Preview tracking control for a class of discrete-time Lipschitz non-linear time-delay systems

Preview tracking control for a class of discrete-time Lipschitz non-linear time-delay systems Abstract The design of a preview tracking controller for a class of discrete-time Lipschitz non-linear time-delay systems is considered in this paper. First, based on preview control theory, we introduce a difference operator and deduce an augmented error system which contains previewable information on the reference signal. The original tracking problem is thus transformed into a regulation problem. Second, we design a memory state feedback controller for the augmented error system. Applying Lyapunov’s second method, the sufficient condition for stability of the closed-loop system is derived in the form of linear matrix inequality (LMI), and the preview controller is obtained accordingly. Finally, to show the superiority of the controller proposed, two numerical examples are presented. 1. Introduction Preview control can improve the performance of a control system by fully utilizing the known future information of the reference signals or the disturbances, and it has been successfully applied in many fields of engineering (Hess & Chan, 1988; Peng & Tomizuka, 1993; Laks et al., 2011; Shimmyo et al., 2013). In comparison with other control methods, preview control has the following advantages: first, by exploiting the previewable future information, the purpose of enhancing the tracking performance of the closed-loop system can be achieved. Second, preview feedforward action can be added to the existing feedback control system expediently (Tomizuka & Whitney, 1975; Tsuchiya & Egami, 1994; Zhen, 2016). By now, preview control has already formed a relatively complete theoretical system in areas such as multirate systems, and linear discrete- or continuous-time systems together with descriptor systems (Tomizuka, 1975; Katayama et al., 1985; Katayama & Hirono, 1987; Takaba, 2000; Liao et al., 2013; Cao & Liao, 2015; Li & Liao, 2017).However, there are few research achievements on preview control for non-linear systems. In fact, the systems in practical engineering are mostly non-linear, and also, linear systems in the strict sense do not exist. The design of a preview controller for non-linear systems, therefore, is of great importance. The so-called Lipschitz non-linear system refers to the system where the non-linearity satisfies the Lipschitz condition. According to the viewpoints in the studies by Raghavan & Hedrick (1994), Zemouche & Boutayeb (2013), Miao & Li (2015) and Defoort et al. (2016), this kind of system is extensively present in practical problems. For a Lipschitz non-linear system, research work at present mainly focuses on observer design (Ibrir et al., 2005; Ibrir et al., 2006; Zemouche et al., 2008; Xu et al., 2009), feedback stabilization (Sun & Liu, 2007; Fu et al., 2013; Zuo et al., 2016) and tracking control (Rehan et al., 2011; Mobayen & Tchier, 2016). For example, the problem of designing an observer as well as an observer-based feedback controller for a class of discrete-time Lipschitz non-linear systems was investigated in the study by Ibrir et al. (2005). Subsequently, some results on the observer design for general discrete-time non-linear systems were put forward in the studies by Xiao (2006), Karafyllis & Kravaris (2007) and Califano et al. (2009). For a continuous-time Lipschitz non-linear system, the authors in the study by Rehan et al. (2011) developed a non-linear feedback controller under which the output of the closed-loop system can track the reference signal accurately. Based on the system discussed in the study by Ibrir et al. (2005), Sun & Liu (2007) studied the state feedback along with output feedback control of a class of continuous-time non-linear systems. However, the results on preview control for Lipschitz non-linear systems have not been reported in the literature thus far. This has motivated our study in this paper. In addition, the time-delay phenomenon frequently appears in practical engineering systems such as communication systems, electric power systems and biological systems. Its presence can degrade performance of the control system or even lead to instability. In the study of controller design for time-delay systems, it is known that the memoryless feedback controller has an advantage of easy implementation in physics. However, its control performance cannot be better than a memory feedback controller when the information on the size of time-delay is available. The memory controller can simultaneously utilize current and past stored state information, thereby achieving superior system performance. Some results related to this issue were reported in the studies by Moon et al. (2001), Azuma et al. (2002) and Ji et al. (2009). In this paper, our objective is to design a preview tracking controller for a class of discrete-time Lipschitz non-linear time-delay systems. We assume that the reference signal is previewable. First, by applying the difference method in the study by Tsuchiya & Egami (1994), an augmented error system is deduced, so that the original tracking problem is transformed into a stabilization problem of a discrete-time linear system. Second, for the augmented error system obtained, we design a memory state feedback controller by employing current and past state information. Based on Lyapunov’s second method, the sufficient condition to ensure asymptotical stability of the closed-loop system is developed. The preview controller of the original system will then be derived accordingly. Finally, numerical examples are presented, to reveal the superiority of the controller proposed. It is assumed throughout this paper that the state is available in real-time and the past states can be stored in memory. Notations Throughout this paper, R is the set of real numbers. For a matrix P, P > 0 means P is symmetric positive definite. For matrices P and Q, P > Q means P − Q > 0. I and 0 denote the identity matrix and the zero matrix with appropriate dimension, respectively. $$f{{^\prime }}\!(x)$$ represents the Jacobi matrix of the vector function $$f\!\left (x\right )$$. 2. Problem formulation Consider the discrete-time non-linear system \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+A_{d} x\!\left(k-d\right)+f\!\left(x\!\left(k\right)\right)+Bu\!\left(k\right)} \\{y\!\left(k\right)=Cx\!\left(k\right)} \end{cases} \end{equation} (1) where $$x(k)\in \mathbf{R}^{n}, u(k)\in \mathbf{R}^{m}, y(k)\in \mathbf{R}^{q} $$ represent the state, control input and output of the system, respectively; matrices $$A,\,A_{d} \in \mathbf{R}^{n\times n} ,\,B\in \mathbf{R}^{n\times m} ,C\in \mathbf{R}^{q\times n} $$ are known and constant. d > 0 is a positive integer which stands for the time-delay of the system. The non-linearity $$f\!\left (x\right )\!\in\! \mathbf{R}^{n} $$ is a time-varying vector. The non-linearity of system (1) satisfies the following condition. A1. The non-linearity f(x) is differentiable, $$f{^{\prime }}\!(x)$$ has the form of \begin{equation} f{^{\prime} }\!(x)=MF(x)N \end{equation} (2) for some $$M\in \mathbf{R}^{n\times r},\,N\in \mathbf{R}^{s\times n} $$ and $$F(x)\in \mathbf{R}^{r\times s} $$ is a norm-bounded matrix satisfying $$F^{T} \!(x)F(x)\le I$$. If A1 is fulfilled, it can be proved that f(x) satisfies the Lipschitz condition. Thus, f(x) is called a Lipschitz non-linearity, and accordingly, system (1) is called a Lipschitz non-linear system. Remark 1 The Lipschitz condition has been widely used in the design of non-linear systems. This kind of Lipschitz non-linearity verifying A1 does not involve any approximation of non-linearity by its norm, and it is pointed out in the study by Ibrir et al. (2005) that this property will reduce the conservatism of the results obtained. Moreover, Ibrir et al. (2005) and Sun & Liu (2007) all require that M, N be n × n square matrices. This paper reduces the requirement for the order of M, N so that the dimension of the LMI derived later decreases, which will lower the computational cost. Remark 2 In order to deal with the problem more conveniently, for such a Lipschitz non-linear system satisfying A1, the function f(x) is required to be vanishing at the origin (i.e. $$f\!\left (0\right )=0$$) (Ibrir et al., 2005; Ibrir et al., 2006; Sun & Liu, 2007; Xu et al., 2009). But in the present paper, this restriction is dropped by adopting a difference method. This study thus covers a broader class of Lipschitz non-linear systems. To establish a preview tracking controller for system (1), we need the following assumption that the reference signal is previewable. A2. The preview length of reference signal r(k) is $$M_{r} $$, that is, at each time k, the $$M_{r} $$ future values, $$r(k+1),\cdots\!\,,r(k+M_{r})$$ as well as the present and past values of the reference signal, are available. The future values of the reference signal are assumed not to change beyond $$k+M_{r} $$, namely, $$ r(k+i)=r(k+M_{r}),\,i=M_{r} +1,\,M_{r} +2,\cdots \!\,.$$ Remark 3 We should mention that A2 is a basic assumption in preview control theory. Theoretical study and engineering practice have shown that only the previewable signal of a recent period significantly influences the performance of the control system, while a reference signal that exceeds the previewable range has little influence on the system performance (Tsuchiya & Egami, 1994). Therefore, the values beyond the preview length are generally assumed to be constant. Indeed, the ordinary feedback control systems do not fully utilize the previewable signal, or equivalently, the preview length is zero. The tracking error signal e(k) is defined as the difference between the output y(k) and the reference signal r(k), i.e. \begin{equation} e(k)=y(k)-r(k). \end{equation} (3) This paper aims to design a memory state feedback controller with preview action such that the output y(k) of the closed-loop system can track the reference signal r(k) without static error, that is, $$ \mathop{\lim }\limits_{k\to \infty } e\!\left(k\right)=\mathop{\lim }\limits_{k\to \infty } \! \big(y\!\left(k\right)-r\!\left(k\right)\big)=0.$$ Let us now introduce two useful lemmas for the development of our work. Lemma 1 (de Souza & Li, 1999) Let M, N and F(t) be real matrices of appropriate dimensions, with F(t) satisfying $$F^{T}\!(t)F(t)\le I$$. Then, the following inequality holds for any constant $$\mu>0$$: $$ MF\!\left(t\right)N+\big[MF\!\left(t\right)N\big]^{T} \le \mu^{-1} MM^{T} +\mu N^{T} N.$$ Lemma 2 (Boyd et al., 1994 (Schur complement lemma)) Symmetric matrix $$ \left [{{S_{11}} \atop {S_{12}^{T}}} \quad {{S_{12}} \atop {S_{22}}} \right ] < 0$$ if and only if one of the following two conditions is satisfied: (i) $$S_{11} <0,S_{22} -S_{12}^{T} S_{11}^{-1} S_{12} <0$$; (ii) $$S_{22} <0,S_{11} -S_{12} S_{22}^{-1} S_{12}^{T} <0$$. 