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Robust convergence analysis of iterative learning control for impulsive RiemannLiouville fractionalorder systems
Advances in Difference Equations volume 2016, Article number: 256 (2016)
Abstract
In this paper, we explore Ptype learning laws for impulsive RiemannLiouville fractionalorder controlled systems \((0<\alpha<1)\) with initial state offset bounded to track the varying reference accurately by using a few iterations in a finite time interval. By using the Gronwall inequality and fundamental inequalities, we obtain openloop and closedloop Ptype robust convergence results in the sense of \((PC_{1\alpha}, \lambda)\)norm \(\\cdot\_{PC_{1\alpha},\lambda}\). Finally, numerical examples are given to illustrate our theoretical results.
1 Introduction
Since Uchiyama and Arimoto put forward the concept of iterative learning control (ILC for short), ILC has been extended to tracking tasks with iteratively varying reference trajectories [1–3] extensively. Up to now, a wide variety of iterative learning control problems and related issues have been proposed and studied in many fields. For example, ILC for fractional differential systems [4–6], ILC for impulsive differential systems [7, 8], research on the robustness of ILC [9–11], and so on.
Recently, the fractionalorder differential system has played an important role in various fields such as electricity, signal and image processing, neural networks [12–14], and control problems [15]. Furthermore, the qualitative theory of fractional differential systems has been studied extensively. The existence theory of solutions to fractionalorder differential equations involving RiemannLiouville and Caputo derivatives has been investigated in [16–25]. Meanwhile, it is remarkable that some interesting existence and controllability results have been obtained for fractional controlled systems involving the Caputo derivative [26–29]. Moreover, the concept and existence of solutions for impulsive fractional differential equations involving RiemannLiouville and Caputo derivatives have been studied in [30–34]. There are few papers on ILC for integerorder and Caputo type fractionalorder impulsive differential systems [35–44]. Since RiemannLiouville fractionalorder systems play the same important role in theory analysis and application, it is necessary to deal with ILC problems for RiemannLiouville type fractional impulsive differential systems.
In this paper, we discuss ILC for impulsive RiemannLiouville fractional controlled systems with initial state offset bounded and present the robust convergence analysis results. More precisely, we study
where \(D_{0,t}^{\alpha}\) denotes RiemannLiouville fractional derivatives of the order \(\alpha\in(0,1)\) from lower limit zero and \(I_{0,t}^{1\alpha}\) denotes RiemannLiouville fractional integral the order \(1\alpha\) from lower limit zero (see Definition 2.1), k denotes the kth learning iteration, T denotes prefixed iteration domain length, impulsive term
where \(I_{0,t_{j}^{+}}^{1\alpha}x(t_{j}^{+})\) and \(I_{0,t_{j}^{}}^{1\alpha}x(t_{j}^{})\) denote the right and the left limits of \(I_{0,t}^{1\alpha}x(t)\) at \(t_{j}\in\{t_{1},\ldots,t_{m}\}\). For more details on \(\Theta(I_{0,t_{j}}^{1\alpha}x)(t_{j})\), one can see [16], Lemma 3.2, Chapter 3. Also, \(t_{j}\), \(j=1,2,\ldots, m\), denotes the jth impulsive points satisfying \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=T\). The nonlinear terms \(f: J\times R^{n} \times R^{n} \rightarrow R^{n}\) and \(G_{j},g: J\times R^{n}\rightarrow R^{n}\) are given functions. The functions \(\xi_{k}\), \(\eta_{k}: J\to R^{n}\) represent the state interference and output disturbance, respectively. The variables \(x_{k}, u_{k}, y_{k}\in R^{n}\) denote state, input, and output, respectively. Moreover, B is a \(n\times n\) real matrix.
According to [32], (2.5), the continuous solution of the system (1) can be formulated by the solution of the fractional integral equations
where \(E_{\alpha,\alpha}\) denotes MittagLeffler type function (see Definition 2.2).
The ILC problems for RiemannLiouville type fractional impulsive differential systems have not been studied extensively. The main difficulties are the following two facts:

(i)
The initial value involving singular term in RiemannLiouville fractional differential equations of order \(\alpha\in (0,1)\) is much different from Caputo fractional differential equations with the same order.

