1 Introduction

In this paper, we focus on the design of sampled-data controllers for a class of high-order nonlinear systems described by

$$\begin{aligned} \left\{ \begin{array}{rcl} {\dot{x}}_1&{}=&{}x_2^{p_1}+\phi _1(x_1)\\ {\dot{x}}_2&{}=&{}x_3^{p_2}+\phi _2(x_1,x_2)\\ &{}\vdots &{}\\ {\dot{x}}_n&{}=&{}u^{p_n}+\phi _n(x_1,\ldots ,x_n) \end{array}\right. \end{aligned}$$
(1)

where the system state \({\bar{x}}=(x_1,\ldots ,x_n)^\mathrm{T}\in R^n\) is measurable and \(u\in R\) is the control input. \(p_i, i=1,\ldots ,n,\) are arbitrarily odd positive integers. The functions \(\phi _i:R^i\rightarrow R, i=1,\ldots ,n,\) are smooth in state \({\bar{x}}\) with \(\phi _i(0)=0\).

System (1) stands for a wide class of practical systems such as underactuated systems with weak coupling [1] and single-link robotic manipulator systems [2]. Notably, it is not feedback linearizable if there exist some \(i\in \{1,2,\ldots ,n\}\) such that \(p_i>1\) [3, 4]. So, the control problem for system (1) is a long-standing active subject in the field of nonlinear control, and more and more researchers are devoted to this area. Over the past few decades, based on the adding a power integrator approach [3], this problem has been intensively investigated, see [3,4,5,6,7,8,9,10,11] etc. Meanwhile, the study is extended to many other extensive forms of system (1) in recent years. For example, in [12], the authors solved the problem of designing \(C^1\) and \(C^{\infty }\) controllers for stochastic high-order nonlinear systems. In [13], a globally stabilizing output-feedback controller was proposed for high-order nonlinear systems with multiple time delays. For other related results, see [14,15,16,17,18,19,20,21,22,23,24,25] and the reference therein. However, all the above-mentioned results are under the continuous-time framework. In practice, controllers are often implemented digitally using computers. In this paper, we aim to further investigate the global stabilization problem of the nonlinear systems with the form (1) by using sampled-data control technique.

As is well known, sampled-data control of nonlinear systems is an important research topic in the field of control science, which has received a great deal of attention recently; see, for instance, [26,27,28,29,30,31,32]. In the context of sampled-data systems, a discrete-time controller is often used in feedback with a continuous-time plant. Such a control mechanism will yield a hybrid closed-loop system that contains both continuous-time signals and discrete-time signals and is difficult to handle mathematically. One effective method to solve the problem is the input-delay approach where the sampled-data control system is modeled as a continuous-time system with a time-varying input delay, and then the stability of the system is analyzed by using delay system approach [33,34,35,36]. Another method is the so-called hybrid system approach where the sampled-data system is represented as an impulsive system and the stability of the system is investigated by employing Lyapunov functions with discontinuities [37,38,39]. The third method is the lifting approach which was first presented in [40]. This approach addresses the sampled-data control problem in terms of an equivalent discrete-time system. Using the lifting technique, the sampled-data problem was investigated for linear systems in [41,42,43], but there are few results for nonlinear systems except the recent paper [44]. Based on these stability analysis approaches of sampled-data systems, one of the most commonly used methods to design sampled-data controllers is referred to as emulation [45, 46], which is carried out by discretizing continuous-time controllers directly. This method is attractive for simplicity but may not perform well in practice since the required sampling period must be small enough, which may exceed the hardware limitations [47]. Another is discrete-time design method [48,49,50] under which a sampled-data controller is designed based on an approximate discrete-time model of continuous-time nonlinear systems. (The Euler approximate model is often used.) However, in this case, the designed controller can only guarantee local or semi-global practical stabilization due to the existence of the approximate errors which are inevitable for nonlinear systems [51]. Recently, based on the Lie series representations of solutions of ordinary differential equations [52] and the idea of Lyapunov-based redesign [53, 54], a sampled-data design method based on the sampled-data equivalent model was proposed to design sampled-data controllers for some nonlinear systems. Using this method, some control performances of a given continuous-time controller may be preserved. For example, Ref. [55] proposed a digital control scheme to maintain the damping performance of the \(L_gV\)-damping controller for synchronous machine systems. Reference [56] introduced two sampled-data control strategies to preserve the passivity for passive SISO input-affine systems, and Ref. [57] proposed a digital control scheme to preserve the performance of a continuous-time backstepping control strategy for strict-feedback systems. But, to the best of our knowledge, there are no results in the existing literature focusing on the sampled-data control problem for high-order nonlinear system (1).

Motivated by the above observation, the main purpose of this paper is to solve the problem of global stabilization by sampled-data feedback control for a class of high-order nonlinear systems, where the powers of the chained integrators are different and larger than 1. Clearly, because of the inherent nonlinearity of the system, it is a nontrivial work to solve the problem. First of all, a new digital control scheme is proposed by using the input-Lyapunov matching approach so that the Lyapunov difference along the solutions of the discrete-time model with the proposed controller is equal to the Lyapunov difference along the sampled solutions of the closed-loop system with a continuous-time controller. Then, the control performance of the approximate solutions of the proposed controller is analyzed theoretically. For details, the main contributions of this paper are summarized as follows.

  • A novel multi-rate sampled-data controller is designed to achieve the global asymptotic stabilization for a class of high-order nonlinear systems. Unlike the design method based on the approximate discrete-time model, the proposed controller is obtained from the exact discrete-time equivalent model and may reproduce the control performance of a continuous-time controller at the sampling instants.

  • The approximate solutions of the proposed controller are proved to be effective in the practical implementation by theoretical analysis and a simulation example. The results show that, compared with the classical emulation controller, our controller may provide faster decrease in the Lyapunov function for each subsystem. This will result in allowing larger sampling periods and enlarging the domain of attraction (DOA) for a given sampling period.

The remainder of this paper is organized as follows. Section 2 introduces some preliminaries and problem formulation. Section 3 presents the main results of this paper, i.e., a novel multi-rate sampled-data control scheme is designed for a class of high-order nonlinear systems, and the performance of the approximate controller is analyzed theoretically. An example is provided to verify the effectiveness of the proposed controller in Sect. 4. The work is concluded in Sect. 5.

2 Preliminaries and problem formulation

2.1 Preliminaries

The following notations and lemmas are to be used throughout this paper.

