- Research
- Open Access
Robust convergence analysis of iterative learning control for impulsive Riemann-Liouville fractional-order systems
- Zijian Luo^{1},
- Wei Wei^{1, 2} and
- JinRong Wang^{1}Email author
https://doi.org/10.1186/s13662-016-0984-2
© Luo et al. 2016
- Received: 14 July 2016
- Accepted: 27 September 2016
- Published: 11 October 2016
Abstract
In this paper, we explore P-type learning laws for impulsive Riemann-Liouville fractional-order 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 open-loop and closed-loop P-type 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.
Keywords
- impulsive fractional-order system
- iterative learning control
- robust convergence
- weighted norm
MSC
- 34A37
- 93C15
- 93C40
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 fractional-order 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 fractional-order differential equations involving Riemann-Liouville 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 Riemann-Liouville and Caputo derivatives have been studied in [30–34]. There are few papers on ILC for integer-order and Caputo type fractional-order impulsive differential systems [35–44]. Since Riemann-Liouville fractional-order systems play the same important role in theory analysis and application, it is necessary to deal with ILC problems for Riemann-Liouville type fractional impulsive differential systems.
- (i)
The initial value involving singular term in Riemann-Liouville 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.
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 P-type ILC updating law with initial state offset bounded.
- (i)
We establish a standard study framework of the ILC problem for an impulsive Riemann-Liouville 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 Riemann-Liouville 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
Next, we recall some basic definitions on fractional calculus.
Definition 2.1
(see [16], Formula (2.1.1))
Definition 2.2
(see [45], (4.1.1))
The following lemmas will be used in the sequel.
Lemma 2.3
(see [34], Lemma 2)
Lemma 2.4
(see [31], Lemma 2.8)
Lemma 2.5
(see [9], Lemma 1)
3 Robust convergence analysis of P-type
In this section, we discuss robust convergence results for (1) via an ILC of an open-loop P-type ILC (3) and closed-loop (4), respectively.
For a start, we impose the following assumptions:
(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 open-loop P-type ILC.
Theorem 3.1
For the system (1), the assumptions (A_{1})-(A_{3}) hold. If \(\|I-BP_{o}\|<1\), 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
In the following, we prove \(\|e_{k+1}\|_{PC_{1-\alpha}, \lambda}\) is uniformly bounded as \(k\rightarrow\infty\).
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 closed-loop P-type 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
Next, we apply the analogy method in Theorem 3.1 to prove that \(\|e_{k+1}\|_{PC_{1-\alpha}, \lambda}\) is uniformly bounded as \(k\rightarrow\infty\).
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
Set \(\alpha=0.5\), \(\mu=-1\), \(f(t,x_{k},u_{k})=t(t-0.5)^{2}u_{k}\), \(G_{j}(t^{-}_{j},x_{k}(t^{-}_{j}))=t_{1}^{-}x_{k}(t_{1}^{-})\), \(j=1\), \(\xi_{k}(t)=\frac{t}{k+1}\), \(\eta_{k}(t)=e^{-kt}, t\in[0,1]\). Obviously, \(L_{f}(t)=0\), \(I_{f}(t)=t(t-0.5)^{2}\), \(L_{G_{j}}(t)=0.5, t\in[0,1]\). 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.
(i) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.3\). Obviously, \(|1-BP_{o}|=0.64<1\). \(\xi_{k}(t)=\frac{t}{k+1}\), \(\eta _{k}(t)=e^{-kt}, t\in[0,1]\). 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 \(t\in[0,1]\).
(ii) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.7\). Obviously, \(|1-BP_{o}|=0.16<1\). Then all the conditions of Theorem 3.1 are satisfied. The \(y_{k}(t)\) is uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
(iii) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.3\). Obviously, \(|1-BP_{o}|=0.64<1\). 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 \(t\in[0,1]\).
(iv) We set \(u_{k}(0)=0\) and \(B=1.2\), \(P_{o}=0.7\). Obviously, \(|1-BP_{o}|=0.16<1\). Then all the conditions of Theorem 3.1 are satisfied. The \(y_{k}(t)\) uniform convergent to \(y_{d}(t)\), for \(t\in[0,1]\).
5 Conclusions
Due to the fact that impulse phenomenon and fractional-order systems widely exist in engineering, we investigated the P-type learning laws for impulsive Riemann-Liouville fractional-order controlled systems (\(0<\alpha<1\)) with initial state offset bounded. We obtain open-loop and closed-loop P-type 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 Riemann-Liouville fractional systems. We would like to point out that it is possible to extend our results to other impulsive fractional-order models such as non-instantaneous impulsive fractional-order systems and so on.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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