(ii)
Impulsive conditions make the formula of solutions to fractional differential equations more complex due to the memory property of the fractional derivative.
After carefully observing, we have to introduce the piecewise continuous space with weighted norm to deal with the singular term appearing in the initial condition via a new singular impulsive Gronwall inequality, which is the main difficult to be solved by us.
For the system (1), we consider an openloop Ptype ILC updating law with the initial state offset bounded,
and a closedloop Ptype ILC updating law learning with initial state offset bounded,
where \(\Delta u_{k}=u_{k+1}u_{k}\), \(e_{k}=y_{d}y_{k}\), \(\Delta x_{k}=x_{k+1}x_{k}\) denote the tracking error and \(y_{d}\) the iteratively varying reference trajectory, and \(d_{0}\) is positive constant, \(P_{o}\) and \(P_{d}\) are unknown \(n\times n\) matrix parameters to be determined.
The main objective of this paper is to generate the control input \(u_{k}\) such that the impulsive fractional system output \(y_{k}\) tracking the iteratively varying reference trajectories \(y_{d}\) (may be continuous or discontinuous) as accurately as possible when \(k\rightarrow\infty\) uniformly on \([0,T]\) in the sense of \((PC_{1\alpha},\lambda)\)norm by adopting a Ptype ILC updating law with initial state offset bounded.
The main contribution of this paper are as follows.

(i)
We establish a standard study framework of the ILC problem for an impulsive RiemannLiouville fractional system associated with an impulsive Gronwall inequality with singular kernel given by us (see [31], Lemma 2.8).

(ii)
Sufficient conditions ensuring the robust convergence of ILC problem for impulsive RiemannLiouville fractional system with order lying in \((0,1)\) are derived.
The rest of this paper is organized as follows. In Section 2, we give some necessary notations, concepts, and lemmas. In Section 3, two sufficient conditions ensuring convergence results of the system (1) are presented. An interesting example is given in the final section to demonstrate the application of our main results.
2 Preliminaries
Set \(J=[0,T]\). Let \(C(J, R^{n})\) be the Banach space of vectorvalue continuous functions from \(J\to R^{n}\) endowed with the standard norm \(\\cdot\\). In order to define the solutions of system (1), we consider a Banach space \(PC(J, R^{n})= \{x: (tt_{j})^{1\alpha}x(t)\in C((t_{j}, t_{j+1}], R^{n}), \mbox{and } \lim_{t\rightarrow t_{j}^{+}}(tt_{j})^{1\alpha}x(t) \mbox{ exists}, j=1,2,\ldots , m\}\) endowed with the \((PC_{1\alpha}, \lambda)\)norm
Next, we recall some basic definitions on fractional calculus.
Definition 2.1
(see [16], Formula (2.1.1))
For a given function f, the RiemannLiouville fractional integral \(I_{a,x}^{\alpha}f\) is defined by
and the RiemannLiouville fractional derivative \(D_{a,x}^{\alpha}f\) is defined by
where \(\Gamma(\cdot)\) is the Gamma function.
Definition 2.2
(see [45], (4.1.1))
The twoparameter MittagLeffler type function is defined by
The following lemmas will be used in the sequel.
Lemma 2.3
(see [34], Lemma 2)
Let \(\alpha\in(0,1]\) and \(\lambda>0\) be arbitrary. The functions \(E_{\alpha}(\cdot)\), \(E_{\alpha,\alpha}(\cdot)\) are nonnegative and
Lemma 2.4
(see [31], Lemma 2.8)
Let \(v\in PC(J, R^{+})\) satisfy the following inequality:
where \(c_{1}(t)\) is nonnegative continuous and nondecreasing on J, and \(c_{2}, \theta_{j}>0\) are constants. Then
where \(\theta=\max\{\theta_{j}: j=1,2,\ldots,m\}\).
Lemma 2.5
(see [9], Lemma 1)
Let \(d_{k}\) be a sequence of real number which converges to the limit \(d_{\infty}\) as \(k\rightarrow\infty\). Suppose that \(a_{k}\) is a sequence of real number such that
Then we have
3 Robust convergence analysis of Ptype
In this section, we discuss robust convergence results for (1) via an ILC of an openloop Ptype ILC (3) and closedloop (4), respectively.
