1 Introduction

There are many kinds of reasons that can lead to time lag in a practical system, such as delayed measurements, signal transmissions and some intrinsic properties of the system. For the sake of reflecting the real time lag and achieving some special objectives, time delays are often used in actual system modeling [1]. Therefore, the study of time-delay dynamic systems is not just of great significance in engineering applications but also poses a fundamental challenge for mathematical theory. There are many important results in this filed. For example, Lyapunov stability theory was extended to time-delay systems by Krasovskii in 1959, which results in the well-known Lyapunov–Krasovskii functional (LKF)-based method [2]. Later, to avoid the manipulation of functionals, another important stability analysis approach called Razumikhin theorem-based method was proposed by Razumikhin [3]. Noteworthily, the early study on time-delay systems mainly focused on linear systems (e.g., [4, 5]). In the last two decades, nonlinear time-delay systems have attracted more attention. Lots of research results have emerged for solving the relevant problems with the help of Krasovskii’s or Razumikhin’s approach. Examples include [6,7,8] and reference therein. Furthermore, because stochastic phenomena exist widely in the real world, stochastic nonlinear time-delay systems (SNTDSs) have gradually become a research hotspot in nonlinear control field in recent years; see, for instance [9,10,11,12,13,14].

As we known, the systems with lower-triangular structure are very important research objectives in nonlinear systems domain due to the fact that not only they are able to model many practical systems but also lots of general nonlinear systems can be transformed into a system with lower-triangular form by a differential homeomorphism transformation under some conditions [15]. The backstepping method proposed by Kokotovic et al. (see [16, 17]), is one of the most useful techniques for solving control problems of the lower-triangular nonlinear systems with the strict-feedback structure in various scenarios including stochastic and time-delay cases [18,19,20,21,22,23]. However, the systems processed by using this technique must meet the requirement of fully or partially feedback linearization and have linear virtual control inputs. The high-order systems (HOS) [24, 25] that have been focused on for many years are a typical class of lower-triangular nonlinear systems not meeting the above requirements. Fortunately, by using the homogeneous domination approach [26] and the adding a power integrator (AAPI) technique [27], many interesting results on the feedback control problem of the HOS with stochastic disturbance have been obtained. Particularly, combining the LKF-based method, stabilization of high-order SNTDSs by state feedback control was considered by authors in [28,29,30,31]. Meanwhile, by introducing state-observer approaches the stabilizing problem of high-order SNTDSs via output feedback control was investigated in [32,33,34,35]. In the present paper, stabilization of low-order SNTDSs will be further studied, which is generally considered to be the nonsmooth counterpart of high-order SNTDSs.

It should be noted that several practical systems can be modeled by using the low-order system (LOS) in engineering practice, such as the liquid-level system with interaction [36], the regenerative chatter system [37] and the cascade chemical system [38]. Because the LOS cannot be feedback linearized, stabilization of such system cannot be handled by using the standard backstepping technique. In addition, due to the powers of chained integrator being less than one, the LOS is only continuous but completely nondifferentiable even not satisfying the Lipschitz condition. From a technique point of view, it is more difficult to stabilize a LOS than stabilize a HOS. Nevertheless, there have been lots of important results focusing on this problem. For example, in [39], Qian et al. described the conditions of achieving global strong stability (GSS) of continuous systems having multiple solutions, and a planar low-order system was used as an illustrative example. Later, for a class of low-order nonlinear systems, Ref. [40] investigated the finite-time stabilization problem by developing an efficient recursive design procedure, Ref. [41] addressed the GSS of a class of low-order nonlinear systems by proposed multi-rate sampling controller, and Ref. [42] dealt with the output feedback control problem of the LOS described with the p-normal form. In the stochastic setting, Ref. [43] discussed the finite-time stabilization for the LOS with stochastic disturbances by making use of the AAPI technique and stochastic stability criterion. The same problem was investigated in Ref. [44] where stochastic inverse dynamics are considered in the system. However, there are no any results on low-order SNTDSs up to now.

