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IMA Journal of Mathematical Control and Information
, Volume Advance Article – Feb 1, 2018

15 pages

/lp/ou_press/iterative-learning-control-for-discrete-singular-time-delay-systems-y7lv6GlN60

- Publisher
- Oxford University Press
- Copyright
- © Crown copyright 2018.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnx061
- Publisher site
- See Article on Publisher Site

Abstract In this paper, the iterative learning control problem is studied for a class of discrete singular time-delay systems. Based on the equivalent restrict decomposition form of singular systems, the original systems are transformed into the difference-algebraic time-delay systems. Then a discrete-time learning algorithm is proposed for such difference-algebraic systems. Furthermore, the sufficient condition for the convergence of the algorithm is presented and analyzed. Moreover, the proposed algorithm is also suitable for discrete singular systems with multiple time-delays. Finally, two numerical examples are given to illustrate the effectiveness of the presented algorithm. 1. Introduction Iterative learning control (ILC) is a kind of intelligent control method with strict mathematical description, which is suitable for repetitive controlled systems in a finite time interval. The study of ILC is of great significance for dynamic systems with complex modeling, uncertainty and strong non-linear coupling, see (Bien & Xu, 1998) and (Xu & Tan, 2003). Since the complete algorithm of ILC was first proposed by (Arimoto et al., 1984), it has become a hot issue in the field of control theory and has attracted broad attention over the past decades. Nowadays, ILC is playing an increasingly important role in controlling repeatable processes. Singular systems have essential differences than the normal systems, due to the fact that singular systems can preserve the structure of physical systems and impulsive elements, and widely applied in many practical control systems such as circuit systems, power systems, robotic systems and economic systems. During the past decades, singular systems have been extensively investigated and many significant researches have been published, see (Berger, 2016; Dai, 1989; Duan, 2010; Karampetakis & Gregoriadou, 2014; Kunkel & Mehrmann, 2008; Lewis & Mertizios, 1990; März, 1995; Mehrmann & Wunderlich, 2009; Reis & Voigt, 2015; Weng & Mao, 2013; Xu et al., 2009) and (Ooi et al., 2017). However, there are only few results which have been reported on the ILC for singular systems. For example, (Piao & Zhang, 2007) analyzed the convergence of D-type and PD-type closed-loop ILC algorithms for linear singular systems in the sense of Frobenius norm. Based on the Weierstrass canonical form of singular systems, (Piao et al., 2007) designed a P-type ILC algorithm for the fast subsystems with impulse. (Tian & Zhou, 2012) proposed a new learning algorithm to study the state tracking problem for a class of singular systems, furthermore, the convergence analysis of the algorithm is given. Recently, (Tian et al., 2016) applied the ILC technique to a class of discrete singular systems. On the other hand, time-delay is often encountered in many practical control problems such as batch processes, remote controlled robots and man-machine systems. The existence of time-delay quite often degrades the performance of the control system, or even leads to the instability of the whole system, see (Niculescu, 2001) for detailed results. Fortunately, ILC has been found to be a good alternative to deal with the control problems of time-delay systems by iterations in a fixed time interval (see e.g. (Li et al., 2008; Liu & Ruan, 2016; Meng et al., 2009; Park et al., 1998; Shen et al., 2011) and the references therein). Thus, the study of ILC for singular time-delay systems is undoubtedly of theoretical and practical significance. In (Xie et al., 1999), the ILC algorithm was applied for the first time to a class of continuous-time singular systems with state delay, then the convergence of the algorithm and the possibility of