Notations R and \(R^n\) denote the set of real numbers and \(n\times 1\) real vectors, respectively. \(|\cdot |\) denotes the Euclidian norm. \(O(\cdot )\) denotes the infinitesimal of same order. \(x'\) denotes the transpose of a vector x, and \(|_x\) denotes the evaluation of a generic map at a point x. The associated dynamics are assumed to be forward complete [58] to guarantee the existence of solutions for all positive time \(t>0\). Given a vector field f, \(L_f=\sum _{i=1}^kf_i(\cdot )\frac{\partial }{\partial x_i}\) denotes the Lie derivative operator and \(e^{L_f}\) (or \(e^f\) when no confusion is possible) denotes Lie series operator, i.e., \(e^f=1+L_f+\frac{1}{2!}L_f^2+\cdots +\frac{1}{q!}L_f^q+\cdots .\) To simplify the notations, \(L_fL_g\) stands for \(L_f\circ L_g\), which means the composition of \(L_f\) and \(L_g\).

Lemma 1

([59]) For \(x\in R, y\in R,\) and any positive real numbers \(a,b,\gamma \), the following inequality holds:

$$\begin{aligned} |x|^a|y|^b \le \frac{a}{a+b}\gamma |x|^{a+b}+\frac{b}{a+b}\gamma ^{-\frac{a}{b}}|y|^{a+b}. \end{aligned}$$

Lemma 2

([59]) For \(x\in R, y\in R,\) and \(\lambda \ge 1\), the following inequality holds:

$$\begin{aligned} |x+y|^\lambda \le 2^{\lambda -1}|x^\lambda +y^\lambda |. \end{aligned}$$

Lemma 3

([60]) Let h(x) be any function which is holomorphic in a neighborhood of x, and \(e^f\) denotes Lie series operator associated with a vector field f. Then, the function sign h of the holomorphic function is commuted with the symbol \(e^f\), i.e.,

$$\begin{aligned} h(e^fx)=e^fh(x), \end{aligned}$$

where \(e^fx\) stands for \(e^fI_d|_x\) with the identity function \(I_d\) on \(R^n\).

2.2 Problem formulation

For the continuous-time nonlinear system

$$\begin{aligned} {\dot{x}}=f(x,u), \end{aligned}$$
(2)

we pay attention to the sampled-data information during the control design. By using the zero-order hold technique, the control input takes the form

$$\begin{aligned} u(t)=u_k=\hbox {constant},\quad t\in [k\delta , (k+1)\delta ),\quad k\in N, \end{aligned}$$

where \(\delta \) represents the sampling period of the system. The discrete-time equivalent model of the sampled-data system is defined by a discrete-time dynamic which takes the form of the exponential Lie series [52]

$$\begin{aligned} x_{k+1}=F^{\delta }(x_k,u_k)=e^{\delta f_{u_k}}x_k. \end{aligned}$$
(3)

If the value of control input changes m times over each sample interval \([k\delta ,(k+1)\delta )\), i.e., u(t) is maintained constant at value \(u_{ik}\) over subintervals \([k\delta +(i-1){\bar{\delta }}, k\delta +i{\bar{\delta }}), i=1,2,\ldots ,m,\) where \({\bar{\delta }}=\delta /m\), then the discrete-time equivalent model of the sampled-data system is described by

$$\begin{aligned} x_{k+1}= & {} F^{{\bar{\delta }}}\left( \cdots (F^{{\bar{\delta }}}(x_k,u_{1k}),\cdots ),u_{mk}\right) \nonumber \\= & {} e^{{\bar{\delta }} f_{u_{mk}}}\circ \cdots \circ e^{{\bar{\delta }} f_{u_{1k}}}x_k. \end{aligned}$$
(4)

The \((u_{1k},\ldots ,u_{mk})\) is called m-rate sampled-data controller. Specifically, when \(m=1\), it is denoted as \(u_k\) and called single-rate sampled-data controller.

Let \(u_c\) denote a continuous-time stabilization controller of the system (2) with an associated control Lyapunov function V(x), and \(u_d\) denote a sampled-data controller (piecewise constant control). \(x^c_k\) denotes the value of the state trajectory of the continuous-time closed-loop system \({\dot{x}}=f(x,u_c)\) at time \(t=k\delta \), and \(x^d_k\) denotes the value of the state trajectory of the sampled-data closed-data system \({\dot{x}}=f(x,u_d)\) at time \(t=k\delta .\) The input-Lyapunov matching means finding a piecewise constant controller \(u_d\) such that

$$\begin{aligned} V(x^d_{k+1})= V(x^c_{k+1}), \end{aligned}$$
(5)

when \(x^d_k=x^c_k.\)

Remark 1

In this paper, we aim to design a sampled-data controller for system (1) such that the origin of the closed-loop system is globally asymptotically stable. Considering the structural characteristics and inherently nonlinearities of system (1), n Lyapunov functions \(V_i,i=1,2,\ldots ,n,\) need to be constructed to design a proper continuous-time control law for each subsystem by using a recursive design procedure. It is noted that the Lyapunov function \(V_i\) describes the stabilizing property of the \((x_1,\ldots ,x_i)\)-subsystem. In order to design a sampled-data controller which may reproduce the performance of the continuous-time stabilizer, we use the input-Lyapunov matching criteria for each \(V_i,i=1,2,\ldots ,n\). That is, find a piecewise constant controller \(u_d\) such that the equality (5) holds for each \(V_i\), \(i=1,2,\ldots ,n\). In this context, we propose a multi-rate sampled-data control scheme for system (1). In this way, a discrete-time equivalent model with more independent inputs is then obtained, which can be profitably employed in the design of the controller \(u_d\) mentioned above.

To this end, we make the following assumptions for system (1).

Assumption 1

For \(i=1,\ldots ,n\),

$$\begin{aligned} |\phi _i(x_1,\ldots ,x_i)|\le (|x_1|^{p_i}+\cdots +|x_i|^{p_i})\rho _i(x_1,\ldots ,x_i), \end{aligned}$$

where \(\rho _i(\cdot )\) is a known nonnegative smooth function.

Assumption 2

\(p_1\ge p_2\ge \cdots \ge p_n\ge 1\).

Remark 2

In order to get a smooth global stabilizer for nonlinear system (1), Assumptions 1 and 2 are commonly used in the continuous-time design framework. It should be pointed out that Assumption 1 can be seen as a natural generalization of the well-known feedback linearizable condition [61, 62]. Their significance and rationality have been discussed in [3, 4, 11, 63].

3 Main results

3.1 Sampled-data controller design

In this section, a single-rate and a multi-rate digital control schemes are proposed to achieve the global asymptotic stabilization of system (1).