For a start, we impose the following assumptions:
(A_{1}) The function \(f: J\times R^{n} \times R^{n} \rightarrow R^{n}\) is continuous and there exist two nonnegative functions \(L_{f}(\cdot)\) and \(I_{f}(\cdot)\) such that
for any \(x,\hat{x},u,\hat{u}\in R^{n}\) and all \(t\in J\).
The function \(g: J\times R^{n}\rightarrow R^{n}\) is continuous and there exists a constant \(L_{g}>0\) such that
for any \(x,\hat{x}\in R^{n}\) and all \(t\in J\).
We have the function \(G_{j}: J\times R^{n}\rightarrow R^{n}\), \(j=1,2,\ldots,m\), and there exists a nonnegative function \(L_{G_{j}}(\cdot)\) such that
for any \(x,\hat{x}\in R^{n}\) and all \(t\in J\).
(A_{2}) For nonnegative functions \(L_{f}(\cdot)\), \(I_{f}(\cdot)\), and \(L_{G_{j}}(\cdot)\), we set
(A_{3}) For uncertainty and disturbance terms \(\xi_{k}(t)\in R^{n}\), \(\eta_{k}(t)\in R^{n}\) and the initial value \(x_{k}(0)\in R^{n}\) are bounded as follows, for all \(t\in(t_{j}, t_{j+1}], j=1,2,\ldots, m\), and for any k, \(\\xi_{k+1}(t)\xi_{k}(t)\\leq d_{\xi}\), \(\\eta_{k+1}(t)\eta_{k}(t)\\leq d_{\eta}\), where \(d_{\xi}\) and \(d_{\eta}\) are positive constants.
Now we are ready to present the robust convergence analysis result for an openloop Ptype ILC.
Theorem 3.1
For the system (1), the assumptions (A_{1})(A_{3}) hold. If \(\IBP_{o}\<1\), then, for arbitrary initial input \(u_{0}\), (3) guarantees that \(y_{k}(t)\) is uniformly bounded for \(t\in J\) as \(k\rightarrow\infty\) in the sense of \((PC_{1\alpha}, \lambda )\)norm. Further, \(y_{k}(t)\) uniform convergent to \(y_{d}(t)\) for \(t\in J\) if disturbance is converge asymptotically to zero.
Proof
Without loss of generality, we only consider \(t\in(t_{j}, t_{j+1}]\), \(j=0,1,2,\ldots, m\). Linking (1) and (3), we have
Taking the norm \(\\cdot\\) on both sides of (5), one can derive that
In the following, we prove \(\e_{k+1}\_{PC_{1\alpha}, \lambda}\) is uniformly bounded as \(k\rightarrow\infty\).
Taking the norm \(\\cdot\\) on both sides of (2), one can apply (A_{1}) and (A_{3}) to derive that
Multiplying \((tt_{j})^{1\alpha}\) on both sides of (7), using (A_{2}) we have
Note that the fact \(\frac{tt_{j}}{tt_{i}}\leq1\) since \(t_{j}\geq t_{i}\) and
Then (8) reduces to
For the inequality (9), we set \(v(t)=(tt_{i})^{1\alpha}\\Delta x_{k}(t)\\). Then one can apply Lemma 2.4 to derive that
where
Multiplying \(e^{\lambda(tt_{j})}\) on both sides of (10) and noting the fact that \(E_{\alpha}(z),z>0\) is an increasing function, we have
where
For (11), one can take the \((PC_{1\alpha}, \lambda)\)norm to derive that
where
where
Submitting (12) into (13), we obtain
where
Note that there exists a large enough λ such that \(\tilde{q}<1\) due to \(\Vert IBP_{o}\Vert <1\). Concerning (14), one can use Lemma 2.5 to derive that
which shows that \(y_{k}(t)\) is uniformly bounded in the sense of \((PC_{1\alpha}, \lambda)\)norm. Further, if the disturbance has asymptotic convergence, which means that \(d_{\xi}\rightarrow0\), \(d_{\eta}\rightarrow0\), and \(d_{0}\rightarrow0\), as \(k\rightarrow\infty\), then \(y_{k}(t)\) uniform convergent to \(y_{d}(t)\) for \(t\in J\) if the disturbance converges asymptotically to zero. □
Remark 3.2
In Theorem 3.1, If we set \(\alpha=0\), \(x_{k}(0)=x_{k+1}(0)\), \(\xi_{k}(t)=\eta_{k}(t)=0\), \(0<\beta_{3}<\frac{\partial g}{\partial x}<\beta_{4}\), \(G_{j}(t,x)=G_{j}(x)\), \(L_{f}(t)=L_{f}\), \(I_{f}(t)=I_{f}\), and \(L_{G_{j}}(t)=L_{G}\), then \(y_{k}(\cdot)\) is uniform convergent to \(y_{d}(\cdot)\) in the sense of \((PC, \lambda)\)norm, which is a parallel result to [40], Theorem 3.1, in the sense of the \(L^{2}\)norm.
