Open Access

Robust convergence analysis of iterative learning control for impulsive Riemann-Liouville fractional-order systems

Advances in Difference Equations20162016:256

https://doi.org/10.1186/s13662-016-0984-2

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 [13] 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 [46], ILC for impulsive differential systems [7, 8], research on the robustness of ILC [911], 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 [1214], 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 [1625]. Meanwhile, it is remarkable that some interesting existence and controllability results have been obtained for fractional controlled systems involving the Caputo derivative [2629]. Moreover, the concept and existence of solutions for impulsive fractional differential equations involving Riemann-Liouville and Caputo derivatives have been studied in [3034]. There are few papers on ILC for integer-order and Caputo type fractional-order impulsive differential systems [3544]. 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.

In this paper, we discuss ILC for impulsive Riemann-Liouville fractional controlled systems with initial state offset bounded and present the robust convergence analysis results. More precisely, we study
$$ \left \{ \textstyle\begin{array}{@{}l} (D_{0,t}^{\alpha}x_{k})(t)=\mu x_{k}(t)+f(t, x_{k}(t), u_{k}(t))+\xi_{k}(t),\quad t\in[0,T]\setminus\{t_{1},\ldots ,t_{m}\}, \mu< 0,\\ \lim_{t\to0+}(I_{0,t}^{1-\alpha}x_{k})(t)=x_{k}(0), \\ \Theta(I_{0,t_{j}}^{1-\alpha }x_{k})(t_{j})=G_{j}(t^{-}_{j},x_{k}(t^{-}_{j})),\quad t_{j}\in\{t_{1},\ldots,t_{m}\},\\ y_{k}(t)=g(t, x_{k}(t))+Bu_{k}(t)+\eta_{k}(t), \end{array}\displaystyle \right . $$
(1)
where \(D_{0,t}^{\alpha}\) denotes Riemann-Liouville fractional derivatives of the order \(\alpha\in(0,1)\) from lower limit zero and \(I_{0,t}^{1-\alpha}\) denotes Riemann-Liouville fractional integral the order \(1-\alpha\) from lower limit zero (see Definition 2.1), k denotes the kth learning iteration, T denotes pre-fixed iteration domain length, impulsive term
$$\Theta \bigl(I_{0,t_{j}}^{1-\alpha}x\bigr) (t_{j}):= I_{0,t_{j}^{+}}^{1-\alpha}x\bigl(t_{j}^{+} \bigr)-I_{0,t_{j}^{-}}^{1-\alpha}x\bigl(t_{j}^{-}\bigr)= \Gamma(\alpha) \Bigl[\lim_{t\rightarrow t_{j}^{+}}(t-t_{j})^{1-\alpha}x(t)- \lim_{t\rightarrow t_{j}^{-}}(t-t_{j})^{1-\alpha}x(t) \Bigr], $$
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
$$ x_{k}(t)= \left \{ \textstyle\begin{array}{@{}l} t^{\alpha-1}E_{\alpha,\alpha}(\mu t^{\alpha})x_{k}(0)\\ \quad{}+\int_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(\mu(t-s)^{\alpha})[f(s, x_{k}(s), u_{k}(s))+\xi_{k}(s)]\,ds,\quad t\in[0,t_{1}],\\ t^{\alpha-1}E_{\alpha,\alpha}(\mu t^{\alpha})x_{k}(0)\\ \quad{}+\int_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(\mu(t-s)^{\alpha})[f(s, x_{k}(s), u_{k}(s))+\xi_{k}(s)]\,ds\\ \quad{}+\sum_{i=1}^{j}E_{\alpha,\alpha}(\mu(t-t_{i})^{\alpha })(t-t_{i})^{\alpha-1}G_{i}(t_{i}^{-},x_{k}(t_{i}^{-})),\\ \qquad t\in(t_{j},t_{j+1}], j=1,2,\ldots, m, \end{array}\displaystyle \right . $$
(2)
where \(E_{\alpha,\alpha}\) denotes Mittag-Leffler type function (see Definition 2.2).
The ILC problems for Riemann-Liouville type fractional impulsive differential systems have not been studied extensively. The main difficulties are the following two facts:
  1. (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.