Theorem 1

Consider system (1) under Assumptions 1 and 2. There exist \(T^*>0\) and an m-rate \((m=1\ or\ n)\) sampled-data controller \(u_d=(u_{1k},\ldots ,u_{mk})\) where

$$\begin{aligned} u_{ik}=\sum _{j=0}^{\infty }a_{ikj}\delta ^{j},\quad i=1,2,\ldots ,m \end{aligned}$$
(6)

such that the equilibrium \({\bar{x}}=0\) of the sampled-data closed-loop system is globally asymptotically stable for any \(\delta \in [0,T^*)\).

Proof

Initial step For \(x_1\)-subsystem of system (1), we choose a smooth Lyapunov function

$$\begin{aligned} V_1(x_1)=\frac{1}{2}\alpha _1 x_1^2, \alpha _1> 0. \end{aligned}$$

Let \(\xi _1=x_1\). Based on Assumption 1, the virtual controller is designed as \(x_2=x_2^*=-b_1(\xi _1)\xi _1\), where \(b_1(\xi _1)=(n+\rho _1(\xi _1))^{\frac{1}{p_1}}\). Then, we have

$$\begin{aligned} {\dot{V}}_1\le -\,n\alpha _1 \xi _1^{p_1+1}. \end{aligned}$$
(7)

\(\square \)

Inductive step For \((x_1,\ldots ,x_i)\)-subsystem of system (1), let \(\xi _j=x_j-x_j^*,j=1,2,\ldots ,i\). It can be transformed into a system of the form

$$\begin{aligned} \left\{ \begin{array}{rcl} {\dot{\xi }}_j&{}=&{}f_j(\xi _1,\ldots ,\xi _{j+1}),\quad j=1,2,\ldots ,i-1, \\ {\dot{\xi }}_{i}&{}=&{}x_{i+1}^{p_i}+g_{i}(\xi _1,\ldots ,\xi _{i}),\\ \end{array} \right. \end{aligned}$$
(8)

where \(f_j,j=1,2,\ldots ,i-1\) and \(g_i\) are bounded by \(|f_j|\le (|\xi _1|^{p_j}+\cdots +|\xi _{j+1}|^{p_j})\beta _{j}(\xi _1,\ldots ,\xi _{j+1})\) and \(|g_i|\le (|\xi _1|^{p_i}+\cdots +|\xi _i|^{p_i})\gamma _i(\xi _1,\ldots ,\xi _i)\), respectively. Consider a smooth Lyapunov function

$$\begin{aligned} V_i(\xi _1,\ldots ,\xi _i)=\sum _{j=1}^{i}\frac{\alpha _i\xi _j^{p_1-p_j+2}}{p_1-p_j+2}, \alpha _i>0. \end{aligned}$$

Suppose there exists a virtual controller \(x_{i+1}=x_{i+1}^*=-b_{i}(\xi _1,\!\ldots \!,\xi _{i})\xi _{i}\) with \(b_{i}(\xi _1,\!\ldots \!,\xi _{i})=\left( \frac{n-i+1+{\bar{\rho }}_{i}(\xi _1,\ldots ,\xi _{i})}{\alpha _{i}}\right) \!^{\frac{1}{p_{i}}}\) such that

$$\begin{aligned} {\dot{V}}_i\le -(n-i+1)\alpha _1\left( \xi _1^{p_1+1}+\cdots +\xi _i^{p_1+1}\right) . \end{aligned}$$
(9)

It will be shown that (9) also holds for \((x_1,\ldots , x_{i+1})\)-subsystem. Let \(\xi _{i+1}=x_{i+1}-x_{i+1}^*\). Together with system (8), the \((x_1,\ldots ,x_{i+1})\)-subsystem can be transformed into

$$\begin{aligned} \left\{ \begin{array}{rcl} \!\!\!\!&{}&{}{\dot{\xi }}_j=f_j(\xi _1,\ldots ,\xi _{j+1}),\quad j=1,2,\ldots ,i, \\ \!\!\!\!&{}&{}{\dot{\xi }}_{i+1}=x_{i+2}^{p_{i+1}}+g_{i+1}(\xi _1,\ldots ,\xi _{i+1}),\\ \end{array} \right. \end{aligned}$$
(10)

where \(f_i(\xi _1,\ldots ,\xi _{i+1})=(\xi _{i+1}+x^*_{k+1})^{p_i}+g_i(\xi _1,\ldots ,\xi _i)\) and \(g_{i+1}(\xi _1,\ldots ,\xi _{i+1})\!=\!\phi _{i+1}(\xi _1,\ldots ,\xi _{i+1}) \,- \,\sum ^{i}_{j=1}\frac{\partial x^*_{i+1}}{\partial \xi _j} f_j(\xi _1,\ldots ,\xi _{j+1}).\) We constructed a smooth Lyapunov function defined by

$$\begin{aligned}&V_{i+1}(\xi _1,\ldots ,\xi _{i+1})\\&\quad =V_i+\frac{\alpha _{i+1}\xi _{i+1}^{p_1-p_{i+1}+2}}{p_1-p_{i+1}+2}, \quad \alpha _{i+1}>0. \end{aligned}$$

First, according to Lemma 2, we have

$$\begin{aligned} |f_i|\le & {} |\xi _{i+1}+x^*_{i+1}|^{p_i}+|g_i(\xi _1,\ldots ,\xi _i)|\\\le & {} 2^{p_k-1}\left( |\xi _{i+1}|^{p_i}\!+\!(n-k+1+{\bar{\rho }}_i(\xi _1,\ldots ,\xi _i))|\xi _i|^{p_i}\right) \\&\quad +\,\left( |\xi _1|^{p_i}+\cdots +|\xi _i|^{p_i}\right) \gamma _i(\xi _1,\ldots ,\xi _i)\\\le & {} \left( |\xi _1|^{p_i}+\cdots +|\xi _{i+1}|^{p_i}\right) \beta _i(\xi _1,\ldots ,\xi _{i+1}), \end{aligned}$$

where \(\beta _i(\xi _1,\ldots ,\xi _{i+1})=2^{p_i-1}(n-i+1+{\bar{\rho }}_i+\gamma _i).\) Furthermore, under Assumption 2, we obtain that

$$\begin{aligned} |g_{i+1}|\le & {} |\phi _{i+1}|+\sum _{j=1}^{i}\left| \frac{\partial {x^*_{i+1}}}{\partial {\xi _j}}\right| \left| f_j(\xi _1,\ldots ,\xi _{j+1})\right| \\\le & {} (|\xi _1|^{p_{i+1}}+\cdots +|\xi _{i+1}|^{p_{i+1}})\gamma _{i+1}\\&\times (\xi _1,\ldots ,\xi _{i+1}), \end{aligned}$$