3. Construction of the augmented error system We adopt the augmented error system method to study the problem of controller design for system (1). The difference operator is defined as $$ \Delta x(k)=x(k)-x(k-1).$$ Taking the difference operator on both sides of the first equation in (1) leads to the following: \begin{equation} \Delta x\!\left(k+1\right)=A\Delta x\!\left(k\right)+A_{d} \Delta x\!\left(k-d\right)+f\!\left(x\left(k\right)\right)-f\!\left(x\left(k-1\right)\right)+B\Delta u\!\left(k\right)\!. \end{equation} (4) Applying the Mean-Value Theorem (Ortega & Rheinboldt, 2000), one can obtain $$ f\!\left(x\!\left(k\right)\right)-f\!\left(x\!\left(k-1\right)\right)={{\int_{0}^{1}}}f{^{\prime} }\!\left(x\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \Delta x\!\left(k\right)\,\mathrm{d}\lambda .$$ Then, equation (4) becomes \begin{equation} \Delta x\!\left(k+1\right)=\left(A+A_{1} \right)\Delta x\!\left(k\right)+A_{d} \Delta x\!\left(k-d\right)+B\Delta u\!\left(k\right)\!, \end{equation} (5) where \begin{equation} A_{1} ={{\int_{0}^{1}}}MF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda . \end{equation} (6) Taking the same difference operator on the error signal results in $$\Delta e(k)=\Delta y(k)-\Delta r(k)$$, and combining (1) and (5) yields \begin{align*} {\Delta e\!\left(k+1\right)} &= {C\Delta x\!\left(k+1\right)-\Delta r\!\left(k+1\right)} \\ &= {C\left(A+A_{1} \right)\Delta x\!\left(k\right)+CA_{d} \Delta x\!\left(k-d\right)+CB\Delta u\!\left(k\right)-\Delta r\!\left(k+1\right)\!.} \end{align*} Since $$\Delta e\!\left (k+1\right )=e\!\left (k+1\right )-e\!\left (k\right )$$, we have \begin{equation} e\!\left(k+1\right)=e\!\left(k\right)+C\!\left(A+A_{1} \right)\Delta x\!\left(k\right)+CA_{d} \Delta x\!\left(k-d\right)+CB\Delta u\!\left(k\right)-\Delta r\!\left(k+1\right)\!. \end{equation} (7) Considering (5) in conjunction with (7), it follows that \begin{equation} \tilde{x}\!\left(k+1\right)=\big(\tilde{A}+\tilde{A}_{1} \big)\tilde{x}\!\left(k\right)+\tilde{A}_{d} \tilde{x}\!\left(k-d\right)+\tilde{B}\Delta u\!\left(k\right)+G\Delta r\!\left(k+1\right)\!, \end{equation} (8) where $$ \tilde{x}\!\left(k\right)=\left[\begin{array}{@{}c@{}}{e\!\left(k\right)} \\{\Delta x\!\left(k\right)} \end{array}\right]\!,\,\tilde{A}=\left[\begin{array}{@{}cc@{}} I &{CA} \\{0} & A \end{array}\right]\!,\,\tilde{A}_{1} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{1} } \\{0} &{A_{1} } \end{array}\right]\!,\,\tilde{A}_{d} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{d} } \\{0} &{A_{d} } \end{array}\right]\!,\,\tilde{B}=\left[\begin{array}{@{}c@{}}{CB} \\ B \end{array}\right]\!,\,G=\left[\begin{array}{@{}c@{}}{-I} \\{0} \end{array}\right]\!.$$ Specifically, one can notice from (6) that $$ \tilde{A}_{1} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{1} } \\{0} &{A_{1} } \end{array}\right]=\left[\begin{array}{@{}cc@{}}{0} &{{{\int_{0}^{1}}}CMF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda } \\{0} &{{{\int_{0}^{1}}}MF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda } \end{array}\right]={{\int_{0}^{1}}}\left[\begin{array}{@{}c@{}}{CM} \\ M \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \left[\begin{array}{@{}cc@{}}{0} & N \end{array}\right]\,\mathrm{d}\lambda. $$ Denoting $$\tilde{M}=\left [{{CM} \atop M}\right ],\tilde{N}= [0 \ N]$$, then we obtain \begin{equation} \tilde{A}_{1} ={{\int_{0}^{1}}}\tilde{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \tilde{N}\,\mathrm{d}\lambda . \end{equation} (9) Equation (8) is the constructed error system. In the following, let us deduce the augmented error system. To bring in the previewable reference signal, we define the following vector: $$ x_{r} \left(k\right)=\left[\begin{array}{@{}c@{}}{\Delta r\!\left(k+1\right)} \\{\Delta r\!\left(k+2\right)} \\{\vdots } \\{\Delta r\!\left(k+M_{r} \right)} \end{array}\right]\!.$$ By exploiting A1, we get \begin{equation} x_{r} \!\left(k+1\right)=A_{r} x_{r} \!\left(k\right)\!, \end{equation} (10) where $$ A_{r} =\left[\begin{array}{@{}ccccc@{}}{0} & I &{0} &{\cdots } &{0} \\{0} &{0} & I &{\cdots } &{0} \\{\vdots } &{\vdots } &{\vdots } &{} &{\vdots } \\{0} &{0} &{0} &{\cdots } & I \\{0} &{0} &{0} &{\cdots } &{0} \end{array}\right]\!.$$ Combining (8) and (10), we obtain \begin{equation} \bar{x}\!\left(k+1\right)=\left(\bar{A}+\bar{A}_{1} \right)\bar{x}\!\left(k\right)+\bar{A}_{d} \bar{x}\!\left(k-d\right)+\bar{B}\Delta u\!\left(k\right)\!. \end{equation} (11) This is the augmented error system constructed in this paper, where $$ \bar{x}\!\left(k\right)=\left[\begin{array}{@{}c@{}}{\tilde{x}\!\left(k\right)} \\{x_{r} \!\left(k\right)} \end{array}\right]\!,\,\tilde{G}=\left[\begin{array}{@{}cccc@{}} G &{0} &{\cdots } &{0} \end{array}\right]\!,\,\bar{A}=\left[\begin{array}{@{}cc@{}}{\tilde{A}} &{\tilde{G}} \\{0} &{A_{r} } \end{array}\right]\!,\,\bar{A}_{1} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{1} } &{0} \\{0} &{0} \end{array}\right]\!,\,\bar{A}_{d} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{d} } &{0} \\{0} &{0} \end{array}\right]\!,\,\bar{B}=\left[\begin{array}{@{}c@{}}{\tilde{B}} \\{0} \end{array}\right]\!.$$ From (9) it follows that $$ \bar{A}_{1} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{1} } &{0} \\{0} &{0} \end{array}\right]=\left[\begin{array}{@{}cc@{}}{{{\int_{0}^{1}}}\tilde{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \tilde{N}\,\mathrm{d}\lambda } &{0} \\{0} &{0} \end{array}\right]={{\int_{0}^{1}}}\left[\begin{array}{@{}c@{}}{\tilde{M}} \\{0} \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \left[\begin{array}{@{}cc@{}}{\tilde{N}} &{0} \end{array}\right]\,\mathrm{d}\lambda .$$ Denoting $$\bar{M}=\left [{{\tilde{M}} \atop {0}} \right ],\bar{N}= [\tilde{N} \ 0] $$, then we get \begin{equation} \bar{A}_{1} ={{\int_{0}^{1}}}\bar{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \bar{N}\,\mathrm{d}\lambda . \end{equation} (12) For the augmented error system (11), let us design the following memory state feedback controller \begin{equation} \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)\!, \end{equation} (13) where the feedback gain matrices K, L will be determined later. Substituting this controller into system (11), we obtain the closed-loop system \begin{equation} \bar{x}\!\left(k+1\right)=\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)\bar{x}\!\left(k\right)+\left(\bar{A}_{d} +\bar{B}L\right)\bar{x}\!\left(k-d\right)\!. \end{equation} (14) Hereafter, we will determine the gain matrices K, L via the LMI method. 4. Design of a memory state feedback controller with preview action Theorem 1 Under A1–A2, the closed-loop system (14) is globally asymptotically stable if there exist matrices P > 0, S > 0 and matrices K, L so that the following matrix inequality condition holds: \begin{equation} \left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{A}_{1} +\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]<0. \end{equation} (15) Proof Consider the Lyapunov–Krasovskii functional \begin{equation} V\!\left(\bar{x}\left(k\right)\right)=\bar{x}^{T} \!\left(k\right)P\bar{x}\!\left(k\right)+\sum_{i=k-d}^{k-1}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right) \end{equation} (16) It is clear that V is positive definite due to P > 0, S > 0. Taking the difference of V along the trajectories of system (14), it follows that \begin{align*} \Delta V\!\left(\bar{x}\left(k\right)\right)&=V\!\left(\bar{x}\left(k\right)\right)-V\!\left(\bar{x}\left(k-1\right)\right)\\ &=\bar{x}^{T} \!\left(k\right)P\bar{x}\!\left(k\right)+\sum_{i=k-d}^{k-1}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right)-\bar{x}^{T} \!\left(k-1\right)P\bar{x}\!\left(k-1\right)-\sum_{i=k-d-1}^{k-2}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right)\\ &=\bar{x}^{T} \!\left(k-1\right)\left[\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)-P+S\right]\bar{x}\!\left(k-1\right)\\ &\quad+2\bar{x}^{T} \!\left(k-1\right)\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)\bar{x}\!\left(k-d-1\right)\\ &\quad+\bar{x}^{T} \!\left(k-d-1\right)\left[\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)-S\right]\bar{x}\!\left(k-d-1\right)\\ &=\left[ \begin{array}{@{}c@{}} {\bar{x}\left(k-1\right)}\\{\bar{x}\left(k-d-1\right)} \end{array} \right]^{T} \Omega\! \left[\begin{array}{@{}c@{}}{\bar{x}\left(k-1\right)} \\{\bar{x}\left(k-d-1\right)} \end{array}\right]\!, \end{align*} where $$ \Omega =\left[\begin{array}{@{}cc@{}}{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)-P+S} &{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)} &{\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)-S} \end{array}\right]\!.$$ Now we prove $$\Omega <0$$. Notice that $$ \left[\begin{array}{@{}cc@{}}{-P^{-1} } &{0} \\{0} &{-S^{-1} } \end{array}\right]<0.$$ By applying the Schur complement lemma (Lemma 2), inequality (15) implies $$ \left[\begin{array}{@{}cc@{}}{-P} &{0} \\{0} &{-S} \end{array}\right]-\left[\begin{array}{@{}cc@{}}{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} } & I \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} \end{array}\right]\left[\begin{array}{@{}cc@{}}{-P^{-1} } &{0} \\{0} &{-S^{-1} } \end{array}\right]^{-1} \left[\begin{array}{@{}cc@{}}{\bar{A}+\bar{A}_{1} +\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\ I &{0} \end{array}\right]<0.$$ The left side of the above inequality is $$\Omega $$ exactly, thus $$\Delta V$$ is negative definite. Moreover, $$V\!\left (\bar{x}\left (k\right )\right )$$ is radially unbounded, that is, when $$\left \| \bar{x}\!\left (k\right )\right \| \to \infty $$, we have $$V\!\left (\bar{x}\left (k\right )\right )\to \infty $$, hence the closed-loop system (14) is globally asymptotically stable. This ends the proof. Due to the fact that the element $$\bar{A}_{1} $$ in (15) contains an integral term related to the non-linearity, it is difficult to apply Theorem 1 to practical problems. Therefore, we present a computationally tractable LMI form in what follows. Theorem 2 Under A1–A2, the system (14) is globally asymptotically stable if there exist a constant $$0<\varepsilon <1$$, matrices X > 0, Y > 0 and matrices Q, H so that the following LMI condition holds: \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-X} &{0} &{\bar{A}X+\bar{B}Q} &{\bar{A}_{d} Y+\bar{B}H} &{\bar{M}} &{0} \\{0} &{-Y} & X &{0} &{0} &{0} \\{X\bar{A}^{T} +Q^{T} \bar{B}^{T} } & X &{-X} &{0} &{0} &{X\bar{N}^{T} } \\{Y\bar{A}_{d}^{T} +H^{T} \bar{B}^{T} } &{0} &{0} &{-Y} &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}X} &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (17) Then, the controller is determined by (13), and the gain will be given by: $$K=QX^{-1},L=HY^{-1} $$. Proof Let the matrix on the left side of (15) be $$\Phi $$. We can always partition matrix $$\Phi $$ as $$ \Phi =\left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}+\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]+\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{A}_{1} } &{0} \\{0} &{0} &{0} &{0} \\{\bar{A}_{1}^{T} } &{0} &{0} &{0} \\{0} &{0} &{0} &{0} \end{array}\right].