     
  2. (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 open-loop P-type ILC updating law with the initial state offset bounded,
$$ \Delta u_{k}=P_{o}e_{k}(t),\qquad \bigl\Vert \Delta x_{k}(0)\bigr\Vert \leq d_{0}, $$
(3)
and a closed-loop P-type ILC updating law learning with initial state offset bounded,
$$ \Delta u_{k}=P_{d}e_{k+1}(t),\qquad \bigl\Vert \Delta x_{k}(0)\bigr\Vert \leq d_{0}, $$
(4)
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 P-type ILC updating law with initial state offset bounded.

The main contribution of this paper are as follows.
  1. (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).

     
  2. (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

Set \(J=[0,T]\). Let \(C(J, R^{n})\) be the Banach space of vector-value 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 \(PC(J, R^{n})= \{x: (t-t_{j})^{1-\alpha}x(t)\in C((t_{j}, t_{j+1}], R^{n}), \mbox{and } \lim_{t\rightarrow t_{j}^{+}}(t-t_{j})^{1-\alpha}x(t) \mbox{ exists}, j=1,2,\ldots , m\}\) endowed with the \((PC_{1-\alpha}, \lambda)\)-norm
$$\Vert x\Vert _{PC_{1-\alpha},\lambda}=\max \bigl\{ (t-t_{j})^{1-\alpha}e^{-\lambda (t-t_{j})} \bigl\Vert x(t)\bigr\Vert : j=0,1,\ldots, m \bigr\} . $$

Next, we recall some basic definitions on fractional calculus.

Definition 2.1

(see [16], Formula (2.1.1))

For a given function f, the Riemann-Liouville fractional integral \(I_{a,x}^{\alpha}f\) is defined by
$$\begin{aligned} \bigl(I_{a,x}^{\alpha}f\bigr) (x):=\frac{1}{\Gamma(\alpha)} \int_{a}^{x}\frac {f(t)}{(x-t)^{1-\alpha}}\,dt,\quad x>a; 0< \alpha< 1, \end{aligned}$$
and the Riemann-Liouville fractional derivative \(D_{a,x}^{\alpha}f\) is defined by
$$\begin{aligned} \bigl(D_{a,x}^{\alpha}f\bigr) (x):=\frac{d}{dx} \bigl(I_{a,x}^{1-\alpha}f\bigr) (x)=\frac {1}{\Gamma(1-\alpha)}\frac{d}{dx} \int_{a}^{x}\frac{f(t)}{(x-t)^{\alpha}}\,dt, \end{aligned}$$
where \(\Gamma(\cdot)\) is the Gamma function.

Definition 2.2

(see [45], (4.1.1))

The two-parameter Mittag-Leffler type function is defined by
$$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty} \frac{z^{k}}{\Gamma(\alpha k+\beta)}, \quad\alpha>0, \beta\in R, z\in R. $$

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
$$E_{\alpha}\bigl(-t^{\alpha}\lambda\bigr)\leq1,\qquad E_{\alpha,\alpha} \bigl(-t^{\alpha}\lambda\bigr)\leq \frac{1}{\Gamma(\alpha)}. $$

Lemma 2.4

(see [31], Lemma 2.8)

Let \(v\in PC(J, R^{+})\) satisfy the following inequality:
$$\begin{aligned} v(t)\leq c_{1}(t)+c_{2} \int_{0}^{t}(t-s)^{\beta-1}v(s)\,ds+\sum _{j=1}^{k}\theta_{j}v \bigl(t_{j}^{-}\bigr), \end{aligned}$$
where \(c_{1}(t)\) is nonnegative continuous and nondecreasing on J, and \(c_{2}, \theta_{j}>0\) are constants. Then
$$\begin{aligned} v(t)\leq c_{1}(t) \bigl(1+\theta E_{\beta} \bigl(c_{2}\Gamma(\beta)t^{\beta}\bigr) \bigr)^{j}E_{\beta} \bigl(c_{2}\Gamma(\beta)t^{\beta}\bigr),\quad \textit{for }t \in(t_{j}, t_{j+1}], \end{aligned}$$
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
$$pa_{k}+qa_{k-1}\leq d_{k},\quad p>-q\geq0. $$
Then we have
$$\limsup_{k\rightarrow\infty}a_{k}\leq\frac{d_{\infty}}{p+q}. $$