the state tracking were analyzed. (Hu et al., 2014) further studied the ILC problem for a class of continuous-time singular systems with multiple time-delays. In recent years, discrete singular time-delay systems have drawn extensive attention from the mathematics and control community, due to the fact that such kind of systems frequently appear in dynamic input-output economic systems (Shao, 2013; Shao et al., 2014) and macro-economic systems (Ying, 2003). Now, we give an example with practical application. Consider the following dynamic Leontief input-output system: \begin{align} x(t) = Ax(t) + B[x(t + 1) - x(t)] + d(t), \end{align} (1.1) where x(t) is the total output vector, d(t) is the final net product vector. A is the direct consumption coefficient matrix, B is the capital coefficient matrix. Here, x(t) can be treated as state vector, d(t) can be considered as the system’s control vector because we can affect the quantity of final net product by controlling the scale of investment. When the total output for the tth year is related to the (t − τ)th year investment, the system (1.1) can be modified as follows: $$ x(t) = Ax(t) +{B_{0}}[x(t + 1) - x(t)] +{B_{1}}[x(t) - x(t - \tau )] + d(t), \nonumber $$ which can be rewritten as \begin{align} {B_{0}}x(t + 1) = (I - A +{B_{0}} -{B_{1}})x(t) +{B_{1}}x(t - \tau ) - d(t). \end{align} (1.2) In economics, the product of some sectors can not be treated as capital product and applied to invest, that is, the capital coefficient matrix B0 is often singular. In this sense, the system (1.2) is a typical discrete singular time-delay system. Correspondingly, many efforts for such systems have been made including stability (Chen, 2003; Stojanovic et al., 2015), admissibility (Feng et al., 2015), $$H_{\infty }$$ control (Ma et al., 2008) etc. However, to the best of our knowledge, the study on the controller design for trajectory tracking of discrete singular time-delay systems is still rare. Motivated by the above considerations, this paper is concerned with the problem of ILC for a class of discrete singular time-delay systems. The main contributions of this paper can be summarized as follows. First, different from our former work (Tian et al., 2016), the systems we consider are a class of discrete singular time-delay systems. Second, the convergence of the state tracking error is investigated using the α-norm, which makes the convergence proof simpler than using the ordinary norms. The organization of this paper is as follows. In Section 2, the notations of this paper are given and the problem of ILC for a class of discrete singular time-delay systems is proposed. In Section 3, a discrete-time learning algorithm is constructed and the convergence result is presented, then the extension to a class of more general discrete singular time-delay systems is shown. In order to illustrate the effectiveness of the proposed algorithm, two numerical examples are constructed in Section 4. Finally, a conclusion is drawn in Section 5. 2. Problem formulation Throughout this paper, I and 0 denote the identity matrix and zero matrix with appropriate dimensions, respectively. For a given vector or matrix X, ∥X∥ denotes its Euclidean norm. For a discrete system, t ∈ [0, T] denotes the integer sequence t = 0, 1, 2, ⋯ , T. For a function h: [0, T] → Rn and a real number α ⩾ 1, ∥h∥α denotes the α-norm defined by $${\| h \|_{\alpha } } = \mathop{\sup }\limits _{t \in [0,T]}{{\alpha }^{-t}}\|{h(t)} \|$$; ∥h∥s denotes the supreme norm defined by $${\| h \|_{s}} = \mathop{\sup }\limits _{t \in [0,T]} \|{h(t)} \|$$. From (Wang, 2001), we know that ∥h∥α and ∥h∥s are equivalent, i.e., either of the norms can be used to prove the convergence. Consider the following discrete singular time-delay system: \begin{align} E{x_{k}}(t + 1) = A{x_{k}}(t) + D{x_{k}}(t - \tau ) + B{u_{k}}(t), \end{align} (2.1) where k denotes the iteration index, t ∈ [0, T] denotes the time index, τ is a known positive integer time delay, E ∈ Rn×n is a singular matrix. Assume that the pair (E, A) is regular and 0 < rank(E) = r < n. Thus, there exist non-singular