$$ By virtue of (12), we get \begin{multline} \Phi \!=\!\left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}\!+\!\bar{B}K} &{\bar{A}_{d} \!+\!\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}\!+\!\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} \!+\!\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]\!\\+\!{{\int_{0}^{1}}}\!\left\{\!\left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]\!F\!\left(x\!\left(k\right)\!-\!\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]\!\right. \left.+\!\left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]\!F^{T} \!\!\left(x\!\left(k\right)\!-\!\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right]\right\}\,\mathrm{\!d}\lambda. \end{multline} (18) Applying Lemma 1 to the integrand, it follows that \begin{multline*}{\left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]+\left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]F^{T} \!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right]} \\{\le \varepsilon \left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right] +\varepsilon^{-1} \left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]} \end{multline*} Substituting the above into (18) and taking the integral properties into account, one can obtain $$\Phi \le \left [\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left (\bar{A}+\bar{B}K\right )^{T} } & I &{-P} &{0} \\{\left (\bar{A}_{d} +\bar{B}L\right )^{T} } &{0} &{0} &{-S} \end{array}\right ]+\left [\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right ]\left [\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right ]+\frac{1}{\varepsilon } \left [\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right ]\left [\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right ]=\Psi .$$ Thus, $$\Phi <0$$ if $$\Psi <0$$. By employing the Schur complement lemma (Lemma 2) again, $$\Psi <0$$ is equivalent to \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} &{\bar{M}} &{0} \\{0} &{-S^{-1} } & I &{0} &{0} &{0} \\{\left(\bar{A}+\bar{B}K\right)^{T} } & I &{-P} &{0} &{0} &{\bar{N}^{T} } \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}} &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (19) Now, performing the congruence transformation to inequality (19) with $$diag\left (I,I,P^{-1},S^{-1},I,I\right )$$ leads to \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-P^{-1} } &{0} &{\bar{A}P^{-1} +\bar{B}KP^{-1} } &{\bar{A}_{d} S^{-1} +\bar{B}LS^{-1} } &{\bar{M}} &{0} \\{0} &{-S^{-1} } &{P^{-1} } &{0} &{0} &{0} \\{P^{-1} \bar{A}^{T} +P^{-1} K^{T} \bar{B}^{T} } &{P^{-1} } &{-P^{-1} } &{0} &{0} &{P^{-1} \bar{N}^{T} } \\{S^{-1} \bar{A}_{d}^{T} +S^{-1} L^{T} \bar{B}^{T} } &{0} &{0} &{-S^{-1} } &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}P^{-1} } &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (20) By letting $$X=P^{-1},Y=S^{-1},Q=KP^{-1},H=LS^{-1} $$, LMI (17) is obtained. We thus deduce that inequality (15) is fulfilled if LMI (17) holds. The conclusion to be proved is derived immediately by Theorem 1. This ends the proof. In fact, if there exists a feasible solution to LMI (17), then the state feedback can be shown as $$ \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)$$ It follows from Theorem 2 that the closed-loop system (14) of the augmented error system (11) is asymptotically stable. As a consequence, we have $$ \mathop{\lim }\limits_{k\to \infty } e\!\left(k\right)=\mathop{\lim }\limits_{k\to \infty } \!\left(y\!\left(k\right)-r\!\left(k\right)\right)=0.$$ Namely, the output $$y\!\left (k\right )$$ of the closed-loop system can track the reference signal $$r\!\left (k\right )$$ without static error. Partition the control gain matrices K and L so that \begin{equation} K=\left[\begin{array}{@{}ccc@{}}{K_{e} } &{K_{\Delta x} } &{K_{r} } \end{array}\right]\!,\qquad\qquad\quad \end{equation} (21) \begin{equation} K_{r} =\left[\begin{array}{@{}cccc@{}}{k_{r} \!\left(1\right)} &{k_{r} \!\left(2\right)} &{\cdots } &{k_{r} \!\left(M_{r} \right)} \end{array}\right]\!, \end{equation} (22) \begin{equation} L=\left[\begin{array}{@{}ccc@{}}{L_{e} } &{L_{\Delta x} } &{L_{r} } \end{array}\right]\!,\qquad\qquad\quad\ \end{equation} (23) \begin{equation} L_{r} =\left[\begin{array}{@{}cccc@{}}{l_{r} \!\left(1\right)} &{l_{r} \!\left(2\right)} &{\cdots } &{l_{r} \!\left(M_{r} \right)} \end{array}\right]\!.\ \, \end{equation} (24) Then $$\Delta u\left (k\right )$$ is explicitly given by \begin{multline*}\Delta u\!\left (k\right )=K_{e} e\!\left (k\right )+K_{\Delta x} \Delta x\!\left (k\right )+\sum \limits _{i=1}^{M_{r} }k_{r} \!\left (i\right )\Delta r\!\left (k+i\right )\\[-2pt] +L_{e} e\!\left (k-d\right )+L_{\Delta x} \Delta x\!\left (k-d\right )+\sum \limits _{i=1}^{M_{r} }l_{r} \!\left (i\right )\Delta r\!\left (k+i-d\right )\!.\end{multline*} From the definition of the difference operator, it follows that $$ \Delta u\!\left(k\right)=u\!\left(k\right)-u\!\left(k-1\right)\!.$$ Summarizing the statements above, the main theorem in this paper is thus obtained. Theorem 3 Under A1–A2, if LMI (17) is solvable, then the memory feedback controller with preview action for system (1) is given by \begin{multline} u\!\left(k\right)=u\!\left(k-1\right)+K_{e} e\!\left(k\right)+K_{\Delta x} \Delta x\!\left(k\right)+\sum\limits_{i=1}^{M_{r} }k_{r} \!\left(i\right)\Delta r\!\left(k+i\right) \\[-2pt] +\,L_{e} e\!\left(k-d\right)+L_{\Delta x} \Delta x\!\left(k-d\right)+\sum\limits_{i=1}^{M_{r} }l_{r} \!\left(i\right)\Delta r\!\left(k+i-d\right)\!, \end{multline} (25) where $$K_{e},K_{\Delta x}, K_{r}, L_{e}, L_{\Delta x},L_{r}$$ are determined by (21), (22), (23) and (24), respectively. Under this controller, the output $$y\!\left (k\right )$$ can track the reference signal $$r\!\left (k\right )$$ without static error. Remark 4 Note that $$\sum _{i=1}^{M_{r} }k_{r} \!\left (i\right )\Delta r\!\left (k+i\right ) $$ and $$\sum _{i=1}^{M_{r} }l_{r} \!\left (i\right )\Delta r\!\left (k+i-d\right ) $$ in (25) are reference preview compensation terms. Thus, we say that Theorem 3 gives a controller with preview action for system (1). Remark 5 In particular, using the preceding method of retaining the state delay in the error system, Theorem 3 can also be applied to the Lipschitz system without time-delay \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+f\!\left(x\left(k\right)\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!.} \end{cases} \end{equation} (26) In this case, we only need to take $$\bar{A}_{d} $$ as a zero matrix. At this time the controller can still be taken as \begin{equation} \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)\!, \end{equation} (27) where the positive integer d can be selected in accordance with design needs. Note that d is not the state delay here. The main results in the studies by Lee et al. (2015) and Lee et al. (2016) focus on this case exactly. Remark 6 To show that the results proposed in this paper have broad applicability, let us consider another special case. If $$f\!\left (x\right )$$ in system (1) is linear, namely, $$f\!\left (x\right )=A_{l} x$$, in this case, we have the following two handling methods: (i) In A1, take $$M=A_{l} $$, $$N=F\!\left (x\right )=I$$, then solve LMI (17). (ii) Merging $$f\!\left (x(k)\right )=A_{l} x(k)$$ with Ax(k), system (1) is then rewritten as $$\begin{cases}{x\!\left(k+1\right)=\left(A+A_{l} \right)x\!\left(k\right)+A_{d} x\!\left(k-d\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!.} \end{cases} $$ In the design of the controller, we only need to replace A with $$A+A_{l} $$ in LMI (17) and take M = N = 0 in A1. Comparing the two approaches above, the second one is superior to the first. The reasons are as follows: (i) The second method reduces the dimension of LMI (17) and decreases the number of variables to be solved. The last two lines and two columns of the matrix on the left side of LMI (17) can be removed directly, and also there is no longer any need to solve the variable $$\varepsilon $$. (ii) If we adopt the first method, Lemma 1 will be used to amplify the matrix on the left side of (15) in the proof of Theorem 2 so as to obtain the LMI condition. This will increase the conservatism of the results obtained. 5. Numerical examples Example 1 Consider the popular single-link flexible joint robot system (Raghavan & Hedrick, 1994; Ibrir et al., 2005; Zemouche et al., 2008; Grandvallet et al., 2013; Defoort et al., 2016; Nguyen & Trinh, 2016) \begin{equation} \begin{cases}{\dot{x}\!\left(t\right)=Tx\!\left(t\right)+g\!\left(x\!\left(t\right)\right)+Du\!\left(t\right)\!,} \\{y\!\left(t\right)=Cx\!\left(t\right)\!,} \end{cases} \end{equation} (28) where $$ T=\left [\begin{array}{@{}cccc@{}}{0} &{1} &{0} &{0} \\{-48.6} &{-1.25} &{48.6} &{0} \\{0} &{0} &{0} &{1} \\{19.5} &{0} &{-19.5} &{0} \end{array}\right ]\!,\,g\!\left (x\!\left (t\right )\right )=\left [\begin{array}{@{}c@{}}{0} \\{0} \\{0} \\{-3.33\sin \left (x_{3} \!\left (t\right )\right )} \end{array}\right ]\!,\,D=\left [\begin{array}{@{}c@{}}{0} \\{21.6} \\{0} \\{0} \end{array}\right ]\!,\,C=\left [\begin{array}{@{}c@{}}{2.3} \\{0} \\{9.7} \\{0} \end{array}\right ]^{T}\!.$$ Due to the existence of some factors such as aging components, signal transmission delay and measurement lag, the time-delay issue frequently emerges in realistic physical systems. We now suppose time-delay exists in the state of system (28). Under the above assumption, system (28) can be rewritten under the form \begin{equation} \begin{cases}{\dot{x\!}\left(t\right)=Tx\!\left(t\right)+A_{\tau } x\!\left(t-\tau \right)+g\!\left(x\left(t\right)\right)+Du\!\left(t\right)\!,} \\{y\!\left(t\right)=Cx\!\left(t\right)\!,} \end{cases} \end{equation} (29) where $$\tau =0.03$$, $$A_{\tau } =\left [\begin{array}{@{}cccc@{}}{0} &{0} &{0} &{0} \\{0.27} &{0} &{0} &{0.5} \\{0} &{0} &{0} &{0} \\{0.16} &{0} &{-9.2} &{0} \end{array}\right ]$$, parameters $$T,g\!\left (x\left (t\right )\right )\!,D,C$$ are defined as described previously. For system (29), by Euler discretization with the sample period $$\delta $$, we can obtain the discrete-time model \begin{cases}{x\!\left(\delta (k+1)\right)=Ax\!\left(\delta k\right)+A_{d} x\!\left(\delta \!\left(k-d\right)\right)+f\!\left(x\!\left(\delta k\right)\right)+Bu\!\left(\delta k\right)\!,} \\{y\!\left(\delta k\right)=Cx\!\left(\delta k\right)\!,} \end{cases} where $$A=I+\delta T,\,f\!\left (x\right )=\delta g\!\left (x\right )\!,B=\delta D,A_{d} =\delta A_{\tau },d=\frac{\tau }{\delta } $$. Assume the previewable reference signal $$r\!\left (t\right )$$ is \begin{equation} r\!\left(t\right)=\begin{cases}0,&\quad t<0.3, \\15\!