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:

(A1) 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
$$\begin{aligned} \bigl\Vert f(t, x, u)-f(t, \hat{x}, \hat{u})\bigr\Vert \leq L_{f}(t)\Vert x-\hat{x}\Vert +I_{f}(t)\Vert u-\hat{u} \Vert , \end{aligned}$$
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
$$\begin{aligned} \bigl\Vert g(t, x)-g(t, \hat{x})\bigr\Vert \leq L_{g} \|x-\hat{x}\|, \end{aligned}$$
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
$$\begin{aligned} \bigl\Vert G_{j}(t,x)-G_{j}(t,\hat{x})\bigr\Vert \leq L_{G_{j}}(t)\Vert x-\hat{x}\Vert , \end{aligned}$$
for any \(x,\hat{x}\in R^{n}\) and all \(t\in J\).
(A2) For nonnegative functions \(L_{f}(\cdot)\), \(I_{f}(\cdot)\), and \(L_{G_{j}}(\cdot)\), we set
$$\begin{aligned}& M_{1}=\max \biggl\{ \sup_{t\in(t_{j}, t_{j+1}]}\frac{L_{f}(t)}{(t-t_{j})^{1-\alpha}}, \sup_{t\in(t_{j}, t_{j+1}]}\frac{I_{f}(t)}{(t-t_{j})^{1-\alpha}}, j=0,1,\ldots, m \biggr\} , \\& M_{2}=\max\bigl\{ L_{G_{j}}(t_{j}),j=1,\ldots,m\bigr\} , \\& M_{L}=\max\{M_{1}, M_{2}\}. \end{aligned}$$

(A3) 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 (A1)-(A3) 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

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
$$\begin{aligned} e_{k+1}(t)=(I-BP_{o})e_{k}(t)+g \bigl(t,x_{k}(t)\bigr)-g\bigl(t,x_{k+1}(t)\bigr)+\eta _{k}(t)-\eta_{k+1}(t). \end{aligned}$$
(5)
Taking the norm \(\|\cdot\|\) on both sides of (5), one can derive that
$$\begin{aligned} \bigl\Vert e_{k+1}(t)\bigr\Vert \leq \Vert I-BP_{o}\Vert \bigl\Vert e_{k}(t)\bigr\Vert +L_{g}\bigl\Vert \Delta x_{k}(t)\bigr\Vert +d_{\eta}. \end{aligned}$$
(6)