matrices M and N such that $$ MEN =\left[ \begin{array}{@{}cc@{}}{I_{r}}&0\\ 0&0 \end{array}\right],\ MAN =\left[ \begin{array}{@{}cc@{}}{A_{11}}&{A_{12}}\\{A_{21}}&{A_{22}} \end{array}\right],\ M{D}N = \left[\begin{array}{@{}cc@{}}{D_{11}}&{D_{12}}\\{D_{21}}&{D_{22}} \end{array} \right],\ MB = \left[ \begin{array}{@{}c@{}}{{B_{1}}}\\{{B_{2}}} \end{array}\right]. $$ By introducing the state transformation $$\left [ {{x_{1k}}(t)\atop {x_{{\mathrm{2}}k}}(t)} \right ] ={N^{ - 1}}{x_{k}}(t)$$, then the system (2.1) can be rewritten as the following difference-algebraic time-delay system: \begin{align} \begin{cases} {x_{1k}}(t + 1)={A_{11}}{x_{1k}}(t)+{A_{12}}{x_{2k}}(t)+D_{11}{x_{1k}}(t - \tau ) + D_{12}{x_{2k}}(t - \tau ) +{B_{1}}{u_{k}}(t),\\ 0 ={A_{21}}{x_{1k}}(t) + {A_{22}}{x_{2k}}(t) + D_{21}{x_{1k}}(t- \tau )+ D_{22}{x_{2k}}(t - \tau ) +{B_{2}}{u_{k}}(t), \end{cases} \end{align} (2.2) where $$\left [{{x_{1k}}(t) \atop {x_{{\mathrm{2}}k}}(t)} \right ] \in R^{n}$$, uk(t) ∈ Rm represent the state and control input of the system respectively, and x1k(t) ∈ Rr, x2k(t) ∈ Rn−r; A11, A12, A21, A22, D11, D12, D21, D22, B1 and B2 are real matrices with appropriate dimensions. Let $$\left [ {{x_{1k}}(t) \atop {x_{2k}}(t)} \right ] =\left [ {{\psi _{1k}}(t) \atop {\psi _{2k}}(t)} \right ],t \in [-\tau ,0]$$ and $$\left [ {{\psi _{1k}}(t) \atop {\psi _{2k}}(t)} \right ]$$ is the initial function of the system (2.2). Before giving our ILC law, basic assumptions for the system (2.2) are first given as follows: Assumption 2.1 For the given desired state trajectory $$\left [ {{x_{1d}}(t) \atop {x_{2d}}(t)} \right ]$$, there exists a desired control input ud(t) such that \begin{cases} {x_{1d}}(t + 1) ={A_{11}}{x_{1d}}(t)+{A_{12}}{x_{2d}}(t) +D_{11}{x_{1d}}(t - \tau ) + D_{12}{x_{2d}}(t - \tau ) +{B_{1}}{u_{d}}(t),\\ 0={A_{21}}{x_{1d}}(t)+ {A_{22}}{x_{2d}}(t) +D_{21}{x_{1d}}(t- \tau )+ D_{22}{x_{2d}}(t - \tau ) +{B_{2}}{u_{d}}(t). \end{cases} Assumption 2.2 The initial resetting condition holds for all iterations, i.e., $$ \left[ {{\psi_{1k}}(t) \atop \psi_{2k}}(t) \right] = \left[ {{\psi_{1d}}(t) \atop {\psi_{2d}}(t)} \right], \ \quad t \in [-\tau,0],\quad k = 0,1,2, \cdots, $$ where $$\left [ {{\psi _{1d}(t) \atop \psi _{2d}}(t)} \right ]$$ is the desired initial function. Assumption 2.3 The matrix A22 is non-singular. Given a desired state trajectory $$\left [ {{x_{1d}}(t) \atop {x_{2d}}(t)} \right ]$$, t ∈ [0, T + 1], the target of learning control is to find a control input such that the system state follows the desired state trajectory. 3. Convergence analysis of the algorithm Construct the discrete-time learning algorithm for the system (2.2) as follows: \begin{align} {u_{k + 1}}(t) ={u_{k}}(t) +{\varGamma_{1}}{e_{1k}}(t + 1) +{\varGamma_{2}}{e_{2k}}(t), \end{align} (3.1) where $$\varGamma$$1 ∈ Rm×r, $$\varGamma$$2 ∈ Rm×(n−r) are the learning gain matrices, and e1k(t) = x1d(t) − x1k(t), e2k(t) = x2d(t) − x2k(t). Denote $${e_{k}}(t) = \left [ {{e_{1k}}(t) \atop {e_{2k}}(t)} \right ]$$ to represent the state tracking error. Then we have the following result: Theorem 3.1 Consider the system (2.2) satisfying Assumptions 2.1–2.3. If there exist the gain matrices $$\varGamma$$1 ∈ Rm×r, $$\varGamma$$2 ∈ Rm×(n−r) such that \begin{align} \rho = \|{I -{\varGamma_{1}}{{\tilde B}_{1}} -{\varGamma_{2}}{{\tilde B}_{2}}}\| < 1, \end{align} (3.2) where $${\tilde B_{1}}{\mathrm{ = }}{B_{\mathrm{1}}}{\mathrm{ \;+\; }}{A_{12}}{\tilde B_{2}}$$, $${\tilde B_{2}}{\mathrm{ = }} - A_{22}^{ - 1}{B_{2}}$$. Then under the action of the learning algorithm (3.1), the system state can converge to the desired state trajectory on [0, T], i.e., $$\mathop{\lim }\limits _{k \to \infty }{\|{e_{k}} \|_{s}} = 0$$. Proof. Denote $$\varDelta$$uk(t) = ud(t) − uk(t). From (2.2), Assumptions 2.1–2.3, we have \begin{align} {e_{2k}}(t) ={\tilde A_{21}}{e_{1k}}(t)+\tilde D_{21}{e_{1k}}(t - \tau ) + \tilde D_{22}{e_{2k}}(t - \tau ) +{\tilde B_{2}}\varDelta{u_{k}}(t), \end{align} (3.3) where $$ {\tilde A_{21}} = - A_{22}^{ - 1}{A_{21}},\ \tilde D_{21}= - A_{22}^{ - 1}D_{21},\ \tilde D_{22}= - A_{22}^{ - 1}D_{22},\ {\tilde B_{2}}{\mathrm{ = }} - A_{22}^{ - 1}{B_{2}}.