\left(t-0.3\right),&0.3\le t\le 0.5, \\3,&\quad t>0.5. \end{cases} \end{equation} (30) The preview length is assumed to be $$l_{r} $$, and denote $$M_{r} =\frac{l_{r} }{\delta } $$. Some matrices on the right side of equation (2) are obtained as follows: $$ M=\left[\begin{array}{@{}c@{}}{0} \\{0} \\{0} \\{\sqrt{3.33} } \end{array}\right]\!,\,N=\delta \left[\begin{array}{@{}cccc@{}}{0} &{0} &{-\sqrt{3.33} } &{0} \end{array}\right]\!,\,F(x)=\cos \left(x_{3} \right)$$ Thus, in this example, A1 and A2 are satisfied. The sample period is selected as $$\delta =0.01s$$. In order to observe the effectiveness of the proposed design methodology, we now consider four cases, including $$l_{r} =0s\,(\mathrm{i.e.}\ M_{r} =0)$$, $$l_{r} =0.1s\,\left (M_{r} =10\right ),\,l_{r} =0.3s\,\left (M_{r} =30\right )$$ and $$l_{r} =0.5s\,\left (M_{r} =50\right )$$. To obtain feedback gain matrices, LMI (17) must be solved according to Theorem 2. Then the output of the closed-loop system is derived. The gain matrices for different cases are given in Appendix A. In the simulation, the initial state is assumed to be zero. In Fig. 1, we can clearly see the output response of the closed-loop system in different cases. Fig. 1. View largeDownload slide Output response of system (29). Fig. 1. View largeDownload slide Output response of system (29). It can be seen from Fig. 1 that under the memory feedback controller, the closed-loop output accurately tracks the reference signal (30) whether there is preview action or not. Compared with no preview, output response with preview action has the following advantages: (i) Respond in advance. With the influence of preview action, the closed-loop output is able to perceive that the reference signal will change nearby and thus make a response in advance to the future changes. In Fig. 1, the reference signal begins to change in 0.3s. The output with preview action responds before 0.3s, while the output response without preview action is lagging, namely, the output responds after 0.3s. (ii) The overshoot is drastically reduced, and the settling time is significantly decreased. As the preview length increases, the settling time is gradually shortened. In particular, when the preview length is increased to $$M_{r} =50$$, the overshoot becomes very small. This fully demonstrates the validity and superiority of the preview controller in improving the transient performance of the closed-loop system. In addition, we point out that if there is no time-delay phenomenon in system (29), namely, $$A_{\tau } =0$$, then system (29) reduces to system (28). By Euler discretization and according to Remark 5, our results remain valid in this case. We choose d = 1, then the simulation is done in Fig. 2. It can be seen similarly that adding preview action of the reference signal can diminish the overshoot and shorten the settling time. In this case, the feedback gain matrices obtained through computing LMI (17) are omitted here. Fig. 2. View largeDownload slide Output response of system (28). Fig. 2. View largeDownload slide Output response of system (28). The engineering example above verifies the advantage of preview control theory in improving the tracking performance of the closed-loop system. A continuous-time system is provided in Example 1, where the non-linearity is relatively simple and satisfies $$f\!\left (0\right )=0$$. Further, in order to show that the proposed methodology can be applied to a broader class of Lipschitz non-linear systems (including the case of $$f\!\left (0\right )\ne 0$$), a discrete-time system will be considered below. Example 2 Consider system \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+A_{d} x\!\left(k-d\right)+f\!\left(x\left(k\right)\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!,} \end{cases}\end{equation} (31) where \begin{align*} A&=\left[\begin{array}{@{}cc@{}}{-0.98} &{-3.5} \\{0} &{0.2} \end{array}\right]\!,\,A_{d} =\left[\begin{array}{@{}cc@{}}{0.7} &{0.3} \\{0} &{0.5} \end{array}\right]\!,\,B=\left[\begin{array}{@{}c@{}}{0} \\{0.6} \end{array}\right]\!,\,C=\left[\begin{array}{@{}cc@{}}{-0.58} &{0.16} \end{array}\right]\!,\\f\left(x\right)&=\left[\begin{array}{@{}c@{}}{0.02\arctan \left(x_{2} \right)} \\{-0.01\cos \left(x_{1} \right)+0.04x_{2} } \end{array}\right]\!,\,d=2.\end{align*} Note that $$f\!\left (0\right )\ne 0$$, and one computes that $$ f^{\prime} \!\left(x\right)=MF\!\left(x\right)N,$$ where $$ M=\left[\begin{array}{@{}cc@{}}{0} &{0.1} \\{0.2} &{0} \end{array}\right]\!,\,F\!\left(x\right)=\left[\begin{array}{@{}cc@{}}{0.5\sin \left(x_{1} \right)} &{0.2} \\{0} &{\frac{0.2}{1+{{x_{2}^{2}}} } } \end{array}\right]\!,\,N=\left[\begin{array}{@{}cc@{}}{0.1} &{0} \\{0} &{1} \end{array}\right].$$ It can be verified that $$F^{T} \!\left (x\right )F\!\left (x\right )\le I$$, A1 is thus satisfied. Moreover, assume the previewable reference signal is \begin{equation} r\!\left(k\right)=\begin{cases}3,&k\ge 30, \\0,&k<30. \end{cases} \end{equation} (32) Then A2 is satisfied. Now we discuss three cases, including $$M_{r} =0$$, $$M_{r} =7$$, $$M_{r} =11$$. By solving LMI (17), the feedback gain matrices are obtained. Then the output of the closed-loop system is derived accordingly. The gain matrices for different cases are given in Appendix B. The simulation, with zero initial state, is done in Fig. 3. Fig. 3. View largeDownload slide Output response of system (31). Fig. 3. View largeDownload slide Output response of system (31). Figure 3 shows that the closed-loop output, with or without preview, can track the reference signal (32) accurately. By observing the response curves between time 20 and time 35, we see that the output with preview action makes a response before the reference signal changes, while the output response without preview action is lagging. Also, by adding the preview action, the overshoot of the output is reduced and the settling time is shortened. Thus, the proposed preview controller achieves better transient performance. 6. Conclusion In this paper, we study the design of the tracking controller with preview action for a class of discrete-time Lipschitz non-linear time-delay systems. First, according to the error system method in preview control theory, the augmented error system containing previewable reference information is constructed. Second, the memory state feedback controller is designed, and the LMI condition for the stability of the closed-loop system is formulated. Furthermore, a concrete form of the preview tracking controller for the original system is derived. Finally, two illustrative examples show the advantage of the proposed preview controller in enhancing the tracking performance of the closed-loop system. 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Google Scholar CrossRef Search ADS Appendix A The gain matrices for different cases are as follows: When $$M_{r} =0$$, we obtain $${K_{e} = {-4.5493,}}$$ $${K_{\varDelta x} =(-177.3916\ -\!6.2611\ -\!916.9090\ -\!87.0118),}$$ $${L_{e} = {0.0035,}} $$ $${L_{\varDelta x} =( -0.1486\ -\!0.0027\ \ 6.7613\ -\!0.0360).}$$ When $$M_{r} =10$$, we obtain $${K_{e} = {-4.6625,}}$$ $${K_{\varDelta x} =( {-182.5432\ -\!6.2799\ -\!981.3317\ -\!91.2319}),}$$ $${K_{r} =( {4.6392\ \ 4.4755\ \ 4.4547\ \ 4.4607\ \ 4.4259\ \ 4.3713\ \ 4.3036\ \ 4.2215\ \ 4.1391\ \ 4.0633}),}$$ $${L_{e} = {-2.7169}\times 10^{-4},}$$ $$L_{\varDelta x} =( -0.1410\ \ 0.0003\ \ 7.3721\ -\!0.0197),$$ $${L_{r} =10^{-3} ( 0.8118\ \ 0.4451\ \ 0.2516\ \ 0.2111\ \ 0.1452\ \ 0.1153\ \ 0.0851\ \ 0.0453\ \ 0.0114\ -\!0.0030).}$$ When $$M_{r} =30$$, we obtain $$K_{e} =-3.5939,$$ $$K_{\varDelta x} =(-160.7484\ -\!6.0741\ -\!720.9379\ -\!75.7690),$$ $$K_{r} =(3.5820\ \ 3.4711\ \ 3.4574\ \ 3.4273\ \ 3.3713\ \ 3.2875\ \ 3.1875\ \ 3.0816\ \ 2.9757\ \ 2.8720\ \ 2.7745 2.6855\ \ 2.6062\ \ 2.5372\ \ 2.4782\ \ 2.4283\ \ 2.3863\ \ 2.3509\ \ 2.3202\ \ 2.2925\ \ 2.2658\ \ 2.2382\ \ 2.2081 2.1736\ \ 2.1332\ \ 2.0854\ \ 2.0285\ \ 1.9608\ \ 1.8790\ \ 1.7750),$$ $$L_{e} =-0.0013,$$ $$L_{\varDelta x} =(-0.1224\ \ 0.0009\ \ 6.2895\ -\!0.0227),$$ $$L_{r} =(0.0015\ \ 0.0006\ \ 0.0004\ \ 0.0001\ -\!0.0001\ -\!0.0003\ -\!0.0004\ -\!0.0004\ -\!0.0004\ -\!0.0003 -\!0.0002\ -\!0.0002\ -\!0.0001\ -\!0.0000\ \ 0.0000\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001 0.0001\ \ 0.0001\ \ 0.0000\ \ 0.0000\ \ 0.0000\ -\!0.0000\ -\!0.0000\ -\!0.0000\ -\!0.0000).$$ When $$M_{r} =50$$, we obtain $$K_{e} =-4.7532,$$ $$K_{\varDelta x} =10^{3} (-0.1948\ -\!0.0065\ -\!1.0491\ -\!0.0994),$$ $$K_{r} =(4.7352\ \ 4.6054\ \ 4.6155\ \ 4.6016\ \ 4.5566\ \ 4.4938\ \ 4.4143\ \ 4.3259\ \ 4.2398\ \ 4.1551\ \ 4.0775 4.0097\ \ 3.9520\ \ 3.9034\ \ 3.8620\ \ 3.8268\ \ 3.7967\ \ 3.7692\ \ 3.7420\ \ 3.7127\ \ 3.6790\ \ 3.6383\ \ 3.5883 3.5274\ \ 3.4539\ \ 3.3667\ \ 3.2649\ \ 3.1480\ \ 3.0159\ \ 2.8687\ \ 2.7068\ \ 2.5310\ \ 2.3422\ \ 2.1415\ \ 1.9303 1.7100\ \ 1.4823\ \ 1.2488\ \ 1.0116\ \ 0.7723\ \ 0.5329\ \ 0.2953\ \ 0.0615\ -\!0.1668\ -\!0.3877\ -\!0.5994 -\!0.8003 \ -\!0.9886\ -\!1.1625\ -\!1.3199),$$ $$L_{e} =0.0033, $$ $$L_{\varDelta x} =(-0.1465\ -\!0.0012\ \ 7.7215\ -\!0.0232),$$ $$L_{r} =(-0.0007\ -\!0.0013\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012 -\!0.0012\ -\!0.0011\ -\!0.0011\ -\!0.0010\ -\!0.0009\ -\! 0.0009\ -\! 0.0008\ -\! 0.0007\ -\! 0.0007 -\! 0.0006\ -\! 0.0005\ -\! 0.0005\ -\! 0.0004\ -\! 0.0004\ -\! 0.0003\ -\! 0.0003\ -\! 0.0002\ -\! 0.0002\ -\! 0.0002 -\! 0.0001\ -\! 0.0001\ -\! 0.0000\ -\! 0.0000\ \ 0.0000\ \ 0.0000\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0000\ \ 0.0000\ \ 0.0000).$$ Appendix B The gain matrices for different cases are as follows: When $$M_{r} =0$$, we get $$K_{e} = -0.1449,$$ $$K_{\varDelta x} =( 0.3531\ \ 0.9324),$$ $$L_{e} = -1.0399\times 10^{-4},$$ $$L_{\varDelta x} =( -0.2493\ -\!0.9401).$$ When $$M_{r} =7$$, we get $$K_{e} = -0.1138,$$ $$K_{\varDelta x} =(0.3654\ \ 0.9711),$$ $$K_{r} = (0.1139\ \ 0.1136\ \ 0.1047\ \ 0.0950\ \ 0.0869\ \ 0.0794\ \ 0.0716),$$ $$L_{e} = -4.0142\times 10^{-5},$$ $$L_{\varDelta x} =( -0.2598\ -\!0.9445),$$ $$L_{r} =10^{-4} (0.3006\ \ 0.5208\ \ 0.5602\ \ 0.4086\ \ 0.3412\ \ 0.2357\ \ 0.1076).$$ When $$M_{r} =11$$, we get $$K_{e} = -0.1465,$$ $$K_{\Delta x} = (0.3470\ \ 0.9047),$$ $$K_{r} = (0.1465\ \ 0.1459\ \ 0.1312\ \ 0.1147\ \ 0.1019\ \ 0.0893\ \ 0.0785\ \ 0.0682\ \ 0.0588\ \ 0.0506\ \ 0.0434),$$ $$L_{e} = -1.3983\times 10^{-4},$$ $$L_{\Delta x} =( -0.2497\ -\!0.9401),$$ $$L_{r} =10^{-3} ( 0.1889\ \ 0.2165\ \ 0.1660\ \ 0.0601\ \ 0.0095\ \ 0.0008\ -\!0.0049\ -\!0.0093\ -\!0.0125 -\!0.0137\ -\!0.0110).$$ © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Preview tracking control for a class of discrete-time Lipschitz non-linear time-delay systems

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Oxford University Press
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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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Abstract