where \(\gamma _{i+1}=\max \limits _{1<j<i}\{2^{p_{i+1}-1}(1+b_j(\xi _1,\ldots ,\xi _{j}))\} + \sum _{j=1}^{i}\) \(|\frac{\partial {x^*_{i+1}}}{\partial {\xi _j}}|(|\xi _1|^{p_j-p_{i+1}}+\cdots +|\xi _{j+1}|^{p_j-p_{i+1}})\). Second, according to Lemma 1, it can be deduced that there are nonnegative functions \({\tilde{\rho }}_{i+1}(\xi _1,\ldots ,\xi _{i+1})\) and \({\hat{\rho }}_{i+1}(\xi _1,\!\ldots \!, \xi _{i+1})\) such that

$$\begin{aligned}&\alpha _i\xi _i^{p_1-p_i+1}\left( (\xi _{i+1}+x^*_{i+1})^{p_i}-x_{i+1}^{*p_i}\right) \\&\le \frac{1}{2}(|\xi _1|^{p_1+1}+\cdots +|\xi _{i}|^{p_1+1})\\&\quad +\xi _{i+1}^{p_1+1}{\tilde{\rho }}_{i+1}(\xi _1,\!\ldots \!,\xi _{i+1}), \end{aligned}$$

and

$$\begin{aligned}&\alpha _{i+1}\xi _{i+1}^{p_1-p_{i+1}+1}g_{i+1}(\xi _1,\ldots ,\xi _{i+1})\\&\le \frac{1}{2}(|\xi _1|^{p_1+1}+\cdots +|\xi _{i}|^{p_1+1})\\&\quad +\xi _{i+1}^{p_1+1}{\hat{\rho }}_{i+1}(\xi _1,\!\ldots \!,\xi _{i+1}). \end{aligned}$$

Then, the time derivation of \(V_{i+1}\) along the trajectories of system (10) can be expressed as

$$\begin{aligned} {\dot{V}}_{i+1}\le & {} -\,(n-i)\alpha _1(\xi _1^{p_1+1}+\cdots +\xi _i^{p_1+1})\\&+\,\xi _{i+1}^{p_1+1}{\bar{\rho }}_{i+1}(\xi _1,\ldots ,\xi _{i+1})\\&+\,\alpha _{i+1}x_{i+1}^{p_1\!-\!p_{i+1}+1}x_{i+2}^{p_{i+1}}, \end{aligned}$$

where \({\bar{\rho }}_{i+1}={\tilde{\rho }}_{i+1}(\xi _1,\ldots ,\xi _{i+1})+{\hat{\rho }}_{i+1}(\xi _1,\ldots ,\xi _{i+1})\). Obviously, by designing the virtual controller \(x_{i+2}=x_{i+2}^*=-b_{i+1}(\xi _1\ldots ,\xi _{i+1})\xi _{i+1}\) with \(b_{i+1}(\xi _1,\ldots ,\xi _{i+1})= \left( \frac{n-i+{\bar{\rho }}_{i+1}(\xi _1,\ldots ,\xi _{i+1})}{\alpha _{i+1}}\right) ^{\frac{1}{p_{i+1}}}\), we can get

$$\begin{aligned} {\dot{V}}_{i+1}\le -(n-i)\alpha _1\left( \xi _1^{p_1+1}+\cdots +\xi _{i+1}^{p_1+1}\right) . \end{aligned}$$
(11)

Step n: The above inductive proof shows that the inequality (9) is valid for all \(i=1,2,\ldots ,n\). So, for system (1), we can construct a global change of coordinates \(\xi _1=x_1,\ldots ,\xi _n=x_n-x_n^*\) transforming system (1) into a system of the form

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{\xi }}_j=f_j(\xi _1,\ldots ,\xi _{j+1}),\quad j=1,2,\ldots ,n-1,\\ {\dot{\xi }}_{n}=u^{p_n}+g_{n}(\xi _1,\ldots ,\xi _{n}). \end{array} \right. \nonumber \\ \end{aligned}$$
(12)

Choosing \(V_n(\xi _1,\ldots ,\xi _n)=\sum _{j=1}^n\alpha _j\frac{1}{p_1-p_j+2}\xi _n^{p_1-p_j+2}, \alpha _i>0\), as the whole Lyapunov function, a global smooth stabilizer can be designed as \(u=u_c=-b_n(\xi _1\!\ldots \!,\xi _n)\xi _n\) with \(b_n(\xi _1\ldots ,\xi _n)=(1+{{\bar{\rho }}}_n(\xi _1,\ldots ,\xi _n))^{\frac{1}{p_n}}\) such that

$$\begin{aligned} {\dot{V}}_n\le -\,\alpha _1\left( \xi _1^{p_1+1}+\cdots +\xi _n^{p_1+1}\right) . \end{aligned}$$
(13)

Next, we use the global continuous-time stabilizer \(u_c\) designed above as providing an reference response to construct a sampled-data controller \(u_d\) with the form (6). Consider the transformed system (12). By letting \(x=(\xi _1\ldots ,\xi _n)'\), the system can be rewritten in a compact form as

$$\begin{aligned} {\dot{x}}=f(x)+u^{p_n}g(x), \end{aligned}$$
(14)

where f and g are two known continuous vector-value functions. Let \(x_k^c\) and \(x_k^d\) denote the values of the state trajectory of the closed-loop system (14) with \(u_c\) and \(u_d\) at time \(t=k\delta \), respectively.

