Uniform design and analysis of iterative learning control for a class of impulsive firstorder distributed parameter systems
 Xiulan Yu^{1} and
 JinRong Wang^{2}Email author
https://doi.org/10.1186/s1366201505971
© Yu and Wang 2015
Received: 26 March 2015
Accepted: 10 August 2015
Published: 22 August 2015
Abstract
In this paper, we provide uniform design and analysis framework for iterative learning control of a class of impulsive firstorder distributed parameter systems in the time domain. In particular, Ptype and Dtype iterative learning controls with initial state learning are considered. We present convergence results for openloop iterative learning schemes in the sense of the \(L^{p}\)norm and λnorm, respectively. Finally, an example is given to illustrate our theoretical results.
Keywords
1 Introduction
In 1984, Arimoto et al. [1] propose basic theories and algorithms on iterative learning control (ILC) and point out that ILC is a useful practical control approach for systems which perform tasks repetitively over a finite time interval. The performance improvement can be made step by step, and the output trajectory can be realized to track the desired one, through updating the input signal by the error data. Through three decades of study and development, one has achieved significant progress in both theories and applications, and become one of the most active fields in intelligent control. For more details on the contributions for linear and nonlinear ordinary differential equations, the reader is referred to the monographs [2–6], and [7–16].
The issue on designing and analyzing an ILC for impulsive differential equations, discontinuous systems [17, 18], distributed parameter systems or PDEs has not been fully investigated, and only a limited number of results [19–22] are available so far. In [19], Liu et al. explore Ptype iterative learning control law with initial state learning for impulsive ordinary differential equations to tracking the discontinuous output desired trajectory. In [20], Xu et al. study Ptype and Dtype ILC for linear firstorder distributed parameter systems in the sense of the supnorm. In [21], Huang and Xu apply a Ptype steadystate ILC scheme to the boundary control of PDEs. In [22], Huang et al. construct a uniform design and analysis framework for iterative learning control of linear inhomogeneous distributed parameter systems.
When dealing with impulsive distributed parameter systems, the ILC design and property analysis become far more challenging. The existing ILC design and analysis should be improved. The main objective of this paper is extended [20] to a study of the ILC for impulsive nonlinear firstorder distributed parameter systems with initial error in the sense of the λnorm (the symbol of λnorm is introduced by Arimoto et al. [1]; cf. [23] and [24]) via semigroup theory.
This paper is a continuation of our recent related papers [19, 20]. The main contributions of the paper are summarized as follows.
(i) A uniform design and analysis framework is presented for ILC of a class of impulsive firstorder distributed parameter systems in the time domain. Nevertheless, [19] consider the ILC of impulsive ordinary differential equations in finite dimensional spaces.
(ii) Instead of considering ILC of linear firstorder distributed parameter systems without initial error as in [20], we consider a class of impulsive nonlinear firstorder distributed parameter systems with initial error and more general discontinuous output tracking problem.
(iii) Instead of simplifying ILC updating law without initial state learning as in [20], we consider ILC updating law with initial state learning.
(iv) Instead of choosing the supnorm as in [20], we use the \(L^{p}\)norm and λnorm, respectively.
2 System description and problem statement
Denote \(PC(J,X)\) := {\(x : J\rightarrow X : x\), continuous at \(t\in J\backslash\mathbb{D}\), and x is continuous from the left and has right hand limits at \(t\in\mathbb{D}\)} endowed with the λnorm \(\x\_{\lambda}=\sup_{t \in J}e^{\lambda t}\x(t)\ _{X}\) for some \(\lambda>0\). Define \(L^{p}(J,X):= \{x: J\rightarrow X \mbox{ is strongly measurable}: \int_{0}^{a}\x(s)\_{X}^{p}\,ds<\infty \}\), endowed with the norm \(\x\_{L^{p}}= (\int_{0}^{a}\x(t)\_{X}^{p}\,dt )^{\frac{1}{p}}\), \(p\in(1,\infty)\). Obviously, \((PC(J,X),\\cdot\_{\lambda})\) and \((L^{p}(J,X),\\cdot\_{L^{p}})\), \(p\in(1,\infty)\) are Banach spaces.
By adopting the same methods as in [25], Theorem 2.1, one can obtain the existence and uniqueness of a mild solution of (1) with \(x(0)=x_{0}\) when f and \(g_{i}\) satisfy the standard Lipschitz conditions.
Concerning the system (1), we will design Ptype and Dtype iterative learning schemes to generate the control input \(u_{k}(\cdot)\) such that the system piecewise continuous output \(y_{k}(\cdot)\) tracks the discontinuous desired output trajectory \(y_{d}(\cdot)\) as accurately as possible as \(k\to\infty\) for \(t\in J\) in the sense of suitable norms. We shall give two convergence results for openloop iterative learning schemes in the sense of the \(L^{p}\)norm and λnorm, respectively, in the next sections.
3 Convergence results for Ptype ILC updating law
 (H_{0}):

A: \(D(A)\subseteq X \to X\) is the generator of a \(C_{0}\)semigroup \(T(t)\), \(t\geq0\) on X. Denote \(M:=\sup_{t\in J}\T(t)\_{L(X,X)}\).
 (H_{1}):