Next, we present the robust convergence analysis result for a closedloop Ptype ILC.
Theorem 3.3
For the system (1), the assumptions (A_{1})(A_{3}) hold. If \((I+BP_{d})^{1}\) exists and \(\(I+BP_{d})^{1}\<1\), (4) guarantees that \(y_{k}(t)\) is uniformly bounded for \(t\in J\) as \(k\rightarrow\infty\) in the sense of the \((PC_{1\alpha}, \lambda)\)norm. Moreover, if the disturbance converges asymptotically to zero, then \(y_{k}(t)\) is uniform convergent to \(y_{d}(t)\) for \(t\in J\).
Proof
Similar to the proof of Theorem 3.1, we consider \(t\in(t_{j}, t_{j+1}]\), \(j=1,2,\ldots, m\). Linking (1) and (4), we have
Taking the norm \(\\cdot\\) on both sides of (15), we have
Next, we apply the analogy method in Theorem 3.1 to prove that \(\e_{k+1}\_{PC_{1\alpha}, \lambda}\) is uniformly bounded as \(k\rightarrow\infty\).
By repeating the procedure to derive (12), one has
where \(N_{j}\), θ, and \(N_{\max}\) are defined in Theorem 3.1.
Substituting (16) into (17), we have
Taking (17) into (18), we have
which implies that
where
Let
Then (19) reduces to
Note that \(\(I+BP_{d})^{1}\<1\). It is not difficult to see that \(\tilde{q}<1\) for some large enough λ. By Lemma 2.5, we have
Thus, the demised results are obtained immediately. The proof is finished. □
Remark 3.4
In Theorem 3.3, if we set \(\alpha=0\), \(x_{k}(0)=x_{k+1}(0)\), \(\xi_{k}(t)=\eta_{k}(t)=0\), \(0<\beta_{3}<\frac{\partial g}{\partial x}<\beta_{4}\), \(G_{j}(t,x)=G_{j}(x)\), \(L_{f}(t)=L_{f}\), \(I_{f}(t)=I_{f}\), and \(L_{G_{j}}(t)=L_{G}\), then \(y_{k}(\cdot)\) is uniform convergent to \(y_{d}(\cdot)\) in the sense of \((PC, \lambda)\)norm, which is another parallel result to [40], Theorem 3.2, in the sense of \(L^{2}\)norm.
4 Simulation examples
In this section, one numerical example is presented to demonstrate the validity of the designed method. In order to describe the stability of the system which is associated with the increase of the iterations, we denote the total energy in the kth iteration as \(\mathscr{E}_{k}=\u_{k}\_{\infty}=\max_{t\in[0,T]}\u_{k}(t)\\).
Example 4.1
Consider the following impulsive fractional controlled systems:
and the Ptype ILC
Set \(\alpha=0.5\), \(\mu=1\), \(f(t,x_{k},u_{k})=t(t0.5)^{2}u_{k}\), \(G_{j}(t^{}_{j},x_{k}(t^{}_{j}))=t_{1}^{}x_{k}(t_{1}^{})\), \(j=1\), \(\xi_{k}(t)=\frac{t}{k+1}\), \(\eta_{k}(t)=e^{kt}, t\in[0,1]\). Obviously, \(L_{f}(t)=0\), \(I_{f}(t)=t(t0.5)^{2}\), \(L_{G_{j}}(t)=0.5, t\in[0,1]\). It is not difficult to verify that \(M_{1}=0.25\), \(M_{2}=0.5\). Set \(M_{L}=0.5\). Then (A_{1})(A_{3}) are satisfied.