$$\begin{aligned}&Q(0,u_{1k},\ldots ,u_{nk}) \nonumber \\&\quad =\left[ \begin{array}{c} \left( \frac{1}{n}a_{1k0}^{p_n}+\frac{1}{n}a_{2k0}^{p_n}+\cdots +\frac{1}{n}a_{nk0}^{p_n}-u_c^{p_n}\right) L_gV_n(x^d_k)\\ \left( \frac{1}{2!\times n^2}a_{1k0}^{p_n}+\frac{3}{2!\times n^2}a_{2k0}^{p_n}+\cdots +\frac{n^2-(n-1)^2}{2!\times n^2}a_{nk0}^{p_n}-\frac{1}{2!}u_c^{p_n}\right) L_gL_fV_{n-1}(x^d_k)\\ \vdots \\ \left( \frac{1}{n!\times n^n}a_{1k0}^{p_n}+\frac{2^n-(2-1)^n}{n!\times n^n}a_{2k0}^{p_n}+\cdots +\frac{n^n-(n-1)^n}{n!\times n^n}a_{nk0}^{p_n}-\frac{1}{n!}u_c^{p_n}\right) L_gL_f^{n-1}V_{1}(x^d_k) \end{array} \right] . \end{aligned}$$
(15)
$$\begin{aligned}&\frac{\partial (Q_1,\ldots ,Q_n)}{\partial (u_{1k},\ldots ,u_{nk})}\bigg |_{\delta =0}\nonumber \\&\qquad =\left| \begin{array}{cccc} \frac{1}{n}p_na_{1k0}^{p_n-1}L_gV_n(x^d_k)&{}\cdots &{}\frac{1}{n}p_na_{nk0}^{p_n-1}L_gV_n(x^d_k)\\ \frac{1}{2!\times n^2}p_na_{1k0}^{p_n-1}L_gL_fV_{n-1}(x^d_k)&{}\cdots &{}\frac{n^2-(n-1)^2}{2!\times n^2}p_na_{nk0}^{p_n-1}L_gL_fV_{n-1}(x^d_k)\\ \vdots &{}\vdots &{}\vdots \\ \frac{1}{n!\times n^n}p_na_{1k0}^{p_n-1}L_gL_f^{n-1}V_1(x^d_k)&{}\cdots &{}\frac{n^n-(n-1)^2}{n!\times n^n}p_na_{nk0}^{p_n-1}L_gL_f^{n-1}V_1(x^d_k) \end{array} \right| \nonumber \\&\qquad =\left| \begin{array}{cccc} \frac{1}{n}&{}\frac{1}{n}&{}\cdots &{}\frac{1}{n}\\ \frac{1}{2!\times n^2}&{}\frac{3}{2!\times n^2}&{}\cdots &{}\frac{n^2-(n-1)^2}{2!\times n^2}\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ \frac{1}{n!\times n^n}&{}\frac{2^n-(2-1)^n}{n!\times n^n}&{}\cdots &{}\frac{n^n-(n-1)^2}{n!\times n^n} \end{array} \right| p_n^nu_c^{np_n-n}\prod _{i=0}^{n-1}L_gL_f^{i}V_{n-i}(x^d_k). \end{aligned}$$
(16)

Let \({\bar{\delta }}=\delta /m\). For the case of \(m=n\), our purpose is to find an n-rate sampled-data controller \(u_d\) with the form (6) where \(a_{1k0}=\cdots =a_{nk0}=u_c|_{t=k{\bar{\delta }}}\), such that input-Lyapunov matching condition (5) holds for each \(V_i, i=1,2,\ldots ,n,\) constructed above in the design of continuous-time \(u_c\), i.e., when \(x_k^d=x_k^c\), the following equations hold

$$\begin{aligned} V_i(x_{k+1}^d)=V_i(x_{k+1}^c), ~~i=1,2,\ldots ,n. \end{aligned}$$
(17)

Assuming \(x_k^d=x_k^c\), we construct an algebraic equation set

$$\begin{aligned} Q(\delta ,u_{1k},\ldots ,u_{nk})=0, \end{aligned}$$
(18)

where

$$\begin{aligned} Q= \left[ \begin{array}{c} Q_1\\ Q_2\\ \vdots \\ Q_n \end{array} \right] = \left[ \begin{array}{c} \frac{1}{\delta }\big (V_n(x^d_{k+1})-V_n(x_{k+1}^c)\big )\\ \frac{1}{\delta ^2}\big (V_{n-1}(x^d_{k+1})-V_{n-1}(x_{k+1}^c)\big )\\ \vdots \\ \frac{1}{\delta ^n}\big (V_1(x^d_{k+1})-V_1(x_{k+1}^c)\big ) \end{array} \right] . \end{aligned}$$

Based on formula (4) and Lemma 3 (Lie series commutation theorem), for \(i=1,2,\ldots ,n,\) we have

$$\begin{aligned} V_i(x_{k+1}^c)=V_i\left( e^{\delta (f+u_c^{p_n}g)}x^c_k\right) =e^{\delta (f+u_c^{p_n}g)}V_i(x^c_k), \end{aligned}$$

and

$$\begin{aligned} V_i(x_{k+1}^d)&=V_i\left( e^{{\bar{\delta }}(f+u_{nk}^{p_n}g)}\!\circ \cdots \!\circ e^{{\bar{\delta }}(f+u_{1k}^{p_n}g)}x^d_k\right) \\&=e^{{\bar{\delta }}(f+u_{nk}^{p_n}g)}\!\circ \cdots \!\circ e^{{\bar{\delta }}(f+u_{1k}^{p_n}g)}V_i(x^d_k). \end{aligned}$$

Therefore, by using \(x_k^d=x_k^c\), the left term of equality (18) can be further rewritten as

$$\begin{aligned}&Q(\delta ,u_{1k},\ldots ,u_{nk})\\&=\left[ \begin{array}{c} \frac{1}{\delta }\big (e^{{\bar{\delta }}(f+u_{nk}^{p_n}g)}\circ \cdots \circ e^{{\bar{\delta }}(f+u_{1k}^{p_n}g)}-e^{\delta (f+u^{p_n}_cg)}\big )V_n(x^d_k)\\ \frac{1}{\delta ^2}\big (e^{{\bar{\delta }}(f+u_{nk}^{p_n}g)}\circ \cdots \circ e^{{\bar{\delta }}(f+u_{1k}^{p_n}g)}-e^{\delta (f+u^{p_n}_cg)}\big )V_{n-1}(x^d_k)\\ \vdots \\ \frac{1}{\delta ^n}\big (e^{{\bar{\delta }}(f+u_{nk}^{p_n}g)}\circ \cdots \circ e^{{\bar{\delta }}(f+u_{1k}^{p_n}g)}-e^{\delta (f+u^{p_n}_cg)}\big )V_1(x^d_k) \end{array} \right] . \end{aligned}$$

It is noted that the n-rate sampled-data controller \(u_d\) refers to the continuous-time stabilizer \(u_c\) when \(\delta =0\). Thus, supposing that \(\delta =0\), we have \(Q(0,u_{1k},\ldots ,u_{nk}) =0\) as shown in Eq. (15) at the top of next page.

Consider the Jacobian of Q at \((0,a_{1k0},\ldots ,a_{nk0})\). According to the computation as shown in the equation (16) at the top of next page, it follows that [2] \(\frac{\partial (Q_1,\ldots ,Q_n)}{\partial (u_{1k},\ldots ,u_{nk})}\bigg |_{\delta =0}\) is nonzero, since \(V_i\) has relative degree \(n+1-i\), \(L_gL^i_f V_{n-i}(x_k^d)\ne 0\) for \(i=0,1,\ldots ,n-1.\) By the implicit function theorem, the equation set (18) has a unique set of solutions with the form (6) in the neighborhood of \((0,a_{1k0},\ldots ,a_{nk0})\) (assuming \(\delta <T^*\)), where \(a_{ikj}\) can be determined by solving (18).