Table 1 A brief summarization of the literature review
Full size table

The literature review on the study of SNTDSs mentioned above is summarized in Table 1. Based on this observation, we focus on an interesting problem in the present work. That is, how to construct an output feedback control law to guarantee stability of low-order SNTDSs where each system state is affected by multiple time-varying delays? Clearly, because of the intrinsic characteristics of low-order nonlinear systems, it is a difficult task to address the problem. To be specific, the following three difficulties will be encountered when designing an output feedback controller. First, the low-order nonlinear terms lead to that the considered system in our paper is merely continuous but nonsmooth. Thus, the existing methods developed in Refs. [32,33,34,35] cannot be directly used to address the problem because some smoothness of systems is required in these methods. Second, compared with delay-free nonlinear systems, there are more difficulties to stabilize a nonlinear system with time delays. Although the some stabilization results with regard to the LOS have been obtained in Refs. [40,41,42,43,44], it is clearly that they cannot be suitable for the LOS with time delays. Third, there is no doubt that the simultaneous appearance of multiple time-varying delays, unmeasured state variables, stochastic disturbances and low-order nonlinear terms will greatly increase the difficulties of solving the problem. In this technique note, we prepare for developing a new design technique to achieve the stabilization of low-order SNTDSs by overcoming the difficulties mentioned above.

The main contributions of our paper include:

  • The output feedback stabilization of low-order SNTDSs is first considered in the present paper. Based on the AAPI technique of lower-triangular systems and the stability theory of stochastic continuous systems, an output feedback stabilizer is constructed via developing an efficient recursive design method for low-order SNTDSs.

  • For the purpose of counteracting the negative effect of multiple time-varying delays, we skillfully construct an appropriate LKF in the design process which plays a crucial role in system analysis. Note that it is different from the results developed for high-order SNTDSs in [28,29,30,31,32,33,34,35].

  • The powers of chained integrators for the low-order systems studied in the published papers, such as [40, 41, 43, 44], are restricted to the positive odd rational numbers. The present paper removes this restriction. That is, the powers of integrators can be any real numbers between 0 and 1.

  • Motivated by the work of [40, 42], a new reduced-order observer is proposed to produce the estimated values of unmeasured system states. In addition, a recursive selection procedure is developed, and the desired observer gains can be determined by following the procedure.

The present work is organized as follows. The low-order SNTDS considered in our paper, the concepts related to GSASP and some useful lemmas are introduced in Sect. 2. The detailed design process of our output feedback stabilizer and the stability analysis of the closed-loop system are provided in Sect. 3. An example of numerical simulation and an example of application to a practical system are given for the proposed methods in Sect. 4. Some conclusions and future works are stated in Sect. 5.

Throughout the paper, the symbols \({\mathbb {R}},{\mathbb {R}}^m\) and \(\mathbb {R^+}\) are used to represent the real number set, the m-dimensional Euclidian space and the positive real number set, respectively. For a matrix A, the symbol \(A^T\) represents the transpose of A. Furthermore, Tr(A) is used for denoting the trance of A if A is square. The symbol \(||\cdot ||\) denotes the Frobenius norm of matrixes or vectors, and especially we use the symbol \(|\cdot |\) to denote the absolute value. \([\cdot ]^r\) is defined as \(\textrm{sgn}(\cdot )|\cdot |^r\) for \(r\in {\mathbb {R}}\). \({\mathcal {C}}_{{\mathcal {F}}_0}^b([-h,0]; {\mathbb {R}}^n)\) denotes the family of all \({\mathcal {F}}_0\)-measurable and continuous \({\mathbb {R}}^n\)-value random variables \(X=\{X(\omega ): -h\le \omega \le 0\}\). The symbols \({\mathcal {I}}_k\) and \({\mathcal {N}}_k\) denote the index sets \(\{2,3,\ldots ,k\}\) and \(\{1,2,\ldots ,k\}\), respectively. To be simple, the argument of a function is sometime omitted. For example, we sometimes use \(d(\cdot )\) or d to denote d(s).

2 Problem statement and preliminaries

The SNTDS with the following It\(\hat{\textrm{o}}\) formalism is considered in this paper

$$\begin{aligned} \left\{ \begin{array}{lll} dx_1=\big ([x_2]^r+f_1(\bar{x}_1,\bar{x}_{1d})\big )dt + g^T_1(\bar{x}_1,\bar{x}_{1d})dw,\\ dx_2=\big ([x_3]^r+f_2(\bar{x}_2,\bar{x}_{2d})\big )dt + g_2^T(\bar{x}_2,\bar{x}_{2d})dw,\\ \ \ \ \ \ \ \ \vdots \\ dx_n=\big ([u]^r+f_n(\bar{x}_n,\bar{x}_{nd})\big )dt + g_n^T(\bar{x}_n,\bar{x}_{nd})dw,\\ y=x_1, \end{array} \right. \end{aligned}$$
(1)

