2CAbα1/pC0Mqα1+1qωLp+r2γ3C+r2γ4CAbα1/pC0Mqα1+1qωLp.

For brevity, note that(23)K1=r1+r1γ1D+r1γ2CB+r1γ1C+r1γ2CAbα1/pC0MBqα1+1q,K2=r2+r2γ3D+r2γ4CB+r2γ3C+r2γ4CAbα1/pC0MBqα1+1q,K3=r1γ1C+r1γ2CA+r2γ3C+r2γ4CAbα1/pC0Mqα1+1qωLp,and one can deduce Δuk+1LpK1ΔukLp+K2Δuk1Lp+K3.

There exists a constant p>0, which satisfies K1+K2<1 and K30. 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. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M105"><mml:mi>P</mml:mi><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>-Type ILC with Initial State Error

Consider the following one-dimensional systems as follows:(24)Dt0.6RLxt=xk2t+0.1ut+ωkt,tJ=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,

where Ax=x2. Now, we can choose α=0.6, B=0.1, C=1, p=2, γ1=γ2=0.5, ωt=103sin0.001t, and νt=105t. For the system, we use the PDα-type ILC algorithm and set the initial control u0=0, ydt=5sinet2, and t0,2. One can calculate M3>0,κ1=0.47,κ2=0.17, and κ3<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.

Simulation results of output yk.

.Numerical simulation of the output of the system in Section 5.1 and the desired trajectory.

kykydtkkykydtkkykydtk
12.08182.053918−4.9051−4.9774355.01834.9945
21.25801.245419−0.9237−0.964536−4.0133−4.0915
30.32940.3088204.76844.791037−4.6198−4.7037
4−0.7118−0.7414210.81930.791238−3.3214−3.3956
5−1.8284−1.867722−4.7823−4.8564393.22063.1983
6−2.9501−2.9989234.10164.085740−4.7914−4.8791
7−3.9613−4.019024−2.2443−2.2993414.93184.9216
8−4.6935−4.7576252.18032.1603422.11762.0840
9−4.9269−4.993626−3.9964−4.066943−4.8627−4.9535
10−4.4206−4.4837274.35424.3339444.59434.5790
11−2.9888−3.0410283.84353.7982454.25924.2403
12−0.6484−0.6824292.67052.651546−2.3747−2.4479
132.17382.1620304.94694.904847−4.3241−4.4139
144.44564.475531−1.9316−1.989148−2.2423−2.3162
154.75334.7927324.48464.4505493.33873.3090
162.08442.0698333.10673.0883502.89452.8603
17−2.5569−2.6095342.10992.0826

The tracking error of the systems.

5.2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M130"><mml:mi>P</mml:mi><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>-Type ILC with Random Disturbance

Consider a two-dimensional ILC system; we set α=0.7, ωt=1010sinπt/1000,νt=103, Ax1x2=2x12x22,B=0111,C=1101, and D=0111 and construct the second-order PDα-type ILC algorithm as follows:(26)uk+1t=0.1ukt+0.1ekt+0.1ekαt+0.1uk1t+0.2ek1t+0.2ek1αt,k=2,3,

We also select other parameters and initial values of the algorithm as follows: u0=0, ydt=y1dty2dt=5sint4t3,t0,1.9, r1=1,r2=0.5,γ1=1, and γ2=0.5. It is easy to show that M3>0,K1=0.264,K2=0.428, and K30, 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.

Simulation results of output yk.

The tracking error of the system.

From Figure 6 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.

Numerical simulation of the output of the system in Section 5.2 and the desired trajectory.

kykydtkkykydtk
100100
22.38082.397120.01810.0040
34.17824.207330.05340.0320
44.95384.987440.13490.1080
54.51824.546450.28600.2560
62.97842.992360.52990.5000
70.73780.705670.88980.8640
8−1.7256−1.753981.38871.3720
9−3.7370−3.784092.09002.0480
10−4.8293−4.8876102.89582.9160
11−4.7343−4.7946113.95004.0000
12−3.4746−3.5277125.30635.3240
13−1.3580−1.3970136.88466.9120
141.09811.0755148.74888.7880
153.32593.28491510.922710.9760
164.70194.68991613.430013.5000
174.95734.94671716.294616.3840
184.00603.99241819.540319.6520
192.09362.06051923.190923.3280
20−0.3191−0.37572027.270227.4360
6. Conclusion

In this paper, we show the concept of mild solutions of the R-L fractional system and considered two cases of the PDα-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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The authors contributed equally to this work, and all authors read and approved the final manuscript.

Acknowledgments

This work was supported by the NSF of China (no. 11661084) and Guizhou Province Department of Education Fund (046, Qian Jiao He KY093, and Qian Ke He Ping Tai Ren Cai 5784-08).

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