Abstract The design of a preview tracking controller for a class of discrete-time Lipschitz non-linear time-delay systems is considered in this paper. First, based on preview control theory, we introduce a difference operator and deduce an augmented error system which contains previewable information on the reference signal. The original tracking problem is thus transformed into a regulation problem. Second, we design a memory state feedback controller for the augmented error system. Applying Lyapunov’s second method, the sufficient condition for stability of the closed-loop system is derived in the form of linear matrix inequality (LMI), and the preview controller is obtained accordingly. Finally, to show the superiority of the controller proposed, two numerical examples are presented. 1. Introduction Preview control can improve the performance of a control system by fully utilizing the known future information of the reference signals or the disturbances, and it has been successfully applied in many fields of engineering (Hess & Chan, 1988; Peng & Tomizuka, 1993; Laks et al., 2011; Shimmyo et al., 2013). In comparison with other control methods, preview control has the following advantages: first, by exploiting the previewable future information, the purpose of enhancing the tracking performance of the closed-loop system can be achieved. Second, preview feedforward action can be added to the existing feedback control system expediently (Tomizuka & Whitney, 1975; Tsuchiya & Egami, 1994; Zhen, 2016). By now, preview control has already formed a relatively complete theoretical system in areas such as multirate systems, and linear discrete- or continuous-time systems together with descriptor systems (Tomizuka, 1975; Katayama et al., 1985; Katayama & Hirono, 1987; Takaba, 2000; Liao et al., 2013; Cao & Liao, 2015; Li & Liao, 2017).However, there are few research achievements on preview control for non-linear systems. In fact, the systems in practical engineering are mostly non-linear, and also, linear systems in the strict sense do not exist. The design of a preview controller for non-linear systems, therefore, is of great importance. The so-called Lipschitz non-linear system refers to the system where the non-linearity satisfies the Lipschitz condition. According to the viewpoints in the studies by Raghavan & Hedrick (1994), Zemouche & Boutayeb (2013), Miao & Li (2015) and Defoort et al. (2016), this kind of system is extensively present in practical problems. For a Lipschitz non-linear system, research work at present mainly focuses on observer design (Ibrir et al., 2005; Ibrir et al., 2006; Zemouche et al., 2008; Xu et al., 2009), feedback stabilization (Sun & Liu, 2007; Fu et al., 2013; Zuo et al., 2016) and tracking control (Rehan et al., 2011; Mobayen & Tchier, 2016). For example, the problem of designing an observer as well as an observer-based feedback controller for a class of discrete-time Lipschitz non-linear systems was investigated in the study by Ibrir et al. (2005). Subsequently, some results on the observer design for general discrete-time non-linear systems were put forward in the studies by Xiao (2006), Karafyllis & Kravaris (2007) and Califano et al. (2009). For a continuous-time Lipschitz non-linear system, the authors in the study by Rehan et al. (2011) developed a non-linear feedback controller under which the output of the closed-loop system can track the reference signal accurately. Based on the system discussed in the study by Ibrir et al. (2005), Sun & Liu (2007) studied the state feedback along with output feedback control of a class of continuous-time non-linear systems. However, the results on preview control for Lipschitz non-linear systems have not been reported in the literature thus far. This has motivated our study in this paper. In addition, the time-delay phenomenon frequently appears in practical engineering systems such as communication systems, electric power systems and biological systems. Its presence can degrade performance of the control system or even lead to instability. In the study of controller design for time-delay systems, it is known that the memoryless feedback controller has an advantage of easy implementation in physics. However, its control performance cannot be better than a memory feedback controller when the information on the size of time-delay is available. The memory controller can simultaneously utilize current and past stored state information, thereby achieving superior system performance. Some results related to this issue were reported in the studies by Moon et al. (2001), Azuma et al. (2002) and Ji et al. (2009). In this paper, our objective is to design a preview tracking controller for a class of discrete-time Lipschitz non-linear time-delay systems. We assume that the reference signal is previewable. First, by applying the difference method in the study by Tsuchiya & Egami (1994), an augmented error system is deduced, so that the original tracking problem is transformed into a stabilization problem of a discrete-time linear system. Second, for the augmented error system obtained, we design a memory state feedback controller by employing current and past state information. Based on Lyapunov’s second method, the sufficient condition to ensure asymptotical stability of the closed-loop system is developed. The preview controller of the original system will then be derived accordingly. Finally, numerical examples are presented, to reveal the superiority of the controller proposed. It is assumed throughout this paper that the state is available in real-time and the past states can be stored in memory. Notations Throughout this paper, R is the set of real numbers. For a matrix P, P > 0 means P is symmetric positive definite. For matrices P and Q, P > Q means P − Q > 0. I and 0 denote the identity matrix and the zero matrix with appropriate dimension, respectively. $$f{{^\prime }}\!(x)$$ represents the Jacobi matrix of the vector function $$f\!\left (x\right )$$. 2. Problem formulation Consider the discrete-time non-linear system \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+A_{d} x\!\left(k-d\right)+f\!\left(x\!\left(k\right)\right)+Bu\!\left(k\right)} \\{y\!\left(k\right)=Cx\!\left(k\right)} \end{cases} \end{equation} (1) where $$x(k)\in \mathbf{R}^{n}, u(k)\in \mathbf{R}^{m}, y(k)\in \mathbf{R}^{q} $$ represent the state, control input and output of the system, respectively; matrices $$A,\,A_{d} \in \mathbf{R}^{n\times n} ,\,B\in \mathbf{R}^{n\times m} ,C\in \mathbf{R}^{q\times n} $$ are known and constant. d > 0 is a positive integer which stands for the time-delay of the system. The non-linearity $$f\!\left (x\right )\!\in\! \mathbf{R}^{n} $$ is a time-varying vector. The non-linearity of system (1) satisfies the following condition. A1. The non-linearity f(x) is differentiable, $$f{^{\prime }}\!(x)$$ has the form of \begin{equation} f{^{\prime} }\!(x)=MF(x)N \end{equation} (2) for some $$M\in \mathbf{R}^{n\times r},\,N\in \mathbf{R}^{s\times n} $$ and $$F(x)\in \mathbf{R}^{r\times s} $$ is a norm-bounded matrix satisfying $$F^{T} \!(x)F(x)\le I$$. If A1 is fulfilled, it can be proved that f(x) satisfies the Lipschitz condition. Thus, f(x) is called a Lipschitz non-linearity, and accordingly, system (1) is called a Lipschitz non-linear system. Remark 1 The Lipschitz condition has been widely used in the design of non-linear systems. This kind of Lipschitz non-linearity verifying A1 does not involve any approximation of non-linearity by its norm, and it is pointed out in the study by Ibrir et al. (2005) that this property will reduce the conservatism of the results obtained. Moreover, Ibrir et al. (2005) and Sun & Liu (2007) all require that M, N be n × n square matrices. This paper reduces the requirement for the order of M, N so that the dimension of the LMI derived later decreases, which will lower the computational cost. Remark 2 In order to deal with the problem more conveniently, for such a Lipschitz non-linear system satisfying A1, the function f(x) is required to be vanishing at the origin (i.e. $$f\!\left (0\right )=0$$) (Ibrir et al., 2005; Ibrir et al., 2006; Sun & Liu, 2007; Xu et al., 2009). But in the present paper, this restriction is dropped by adopting a difference method. This study thus covers a broader class of Lipschitz non-linear systems. To establish a preview tracking controller for system (1), we need the following assumption that the reference signal is previewable. A2. The preview length of reference signal r(k) is $$M_{r} $$, that is, at each time k, the $$M_{r} $$ future values, $$r(k+1),\cdots\!\,,r(k+M_{r})$$ as well as the present and past values of the reference signal, are available. The future values of the reference signal are assumed not to change beyond $$k+M_{r} $$, namely, $$ r(k+i)=r(k+M_{r}),\,i=M_{r} +1,\,M_{r} +2,\cdots \!\,.$$ Remark 3 We should mention that A2 is a basic assumption in preview control theory. Theoretical study and engineering practice have shown that only the previewable signal of a recent period significantly influences the performance of the control system, while a reference signal that exceeds the previewable range has little influence on the system performance (Tsuchiya & Egami, 1994). Therefore, the values beyond the preview length are generally assumed to be constant. Indeed, the ordinary feedback control systems do not fully utilize the previewable signal, or equivalently, the preview length is zero. The tracking error signal e(k) is defined as the difference between the output y(k) and the reference signal r(k), i.e. \begin{equation} e(k)=y(k)-r(k). \end{equation} (3) This paper aims to design a memory state feedback controller with preview action such that the output y(k) of the closed-loop system can track the reference signal r(k) without static error, that is, $$ \mathop{\lim }\limits_{k\to \infty } e\!\left(k\right)=\mathop{\lim }\limits_{k\to \infty } \! \big(y\!\left(k\right)-r\!\left(k\right)\big)=0.$$ Let us now introduce two useful lemmas for the development of our work. Lemma 1 (de Souza & Li, 1999) Let M, N and F(t) be real matrices of appropriate dimensions, with F(t) satisfying $$F^{T}\!(t)F(t)\le I$$. Then, the following inequality holds for any constant $$\mu>0$$: $$ MF\!\left(t\right)N+\big[MF\!\left(t\right)N\big]^{T} \le \mu^{-1} MM^{T} +\mu N^{T} N.$$ Lemma 2 (Boyd et al., 1994 (Schur complement lemma)) Symmetric matrix $$ \left [{{S_{11}} \atop {S_{12}^{T}}} \quad {{S_{12}} \atop {S_{22}}} \right ] < 0$$ if and only if one of the following two conditions is satisfied: (i) $$S_{11} <0,S_{22} -S_{12}^{T} S_{11}^{-1} S_{12} <0$$; (ii) $$S_{22} <0,S_{11} -S_{12} S_{22}^{-1} S_{12}^{T} <0$$. 3. Construction of the augmented error system We adopt the augmented error system method to study the problem of controller design for system (1). The difference operator is defined as $$ \Delta x(k)=x(k)-x(k-1).