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 (A1) and (A3) to derive that
$$\begin{aligned} \bigl\| \Delta x_{k}(t)\bigr\| \leq&\frac{t^{\alpha-1}}{\Gamma(\alpha)}\bigl\| \Delta x_{k}(0)\bigr\| + \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl[L_{f}(s)\bigl\| \Delta x_{k}(s)\bigr\| +I_{f}(s)\bigl\| \Delta u_{k}(s)\bigr\| \bigr]\,ds \\ &{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}d_{\xi}\,ds+ \frac {1}{\Gamma(\alpha)}\sum_{i=1}^{j}(t-t_{i})^{\alpha-1}L_{G_{i}}(t_{i}) \bigl\| \Delta x_{k}\bigl(t_{i}^{-}\bigr)\bigr\| . \end{aligned}$$
(7)
Multiplying \((t-t_{j})^{1-\alpha}\) on both sides of (7), using (A2) we have
$$\begin{aligned} &(t-t_{j})^{1-\alpha}\bigl\| \Delta x_{k}(t)\bigr\| \\ &\quad\leq\frac{(1-\frac{t_{j}}{t})^{1-\alpha}}{\Gamma(\alpha)}d_{0} +\frac{(t-t_{j})^{1-\alpha}e^{\lambda(t-t_{j})}M_{L}}{\lambda^{\alpha }} \|P_{o}\|\|\Delta e_{k}\|_{PC_{1-\alpha},\lambda} \\ &\qquad{}+\frac{(t_{j+1}-t_{j})^{1-\alpha}M_{L}}{\Gamma(\alpha)} \int _{0}^{t}(t-s)^{\alpha-1}(s-t_{j})^{1-\alpha} \bigl\| \Delta x_{k}(s)\bigr\| \,ds +\frac{(t-t_{j})^{1-\alpha}t^{\alpha}}{\Gamma(\alpha+1)}d_{\xi } \\ &\qquad{}+\frac{(t-t_{j})^{1-\alpha}}{\Gamma(\alpha)} \sum_{i=1}^{j}(t-t_{i})^{\alpha-1}(t_{i}-t_{i-1})^{\alpha -1}L_{G_{i}}(t_{i}) (t_{i}-t_{i-1})^{1-\alpha}\bigl\| \Delta x_{k} \bigl(t_{i}^{-}\bigr)\bigr\| . \end{aligned}$$
(8)
Note that the fact \(\frac{t-t_{j}}{t-t_{i}}\leq1\) since \(t_{j}\geq t_{i}\) and
$$(t-t_{j})^{1-\alpha}(t-t_{i})^{\alpha-1}= \biggl( \frac {t-t_{j}}{t-t_{i}} \biggr)^{1-\alpha}\leq 1. $$
Then (8) reduces to
$$\begin{aligned} &(t-t_{j})^{1-\alpha}\bigl\| \Delta x_{k}(t)\bigr\| \\ &\quad\leq\frac{(1-\frac{t_{j}}{t})^{1-\alpha}}{\Gamma(\alpha)}d_{0} +\frac{(t-t_{j})^{1-\alpha}e^{\lambda(t-t_{j})}M_{L}}{\lambda^{\alpha }} \|P_{o}\|\|\Delta e_{k}\|_{PC_{1-\alpha},\lambda} \\ &\qquad{}+\frac{(t_{j+1}-t_{j})^{1-\alpha}M_{L}}{\Gamma(\alpha)} \int _{0}^{t}(t-s)^{\alpha-1}(s-t_{j})^{1-\alpha} \bigl\| \Delta x_{k}(s)\bigr\| \,ds+\frac {(t-t_{j})^{1-\alpha}t^{\alpha}}{\Gamma(\alpha+1)}d_{\xi} \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \sum_{i=1}^{j}(t_{i}-t_{i-1})^{\alpha-1}M_{L}(t_{i}-t_{i-1})^{1-\alpha} \bigl\| \Delta x_{k}\bigl(t_{i}^{-}\bigr)\bigr\| . \end{aligned}$$
(9)
For the inequality (9), we set \(v(t)=(t-t_{i})^{1-\alpha}\|\Delta x_{k}(t)\|\). Then one can apply Lemma 2.4 to derive that
$$\begin{aligned} &(t-t_{j})^{1-\alpha}\bigl\| \Delta x_{k}(t)\bigr\| \\ &\quad\leq \biggl(\frac{(1-\frac{t_{j}}{t})^{1-\alpha}}{\Gamma(\alpha)}d_{0} +\frac{(t-t_{j})^{1-\alpha}M_{L}e^{\lambda(t-t_{j})}}{\lambda^{\alpha }} \|P_{o}\|\|\Delta e_{k}\|_{PC_{1-\alpha},\lambda} \\ &\qquad{}+\frac{(t-t_{j})^{1-\alpha}t^{\alpha}}{\Gamma(\alpha+1)}d_{\xi} \biggr) \bigl(1+\theta E_{\alpha}\bigl((t_{j+1}-t_{j})^{1-\alpha}t^{\alpha }M_{L} \bigr) \bigr)^{j} E_{\alpha}\bigl((t_{j+1}-t_{j})^{1-\alpha}t^{\alpha}M_{L} \bigr), \end{aligned}$$