where the power of the chained integrators \(r\ (0<r<1)\) is an arbitrary real number, \(w\in {\mathbb {R}}^m\) represents a standard Wiener process (SWP), and u is control input. \(x=(x_1,x_2,\ldots ,x_n)^T\), which is unmeasurable except for \(x_1\), represents the system state. For every \(i\in {\mathcal {N}}_n\), \(\bar{x}_i=(x_1,x_2,\ldots ,x_i)^T\), \(\bar{x}_{id}=(x_{1d},x_{2d},\ldots ,x_{id})^T\), where the symbol \(x_{id}\) represents the delayed state \(x_i\big (t-d_i(t)\big )\). The functions \(d_i(t):{\mathbb {R}}^+\rightarrow [0,h_i]\), satisfying the conditions \(\dot{d}_i(t)\le v_i<1\) with the positive constant numbers \(v_i\), \(i\in {\mathcal {N}}_n\), denote the time-varying state delays. The initial data of the system are taken as \(x(\theta )=\{\phi (\theta ): -h\le \theta \le 0\}\in {\mathcal {C}}^{b}_{{\mathcal {F}}_0}([-h,0];{\mathbb {R}}^n)\), where \(h=\max \{h_1,h_2,\ldots ,h_n\}\). At the end, for every \(i\in {\mathcal {N}}_n\), \(f_i:{\mathbb {R}}^i\times {\mathbb {R}}^i\rightarrow {\mathbb {R}}\) satisfying \(f_i(0,0)=0\) is continuous and called the drift term, and \(g_i:{\mathbb {R}}^i\times {\mathbb {R}}^i\rightarrow {\mathbb {R}}^m\) satisfying \(g_i(0,0)=0\) is continuous and called the diffusion term.

If \(g_i(\cdot )=0\) and \(d_i(t)=0\), system (1) will be a deterministic LOS, whose chained integrators have powers greater than 0 but less than 1. Notably, it is a nonsmooth counterpart of a deterministic HOS. Since the powers of chained integrator are between 0 and 1, the system is not feedback linearizable and differentiable. This characteristic coupling with the effects of stochastic disturbances, unmeasured states and time delays will make the controller design more difficult. In this paper, a new design technique is developed to overcome the difficulties and achieve the output feedback control of system (1). For achieving this aim, we have Assumption 1 for system (1).

Assumption 1

There are constants \(a_i>0\) and \(b_i>0\), \(i\in {\mathcal {N}}_n\), such that

$$\begin{aligned}&|f_i(\bar{x}_i,\bar{x}_{id})|\le a_i\sum _{j=1}^{i}(|x_j|^r+|x_{jd}|^r),\\&||g_i(\bar{x}_i,\bar{x}_{id})||\le b_i\sum _{j=1}^{i}(|x_j|^{\frac{r+1}{2}}+|x_{jd}|^{\frac{r+1}{2}}). \end{aligned}$$

Remark 1

In Liu [43] and Huang [45], the drift terms \(f_i(\bar{x}_i, \bar{x}_{id})=f_i(\bar{x}_i)\) and the diffusion terms \(g_i(\bar{x}_i, \bar{x}_{id})=g_i(\bar{x}_i)\), which are independent of time delays. That is, Assumption 3.1 in [43] and Assumption 2 in [45], which is called the nonlinear growth condition (NGC), are special cases of Assumption 1 of the present paper when \(x_{id}=0, i\in {\mathcal {N}}_n\). Clearly, our assumption is capable of being used for more general stochastic systems and it is a more general NGC. It should be noted that NGC are commonly used in the study of nonlinear systems. Many papers have discussed the significance and rationality of the NGC, to name just a few, see [43, 45, 46] etc.

Consider the general SNTDS

$$\begin{aligned} dx(t)&=\mu \big (x(t),x(t-\tau (t)),t\big )dt \nonumber \\&\quad + \sigma ^T\big (x(t),x(t-\tau (t)),t\big )dw(t), \quad t\ge 0, \end{aligned}$$
(2)

with \(\tau (t)\) satisfying \(0\le \tau (t)\le \tau _M\). w(t) is an m-dimensional SWP. \(\mu :{\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) and \(\sigma :{\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}^{m\times n}\) are continuous functions and satisfy \(\mu (0,0,t)=0\), \(\sigma (0,0,t)=0\). The initial condition is taken as \(\{x(\theta ): -\tau _M\le \theta \le 0\}=\varphi \in {\mathcal {C}}_{{\mathcal {F}}_0}^{b}([-\tau _M,0]; {\mathbb {R}}^n)\). System (2) always has a solution in the weak sense, but it may not be unique [47] because the (local) Lipschitz condition cannot be satisfied by the coefficients. This implies that the stability problem of system (2) cannot be solved by using the classical stochastic system theory (see [48, 49]), since the uniqueness of solution is always needed in the application of this theory. Inspired by the deterministic analogs in [50, 51] and the delay-free analog in [52, 53], the classical stochastic stability concept has been slightly extended in [54] so that it can be applied to more general SNTDSs. In what follows, GSSP and GSASP represent the abbreviation of “globally strongly stable in probability” and “globally strongly asymptotically stable in probability,” respectively.