\(f:J\times X\rightarrow X\) is strongly measurable for the first variable and continuous for the second variable. Moreover, there exists a \(L_{f}>0\) such that$$\bigl\ f(t,u)f(t,v)\bigr\ _{X}\leq L_{f}\uv\_{X}, \quad u,v\in X, t\in J. $$
 (H_{2}):

There exists a \(L_{g}>0\) such that$$\bigl\ g_{i}(u)g_{i}(v)\bigr\ _{X}\leq L_{g} \uv\_{X},\quad u,v\in X, t\in J, i\in \mathbb{M}. $$
 (H_{3}):

Let \(I_{Y}\) be an identity operator in Y and \(I_{Y}D\gamma_{1}CL_{1}\in L(Y,Y)\) satisfy$$ \rho:=\I_{Y}D\gamma_{1}CL_{1} \_{L(Y,Y)}< 1. $$(6)
Now we are ready to give the first convergence result in the sense of the \(L^{p}\)norm.
Theorem 3.1
Proof
Step 1. We prove that \(\e_{k+1}(0)\_{Y}\to0\) as \(k\to\infty\).
Remark 3.2
The condition (7) in Theorem 3.1 seems to be a bit strong since we choose the \(L^{p}\)norm. However, one can choose another suitable norm, the λnorm, to weaken this condition.
Next we give the second convergence result in the sense of the λnorm.
Theorem 3.3
Proof
By our assumptions and Theorem 3.1, we know (11) holds. Next, we only need to prove \(\e_{k+1}\_{\lambda}\to0\) as \(k\to\infty\).
4 Convergence results for Dtype ILC updating law
 (H_{4}):

\(T(t)\) is differentiable for \(t> 0\). Then by [27], Lemma 4.2, \(\frac{d}{dt}T(t)=AT(t)\) is a bounded linear operator, i.e., \(AT(t)\in L(X,X)\).
Theorem 4.1
Proof
In order to prove \(\e_{k+1}\_{\lambda}\to0\) as \(k\to\infty\), we divide our proof into two steps.
5 Example
Let \(X=U=Y=L^{2}(0,1)\). Set \(J=[0,1]\), \(m=1\), and \(t_{1}=\frac{1}{3}\). Define \(A: D(A)\subset X\rightarrow X\) by \(Ax=\frac{\partial ^{2}}{\partial z^{2}}x:=x_{zz} \), where \(D(A)=\{x\in H^{2}((0,1)): x_{z}(0)=x_{z}(1)=0\}\). Then A can be written as \(Ax=\sum_{n=1}^{\infty}n^{2}\langle x,x_{n}\rangle\), \(x\in D(A)\) where \(x_{n}(z) = \sqrt{\frac{2}{\pi}}\cos n\pi z\), \(n = 1,2,\ldots \) . Next, A generates a \(C_{0}\)semigroup \(T(t)\), \(t\geq0\) written as \(T(t)x:=\sum_{n=1}^{\infty}e^{ n^{2}t}\langle x,x_{n}\rangle x_{n}\), with \(\T(t)\_{L(X,X)}\leq e^{t}\leq1=M\). Thus, (H_{0}) holds. Moreover, \(T(t)\) is differentiable for \(t> 0\) and \(\frac{d}{dt}T(t)x=AT(t)x=\sum_{n=1}^{\infty}\frac{n^{2}}{e^{ n^{2}t}}\langle x,x_{n}\rangle x_{n}\) and \(\AT(t)\_{L(X,X)}\leq 1\). Thus, (H_{4}) holds.
Denote \(x(\cdot)(z)=x(\cdot,z)\), \(f(\cdot,x)(z)=l_{1}\cos\cdot\sin x(\cdot,z)\), \(Bu(\cdot)(z)=l_{2}u(\cdot,z)\), \(g_{1}(x(t_{1}^{}))(z)=l_{3} x(t_{1}^{},z)\), then (17) can be abstracted to (1). Thus, (H_{1}) and (H_{2}) hold.
Denote \(y(\cdot)(z)=y(\cdot,z)\) and \(C=cI_{Y}\) and \(D=dI_{Y}\), then (18) can be rewritten as (2). Thus, \((1c)I_{Y}\in L(Y,Y)\), i.e., (H_{3}) holds.

Choosing \(d=1\) and c satisfying \(c<\frac{1}{l_{2}(1+l_{3})e^{l_{1}}}\) implies (7) holds. By Theorem 3.1, \(y_{k}\) tends to \(y_{d}\) as \(k\to\infty\) in the sense of the \(L^{p}\)norm.

Set \(1>d>0\). Then \(1d>0\), which implies (13) holds. By Theorem 3.3, \(y_{k}\) tends to \(y_{d}\) as \(k\to\infty\) in the sense of the λnorm (λ must be a sufficiently large).

Set \(d=0\). Clearly, \(\delta=1cl_{2}<1\) since \(cl_{2}>0\). By Theorem 4.1, \(y_{k}\) tends to \(y_{d}\) as \(k\to\infty\) in the sense of the λnorm (λ must be a sufficiently large).
Declarations
Acknowledgements
The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too. This work is supported by the National Natural Science Foundation of China (11201091) and Outstanding Scientific and Technological Innovation Talent Award of Education Department of Guizhou Province ([2014]240).
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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