$$ Taking the difference operator on both sides of the first equation in (1) leads to the following: \begin{equation} \Delta x\!\left(k+1\right)=A\Delta x\!\left(k\right)+A_{d} \Delta x\!\left(k-d\right)+f\!\left(x\left(k\right)\right)-f\!\left(x\left(k-1\right)\right)+B\Delta u\!\left(k\right)\!. \end{equation} (4) Applying the Mean-Value Theorem (Ortega & Rheinboldt, 2000), one can obtain $$ f\!\left(x\!\left(k\right)\right)-f\!\left(x\!\left(k-1\right)\right)={{\int_{0}^{1}}}f{^{\prime} }\!\left(x\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \Delta x\!\left(k\right)\,\mathrm{d}\lambda .$$ Then, equation (4) becomes \begin{equation} \Delta x\!\left(k+1\right)=\left(A+A_{1} \right)\Delta x\!\left(k\right)+A_{d} \Delta x\!\left(k-d\right)+B\Delta u\!\left(k\right)\!, \end{equation} (5) where \begin{equation} A_{1} ={{\int_{0}^{1}}}MF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda . \end{equation} (6) Taking the same difference operator on the error signal results in $$\Delta e(k)=\Delta y(k)-\Delta r(k)$$, and combining (1) and (5) yields \begin{align*} {\Delta e\!\left(k+1\right)} &= {C\Delta x\!\left(k+1\right)-\Delta r\!\left(k+1\right)} \\ &= {C\left(A+A_{1} \right)\Delta x\!\left(k\right)+CA_{d} \Delta x\!\left(k-d\right)+CB\Delta u\!\left(k\right)-\Delta r\!\left(k+1\right)\!.} \end{align*} Since $$\Delta e\!\left (k+1\right )=e\!\left (k+1\right )-e\!\left (k\right )$$, we have \begin{equation} e\!\left(k+1\right)=e\!\left(k\right)+C\!\left(A+A_{1} \right)\Delta x\!\left(k\right)+CA_{d} \Delta x\!\left(k-d\right)+CB\Delta u\!\left(k\right)-\Delta r\!\left(k+1\right)\!. \end{equation} (7) Considering (5) in conjunction with (7), it follows that \begin{equation} \tilde{x}\!\left(k+1\right)=\big(\tilde{A}+\tilde{A}_{1} \big)\tilde{x}\!\left(k\right)+\tilde{A}_{d} \tilde{x}\!\left(k-d\right)+\tilde{B}\Delta u\!\left(k\right)+G\Delta r\!\left(k+1\right)\!, \end{equation} (8) where $$ \tilde{x}\!\left(k\right)=\left[\begin{array}{@{}c@{}}{e\!\left(k\right)} \\{\Delta x\!\left(k\right)} \end{array}\right]\!,\,\tilde{A}=\left[\begin{array}{@{}cc@{}} I &{CA} \\{0} & A \end{array}\right]\!,\,\tilde{A}_{1} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{1} } \\{0} &{A_{1} } \end{array}\right]\!,\,\tilde{A}_{d} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{d} } \\{0} &{A_{d} } \end{array}\right]\!,\,\tilde{B}=\left[\begin{array}{@{}c@{}}{CB} \\ B \end{array}\right]\!,\,G=\left[\begin{array}{@{}c@{}}{-I} \\{0} \end{array}\right]\!.$$ Specifically, one can notice from (6) that $$ \tilde{A}_{1} =\left[\begin{array}{@{}cc@{}}{0} &{CA_{1} } \\{0} &{A_{1} } \end{array}\right]=\left[\begin{array}{@{}cc@{}}{0} &{{{\int_{0}^{1}}}CMF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda } \\{0} &{{{\int_{0}^{1}}}MF\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) N\,\mathrm{d}\lambda } \end{array}\right]={{\int_{0}^{1}}}\left[\begin{array}{@{}c@{}}{CM} \\ M \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \left[\begin{array}{@{}cc@{}}{0} & N \end{array}\right]\,\mathrm{d}\lambda. $$ Denoting $$\tilde{M}=\left [{{CM} \atop M}\right ],\tilde{N}= [0 \ N]$$, then we obtain \begin{equation} \tilde{A}_{1} ={{\int_{0}^{1}}}\tilde{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \tilde{N}\,\mathrm{d}\lambda . \end{equation} (9) Equation (8) is the constructed error system. In the following, let us deduce the augmented error system. To bring in the previewable reference signal, we define the following vector: $$ x_{r} \left(k\right)=\left[\begin{array}{@{}c@{}}{\Delta r\!\left(k+1\right)} \\{\Delta r\!\left(k+2\right)} \\{\vdots } \\{\Delta r\!\left(k+M_{r} \right)} \end{array}\right]\!.$$ By exploiting A1, we get \begin{equation} x_{r} \!\left(k+1\right)=A_{r} x_{r} \!\left(k\right)\!, \end{equation} (10) where $$ A_{r} =\left[\begin{array}{@{}ccccc@{}}{0} & I &{0} &{\cdots } &{0} \\{0} &{0} & I &{\cdots } &{0} \\{\vdots } &{\vdots } &{\vdots } &{} &{\vdots } \\{0} &{0} &{0} &{\cdots } & I \\{0} &{0} &{0} &{\cdots } &{0} \end{array}\right]\!.$$ Combining (8) and (10), we obtain \begin{equation} \bar{x}\!\left(k+1\right)=\left(\bar{A}+\bar{A}_{1} \right)\bar{x}\!\left(k\right)+\bar{A}_{d} \bar{x}\!\left(k-d\right)+\bar{B}\Delta u\!\left(k\right)\!. \end{equation} (11) This is the augmented error system constructed in this paper, where $$ \bar{x}\!\left(k\right)=\left[\begin{array}{@{}c@{}}{\tilde{x}\!\left(k\right)} \\{x_{r} \!\left(k\right)} \end{array}\right]\!,\,\tilde{G}=\left[\begin{array}{@{}cccc@{}} G &{0} &{\cdots } &{0} \end{array}\right]\!,\,\bar{A}=\left[\begin{array}{@{}cc@{}}{\tilde{A}} &{\tilde{G}} \\{0} &{A_{r} } \end{array}\right]\!,\,\bar{A}_{1} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{1} } &{0} \\{0} &{0} \end{array}\right]\!,\,\bar{A}_{d} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{d} } &{0} \\{0} &{0} \end{array}\right]\!,\,\bar{B}=\left[\begin{array}{@{}c@{}}{\tilde{B}} \\{0} \end{array}\right]\!.$$ From (9) it follows that $$ \bar{A}_{1} =\left[\begin{array}{@{}cc@{}}{\tilde{A}_{1} } &{0} \\{0} &{0} \end{array}\right]=\left[\begin{array}{@{}cc@{}}{{{\int_{0}^{1}}}\tilde{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \tilde{N}\,\mathrm{d}\lambda } &{0} \\{0} &{0} \end{array}\right]={{\int_{0}^{1}}}\left[\begin{array}{@{}c@{}}{\tilde{M}} \\{0} \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \left[\begin{array}{@{}cc@{}}{\tilde{N}} &{0} \end{array}\right]\,\mathrm{d}\lambda .$$ Denoting $$\bar{M}=\left [{{\tilde{M}} \atop {0}} \right ],\bar{N}= [\tilde{N} \ 0] $$, then we get \begin{equation} \bar{A}_{1} ={{\int_{0}^{1}}}\bar{M}F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right) \bar{N}\,\mathrm{d}\lambda . \end{equation} (12) For the augmented error system (11), let us design the following memory state feedback controller \begin{equation} \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)\!, \end{equation} (13) where the feedback gain matrices K, L will be determined later. Substituting this controller into system (11), we obtain the closed-loop system \begin{equation} \bar{x}\!\left(k+1\right)=\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)\bar{x}\!\left(k\right)+\left(\bar{A}_{d} +\bar{B}L\right)\bar{x}\!\left(k-d\right)\!. \end{equation} (14) Hereafter, we will determine the gain matrices K, L via the LMI method. 4. Design of a memory state feedback controller with preview action Theorem 1 Under A1–A2, the closed-loop system (14) is globally asymptotically stable if there exist matrices P > 0, S > 0 and matrices K, L so that the following matrix inequality condition holds: \begin{equation} \left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{A}_{1} +\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]<0. \end{equation} (15) Proof Consider the Lyapunov–Krasovskii functional \begin{equation} V\!\left(\bar{x}\left(k\right)\right)=\bar{x}^{T} \!\left(k\right)P\bar{x}\!\left(k\right)+\sum_{i=k-d}^{k-1}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right) \end{equation} (16) It is clear that V is positive definite due to P > 0, S > 0. Taking the difference of V along the trajectories of system (14), it follows that \begin{align*} \Delta V\!\left(\bar{x}\left(k\right)\right)&=V\!\left(\bar{x}\left(k\right)\right)-V\!\left(\bar{x}\left(k-1\right)\right)\\ &=\bar{x}^{T} \!\left(k\right)P\bar{x}\!\left(k\right)+\sum_{i=k-d}^{k-1}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right)-\bar{x}^{T} \!\left(k-1\right)P\bar{x}\!\left(k-1\right)-\sum_{i=k-d-1}^{k-2}\bar{x}^{T} \!\left(i\right)S\bar{x}\!\left(i\right)\\ &=\bar{x}^{T} \!\left(k-1\right)\left[\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)-P+S\right]\bar{x}\!\left(k-1\right)\\ &\quad+2\bar{x}^{T} \!\left(k-1\right)\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)\bar{x}\!\left(k-d-1\right)\\ &\quad+\bar{x}^{T} \!\left(k-d-1\right)\left[\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)-S\right]\bar{x}\!\left(k-d-1\right)\\ &=\left[ \begin{array}{@{}c@{}} {\bar{x}\left(k-1\right)}\\{\bar{x}\left(k-d-1\right)} \end{array} \right]^{T} \Omega\! \left[\begin{array}{@{}c@{}}{\bar{x}\left(k-1\right)} \\{\bar{x}\left(k-d-1\right)} \end{array}\right]\!, \end{align*} where $$ \Omega =\left[\begin{array}{@{}cc@{}}{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)-P+S} &{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)} &{\left(\bar{A}_{d} +\bar{B}L\right)^{T} P\!\left(\bar{A}_{d} +\bar{B}L\right)-S} \end{array}\right]\!.$$ Now we prove $$\Omega <0$$. Notice that $$ \left[\begin{array}{@{}cc@{}}{-P^{-1} } &{0} \\{0} &{-S^{-1} } \end{array}\right]<0.$$ By applying the Schur complement lemma (Lemma 2), inequality (15) implies $$ \left[\begin{array}{@{}cc@{}}{-P} &{0} \\{0} &{-S} \end{array}\right]-\left[\begin{array}{@{}cc@{}}{\left(\bar{A}+\bar{A}_{1} +\bar{B}K\right)^{T} } & I \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} \end{array}\right]\left[\begin{array}{@{}cc@{}}{-P^{-1} } &{0} \\{0} &{-S^{-1} } \end{array}\right]^{-1} \left[\begin{array}{@{}cc@{}}{\bar{A}+\bar{A}_{1} +\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\ I &{0} \end{array}\right]<0.$$ The left side of the above inequality is $$\Omega $$ exactly, thus $$\Delta V$$ is negative definite. Moreover, $$V\!\left (\bar{x}\left (k\right )\right )$$ is radially unbounded, that is, when $$\left \| \bar{x}\!\left (k\right )\right \| \to \infty $$, we have $$V\!\left (\bar{x}\left (k\right )\right )\to \infty $$, hence the closed-loop system (14) is globally asymptotically stable. This ends the proof. Due to the fact that the element $$\bar{A}_{1} $$ in (15) contains an integral term related to the non-linearity, it is difficult to apply Theorem 1 to practical problems. Therefore, we present a computationally tractable LMI form in what follows. Theorem 2 Under A1–A2, the system (14) is globally asymptotically stable if there exist a constant $$0<\varepsilon <1$$, matrices X > 0, Y > 0 and matrices Q, H so that the following LMI condition holds: \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-X} &{0} &{\bar{A}X+\bar{B}Q} &{\bar{A}_{d} Y+\bar{B}H} &{\bar{M}} &{0} \\{0} &{-Y} & X &{0} &{0} &{0} \\{X\bar{A}^{T} +Q^{T} \bar{B}^{T} } & X &{-X} &{0} &{0} &{X\bar{N}^{T} } \\{Y\bar{A}_{d}^{T} +H^{T} \bar{B}^{T} } &{0} &{0} &{-Y} &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}X} &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (17) Then, the controller is determined by (13), and the gain will be given by: $$K=QX^{-1},L=HY^{-1} $$. Proof Let the matrix on the left side of (15) be $$\Phi $$. We can always partition matrix $$\Phi $$ as $$ \Phi =\left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}+\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]+\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{A}_{1} } &{0} \\{0} &{0} &{0} &{0} \\{\bar{A}_{1}^{T} } &{0} &{0} &{0} \\{0} &{0} &{0} &{0} \end{array}\right].$$ By virtue of (12), we get \begin{multline} \Phi \!=\!