Next, we show that the sampled-data closed-loop system (14) with \(u_d\) is globally asymptotically stable at \({\bar{x}}=0\). Since f(x) and g(x) are continuous functions, the solutions of the sampled-data closed-loop system are uniformly globally bounded over T (UGBT, see [65], in our paper \(T=\delta \)) under the assumption of the forward completeness. The results of [65] indicate that we just need to show the discrete-time equivalent model of the sampled-data closed-loop system is globally asymptotically stable. In fact, according to the input-Lyapunov matching condition (17), for \(\delta \in [0,T^*)\), we have

$$\begin{aligned} V_n(x^d_{k+1})-V_n(x^d_k)= & {} V_n(x_{k+1}^c)-V_n(x_k^c)\\= & {} \int _{k\delta }^{(k+1)\delta }{\dot{V}}_n(x^c(\tau ))d\tau <0. \end{aligned}$$

This completes the proof for the case \(m=n\).

Similarly, we can complete the proof in the case when \(m=1\), the difference is to find a single-rate digital controller \(u_d\) such that input-Lyapunov matching condition (17) holds only for the whole Lyapunov function \(V_n\). \(\square \)

Remark 3

According to Theorem 1, the sampled-data controller (6) is designed based on the exact discrete-time model of the sampled-data system and can be determined by solving the corresponding input-Lyapunov matching equations. This shows that controller (6) will reproduce the stability properties of the continuous-time closed-loop system at the sampling instants. It is not the same as the existing methods aforementioned, such as the input-delay approach, the hybrid system approach or the lifting approach. The stability of the closed-loop sampled-data system (1), (6) is analyzed by using the \({{\mathcal {K}}}{{\mathcal {L}}}\) estimate method (presented in [65]) where the global asymptotic stability of the sampled-data system can be deduced from a uniform inter-sample growth condition plus the global asymptotic stability property for the exact discrete-time model of the sampled-data system.

3.2 Approximate controllers

Since \(u_d\) takes the form of a series expansion, it is difficult to get its exact expression in practice. Consider the approximate solutions of \(u_d\). The r-order approximate single-rate and multi-rate controllers are denoted by \(u^{[r]}_k\) and \(\left( u_{1k}^{[r]},\ldots ,u_{nk}^{[r]}\right) \), respectively, where \(u^{[r]}_k=\sum _{j=0}^{r}a_{kj}\delta ^j\) and \(u_{ik}^{[r]}=\sum _{j=0}^{r}a_{ikj}\delta ^j, i=1,2,\ldots ,n\). Clearly, when \(r=0\), the approximate controller \(u_d =u^{[0]}_k= u_c|_{t=k\delta }\) refers to the emulated controller. If \(\delta \) tends to 0, the continuous-time controller \(u_c\) will be recovered. The following results show that the approximate controllers are effective to stabilize the system (1) and have better control performance than emulated control scheme when \(r\ge 1\).

Corollary 1

Consider the closed-loop system (1) with the r-order approximate multi-rate digital controller \((u_{1k}^{[r]},\ldots ,u_{nk}^{[r]})\). For each \(V_i, i=1,\ldots ,n\), there exists a class-\({\mathcal {K}}\) function \(\alpha _i\) such that \(V_i(x^d_{k+1})-V_i(x^d_k)\le -\delta \alpha _i(||x^d_k||)+O(\delta ^{n+2+r-i}).\)

Proof

Based on Theorem 1 and its proof, we reuse \(x_k^d\) to stand for the state evolution of the closed-loop system (14) with \((u_{1k}^{[r]},\ldots ,u_{nk}^{[r]})\) at time \(t=k\delta \). Suppose \(x^d_k=x_k^c\). For each \(V_i, i=1,\ldots ,n\), let \(E_{V_i}^{[r]}(x^d_k)=V_i(x^d_{k+1})-V_i(x_{k+1}^c)\) denote the matching error of \(V_i\). Since \(V_i\) has relative degree \(n+1-i\), we have

$$\begin{aligned} L_gL_f^jV_i(x_k^d)=0, ~~j=0,1,\ldots ,n-i-1. \end{aligned}$$

By using (4) and (17), further computations yield \(E_{V_i}^{[r]}(x^d_k)=O(\delta ^{n+2+r-i})\).

On the other hand, for \(i=1,2,\ldots ,n\), the difference of \(V_i\) along the trajectories of system (14) with \(u_c\) can be calculated as

$$\begin{aligned}&V_i(x_{k+1}^c)-V_i(x_{k}^c)=e^{\delta (f+u_cg)}V_i(x_k^c)-V_i(x_k^c)\\&\ \ \ \ \ =-\,\delta \bigg (-L_{f+u_cg}V_i(x_k^c)-\frac{\delta }{2}L_{f+u_cg}^2V_i(x_k^c)-O(\delta ^2)\bigg ). \end{aligned}$$

Let \(W_i(x_k^c)=-L_{f+u_cg}V_i(x_k^c)-\frac{\delta }{2}L_{f+u_cg}^2V_i(x_k^c)-O(\delta ^2)\). Since \(V_i(x_{k+1}^c)-V_i(x_{k}^c)<0\), \(W_i(x_k^c)\) is a continuous positive definite function. By Lemma 4.3 in [66], there exists a class-\({\mathcal {K}}\) function \(\alpha _i\) such that \(W_i(x_k^c)\ge \alpha _i(||x_k^c||).\) Then,

$$\begin{aligned} V_i(x^d_{k+1})-V_i(x^d_k)= & {} V_i(x_{k+1}^c)-V_i(x_k^c)+E_{V_i}^{[r]}(x^d_k)\\\le & {} -\delta \alpha _i(||x^d_k||)+O(\delta ^{n+2+r-i}). \end{aligned}$$

\(\square \)

Corollary 2

Consider the closed-loop system (1) with the r-order approximate single-rate digital controller \(u_{k}^{[r]}\). For each \(V_i, i=1,\ldots ,n-1\), there exists a class-\({\mathcal {K}}\) function \(\alpha _i\) such that \(V_i(x^d_{k+1})-V_i(x^d_k)\le -\delta \alpha _i(||x^d_k||)+O(\delta ^{n+2-i}).\) For the whole Lyapunov function \(V_n\), there exists a class-\({\mathcal {K}}\) function \(\alpha _n\) such that \(V_n(x^d_{k+1})-V_n(x^d_k) \le -\delta \alpha _n(||x^d_k||)+O(\delta ^{r+2}).\)