Definition 1

([54]). The trivial solution \(x=0\) of system (2) is GSSP, if there is a class-\({\mathcal {K}}\) function \(\gamma _x(\cdot )\) such that \(P_x\{ |x(t)|\le \gamma _x(||\varphi ||)\}\ge 1-\epsilon \) holds for any \(\epsilon >0\) and weak solution x(t), where \(t\ge 0\), \(\varphi \in {\mathcal {C}}_{{\mathcal {F}}_0}^{b}([-\tau _M,0];{\mathbb {R}}^n)\backslash \{0\}\) with \(||\varphi ||=\sup \{|x(\theta )|:-\tau _M\le \theta \le 0\}\). Furthermore, if \(P_x\{\lim _{t\rightarrow +\infty }||x(t)||=0\}=1\) holds for any weak solution x(t), it is said to be GSASP.

Lemma 1

([54]). Suppose that there exist a positive definite function \(V(x)\in {\mathcal {C}}^2\) satisfying \(V(x)\rightarrow \infty \) as \(||x||\rightarrow \infty \), and a nonnegative function \(W(x)\in {\mathcal {C}}^0\), such that

$$\begin{aligned} \mathscr {L}V(x)&\triangleq \frac{\partial V(x)}{\partial x}\mu \big (x(t),x(t-\tau (t))\big )\nonumber \\&\quad +\frac{1}{2}Tr\bigg \{\sigma \big (x(t),x(t-\tau (t))\big )\frac{\partial ^2V(x)}{\partial x^2}\nonumber \\&\qquad \sigma ^{T}\big (x(t),x(t-\tau (t))\big )\bigg \}\le -W\big (x(t)\big ), \end{aligned}$$
(3)

then system (2) is GSSP at \(x=0\). Furthermore, it is GSASP at \(x=0\) in the case of W(x) being positive definite.

Remark 2

In classical stochastic system theory, there are two types of available approaches to deal with the problem of stability and stabilization for SNTDSs, namely the LKF approach [48] and the Lyapunov–Razumikhin function approach [49]. When one uses these two methods, the existence and uniqueness of strong solutions for SNTDSs are always required, that is, the considered systems must satisfy the (locally) Lipschitz condition. Clearly, they cannot be applied to system (2), because the drift and diffusion terms of system (2) are only continuous but non-Lipschitz. Definition 1 and Lemma 1 can be viewed as an extension of stability theory for SNTDSs developed in [48], which are applicable to more general SNTDSs.

Remark 3

Definition 1 provides two stability notions for nonsmooth stochastic system (2), namely the global strong stability in probability and the global strong asymptotic stability in probability. Both types of stability describe the asymptotic behavior of the trajectories of the system or the stability behavior of the trivial solution \(x=0\) of the system as time goes to infinity. The difference between them is that the latter not only requires each weak solution to satisfy the former, but also requires them to eventually converge to zero.

The following lemmas provide several useful inequalities which are necessary in what follows.

Lemma 2

([55]). Let \(a, b>1\) with \(\frac{1}{a}+\frac{1}{b}=1\). For any \(v,w\in {\mathbb {R}}\), we have

$$\begin{aligned} |vw|\le \frac{1}{a}|v|^{a}+\frac{1}{b}|w|^{b}. \end{aligned}$$
(4)

Lemma 3

([55]). Let \(0<\alpha <1\). For any \(v\ge -1\), we have

$$\begin{aligned} (1+v)^{\alpha }\le 1+\alpha v. \end{aligned}$$
(5)

Lemma 4

([51]). For any \(v,w\in {\mathbb {R}}\) and \(a,b,c\in {\mathbb {R}}^+\), we have

$$\begin{aligned} |v|^a|w|^b\le \frac{a}{a+b}c|v|^{a+b}+\frac{b}{a+b}c^{-\frac{a}{b}}|w|^{a+b}. \end{aligned}$$
(6)