\left[\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}\!+\!\bar{B}K} &{\bar{A}_{d} \!+\!\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left(\bar{A}\!+\!\bar{B}K\right)^{T} } & I &{-P} &{0} \\{\left(\bar{A}_{d} \!+\!\bar{B}L\right)^{T} } &{0} &{0} &{-S} \end{array}\right]\!\\+\!{{\int_{0}^{1}}}\!\left\{\!\left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]\!F\!\left(x\!\left(k\right)\!-\!\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]\!\right. \left.+\!\left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]\!F^{T} \!\!\left(x\!\left(k\right)\!-\!\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right]\right\}\,\mathrm{\!d}\lambda. \end{multline} (18) Applying Lemma 1 to the integrand, it follows that \begin{multline*}{\left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]F\!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]+\left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]F^{T} \!\left(x\!\left(k\right)-\lambda \Delta x\!\left(k\right)\right)\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right]} \\{\le \varepsilon \left[\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right]\left[\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right] +\varepsilon^{-1} \left[\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right]\left[\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right]} \end{multline*} Substituting the above into (18) and taking the integral properties into account, one can obtain $$\Phi \le \left [\begin{array}{@{}cccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} \\{0} &{-S^{-1} } & I &{0} \\{\left (\bar{A}+\bar{B}K\right )^{T} } & I &{-P} &{0} \\{\left (\bar{A}_{d} +\bar{B}L\right )^{T} } &{0} &{0} &{-S} \end{array}\right ]+\left [\begin{array}{@{}c@{}}{\bar{M}} \\{0} \\{0} \\{0} \end{array}\right ]\left [\begin{array}{@{}cccc@{}}{\bar{M}^{T} } &{0} &{0} &{0} \end{array}\right ]+\frac{1}{\varepsilon } \left [\begin{array}{@{}c@{}}{0} \\{0} \\{\bar{N}^{T} } \\{0} \end{array}\right ]\left [\begin{array}{@{}cccc@{}}{0} &{0} &{\bar{N}} &{0} \end{array}\right ]=\Psi .$$ Thus, $$\Phi <0$$ if $$\Psi <0$$. By employing the Schur complement lemma (Lemma 2) again, $$\Psi <0$$ is equivalent to \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-P^{-1} } &{0} &{\bar{A}+\bar{B}K} &{\bar{A}_{d} +\bar{B}L} &{\bar{M}} &{0} \\{0} &{-S^{-1} } & I &{0} &{0} &{0} \\{\left(\bar{A}+\bar{B}K\right)^{T} } & I &{-P} &{0} &{0} &{\bar{N}^{T} } \\{\left(\bar{A}_{d} +\bar{B}L\right)^{T} } &{0} &{0} &{-S} &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}} &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (19) Now, performing the congruence transformation to inequality (19) with $$diag\left (I,I,P^{-1},S^{-1},I,I\right )$$ leads to \begin{equation} \left[\begin{array}{@{}cccccc@{}}{-P^{-1} } &{0} &{\bar{A}P^{-1} +\bar{B}KP^{-1} } &{\bar{A}_{d} S^{-1} +\bar{B}LS^{-1} } &{\bar{M}} &{0} \\{0} &{-S^{-1} } &{P^{-1} } &{0} &{0} &{0} \\{P^{-1} \bar{A}^{T} +P^{-1} K^{T} \bar{B}^{T} } &{P^{-1} } &{-P^{-1} } &{0} &{0} &{P^{-1} \bar{N}^{T} } \\{S^{-1} \bar{A}_{d}^{T} +S^{-1} L^{T} \bar{B}^{T} } &{0} &{0} &{-S^{-1} } &{0} &{0} \\{\bar{M}^{T} } &{0} &{0} &{0} &{-I} &{0} \\{0} &{0} &{\bar{N}P^{-1} } &{0} &{0} &{-\varepsilon I} \end{array}\right]<0. \end{equation} (20) By letting $$X=P^{-1},Y=S^{-1},Q=KP^{-1},H=LS^{-1} $$, LMI (17) is obtained. We thus deduce that inequality (15) is fulfilled if LMI (17) holds. The conclusion to be proved is derived immediately by Theorem 1. This ends the proof. In fact, if there exists a feasible solution to LMI (17), then the state feedback can be shown as $$ \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)$$ It follows from Theorem 2 that the closed-loop system (14) of the augmented error system (11) is asymptotically stable. As a consequence, we have $$ \mathop{\lim }\limits_{k\to \infty } e\!\left(k\right)=\mathop{\lim }\limits_{k\to \infty } \!\left(y\!\left(k\right)-r\!\left(k\right)\right)=0.$$ Namely, the output $$y\!\left (k\right )$$ of the closed-loop system can track the reference signal $$r\!\left (k\right )$$ without static error. Partition the control gain matrices K and L so that \begin{equation} K=\left[\begin{array}{@{}ccc@{}}{K_{e} } &{K_{\Delta x} } &{K_{r} } \end{array}\right]\!,\qquad\qquad\quad \end{equation} (21) \begin{equation} K_{r} =\left[\begin{array}{@{}cccc@{}}{k_{r} \!\left(1\right)} &{k_{r} \!\left(2\right)} &{\cdots } &{k_{r} \!\left(M_{r} \right)} \end{array}\right]\!, \end{equation} (22) \begin{equation} L=\left[\begin{array}{@{}ccc@{}}{L_{e} } &{L_{\Delta x} } &{L_{r} } \end{array}\right]\!,\qquad\qquad\quad\ \end{equation} (23) \begin{equation} L_{r} =\left[\begin{array}{@{}cccc@{}}{l_{r} \!\left(1\right)} &{l_{r} \!\left(2\right)} &{\cdots } &{l_{r} \!\left(M_{r} \right)} \end{array}\right]\!.\ \, \end{equation} (24) Then $$\Delta u\left (k\right )$$ is explicitly given by \begin{multline*}\Delta u\!\left (k\right )=K_{e} e\!\left (k\right )+K_{\Delta x} \Delta x\!\left (k\right )+\sum \limits _{i=1}^{M_{r} }k_{r} \!\left (i\right )\Delta r\!\left (k+i\right )\\[-2pt] +L_{e} e\!\left (k-d\right )+L_{\Delta x} \Delta x\!\left (k-d\right )+\sum \limits _{i=1}^{M_{r} }l_{r} \!\left (i\right )\Delta r\!\left (k+i-d\right )\!.\end{multline*} From the definition of the difference operator, it follows that $$ \Delta u\!\left(k\right)=u\!\left(k\right)-u\!\left(k-1\right)\!.$$ Summarizing the statements above, the main theorem in this paper is thus obtained. Theorem 3 Under A1–A2, if LMI (17) is solvable, then the memory feedback controller with preview action for system (1) is given by \begin{multline} u\!\left(k\right)=u\!\left(k-1\right)+K_{e} e\!\left(k\right)+K_{\Delta x} \Delta x\!\left(k\right)+\sum\limits_{i=1}^{M_{r} }k_{r} \!\left(i\right)\Delta r\!\left(k+i\right) \\[-2pt] +\,L_{e} e\!\left(k-d\right)+L_{\Delta x} \Delta x\!\left(k-d\right)+\sum\limits_{i=1}^{M_{r} }l_{r} \!\left(i\right)\Delta r\!\left(k+i-d\right)\!, \end{multline} (25) where $$K_{e},K_{\Delta x}, K_{r}, L_{e}, L_{\Delta x},L_{r}$$ are determined by (21), (22), (23) and (24), respectively. Under this controller, the output $$y\!\left (k\right )$$ can track the reference signal $$r\!\left (k\right )$$ without static error. Remark 4 Note that $$\sum _{i=1}^{M_{r} }k_{r} \!\left (i\right )\Delta r\!\left (k+i\right ) $$ and $$\sum _{i=1}^{M_{r} }l_{r} \!\left (i\right )\Delta r\!\left (k+i-d\right ) $$ in (25) are reference preview compensation terms. Thus, we say that Theorem 3 gives a controller with preview action for system (1). Remark 5 In particular, using the preceding method of retaining the state delay in the error system, Theorem 3 can also be applied to the Lipschitz system without time-delay \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+f\!\left(x\left(k\right)\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!.} \end{cases} \end{equation} (26) In this case, we only need to take $$\bar{A}_{d} $$ as a zero matrix. At this time the controller can still be taken as \begin{equation} \Delta u\!\left(k\right)=K\bar{x}\!\left(k\right)+L\bar{x}\!\left(k-d\right)\!, \end{equation} (27) where the positive integer d can be selected in accordance with design needs. Note that d is not the state delay here. The main results in the studies by Lee et al. (2015) and Lee et al. (2016) focus on this case exactly. Remark 6 To show that the results proposed in this paper have broad applicability, let us consider another special case. If $$f\!\left (x\right )$$ in system (1) is linear, namely, $$f\!\left (x\right )=A_{l} x$$, in this case, we have the following two handling methods: (i) In A1, take $$M=A_{l} $$, $$N=F\!\left (x\right )=I$$, then solve LMI (17). (ii) Merging $$f\!\left (x(k)\right )=A_{l} x(k)$$ with Ax(k), system (1) is then rewritten as $$\begin{cases}{x\!\left(k+1\right)=\left(A+A_{l} \right)x\!\left(k\right)+A_{d} x\!\left(k-d\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!.} \end{cases} $$ In the design of the controller, we only need to replace A with $$A+A_{l} $$ in LMI (17) and take M = N = 0 in A1. Comparing the two approaches above, the second one is superior to the first. The reasons are as follows: (i) The second method reduces the dimension of LMI (17) and decreases the number of variables to be solved. The last two lines and two columns of the matrix on the left side of LMI (17) can be removed directly, and also there is no longer any need to solve the variable $$\varepsilon $$. (ii) If we adopt the first method, Lemma 1 will be used to amplify the matrix on the left side of (15) in the proof of Theorem 2 so as to obtain the LMI condition. This will increase the conservatism of the results obtained. 5. Numerical examples Example 1 Consider the popular single-link flexible joint robot system (Raghavan & Hedrick, 1994; Ibrir et al., 2005; Zemouche et al., 2008; Grandvallet et al., 2013; Defoort et al., 2016; Nguyen & Trinh, 2016) \begin{equation} \begin{cases}{\dot{x}\!\left(t\right)=Tx\!\left(t\right)+g\!\left(x\!\left(t\right)\right)+Du\!\left(t\right)\!,} \\{y\!\left(t\right)=Cx\!\left(t\right)\!,} \end{cases} \end{equation} (28) where $$ T=\left [\begin{array}{@{}cccc@{}}{0} &{1} &{0} &{0} \\{-48.6} &{-1.25} &{48.6} &{0} \\{0} &{0} &{0} &{1} \\{19.5} &{0} &{-19.5} &{0} \end{array}\right ]\!,\,g\!\left (x\!\left (t\right )\right )=\left [\begin{array}{@{}c@{}}{0} \\{0} \\{0} \\{-3.33\sin \left (x_{3} \!\left (t\right )\right )} \end{array}\right ]\!,\,D=\left [\begin{array}{@{}c@{}}{0} \\{21.6} \\{0} \\{0} \end{array}\right ]\!,\,C=\left [\begin{array}{@{}c@{}}{2.3} \\{0} \\{9.7} \\{0} \end{array}\right ]^{T}\!.$$ Due to the existence of some factors such as aging components, signal transmission delay and measurement lag, the time-delay issue frequently emerges in realistic physical systems. We now suppose time-delay exists in the state of system (28). Under the above assumption, system (28) can be rewritten under the form \begin{equation} \begin{cases}{\dot{x\!}\left(t\right)=Tx\!\left(t\right)+A_{\tau } x\!\left(t-\tau \right)+g\!\left(x\left(t\right)\right)+Du\!\left(t\right)\!,} \\{y\!\left(t\right)=Cx\!\left(t\right)\!,} \end{cases} \end{equation} (29) where $$\tau =0.03$$, $$A_{\tau } =\left [\begin{array}{@{}cccc@{}}{0} &{0} &{0} &{0} \\{0.27} &{0} &{0} &{0.5} \\{0} &{0} &{0} &{0} \\{0.16} &{0} &{-9.2} &{0} \end{array}\right ]$$, parameters $$T,g\!\left (x\left (t\right )\right )\!,D,C$$ are defined as described previously. For system (29), by Euler discretization with the sample period $$\delta $$, we can obtain the discrete-time model \begin{cases}{x\!\left(\delta (k+1)\right)=Ax\!\left(\delta k\right)+A_{d} x\!\left(\delta \!\left(k-d\right)\right)+f\!\left(x\!\left(\delta k\right)\right)+Bu\!\left(\delta k\right)\!,} \\{y\!\left(\delta k\right)=Cx\!\left(\delta k\right)\!,} \end{cases} where $$A=I+\delta T,\,f\!\left (x\right )=\delta g\!