Proof

Since the input-Lyapunov matching condition (17) holds only for the whole Lyapunov function \(V_n\), according to (4), the input-Lyapunov matching errors are calculated as \(E_{V_i}^{[r]}(x^d_k)=O(\delta ^{n+2-i}), i=1,\ldots ,n-1\), and \(E_{V_n}^{[r]}(x^d_k)=O(\delta ^{r+2})\). Similarly, as in the proof of Corollary 1 we can complete the proof. \(\square \)

Remark 4

In our method, the sampling period \(\delta \) is used as a parameter involved in the controller design which can be assigned arbitrarily. This provides an useful strategy to design a sampled-data controller with good performance. In fact, from the discussion above we can see that the proposed controller, well-defined for \(T\in [0,T^*)\), makes Lyapunov difference of the sampled-data closed-loop system more negative by adding appropriate high-order terms to the emulation controller. According to Corollaries 1 and 2, the stability of the closed-loop system (1) under the approximate controller can be guaranteed if the sampling period is small enough, since the term \(O(\delta ^{n+2+r-i})\) or \(O(\delta ^{r+2})\) doesn’t influence the negativity of the \(V_n(x^d_{k+1})-V_n(x_k^d)\) when \(\delta \) is sufficiently small. Clearly, unlike \(u_k^{[r]}\), the multi-rate control law \((u_{1k}^{[r]},\ldots ,u_{nk}^{[r]})\) influences the evolution not only of the whole Lyapunov function \(V_n\), but also of the other Lyapunov functions \(V_i, i=1,\ldots ,n-1\). This indicates that, compared with the approximate single-rate controller \(u_k^{[r]}\) and the emulation controller \(u_k^{[0]}\), the r-order (\(r\ge 1\)) approximate multi-rate digital controller may improve the stability properties of each \((x_1,\ldots ,x_i)\)-subsystem of system (1). On the other hand, the negativity of the \(V_n(x^d_{k+1})-V_n(x_k^d)\) is influenced by \(O(\delta ^{r+2})\) under \(u_k^{[r]}\) and \((u_{1k}^{[r]},\ldots ,u_{nk}^{[r]})\), but it is influenced by \(O(\delta ^2)\) under the emulation controller. So, compared with the emulation controller, the r-order (\(r\ge 1\)) approximate multi-rate and single-rate digital controllers may provide faster decease of the Lyapunov function for the closed-loop sampled-data system (1). These results show that our controllers should be able to enlarge the DOA for a given sampling period and may allow considering larger sampling periods.

Remark 5

It follows from the analysis above that better control performance should be provided by increasing the order of the controller, but it does not mean that we need to compute many terms \(a_{ikj}\) in controller (6). In fact, the complexity of the controller expressions may lead to more numerical errors, which will result in a degradation of the control performance. Thus, considering the trade-off between the control performance and the computation complexity, we just need to compute the first-order or second-order controller in practical implementation. The following simulation example illustrates this point.

For system (1), consider the case when \(n=2\). By solving the algebraic equation set (18), the first terms of the single-rate controller can be computed as follows.

$$\begin{aligned} a_{k0}= & {} u_c|_{t=k\delta },\\ a_{k1}= & {} \frac{1}{2}{\dot{u}}_c|_{t=k\delta },\\ a_{k2}= & {} \frac{1}{6}\ddot{u}_c|_{t=k\delta }+\frac{1}{24u_c}(p_2-1)({\dot{u}}_c)^2\big |_{t=k\delta }\\&\quad +\frac{{\dot{u}}_c}{12L_gV_2(z^2)}(L_fL_g-L_gL_f)V_2(z^2)\big |_{t=k\delta }. \end{aligned}$$

Similarly, we can get the first terms of the 2-rate controller as follows

$$\begin{aligned} a_{1k0}= & {} u_{2k0}=u_c|_{t=k\delta },\\ a_{1k1}= & {} \frac{1}{6}{\dot{u}}_c|_{t=k\delta },\\ a_{2k1}= & {} \frac{5}{6}{\dot{u}}_c|_{t=k\delta },\\ a_{1k2}= & {} -\frac{7(p_2-1)}{36u_c}({\dot{u}}_c)^2\Big |_{t=k\delta },\\ a_{2k2}= & {} \frac{1}{3}\ddot{u}_c|_{t=k\delta }+\frac{19(p_2-1)}{36u_c}({\dot{u}}_c)^2\Big |_{t=k\delta }. \end{aligned}$$

The detailed computation procedure is omitted here.

4 Simulation example

In this section, we provide a numerical simulation to illustrate the theoretical results established above. Consider the following high-order nonlinear plant

$$\begin{aligned} \left\{ \begin{array}{rcl} {\dot{x}}_{1}&{}=&{}x^3_{2}+\hbox {sin}(\lambda x_1^3),\\ {\dot{x}}_2&{}=&{}u, \end{array}\right. \end{aligned}$$
(19)

where \(x_i \in R,i=1,2\), \(\lambda \) is a known constant. It can be verified that Assumptions 1 and 2 are satisfied with \(p_1=3\) and \(p_2=1\). According to the proof of Theorem 1, we first design a continuous-time controller such that the equilibrium (0, 0) of (19) is GAS. Let \(V_1(x_1)=\frac{1}{2} \alpha _1 x^2_1, (\alpha _1>0)\). A virtual stabilizer is designed as \(x^*_2=-x_1 (\frac{3}{4}+\lambda )^{\frac{1}{3}}\) for the \(x_1\)-subsystem of (19). Let \(\xi _1=x_1\) and \(\xi _2=x_2-x_2^*\). System (19) is transformed into

$$\begin{aligned} \left\{ \begin{array}{rcl} {\dot{\xi }}_1&{}=&{}(\xi _2+x_2^{*3})^3+\hbox {sin}(\lambda x_1^3),\\ {\dot{\xi }}_2&{}=&{}u-\frac{\partial x_2^*}{\partial x_1}\dot{x}_1.\end{array}\right. \end{aligned}$$
(20)

Consider the whole Lyapunov function \(V_2(\xi _1,\xi _2)= V_1(\xi _1)+\frac{1}{4}\alpha _2 \xi _2^4,(\alpha _2>0)\), whose time derivative along the trajectories of system (20) satisfies \({\dot{V}}_2\le -\frac{3}{4}\alpha _1 x_1^4+\alpha _1 x_1\big ((\xi _2+x_2^*)^3-x_2^*\big )+\alpha _2\xi _2^3(-\frac{\partial x_2^*}{\partial x_1}+u).\)

Thus, a continuous-time controller is designed as \(u_c=-\,\xi _2\big (p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\big )\) such that \({\dot{V}}_2 < 0\), where \(p=\frac{\alpha _1}{\alpha _2}\big (9+2(8+8\lambda )^4\big ), q=3(\frac{\alpha _2}{4\alpha _1})^{\frac{1}{3}}(\frac{3}{4}+\lambda )^{\frac{4}{9}}\) and \(s=4(\frac{3}{4}+\lambda )^{\frac{1}{3}}.\)