$$ Taking Euclidean norm on both sides of (3.3) yields \begin{align} \|{{e_{2k}}(t)} \| \leqslant \|{{{\tilde A}_{21}}} \|\|{{e_{1k}}(t)} \|{\mathrm{ + }}\|{\tilde D_{21}} \|\|{{e_{1k}}(t - \tau )} \| + \|{\tilde D_{22}} \|\|{{e_{2k}}(t - \tau )} \| + \|{{{\tilde B}_{2}}} \|\|{\varDelta{u_{k}}(t)} \|. \end{align} (3.4) From Assumptions 2.2, for t ∈ [−τ, 0], \begin{align} \|{{e_{ik}}(t)}\| = \|{{\psi_{id}}(t) -{\psi_{ik}}(t)}\| = 0,\quad i=1,2. \end{align} (3.5) According to the definition of α-norm and combining with (3.4) and (3.5), we can get \begin{align*} {\|{{e_{2k}}} \|_{\alpha} }&=\mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{2k}}(t)}\|\\ &\leqslant \|{{{\tilde A}_{21}}}\| \mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\| + \|{{{\tilde D}_{21}}}\|\mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{1k}}(t - \tau )}\|\\ &\quad+\|{{{\tilde D}_{22}}}\|\mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{2k}}(t - \tau )}\| + \|{{{\tilde B}_{2}}}\| \mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|\varDelta{{u_{k}}(t)}\|\\ &= \|{{{\tilde A}_{21}}}\|{\|{{e_{1k}}}\|_{\alpha} }+{\alpha^{-\tau }}\|{{{\tilde D}_{21}}}\|\mathop{\sup }\limits_{t \in [ - \tau,T - \tau ]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\|\\ &\quad+{\alpha^{ - \tau }}\|{{{\tilde D}_{22}}} \|\mathop{\sup }\limits_{t \in [ - \tau,T - \tau ]}{\alpha^{ - t}}\|{{e_{2k}}(t)} \| + \|{{{\tilde B}_{2}}} \|{\|{\varDelta{u_{k}}} \|_{\alpha} }\\ &\leqslant \left({\|{{{\tilde A}_{21}}}\| +{\alpha^{ - \tau }}\|{{{\tilde D}_{21}}}\|} \right){\|{{e_{1k}}} \|_{\alpha} } +{\alpha^{ - \tau }}\|{{{\tilde D}_{22}}}\|{\|{{e_{2k}}}\|_{\alpha} } +\|{{{\tilde B}_{2}}}\|{\|{\varDelta{u_{k}}}\|_{\alpha} }. \end{align*} Taking α so that $${\alpha ^{ - \tau }}\|{\tilde D_{22}}\| < 1$$, it is easy to yield that \begin{align} {\|{{e_{2k}}} \|_{\alpha} } \leqslant{c_{1}}{\|{{e_{1k}}}\|_{\alpha} } +{c_{2}}{\|{\varDelta{u_{k}}}\|_{\alpha} }, \end{align} (3.6) where $$ {c_{1}} = \frac{{\|{{{\tilde A}_{21}}}\| +{\alpha^{ - \tau }}\|{{{\tilde D}_{21}}}\|}}{{1 -{\alpha^{ - \tau }}\|{{{\tilde D}_{22}}}\|}},\quad{c_{2}} = \frac{{\|{{{\tilde B}_{2}}}\|}}{{1 -{\alpha^{ - \tau }}\|{{{\tilde D}_{22}}}\|}}. $$ Similarly, it follows from (2.2) and Assumption 2.1 that $$ {e_{1k}}(t + 1) ={A_{11}}{e_{1k}}(t) +{A_{12}}{e_{2k}}(t)+{D_{11}}{e_{1k}}(t - \tau )+{D_{12}}{e_{2k}}(t - \tau ) +{B_{1}}\varDelta{u_{k}}(t). $$ Substituting (3.3) into the above expression, we have \begin{align} {e_{1k}}(t + 1) ={\tilde A_{11}}{e_{1k}}(t) +{\tilde D_{11}}{e_{1k}}(t - \tau ) +{\tilde D_{12}}{e_{2k}}(t - \tau ) +{\tilde B_{1}}\varDelta{u_{k}}(t), \end{align} (3.7) where $$ {\tilde A_{11}} ={A_{11}} +{A_{12}}{\tilde A_{21}},\ {\tilde D_{11}} ={D_{11}} +{A_{12}}{\tilde D_{21}},\ {\tilde D_{12}} ={D_{12}} +{A_{12}}{\tilde D_{22}},\ {\tilde B_{1}} ={B_{1}} +{A_{12}}{\tilde B_{2}}. $$ Taking Euclidean norm on both sides of (3.7), it yields $$ \|{{e_{1k}}(t + 1)}\| \leqslant \|{{{\tilde A}_{11}}}\|\|{{e_{1k}}(t)} \| + \|{{{\tilde D}_{11}}} \|\|{{e_{1k}}(t - \tau )}\| + \|{{{\tilde D}_{12}}} \|\|{{e_{2k}}(t - \tau )} \| + \|{{{\tilde B}_{1}}} \|\|{\varDelta{u_{k}}(t)}\|. $$ Combining with (3.5) and noting that e1k(0) = 0, we can obtain \begin{align} {\|{{e_{1k}}} \|_{\alpha} }&= \mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ -t}}\|{{e_{1k}}(t)}\|=\mathop{\sup }\limits_{t \in [1,T]}{\alpha^{ -t}}\|{{e_{1k}}(t)}\| = \mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - (t + 1)}}\|{{e_{1k}}(t + 1)}\|\nonumber \\ &\leqslant{\alpha^{ - 1}}\|{{{\tilde A}_{11}}}\|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\| +{\alpha^{ - 1}}\|{{{\tilde D}_{11}}} \|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{{e_{1k}}(t - \tau )}\|\nonumber \\ &\quad +{\alpha^{ - 1}}\|{{{\tilde D}_{12}}} \|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{{e_{2k}}(t - \tau )} \| +{\alpha^{ - 1}}\|{{{\tilde B}_{1}}}\|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{\varDelta{u_{k}}(t)} \|\nonumber \\ &={\alpha^{ - 1}}\|{{{\tilde