Lemma 5

([56]). For any \(v,w\in {\mathbb {R}}\) and \(\gamma \ge 1\), the following inequalities hold

$$\begin{aligned}&v+w|^{\gamma }\le \! 2^{\gamma -1}|v^{\gamma }+w^{\gamma }|, \end{aligned}$$
(7)
$$\begin{aligned}&\quad |[v]^{\frac{1}{\gamma }}-[w]^{\frac{1}{\gamma }}|\le \!2^{1-\frac{1}{\gamma }}|v-w|^{\frac{1}{\gamma }}. \end{aligned}$$
(8)

Lemma 6

([57]). For any \(v_1,v_2,\ldots ,v_n,c\in \mathbb {R^+}\), it is true that

$$\begin{aligned} (v_1\!+\!v_2\!+\!\cdots \!+\!v_n)^c\!\le \!\max \{n^{c\!-\!1},1\}\!(v_1^c\!+\!v_2^c\!+\!\cdots \!+\!v_n^c). \end{aligned}$$
(9)

In particular, when \(c=2\), we have \((v_1+v_2+\cdots +v_n)^2\le n(v_1^2+v_2^2+\cdots +v_n^2)\).

Lemma 7

Let \(\varepsilon \in (0,1)\) and \(v\in {\mathbb {R}}\). For any real number \(s\in (0,1)\), we have

$$\begin{aligned}{}[v]^s+[1-v]^s>-\varepsilon ^2|v|^{1+s}+(2^s-1)\varepsilon ^{1-s}. \end{aligned}$$
(10)

Proof

First, consider the case when \(|v|\le 1\). Letting \(F(v)=[v]^s+[1-v]^s\), we have \(F'(v)=s|v|^{s-1}-s(1-v)^{s-1}\). Clearly, when \(-1\le v <\frac{1}{2}, F'(v)>0\) and when \(\frac{1}{2}\le v<1, F'(v)<0\), that is to say, F(v) is strictly monotonically increasing on \([-1,\frac{1}{2})\) and strictly monotonically decreasing on \([\frac{1}{2}, 1)\). Thus, it follows from \(F(0)=F(1)\) that F(v) have a minimum at \(v=-1\) on the set \(\{v:|v|\le 1\}\). Since \(0<\varepsilon ^{1-s}<1\), a direct calculation yields

$$\begin{aligned}{} & {} F(v)+\varepsilon ^2|v|^{1+s}\ge F(v)\ge F(-1)= 2^s-1\\{} & {} >\varepsilon ^{1-s}(2^s-1). \end{aligned}$$

Namely, inequality (10) holds when \(|v|\le 1\).

Next, we consider the case when \(|v|>1\). According to Lemma 2, we have

$$\begin{aligned}&[v]^s+[1-v]^s+\varepsilon ^2|v|^{s+1} \nonumber \\&=\big (([v]^s+[1-v]^s)^{\frac{1+s}{2}} \big )^{\frac{2}{1+s}}+\big ((\varepsilon ^2|v|^{1+s})^{\frac{1-s}{2}} \big )^{\frac{2}{1-s}} \nonumber \\&\!\ge \! \frac{1\!+\!s}{2}\big (([v]^s\!\!+\!\![1\!-\!\!v]^s)^{\frac{1\!\!+\!\!s}{2}} \big )^{\frac{2}{1\!+\!s}}\!\!+\!\!\frac{1\!-\!s}{2}\big ((\varepsilon ^2|v|^{1\!+\!s})^{\frac{1\!-\!s}{2}} \big )^{\frac{2}{1-s}} \nonumber \\&\ge ([v]^s+[1-v]^s)^{\frac{1+s}{2}}(\varepsilon ^2|v|^{1+s})^{\frac{1-s}{2}} \nonumber \\&= \big (([v]^s+[1-v]^s)|v|^{1-s}\big )^{\frac{1+s}{2}}\varepsilon ^{1-s}. \end{aligned}$$
(11)

Let \(G(v)=([v]^s+[1-v]^s)|v|^{1-s}\). Then, the derivative of G(v) is capable of being calculated as

$$\begin{aligned} G'(v)&=\big (\textrm{sgn}(v)|v|+\textrm{sgn}(1-v)|1-v|^s|v|^{1-s}\big )'\nonumber \\&=1-s\big |\frac{1}{v}-1\big |^{s-1}-(1-s)\big |\frac{1}{v}-1\big |^s. \end{aligned}$$
(12)