Let \(x=(\xi _1,\xi _2)^\mathrm{T}\). Starting from the system (20) with \( f(x)=\left[ \begin{array}{c} (\xi _2+x_2^{*3})^3+sin(\lambda \xi _1^3)\\ -\frac{\partial x_2^*}{\partial \xi _1}{\dot{\xi }}_1 \end{array} \right] \) and \( g(x)=\left[ \begin{array}{c} 0\\ 1\end{array}\right] , \) we compute the 2-order approximate single-rate sampled-data controller \(u_k^{[2]}=a_{k0}+a_{k1}\delta +a_{k2}\delta ^2\), where

$$\begin{aligned} a_{k0}&=\Bigg [-\xi _2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \Bigg ]_{t=k\delta },\\ a_{k1}&=\Bigg [-{\dot{\xi }}_2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \\&\quad -\,\xi _2(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \Bigg ]_{t=k\delta },\\ a_{k2}&=\frac{1}{6}\bigg [-{\dot{\xi }}_2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \\&\quad -\,2{\dot{\xi }}_2(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \\&\quad -\,2\xi _2({\dot{x}}_1^2+x_1\ddot{x}_1+{\dot{\xi }}_2^2+\xi _2\ddot{\xi }_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \\&\quad -\,\frac{4}{9}\xi _2(x_1^2+\xi _2^2)^{-\frac{2}{3}}(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)^2\\&\quad +\,\frac{a_{k1}}{LgV_2}(L_fL_g-L_gL_f)V_2\bigg ]_{t=k\delta }, \end{aligned}$$

and the 2-order approximate 2-rate sampled-data controller \((u_{1k}^{[2]},u_{2k}^{[2]})=(a_{1k0}+a_{1k1}\delta +a_{1k2}\delta ^2, a_{2k0}+a_{2k1}\delta +a_{2k2}\delta ^2)\), where

$$\begin{aligned} a_{1k0}&=a_{1k0}=\big [-\xi _2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \big ]_{t=k\delta },\\ a_{1k1}&=\frac{1}{6}\bigg [-{\dot{\xi }}_2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \\&\quad -\,\xi _2\left( 2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2\right) \left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \bigg ]_{t=k\delta },\\ a_{2k1}&=\frac{5}{6}\bigg [-{\dot{\xi }}_2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \\&\quad -\,\xi _2(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \bigg ]_{t=k\delta },\\ a_{1k2}&=0,\\ a_{2k2}&=\frac{1}{3}\bigg [-{\dot{\xi }}_2\left( p+q(x_1^2+\xi _2^2)^{\frac{4}{3}}+s(x_1^2+\xi _2^2)\right) \\&\quad -\,2{\dot{\xi }}_2(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \\&\quad -\,2\xi _2({\dot{x}}_1^2+x_1\ddot{x}_1+{\dot{\xi }}_2^2+\xi _2\ddot{\xi }_2)\left( \frac{4}{3}q(x_1^2+\xi _2^2)^{\frac{1}{3}}+s\right) \\&\quad -\,\frac{4}{9}\xi _2(x_1^2+\xi _2^2)^{-\frac{2}{3}}(2x_1{\dot{x}}_1+2\xi _2{\dot{\xi }}_2)^2\bigg ]_{t=k\delta }. \end{aligned}$$
Fig. 1
figure 1

State evolutions when the case of \(\delta =0.18 s\); a phase portrait with 1-th approximate controllers, b phase portrait with 2-th approximate controllers

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Fig. 2
figure 2

Control evolutions with \(\delta =0.18 s\); a 1-th approximate controllers, b 2-th approximate controllers

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In the simulation, SRr and DRr (we only take \(r=1,2\)) denote the r-order approximate single-rate and double-rate controllers, respectively. CT and EM denote the continuous-time and the emulation controller, respectively. Let \(\alpha _1=10^{-3}, \alpha _2=10\) and \(\lambda =1/16\). The initial condition is set to be \((x_1(0), x_2(0))=(0.3,-\,0.4)\). The state evolutions of closed-loop systems (19) under CT, EM, SRr and DRr are plotted in Fig. 1 for \(\delta =0.18 s\). For comparison, the CT and EM solutions are plotted in all figures. It is noted that all the digital control strategies can achieve the stabilization of system (19), but the better control performance is obtained by increasing the sampling rate or increasing the approximation order. The control evolutions are depicted in Fig. 2 in the case when \(\delta =0.18 s\), by which we can see that DR2 provides the best approximation for \(u_c\). Furthermore, Fig. 3 shows that, for a large sampling period \(\delta =0.4 s\), the multi-rate controllers DR1 and DR2 still achieve stabilization and present an acceptable performance, while the emulated controller and the single-rate controller cannot guarantee the stability anymore. The advantages of multi-rate digital controller become clear when \(\delta \) increases. It should be noted that the 1-order approximate double-rate controller DR1 may be a good choice in practical applications if we consider the trade-off between the computational complexity and the control performance.

Fig. 3
figure 3

Phase portrait of the system with \(\delta =0.4 s\)

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Fig. 4
figure 4

Estimate of DOA with \(\delta =0.5 s\)

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For a given sampling period \(\delta =0.5 s\), the estimates of DOA with DR2 and EM are depicted in Fig. 4, which are obtained in the following manner. We pick a number of rays starting at the origin and simulated the system with DR2 for a set of initial conditions on these rays. In each simulation, we increase the radius by 0.05. This process is stopped when we obtain an unbounded trajectory and the initial condition from the previous simulation is used as a point on the boundary of DOA indicated by “\(\circ \)” on Fig. 4. Similarly, the same method is applied to getting the points on the boundary of DOA with EM which are denoted by “+”. It can be seen from Fig. 4 that the DOA of system (19) with DR2 is obviously larger than the DOA with EM.

5 Conclusions

In this paper, we presented two sampled-data controllers with the form of a series expansion to achieve the global asymptotic stability for a class of high-order nonlinear systems. The control performance of approximate controllers is analyzed theoretically and demonstrated by a simulation example. As a result, compared with the emulation controller, our controllers should be able to enlarge the DOA for a given sampling period and may allow considering larger sampling periods. There are still some interesting problems on this topic for further study, for example, how to obtain an explicit formula for \(T^*\) mentioned in Theorem 1, how to study the problem of output-feedback sampled-data control of high-order systems using the proposed method and how to extend the result obtained in this paper to the sampled-data systems with time-varying sampling periods.