A}_{11}}}\|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\| +{\alpha^{ - 1}}{\alpha^{ - \tau }}\|{{{\tilde D}_{11}}} \|\mathop{\sup }\limits_{t \in [ - \tau,T - \tau - 1]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\|\nonumber \\ &\quad +{\alpha^{ - 1}}{\alpha^{ - \tau }}\|{{{\tilde D}_{12}}}\|\mathop{\sup }\limits_{t \in [ - \tau,T - \tau - 1]}{\alpha^{ - t}}\|{{e_{2k}}(t)} \| +{\alpha^{ - 1}}\|{{{\tilde B}_{1}}}\|\mathop{\sup }\limits_{t \in [0,T - 1]}{\alpha^{ - t}}\|{\varDelta{u_{k}}(t)}\|\nonumber \\ &\leqslant{\alpha^{ - 1}}{c_{3}}{\|{{e_{1k}}}\|_{\alpha} }+{\alpha^{ - 1}}{c_{4}}{\|{{e_{{\mathrm{2}}k}}}\|_{\alpha} } +{\alpha^{ - 1}}{c_{5}}{\|{\varDelta{u_{k}}} \|_{\alpha} }, \end{align} (3.8) where $$ {c_{3}} = \|{{{\tilde A}_{11}}}\|+{\alpha^{ - \tau }}\|{{{\tilde D}_{11}}} \|,\ {c_{4}} ={\alpha^{ - \tau }}\|{{{\tilde D}_{12}}}\|,\ {c_{5}} = \|{{{\tilde B}_{1}}}\|. $$ Substituting (3.6) into (3.8) results $$ {\|{{e_{1k}}}\|_{\alpha} } \leqslant{\alpha^{ - 1}}({c_{3}} +{c_{1}}{c_{4}}){\|{{e_{1k}}}\|_{\alpha} } +{\alpha^{ - 1}}({c_{5}} +{c_{2}}{c_{4}}){\|{\varDelta{u_{k}}}\|_{\alpha} }. $$ Letting the α of above can also make α−1(c3 + c1c4) < 1. Further, we can get \begin{align} {\|{{e_{1k}}}\|_{\alpha} } \leqslant{\alpha^{ - 1}}{c_{6}}{\|{\varDelta{u_{k}}}\|_{\alpha} }, \end{align} (3.9) where $$ {c_{6}} = \frac{{{c_{5}} +{c_{2}}{c_{4}}}}{{1 -{\alpha^{ - 1}}({c_{3}} +{c_{1}}{c_{4}})}}. $$ It follows from (3.1), (3.3) and (3.7) that \begin{align*} \varDelta{u_{k + 1}}(t) &= \varDelta{u_{k}}(t) - ({u_{k + 1}}(t) -{u_{k}}(t)) = \varDelta{u_{k}}(t) -{\varGamma_{1}}{e_{1k}}(t + 1) -{\varGamma_{2}}{e_{2k}}(t)\\ & = (I -{\varGamma_{1}}{\tilde B_{1}} -{\varGamma_{2}}{\tilde B_{2}})\varDelta{u_{k}}(t) - ({\varGamma_{1}}{\tilde A_{11}} +{\varGamma_{2}}{\tilde A_{21}}){e_{1k}}(t)\\ &\quad- ({\varGamma_{1}}{\tilde D_{11}} +{\varGamma_{2}}{\tilde D_{21}}){e_{1k}}(t - \tau ) - ({\varGamma_{1}}{\tilde D_{12}} +{\varGamma_{2}}{\tilde D_{22}}){e_{2k}}(t - \tau ). \end{align*} Taking Euclidean norm on both sides of the above expression and combining with (3.2), it yields $$ \|{\varDelta{u_{k + 1}}(t)} \| \le \rho \|{\varDelta{u_{k}}(t)}\| +{c_{\mathrm{7}}}\|{{e_{1k}}(t)}\| +{c_{8}}\|{{e_{1k}}(t - \tau )}\| +{c_{9}}\|{{e_{2k}}(t - \tau )}\|, $$ where $$ {c_{7}} = \|{{\varGamma_{1}}{{\tilde A}_{11}} +{\varGamma_{2}}{{\tilde A}_{21}}} \|,\ {c_{8}} = \|{{\varGamma_{1}}{{\tilde D}_{11}} +{\varGamma_{2}}{{\tilde D}_{21}}}\|,\ {c_{9}} = \|{{\varGamma_{1}}{{\tilde D}_{12}} +{\varGamma_{2}}{{\tilde D}_{22}}}\|. $$ Combining with (3.5), (3.6) and (3.9), we can derive \begin{align} {\|{\varDelta{u_{k + 1}}}\|_{\alpha} } &\leqslant \rho{\|{\varDelta{u_{k}}} \|_{\alpha} } +{c_{\mathrm{7}}}{\|{{e_{1k}}} \|_{\alpha} } +{c_{8}}\mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{1k}}(t - \tau )} \| +{c_{9}}\mathop{\sup }\limits_{t \in [0,T]}{\alpha^{ - t}}\|{{e_{2k}}(t - \tau )} \|\nonumber \\ &= \rho{\|{\varDelta{u_{k}}} \|_{\alpha} } +{c_{\mathrm{7}}}{\|{{e_{1k}}} \|_{\alpha} } +{\alpha^{ - \tau }}{c_{8}}\mathop{\sup }\limits_{t \in [ - \tau,T - \tau ]}{\alpha^{ - t}}\|{{e_{1k}}(t)}\| +{\alpha^{ - \tau }}{c_{9}}\mathop{\sup }\limits_{t \in [ - \tau,T - \tau ]}{\alpha^{ - t}}\|{{e_{2k}}(t)}\|\nonumber \\ &\leqslant \rho{\|{\varDelta{u_{k}}}\|_{\alpha} } + ({c_{\mathrm{7}}} +{\alpha^{ - \tau }}{c_{8}}){\|{{e_{1k}}}\|_{\alpha} } +{\alpha^{ - \tau }}{c_{9}}{\|{{e_{2k}}} \|_{\alpha} }\leqslant\hat \rho{\left\|{\varDelta{u_{k}}} \right\|_{\alpha} }, \end{align} (3.10) where $$ \hat \rho = \rho +{\alpha^{ - 1}}{c_{6}}{c_{\mathrm{7}}}+{\alpha^{ - \tau }}{c_{2}}{c_{9}}+{\alpha^{ - (\tau+1) }}({c_{6}}{c_{8}} +{c_{1}}{c_{6}}{c_{9}}). $$ Since 0 ⩽ ρ < 1 by (3.2), it is possible to choose α large enough so that $$\hat \rho < 1$$. Then, (3.10) is a contraction in ∥$$\varDelta$$uk∥α, that is, \begin{align} \mathop{\lim }\limits_{k \to \infty }{\|{\varDelta{u_{k}}} \|_{\alpha} } = 0. \end{align} (3.11) It follows from (3.6), (3.9) and (3.11) that $$\mathop{\lim }\limits _{k \to \infty }{\|{{e_{ik}}} \|_{\alpha } } = 0,i = 1,2,$$ which implies $$ \mathop{\lim }\limits_{k \to \infty }{\|{{e_{k}}}\|_{\alpha} } = 0. $$ Note that $${\left \|{{e_{k}}} \right \|_{s}}\leqslant{\alpha ^{T}}{\|{{e_{k}}}\|_{\alpha } }$$. Therefore, we have $$ \mathop{\lim }\limits_{k \to \infty }{\left\|{{e_{k}}} \right\|_{s}} = 0. $$ This completes the proof. Remark 3.1 In this paper, we only consider the case that the matrix A22 is non-singular, which is the basic condition to ensure the singular system to be impulse-free (for the continuous-time singular system) or causal (for the discrete-time singular system), see (Duan, 2010), (Tian & Zhou, 2012), (Liao et al., 2012) and (Tian et al., 2016). For the general case, the following closed-loop learning algorithm can be adopted for the system (2.1): $$ {u_{k}}(t) ={u_{k - 1}}(t) + \varGamma \left({x_{d}}(t + 1) -{x_{k}}(t + 1)\right), $$ where $$\varGamma$$ ∈ Rm×n is the gain matrix, and xd(t) is the desired state trajectory. The corresponding proof is similar to that for the continuous-time system case (Piao & Zhang, 2007) and is omitted. Remark 3.2 Regarding the selection of the gain matrices $$\varGamma$$1 and $$\varGamma$$2 in the learning algorithm (3.1), Theorem 3.1 gives a theoretical guideline that $$\varGamma$$1 and $$\varGamma$$2 should satisfy (3.2). Note that the matrices $${\tilde B_{1}}\in{{R}^{{r} \times m}}$$ and $${\tilde B_{2}}\in{{R}^{{(n-r)} \times m}}$$, furthermore, we give the following discussion: Case 1: If $${\mathrm{rank}}({\tilde B_{1}})\!=\!m$$ and $${\mathrm{rank}}({\tilde B_{2}})\!<\!m$$, then the matrix $${\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}}$$ is non-singular. Let $${\varGamma _{1}}\!=\!{\alpha _{1}}{({\tilde B_{1}}^{\mathrm{T}}\tilde B_{1})^{\!-\!1}}{\tilde B_{1}}^{\mathrm{T}}$$, $$\varGamma$$2 = 0. We can find α1 ∈ (0, 2) such that $$ \rho = \left\|{I -{\alpha_{1}}{{\left({\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}}\right)}^{ - 1}}{{\tilde B_{1}}^{\mathrm{T}}}{{\tilde B}_{1}}} \right\| = \|{(1 -{\alpha_{1}})I} \| < 1. $$ Case 2: If $${\mathrm{rank}}({\tilde B_{1}})<m$$ and $${\mathrm{rank}}({\tilde B_{2}})=m$$, then the matrix $${\tilde B_{2}}^{\mathrm{T}}{\tilde B_{2}}$$ is non-singular. Let $$\varGamma$$1 = 0, $${\varGamma _{2}}={\alpha _{2}}{({\tilde B_{2}}^\mathrm{T}\tilde B_{2})^{ - 1}}{\tilde B_{2}}^{\mathrm{T}}$$. We can find α2 ∈ (0, 2) such that $$ \rho = \left\|{I -{\alpha_{2}}{{\left({\tilde B_{2}}^{\mathrm{T}}\tilde B_{2}\right)}^{ - 1}}{{\tilde B_{2}}^{\mathrm{T}}}{{\tilde B}_{2}}} \right\| = \|{(1 -{\alpha_{2}})I} \| < 1. $$ Case 3: If $${\mathrm{rank}}({\tilde B_{1}})={\mathrm{rank}}({\tilde B_{2}})=m$$, then the matrices $${\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}}$$ and $${\tilde B_{2}}^{\mathrm{T}}{\tilde B_{2}}$$ are non-singular. Let $${\varGamma _{1}}={\alpha _{1}}{({\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}})^{ - 1}}{\tilde B_{1}}^{\mathrm{T}}$$, $${\varGamma _{2}}={\alpha _{2}}{({\tilde B_{2}}^{\mathrm{T}}{\tilde B_{2}})^{ - 1}}{\tilde B_{2}}^{\mathrm{T}}$$. We can find α1 + α2 ∈ (0, 2) such that $$ \rho = \left\| I -{\alpha_{1}}{{\left({\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}}\right)}^{ - 1}}{{\tilde B_{1}}^{\mathrm{T}}}{{\tilde B}_{1}}-{\alpha_{2}}{{\left({\tilde B_{2}}^{\mathrm{T}}{\tilde B_{2}}\right)}^{ - 1}}{{\tilde B_{2}}^{\mathrm{T}}}{{\tilde B}_{2}}\right\| = \|{(1 -{\alpha_{1}} -{\alpha_{2}})I}\| < 1. $$ In other cases, the gain matrices $$\varGamma$$1 and $$\varGamma$$2 can be chosen by trial and error based on MATLAB technique. Now, we extend the result of Theorem 3.1 to a more general discrete singular time-delay system, which is described by \begin{align} {E{x_{k}}\left({t + 1} \right) = A{x_{k}}\left( t \right) + \sum\limits_{j = 1}^{\tau}{{D_{j}}}{x_{k}}\left({t - j} \right) + B{u_{k}}\left( t \right)}. \end{align} (3.12) Denote $$M{D_{j}}N = \left [ {D_{j11} \atop D_{j21}} \quad {D_{j12} \atop D_{j22}} \right ]$$. Correspondingly, the system (3.12) can be rewritten as follows: \begin{equation} \begin{cases} {x_{1k}}(t + 1)={A_{11}}{x_{1k}}(t)+{A_{12}}{x_{2k}}(t)+{\sum\limits_{j = 1}^{\tau} {{D_{j11}}{x_{1k}}(t - j)} } +{\sum\limits_{j = 1}^{\tau} {{D_{j12}}{x_{2k}}(t - j)} }+{B_{1}}{u_{k}}(t),\\ 0 ={A_{21}}{x_{1k}}(t) + {A_{22}}{x_{2k}}(t) +{\sum\limits_{j = 1}^{\tau} {{D_{j21}}{x_{1k}}(t - j)} } +{\sum\limits_{j = 1}^{\tau}{{D_{j22}}{x_{2k}}(t - j)}}+{B_{2}}{u_{k}}(t). \end{cases} \end{equation} (3.13) For the system (3.13), the following assumption is needed for further analysis: Assumption 3.1 For the given desired state trajectory $$\left [ {{x_{1d}}(t) \atop {x_{2d}}(t)} \right ]$$, there exists a desired control input ud(t) such that \begin{cases} {x_{1d}}(t + 1)={A_{11}}{x_{1d}}(t)+{A_{12}}{x_{2d}}(t) +{\sum\limits_{j = 1}^{\tau} {{D_{j11}}{x_{1d}}(t - j)} } +{\sum\limits_{j = 1}^{\tau} {{D_{j12}}{x_{2d}}(t - j)} }+{B_{1}}{u_{d}}(t),\\ 0 ={A_{21}}{x_{1d}}(t) + {A_{22}}{x_{2d}}(t) +{\sum\limits_{j = 1}^{\tau} {{D_{j21}}{x_{1d}}(t - j)} } +{\sum\limits_{j = 1}^{\tau} }{{D_{j22}}{x_{2d}}(t - j)} +{B_{2}}{u_{d}}(t). \end{cases} Then we have the following theorem: Theorem 3.2 Consider the system (3.13) satisfying Assumptions 2.2, 2.3 and 3.1. If there exist the gain matrices $$\varGamma$$1 ∈ Rm×r, $$\varGamma$$2 ∈ Rm×(n−r) such that the convergence condition (3.2) holds. Then under the action of the learning algorithm (3.1), the system state can converge to the desired state trajectory on [0, T], i.e., $$\mathop{\lim }\limits _{k \to \infty }{\|{e_{k}} \|_{s}} = 0$$. Proof. The proof is similar to that of Theorem 3.1 and is omitted. 4. Numerical examples In order to demonstrate the effectiveness of the proposed ILC algorithm, two numerical examples are constructed in this Section. Example 4.1 Consider the following discrete singular system with one time-delay: $$ E{x_{k}}(t + 1) = A{x_{k}}(t) + D{x_{k}}(t - \tau ) + B{u_{k}}(t), $$ where t ∈ [0, 15], the time delay τ = 2, and $$ E=\left[ \begin{array}{@{}cc@{}} 1&0 \\ 0&0 \end{array}\right], \quad A= \left[ \begin{array}{@{}cc@{}} 0.5&0\\ 1&-1 \end{array}\right], \quad D= \left[ \begin{array}{@{}cc@{}} 0.4&0.1\\ -0.5&0.75 \end{array}\right], \quad B= \left[ \begin{array}{@{}cc@{}} 0.3&-0.2 \\ 0&-0.9 \end{array}\right]. $$ From the above matrices, we know that \begin{align*} A_{11}&=0.5, \quad A_{12}=0, \quad A_{21}=1, \quad A_{22}=-1, \quad D_{11}=0.4, \quad D_{12}=0.1,\\ D_{21}&=-0.5, \quad D_{22}=0.75, \quad B_{1}=[0.3-0.2], \quad B_{2}=[0-0.9]. \end{align*} Furthermore, we can compute that $$ {\tilde B_{1}} ={B_{1}} -{A_{12}}A_{22}^{ - 1}{B_{2}}=[0.3-0.2],\quad{\tilde B_{2}} = - A_{22}^{ - 1}{B_{2}}=[0-0.9]. $$ According to the learning algorithm (3.1), take the gain matrices $$\varGamma _{1} =\left [ {3 \atop 0} \right ],\ $$$${\varGamma _{2}}=\left [ {-1 \atop -0.6} \right ]\!,\ $$then we have $$ \rho = \|{I -{\varGamma_{1}}{{\tilde B}_{1}} -{\varGamma_{2}}{{\tilde B}_{2}}}\|=0.552 < 1, $$ which implies that the convergence condition (3.2) holds. Take the given desired state trajectory as: $$ \left[ \begin{array}{@{}c@{}}{{x_{1d}}(t)}\\{{x_{2d}}(t)} \end{array} \right] = \left[ \begin{array}{@{}c@{}}{{{{\mathrm{e}}}^{0.1t}}}\\{\sin (0.4t)} \end{array} \right]\!. $$ Set the initial state and the initial control as: $$ \left[ \begin{array}{@{}c@{}}{{\psi_{1k}}(t)}\\{{\psi_{2k}}(t)} \end{array}\right] = \left[ \begin{array}{@{}c@{}}1+t\\ t \end{array}\right],\quad t \in [ - 2,0],\quad{u_{0}}(t) = \left[ \begin{array}{@{}c@{}}0\\ 0 \end{array} \right]\!. $$ The simulation results are shown in Figs 1–4. From Figs 1 and 2, we can see that the system state profile at 20th iteration is close to the desired state trajectory. From Figs 3 and 4, we find that the maximum state tracking error is tend to zero as the iteration increases. Therefore, the simulation results show that the learning algorithm (3.1) is effective for the discrete singular time-delay system. Example 4.2 Consider the following discrete singular system with multiple time-delays: $$ {E{x_{k}}\left({t + 1} \right) = A{x_{k}}\left( t \right) + \sum\limits_{j = 1}^{2}{{D_{j}}}{x_{k}}\left({t - j} \right) + B{u_{k}}\left( t \right)}, $$ where t ∈ [0, 15], and $$ E= \left[ \begin{array}{@{}cc@{}}1&0\\0&0 \end{array} \right]\!,\ A= \left[ \begin{array}{@{}cc@{}}0.8&0 \\1&-1 \end{array} \right]\!,\ D_{1}=\left[ \begin{array}{@{}cc@{}}0.1&0\\ 0&1 \end{array} \right]\!,\ D_{2}=\left[ \begin{array}{@{}cc@{}}0.3&0.2\\-1&0 \end{array} \right]\!,\ B= \left[ \begin{array}{@{}c@{}}1\\0 \end{array} \right]. $$ It is easy to see that \begin{align*} &A_{11}=0.8,\ A_{12}=0,\ A_{21}=1,\ A_{22}=-1,\ D_{111}=0.1,\ D_{112}=0,\ D_{121}=0,\\ &D_{122}=1,\ D_{211}=0.3,\ D_{212}=0.2,\ D_{221}=-1,\ D_{222}=0,\ B_{1}=1,\ B_{2}=0. \end{align*} Then we can get $$ {\tilde B_{1}} ={B_{1}} -{A_{12}}A_{22}^{ - 1}{B_{2}}=1,\ {\tilde B_{2}} = - A_{22}^{ - 1}{B_{2}}=0. $$ According to the Remark 3.2 (Case 1), take α1 = 0.6, then the gain matrices $$ {\varGamma_{1}}={\alpha_{1}}{\left({\tilde B_{1}}^{\mathrm{T}}{\tilde B_{1}}\right)^{ - 1}}{{\tilde B_{1}}^{\mathrm{T}}}=0.6,\ {\varGamma_{2}} = 0. $$ Furthermore, we have $$ \rho = \|{I -{\varGamma_{1}}{{\tilde B}_{1}} -{\varGamma_{2}}{{\tilde B}_{2}}}\|=0.4 < 1, $$ that is to say, the convergence condition (3.2) is satisfied. Take the given desired state trajectory as: $$ \left[ \begin{array}{@{}c@{}}{{x_{1d}}(t)}\\{{x_{2d}}(t)} \end{array}\right] = \left[ \begin{array}{@{}c@{}}{0.005t(t-6)}\\{0.01t(t-7)} \end{array}\right]\!. $$ Set the initial state and the initial control as: $$ \left[ \begin{array}{@{}c@{}}{{\psi_{1k}}(t)}\\{{\psi_{2k}}(t)} \end{array}\right] = \left[ \begin{array}{@{}c@{}}{0.005t(t-6)}\\{0.01t(t-7)} \end{array}\right],\ t \in [ - 2,0],\ {u_{0}}(t) = 0. $$ Correspondingly, the simulation results are shown in Figs 5–8. From Figs 5 and 6, we know that the system state can follow the desired state trajectory at 15th iteration. From Figs 7 and 8, we can see that the algorithm (3.1) ensures the state tracking error to converge to zero as the iteration number increases. Fig. 1. View largeDownload slide The trajectories x1d(t) and x1k(t). Fig. 1. View largeDownload slide The trajectories x1d(t) and x1k(t). Fig. 2. View largeDownload slide The trajectories x2d(t) and x2k(t). Fig. 2. View largeDownload slide The trajectories x2d(t) and x2k(t). Fig. 3. View largeDownload slide The tracking error e1k(t). Fig. 3. View largeDownload slide The tracking error e1k(t). Fig. 4. View largeDownload slide The tracking error e2k(t). Fig. 4. View largeDownload slide The tracking error e2k(t). Fig. 5. View largeDownload slide The trajectories x1d(t) and x1k(t). Fig. 5. View largeDownload slide The trajectories x1d(t) and x1k(t). Fig. 6. View largeDownload slide The trajectories x2d(t) and x2k(t). Fig. 6. View largeDownload slide The trajectories x2d(t) and x2k(t). Fig. 7. View largeDownload slide The tracking error e1k(t). Fig. 7. View largeDownload slide The tracking error e1k(t). Fig. 8. View largeDownload slide The tracking error e2k(t). Fig. 8. View largeDownload slide The tracking error e2k(t). 5. Conclusion Based on the equivalent restrict decomposition form of singular systems, the ILC technique is applied to a class of discrete singular time-delay systems in this paper. Then a discrete-time learning algorithm is proposed and the convergence condition of the algorithm is established. It is shown that the algorithm can guarantee the system state converges to the desired trajectory on the whole time interval. And the proposed algorithm is also suitable for a class of discrete singular systems with multiple time-delays. In the end, numerical simulation results illustrate the effectiveness of the presented algorithm. Acknowledgements The authors would like to express their gratitude to the editors and anonymous reviewers for their constructive comments that have greatly improved the quality of this paper. Funding National Natural Science Foundation of China (61374104, 61773170); and Natural Science Foundation of Guangdong Province of China (2016A030313505). References Arimoto , S. , Kawamura , S. & Miyazaki , F. ( 1984 ) Bettering operation of robots by learning . J. Robotic Syst. , 1, 123 -- 140 . Google Scholar CrossRef Search ADS Berger , T . 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**Published: ** Feb 1, 2018

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