(10)
where
$$\theta=\max \biggl\{ \frac{(t_{j+1}-t_{j})^{\alpha-1}M_{L}}{\Gamma(\alpha )}:j=0,1,2,\ldots ,m \biggr\} . $$
Multiplying \(e^{-\lambda(t-t_{j})}\) on both sides of (10) and noting the fact that \(E_{\alpha}(z),z>0\) is an increasing function, we have
$$\begin{aligned} &(t-t_{j})^{1-\alpha}e^{-\lambda(t-t_{j})}\bigl\| \Delta x_{k}(t)\bigr\| \\ &\quad\leq N_{j} \biggl(\frac{d_{0}}{t_{j+1}\Gamma(\alpha)} +\frac{M_{L}}{t_{j}^{\alpha}\lambda^{\alpha}} \|P_{o}\|\|\Delta e_{k}\| _{PC_{1-\alpha},\lambda} + \frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \\ &\qquad{}\times\bigl(1+\theta E_{\alpha }(N_{j}M_{L}) \bigr)^{j}E_{\alpha}(N_{j}M_{L}), \end{aligned}$$
(11)
where
$$N_{j}= \biggl[1-\frac{t_{j}}{t_{j+1}} \biggr]^{1-\alpha}\times t_{j+1}. $$
For (11), one can take the \((PC_{1-\alpha}, \lambda)\)-norm to derive that
$$\begin{aligned} \Vert \Delta x_{k}\Vert _{PC_{1-\alpha},\lambda} \leq&N_{\max} \biggl(\frac{d_{0}}{t_{1}\Gamma(\alpha)} +\frac{M_{L}}{t_{1}^{\alpha}\lambda^{\alpha}} \Vert P_{o}\Vert \Vert \Delta e_{k}\Vert _{PC_{1-\alpha},\lambda} + \frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \\ &{}\times \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha }(N_{\max}M_{L}), \end{aligned}$$
(12)
where
$$N_{\max}=\max\{N_{j}: j=1,2,\ldots, m\}. $$
Linking (6) and (12), we have
$$\begin{aligned} \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda}\leq \Vert I-BP_{o}\Vert \Vert e_{k}\Vert _{PC_{1-\alpha},\lambda}+L_{g} \Vert \Delta x_{k}\Vert _{PC_{1-\alpha},\lambda}+\Delta t_{j\max}^{1-\alpha}d_{\eta}, \end{aligned}$$
(13)
where
$$\Delta t_{j\max}^{1-\alpha}=\max\bigl\{ (t_{j+1}-t_{j})^{1-\alpha }: j=0,1,\ldots, m\bigr\} . $$
Submitting (12) into (13), we obtain
$$\begin{aligned} \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} \leq\tilde{q} \Vert \Delta e_{k}\Vert _{PC_{1-\alpha},\lambda} +\tilde{M}, \end{aligned}$$
(14)
where
$$\begin{aligned}& \tilde{M}= \biggl[\frac{L_{g}N_{\max}d_{0}}{t_{1}\Gamma(\alpha)} +\frac{L_{g}N_{\max}d_{\xi}}{\Gamma(\alpha+1)} \biggr]\bigl(1+\theta E_{\alpha}(N_{\max}M_{L})\bigr)^{m}E_{\alpha}(N_{\max}M_{L})+ \Delta t_{j\max}^{1-\alpha}d_{\eta}, \\& \tilde{q}=\Vert I-BP_{o}\Vert +\frac{L_{g}N_{\max}M_{L}}{t_{1}^{\alpha}\lambda ^{\alpha}} \Vert P_{o}\Vert \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha}(N_{\max}M_{L}). \end{aligned}$$
Note that there exists a large enough λ such that \(\tilde{q}<1\) due to \(\Vert I-BP_{o}\Vert <1\). Concerning (14), one can use Lemma 2.5 to derive that
$$\lim_{k\rightarrow\infty}\sup \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} \leq \frac{\tilde{M}}{1-\tilde{q}}, $$
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 closed-loop P-type ILC.