\left (x\right )\!,B=\delta D,A_{d} =\delta A_{\tau },d=\frac{\tau }{\delta } $$. Assume the previewable reference signal $$r\!\left (t\right )$$ is \begin{equation} r\!\left(t\right)=\begin{cases}0,&\quad t<0.3, \\15\!\left(t-0.3\right),&0.3\le t\le 0.5, \\3,&\quad t>0.5. \end{cases} \end{equation} (30) The preview length is assumed to be $$l_{r} $$, and denote $$M_{r} =\frac{l_{r} }{\delta } $$. Some matrices on the right side of equation (2) are obtained as follows: $$ M=\left[\begin{array}{@{}c@{}}{0} \\{0} \\{0} \\{\sqrt{3.33} } \end{array}\right]\!,\,N=\delta \left[\begin{array}{@{}cccc@{}}{0} &{0} &{-\sqrt{3.33} } &{0} \end{array}\right]\!,\,F(x)=\cos \left(x_{3} \right)$$ Thus, in this example, A1 and A2 are satisfied. The sample period is selected as $$\delta =0.01s$$. In order to observe the effectiveness of the proposed design methodology, we now consider four cases, including $$l_{r} =0s\,(\mathrm{i.e.}\ M_{r} =0)$$, $$l_{r} =0.1s\,\left (M_{r} =10\right ),\,l_{r} =0.3s\,\left (M_{r} =30\right )$$ and $$l_{r} =0.5s\,\left (M_{r} =50\right )$$. To obtain feedback gain matrices, LMI (17) must be solved according to Theorem 2. Then the output of the closed-loop system is derived. The gain matrices for different cases are given in Appendix A. In the simulation, the initial state is assumed to be zero. In Fig. 1, we can clearly see the output response of the closed-loop system in different cases. Fig. 1. View largeDownload slide Output response of system (29). Fig. 1. View largeDownload slide Output response of system (29). It can be seen from Fig. 1 that under the memory feedback controller, the closed-loop output accurately tracks the reference signal (30) whether there is preview action or not. Compared with no preview, output response with preview action has the following advantages: (i) Respond in advance. With the influence of preview action, the closed-loop output is able to perceive that the reference signal will change nearby and thus make a response in advance to the future changes. In Fig. 1, the reference signal begins to change in 0.3s. The output with preview action responds before 0.3s, while the output response without preview action is lagging, namely, the output responds after 0.3s. (ii) The overshoot is drastically reduced, and the settling time is significantly decreased. As the preview length increases, the settling time is gradually shortened. In particular, when the preview length is increased to $$M_{r} =50$$, the overshoot becomes very small. This fully demonstrates the validity and superiority of the preview controller in improving the transient performance of the closed-loop system. In addition, we point out that if there is no time-delay phenomenon in system (29), namely, $$A_{\tau } =0$$, then system (29) reduces to system (28). By Euler discretization and according to Remark 5, our results remain valid in this case. We choose d = 1, then the simulation is done in Fig. 2. It can be seen similarly that adding preview action of the reference signal can diminish the overshoot and shorten the settling time. In this case, the feedback gain matrices obtained through computing LMI (17) are omitted here. Fig. 2. View largeDownload slide Output response of system (28). Fig. 2. View largeDownload slide Output response of system (28). The engineering example above verifies the advantage of preview control theory in improving the tracking performance of the closed-loop system. A continuous-time system is provided in Example 1, where the non-linearity is relatively simple and satisfies $$f\!\left (0\right )=0$$. Further, in order to show that the proposed methodology can be applied to a broader class of Lipschitz non-linear systems (including the case of $$f\!\left (0\right )\ne 0$$), a discrete-time system will be considered below. Example 2 Consider system \begin{equation} \begin{cases}{x\!\left(k+1\right)=Ax\!\left(k\right)+A_{d} x\!\left(k-d\right)+f\!\left(x\left(k\right)\right)+Bu\!\left(k\right)\!,} \\{y\!\left(k\right)=Cx\!\left(k\right)\!,} \end{cases}\end{equation} (31) where \begin{align*} A&=\left[\begin{array}{@{}cc@{}}{-0.98} &{-3.5} \\{0} &{0.2} \end{array}\right]\!,\,A_{d} =\left[\begin{array}{@{}cc@{}}{0.7} &{0.3} \\{0} &{0.5} \end{array}\right]\!,\,B=\left[\begin{array}{@{}c@{}}{0} \\{0.6} \end{array}\right]\!,\,C=\left[\begin{array}{@{}cc@{}}{-0.58} &{0.16} \end{array}\right]\!,\\f\left(x\right)&=\left[\begin{array}{@{}c@{}}{0.02\arctan \left(x_{2} \right)} \\{-0.01\cos \left(x_{1} \right)+0.04x_{2} } \end{array}\right]\!,\,d=2.\end{align*} Note that $$f\!\left (0\right )\ne 0$$, and one computes that $$ f^{\prime} \!\left(x\right)=MF\!\left(x\right)N,$$ where $$ M=\left[\begin{array}{@{}cc@{}}{0} &{0.1} \\{0.2} &{0} \end{array}\right]\!,\,F\!\left(x\right)=\left[\begin{array}{@{}cc@{}}{0.5\sin \left(x_{1} \right)} &{0.2} \\{0} &{\frac{0.2}{1+{{x_{2}^{2}}} } } \end{array}\right]\!,\,N=\left[\begin{array}{@{}cc@{}}{0.1} &{0} \\{0} &{1} \end{array}\right].$$ It can be verified that $$F^{T} \!\left (x\right )F\!\left (x\right )\le I$$, A1 is thus satisfied. Moreover, assume the previewable reference signal is \begin{equation} r\!\left(k\right)=\begin{cases}3,&k\ge 30, \\0,&k<30. \end{cases} \end{equation} (32) Then A2 is satisfied. Now we discuss three cases, including $$M_{r} =0$$, $$M_{r} =7$$, $$M_{r} =11$$. By solving LMI (17), the feedback gain matrices are obtained. Then the output of the closed-loop system is derived accordingly. The gain matrices for different cases are given in Appendix B. The simulation, with zero initial state, is done in Fig. 3. Fig. 3. View largeDownload slide Output response of system (31). Fig. 3. View largeDownload slide Output response of system (31). Figure 3 shows that the closed-loop output, with or without preview, can track the reference signal (32) accurately. By observing the response curves between time 20 and time 35, we see that the output with preview action makes a response before the reference signal changes, while the output response without preview action is lagging. Also, by adding the preview action, the overshoot of the output is reduced and the settling time is shortened. Thus, the proposed preview controller achieves better transient performance. 6. Conclusion In this paper, we study the design of the tracking controller with preview action for a class of discrete-time Lipschitz non-linear time-delay systems. First, according to the error system method in preview control theory, the augmented error system containing previewable reference information is constructed. Second, the memory state feedback controller is designed, and the LMI condition for the stability of the closed-loop system is formulated. Furthermore, a concrete form of the preview tracking controller for the original system is derived. Finally, two illustrative examples show the advantage of the proposed preview controller in enhancing the tracking performance of the closed-loop system. 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Google Scholar CrossRef Search ADS Appendix A The gain matrices for different cases are as follows: When $$M_{r} =0$$, we obtain $${K_{e} = {-4.5493,}}$$ $${K_{\varDelta x} =(-177.3916\ -\!6.2611\ -\!916.9090\ -\!87.0118),}$$ $${L_{e} = {0.0035,}} $$ $${L_{\varDelta x} =( -0.1486\ -\!0.0027\ \ 6.7613\ -\!0.0360).}$$ When $$M_{r} =10$$, we obtain $${K_{e} = {-4.6625,}}$$ $${K_{\varDelta x} =( {-182.5432\ -\!6.2799\ -\!981.3317\ -\!91.2319}),}$$ $${K_{r} =( {4.6392\ \ 4.4755\ \ 4.4547\ \ 4.4607\ \ 4.4259\ \ 4.3713\ \ 4.3036\ \ 4.2215\ \ 4.1391\ \ 4.0633}),}$$ $${L_{e} = {-2.7169}\times 10^{-4},}$$ $$L_{\varDelta x} =( -0.1410\ \ 0.0003\ \ 7.3721\ -\!0.0197),$$ $${L_{r} =10^{-3} ( 0.8118\ \ 0.4451\ \ 0.2516\ \ 0.2111\ \ 0.1452\ \ 0.1153\ \ 0.0851\ \ 0.0453\ \ 0.0114\ -\!0.0030).}$$ When $$M_{r} =30$$, we obtain $$K_{e} =-3.5939,$$ $$K_{\varDelta x} =(-160.7484\ -\!6.0741\ -\!720.9379\ -\!75.7690),$$ $$K_{r} =(3.5820\ \ 3.4711\ \ 3.4574\ \ 3.4273\ \ 3.3713\ \ 3.2875\ \ 3.1875\ \ 3.0816\ \ 2.9757\ \ 2.8720\ \ 2.7745 2.6855\ \ 2.6062\ \ 2.5372\ \ 2.4782\ \ 2.4283\ \ 2.3863\ \ 2.3509\ \ 2.3202\ \ 2.2925\ \ 2.2658\ \ 2.2382\ \ 2.2081 2.1736\ \ 2.1332\ \ 2.0854\ \ 2.0285\ \ 1.9608\ \ 1.8790\ \ 1.7750),$$ $$L_{e} =-0.0013,$$ $$L_{\varDelta x} =(-0.1224\ \ 0.0009\ \ 6.2895\ -\!0.0227),$$ $$L_{r} =(0.0015\ \ 0.0006\ \ 0.0004\ \ 0.0001\ -\!0.0001\ -\!0.0003\ -\!0.0004\ -\!0.0004\ -\!0.0004\ -\!0.0003 -\!0.0002\ -\!0.0002\ -\!0.0001\ -\!0.0000\ \ 0.0000\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001 0.0001\ \ 0.0001\ \ 0.0000\ \ 0.0000\ \ 0.0000\ -\!0.0000\ -\!0.0000\ -\!0.0000\ -\!0.0000).$$ When $$M_{r} =50$$, we obtain $$K_{e} =-4.7532,$$ $$K_{\varDelta x} =10^{3} (-0.1948\ -\!0.0065\ -\!1.0491\ -\!0.0994),$$ $$K_{r} =(4.7352\ \ 4.6054\ \ 4.6155\ \ 4.6016\ \ 4.5566\ \ 4.4938\ \ 4.4143\ \ 4.3259\ \ 4.2398\ \ 4.1551\ \ 4.0775 4.0097\ \ 3.9520\ \ 3.9034\ \ 3.8620\ \ 3.8268\ \ 3.7967\ \ 3.7692\ \ 3.7420\ \ 3.7127\ \ 3.6790\ \ 3.6383\ \ 3.5883 3.5274\ \ 3.4539\ \ 3.3667\ \ 3.2649\ \ 3.1480\ \ 3.0159\ \ 2.8687\ \ 2.7068\ \ 2.5310\ \ 2.3422\ \ 2.1415\ \ 1.9303 1.7100\ \ 1.4823\ \ 1.2488\ \ 1.0116\ \ 0.7723\ \ 0.5329\ \ 0.2953\ \ 0.0615\ -\!0.1668\ -\!0.3877\ -\!0.5994 -\!0.8003 \ -\!0.9886\ -\!1.1625\ -\!1.3199),$$ $$L_{e} =0.0033, $$ $$L_{\varDelta x} =(-0.1465\ -\!0.0012\ \ 7.7215\ -\!0.0232),$$ $$L_{r} =(-0.0007\ -\!0.0013\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012\ -\!0.0012 -\!0.0012\ -\!0.0011\ -\!0.0011\ -\!0.0010\ -\!0.0009\ -\! 0.0009\ -\! 0.0008\ -\! 0.0007\ -\! 0.0007 -\! 0.0006\ -\! 0.0005\ -\! 0.0005\ -\! 0.0004\ -\! 0.0004\ -\! 0.0003\ -\! 0.0003\ -\! 0.0002\ -\! 0.0002\ -\! 0.0002 -\! 0.0001\ -\! 0.0001\ -\! 0.0000\ -\! 0.0000\ \ 0.0000\ \ 0.0000\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0001\ \ 0.0000\ \ 0.0000\ \ 0.0000).$$ Appendix B The gain matrices for different cases are as follows: When $$M_{r} =0$$, we get $$K_{e} = -0.1449,$$ $$K_{\varDelta x} =( 0.3531\ \ 0.9324),$$ $$L_{e} = -1.0399\times 10^{-4},$$ $$L_{\varDelta x} =( -0.2493\ -\!0.9401).$$ When $$M_{r} =7$$, we get $$K_{e} = -0.1138,$$ $$K_{\varDelta x} =(0.3654\ \ 0.9711),$$ $$K_{r} = (0.1139\ \ 0.1136\ \ 0.1047\ \ 0.0950\ \ 0.0869\ \ 0.0794\ \ 0.0716),$$ $$L_{e} = -4.0142\times 10^{-5},$$ $$L_{\varDelta x} =( -0.2598\ -\!0.9445),$$ $$L_{r} =10^{-4} (0.3006\ \ 0.5208\ \ 0.5602\ \ 0.4086\ \ 0.3412\ \ 0.2357\ \ 0.1076).$$ When $$M_{r} =11$$, we get $$K_{e} = -0.1465,$$ $$K_{\Delta x} = (0.3470\ \ 0.9047),$$ $$K_{r} = (0.1465\ \ 0.1459\ \ 0.1312\ \ 0.1147\ \ 0.1019\ \ 0.0893\ \ 0.0785\ \ 0.0682\ \ 0.0588\ \ 0.0506\ \ 0.0434),$$ $$L_{e} = -1.3983\times 10^{-4},$$ $$L_{\Delta x} =( -0.2497\ -\!0.9401),$$ $$L_{r} =10^{-3} ( 0.1889\ \ 0.2165\ \ 0.1660\ \ 0.0601\ \ 0.0095\ \ 0.0008\ -\!0.0049\ -\!0.0093\ -\!0.0125 -\!0.0137\ -\!0.0110).$$ © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Mar 21, 2018

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