Case 1: The original reference trajectory is a piecewise continuous function,
(i) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.3\). Obviously, \(1BP_{o}=0.64<1\). \(\xi_{k}(t)=\frac{t}{k+1}\), \(\eta _{k}(t)=e^{kt}, t\in[0,1]\). All the conditions of Theorem 3.1 are satisfied. Meanwhile, the disturbances have asymptotic convergence, then \(y_{k}(t)\) is uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
⋆ Figure 1 shows the output \(y_{k}\) of equation (21) of the 10th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 1 shows the ∞norm of the tracking error in each iteration and the error is 0.0647.
⋆ Figure 2 shows the output \(y_{k}\) of equation (21) of the 20th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 2 shows the ∞norm of the tracking error in each iteration and the error is 0.0405.
(ii) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.7\). Obviously, \(1BP_{o}=0.16<1\). Then all the conditions of Theorem 3.1 are satisfied. The \(y_{k}(t)\) is uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
⋆ Figure 3 shows the output \(y_{k}\) of equation (21) of the 10th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 3 shows the ∞norm of the tracking error in each iteration and the error is 0.0575.
⋆ Figure 4 shows the output \(y_{k}\) of equation (21) of the 20th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 4 shows the ∞norm of the tracking error in each iteration and the error is 0.0153.
Conclusions:

From Figures 1 and 2 or 3 and 4, we find that the tracking error decreases with k increasing.

From Figures 1 and 3 or 2 and 4, we find that the tracking error decreases with \(P_{o}\) increasing.
Case 2: The second original reference trajectory is continuous,
(iii) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.3\). Obviously, \(1BP_{o}=0.64<1\). All the conditions of Theorem 3.1 are satisfied. Meanwhile, the disturbances are asymptotic convergence, then \(y_{k}(t)\) uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
⋆ Figure 5 shows the output \(y_{k}\) of equation (21) of the 10th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 5 shows the ∞norm of the tracking error in each iteration and the error is 0.01754.
⋆ Figure 6 shows the output \(y_{k}\) of equation (21) of the 20th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 6 shows the ∞norm of the tracking error in each iteration and the error is 0.00033936.
(iv) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.7\). Obviously, \(1BP_{o}=0.16<1\). Then all the conditions of Theorem 3.1 are satisfied. The \(y_{k}(t)\) uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
⋆ Figure 7 shows the output \(y_{k}\) of equation (21) of the 10th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 7 shows the ∞norm of the tracking error in each iteration and the error is 0.00058637.
⋆ Figure 8 shows the output \(y_{k}\) of equation (21) of the 20th iteration and the reference trajectory \(y_{d}\). The lower figure of Figure 8 shows the ∞norm of the tracking error in each iteration and the error is 0.00017541.
Conclusions:
5 Conclusions
Due to the fact that impulse phenomenon and fractionalorder systems widely exist in engineering, we investigated the Ptype learning laws for impulsive RiemannLiouville fractionalorder controlled systems (\(0<\alpha<1\)) with initial state offset bounded. We obtain openloop and closedloop Ptype robust convergence results in the sense of \((PC_{1\alpha}, \lambda)\)norm \(\\cdot\_{PC_{1\alpha},\lambda}\) via an impulsive Gronwall inequality. Furthermore, one example is given to verify the effectiveness and feasibility of the obtained results. The proposed scheme can deal with the robust convergence of impulsive RiemannLiouville fractional systems. We would like to point out that it is possible to extend our results to other impulsive fractionalorder models such as noninstantaneous impulsive fractionalorder systems and so on.
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Acknowledgements
This work is partially supported by NNSFC (No. 11661016;11261011) and Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), 2011 Collaborative Innovation Project of Guizhou Province ([2014]02), Unite Foundation of Guizhou Province ([2015]7640), and Graduate ZDKC ([2015]003).
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This work was carried out in collaboration between all authors. ZJL, WW, and JRW proved the theorems, interpreted the results, and wrote the article. All authors defined the research theme, and they read and approved the manuscript.
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Luo, Z., Wei, W. & Wang, J. Robust convergence analysis of iterative learning control for impulsive RiemannLiouville fractionalorder systems. Adv Differ Equ 2016, 256 (2016). https://doi.org/10.1186/s1366201609842
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DOI: https://doi.org/10.1186/s1366201609842