Theorem 3.3

For the system (1), the assumptions (A1)-(A3) 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
$$\begin{aligned} e_{k+1}(t) =&e_{k}(t)+g\bigl(t,x_{k}(t) \bigr)-g\bigl(t,x_{k+1}(t)\bigr)-BP_{d}e_{k+1}(t)+ \eta _{k}(t)-\eta_{k+1}(t) \\ =&(I+BP_{d})^{-1}e_{k}(t)+(I+BP_{d})^{-1} \bigl(g\bigl(t,x_{k}(t)\bigr)-g\bigl(t,x_{k+1}(t)\bigr)\bigr) \\ &{}+(I+BP_{d})^{-1}\bigl(\eta_{k}(t)- \eta_{k+1}(t)\bigr). \end{aligned}$$
(15)
Taking the norm \(\|\cdot\|\) on both sides of (15), we have
$$\begin{aligned} \bigl\Vert e_{k+1}(t)\bigr\Vert \leq\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert \bigl\Vert e_{k}(t) \bigr\Vert +L_{g}\bigl\Vert (I+BP_{d})^{-1} \bigr\Vert \bigl\Vert \Delta x_{k}(t)\bigr\Vert +\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert d_{\eta}. \end{aligned}$$
(16)

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
$$\begin{aligned} \Vert \Delta x_{k}\Vert _{PC_{1-\alpha},\lambda} \leq&N_{\max} \biggl(\frac{d_{0}}{t_{1}\Gamma(\alpha)} +\frac{M_{L}}{t_{1}^{\alpha}\lambda^{\alpha}} \Vert P_{d}\Vert \Vert \Delta e_{k+1}\Vert _{PC_{1-\alpha},\lambda} + \frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \\ &{}\times \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha }(N_{\max}M_{L}), \end{aligned}$$
(17)
where \(N_{j}\), θ, and \(N_{\max}\) are defined in Theorem 3.1.
Substituting (16) into (17), we have
$$\begin{aligned} \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} \leq&\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert \Vert e_{k}\Vert _{PC_{1-\alpha},\lambda} +L_{g}\bigl\Vert (I+BP_{d})^{-1} \bigr\Vert \Vert \Delta x_{k}\Vert _{PC_{1-\alpha},\lambda} \\ &{}+\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert \Delta t_{j\max}^{1-\alpha}d_{\eta}. \end{aligned}$$
(18)
Taking (17) into (18), we have
$$\begin{aligned} \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} \leq{}&\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert \Vert e_{k}\Vert _{PC_{1-\alpha},\lambda} \\ &{}+L_{g}\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert N_{\max} \biggl(\frac{d_{0}}{t_{1}\Gamma(\alpha)} +\frac{M_{L}}{t_{1}^{\alpha}\lambda^{\alpha}} \Vert P_{d}\Vert \Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} + \frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \\ &{}\times \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha }(N_{\max}M_{L}) +\bigl\Vert (I+BP_{d})^{-1}\bigr\Vert \Delta t_{j\max}^{1-\alpha}d_{\eta}, \end{aligned}$$
which implies that
$$\begin{aligned} &\Vert e_{k+1}\Vert _{PC_{1-\alpha},\lambda} \\ &\quad\leq\frac{\Vert (I+BP_{d})^{-1}\Vert }{H}\Vert e_{k}\Vert _{PC_{1-\alpha},\lambda} + \frac{L_{g}\Vert (I+BP_{d})^{-1}\Vert N_{\max}}{H} \biggl(\frac {d_{0}}{t_{1}\Gamma(\alpha)} +\frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \\ &\qquad{}\times \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha }(N_{\max}M_{L}) + \frac{\Vert (I+BP_{d})^{-1}\Vert }{H}\Delta t_{j\max}^{1-\alpha}d_{\eta }, \end{aligned}$$
(19)
where
$$H=1-\frac{L_{g}\|(I+BP_{d})^{-1}\|M_{L}}{t_{1}^{\alpha}\lambda^{\alpha }}\|P_{d}\|\bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha}(N_{\max}M_{L}). $$
Let
$$\begin{aligned}& \begin{aligned}[b] \bar{M}={}&\frac{L_{g}\|(I+BP_{d})^{-1}\|N_{\max}}{H} \biggl(\frac {d_{0}}{t_{1}\Gamma(\alpha)} + \frac{d_{\xi}}{\Gamma(\alpha+1)} \biggr) \bigl(1+\theta E_{\alpha}(N_{\max}M_{L}) \bigr)^{m}E_{\alpha}(N_{\max}M_{L}) \\ &{}+\frac{\|(I+BP_{d})^{-1}\|}{H}\Delta t_{j\max}^{1-\alpha}d_{\eta}, \end{aligned} \\& \bar{q}=\frac{\|(I+BP_{d})^{-1}\|}{H} . \end{aligned}$$
Then (19) reduces to
$$\|e_{k+1}\|_{PC_{1-\alpha},\lambda}\leq\bar{q} \|e_{k} \|_{PC_{1-\alpha },\lambda}+ \bar{M}. $$
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
$$\begin{aligned} \lim_{k\rightarrow\infty}\sup\|e_{k+1} \|_{PC_{1-\alpha},\lambda}\leq \frac{\tilde{M}}{1-\tilde{q}}. \end{aligned}$$
(20)
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:
$$ \left \{ \textstyle\begin{array}{@{}l} (D_{0,t}^{0.5}x_{k})(t)=-x_{k}(t)+t(t-0.5)^{2}u_{k}(t)+\frac {t}{k+1},\quad t\in[0,1]\setminus\{0.5\},\\ \lim_{t\rightarrow0+}(I_{0,t}^{0.5}x_{k})(t)=0, \\ \Theta (I_{0,t_{1}}^{0.5}x)(t_{1}^{-})=t_{1}^{-}x_{k}(t_{1}^{-}),\quad t_{1}=0.5,\\ y_{k}(t)=x_{k}(t)+1.2u_{k}(t)+e^{-kt}, \end{array}\displaystyle \right . $$
(21)
and the P-type ILC
$$\begin{aligned} u_{k+1}(t)=u_{k}(t)+P_{o}e_{k}(t). \end{aligned}$$

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 (A1)-(A3) are satisfied.

Case 1: The original reference trajectory is a piecewise continuous function,
$$y_{d}(t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1}{2}t^{3}+1,& t\in[0, 0.3],\\ \frac{1}{2}t^{3}+(t^{4}+0.3)+1, & t\in(0.3, 0.6],\\ \frac{1}{2}t^{3}+(t^{4}+0.3)+(t^{5}+0.6)+1, & t\in(0.6, 1]. \end{array}\displaystyle \right . $$

(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 \(t\in[0,1]\).

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 1

The system output and the tracking error.

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.
Figure 2

The system output and the tracking error.

(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 \(t\in[0,1]\).

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 3

The system output and the tracking error.

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.
Figure 4

The system output and the tracking error.

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,
$$\begin{aligned} y_{d}(t)=6t^{2}(1-t)+t-1. \end{aligned}$$

(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 \(t\in[0,1]\).

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 5

The system output and the tracking error.

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.
Figure 6

The system output and the tracking error.

(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 \(t\in[0,1]\).

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 7

The system output and the tracking error.

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.
Figure 8

The system output and the tracking error.

Conclusions:
  • From Figures 5 and 6 or 7 and 8, we can see that the tracking error decreases with k increasing.

  • From Figures 5 and 7 or 6 and 8, we can see that the tracking error decreases with \(P_{o}\) increasing.

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

(1)
College of Mathematics and Statistics, Department of Mathematics, Guizhou University
(2)
College of Science, Guizhou Minzu University

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