Since \(\frac{1}{v}-1<0\), Eq. (12) can be further calculated as \(G'(v)=1+|\frac{1}{v}-1|^{s-1}(\frac{1}{v}-\frac{s}{v}-1)\). From Lemma 3, it is easy to see \((1-\frac{1}{v})^{1-s}\le 1-\frac{1}{v}+\frac{s}{v}\). Thus, we can get that \(G'(v)<0\) on the set \(\{v:|v|>1\}\). This implies that \(G(v)\ge \min \{G(-1),G(+\infty )\}=\min \{2^s-1, s\}\). Clearly, Lemma 3 yields \(2^s-1<s\). So, we have \(G(v)\ge 2^s-1\). Applying this result to inequality (11), we can get that inequality (10) holds when \(|v|>1\). \(\square \)

Remark 4

It should be pointed that an inequality similar to inequality (10) has been introduced in Ref. [58] where the power s must be expressed as the fraction form with positive odd numerators and denominators. In Lemma 7, we provide a new version without this requirement on the power s and give the corresponding proof. Inequality (10) is crucial in the observer design for system (1).

3 Dynamic output feedback stabilizer design

Because only output signal \(y=x_1\) is measured, we will construct an observer-based stabilizer to deal with the stabilization of low-order SNTDS (1) in this section. Under Assumption 1, the construction process of the controller is divided into three parts.

3.1 State feedback design

In the first part, a state feedback control law is designed by extending the AAPI technique. Notably, to solve the obstacles arisen from the multiple delays, a novel LKF will be introduced in the controller design.

For the \(x_1\)-subsystem of (1), choose a Lyapunov–Krasovskii functional \(V_1(x_1)=\frac{k_1}{2}x_1^2+\frac{a_1}{1-v_1}\int _{t-d_1}^{t}|x_1(s)|^{r+1}ds\), \(k_1>0\), and two coordinate transformations \(\xi _1=x_1\), \(\xi _{1d}=x_{1d}\). According to Lemmas 2 and 6, we have

$$\begin{aligned}&\mathscr {L}V_1(x_1)\nonumber \\&=k_1\xi _1\big ([x_2]^r\!\!+\!\!f_1\big )\!\!+\!\!\frac{1}{2}k_1||g_1||^2\nonumber \\&\quad \!\!+\!\!\frac{a_1}{1\!\!-\!\!v_1} \big (|\xi _1|^{r\!+\!1}\!\!-\!\!|\xi _{1d}|^{r\!+\!1}(1\!\!-\!\!\dot{d}_1)\big )\nonumber \\&\le k_1\xi _1\big ([x_2]^r\!-\![x_2^*]^r\big ) +k_1\xi _1[x_2^*]^r\nonumber \\&\quad +k_1a_1|\xi _1|\big (|\xi _1|^r\!+\!|\xi _{1d}|^r\big )\nonumber \\&\quad +\frac{1}{2}k_1b_1^2\big (|\xi _1|^{\frac{r+1}{2}}+|\xi _{1d}|^{\frac{r+1}{2}}\big )^2 \!\!+\!\!\frac{a_1}{1\!\!-\!\!v_1}|\xi _1|^{r+1}\! -\!a_1|\xi _{1d}|^{r+1}\nonumber \\&\le k_1\xi _1\big ([x_2]^r\!\!-\!\![x_2^*]^r\big )\!\! +\!\!k_1\xi _1[x_2^*]^r\!+\!(k_1a_1\!\!+\!\!k_1b_1^2\!\! +\!\!\frac{a_1}{1\!\!-\!\!v_1})|\xi _1|^{r\!+\!1}\nonumber \\&\quad +k_1a_1|\xi _1||\xi _{1d}|^r-P_{11}|\xi _{1d}|^{r+1}\nonumber \\&\le k_1\xi _1\big ([x_2]^r-[x_2^*]^{r}\big ) + k_1\xi _1[x_2^*]^r+l_{111}|\xi _1|^{r+1}, \end{aligned}$$
(13)

where \(P_{11}=k_1b_1^2-a_1\) and \(l_{111}=k_1a_1+k_1b_1^2+\frac{a_1}{1-v_1}+\frac{k_1a_1}{1+r} (\frac{(a_1-k_1b_1^2)(1+r)}{rk_1a_1})^{-r}\). Design \(x_2^*=-\beta _1\xi _1\) with \(\beta _1=\big (\frac{c_{11}+l_{111}}{k_1}\big )^{\frac{1}{r}}>0\) and the arbitrary constant \(c_{11}>0\). Since \(k_1\xi _1[x_2^*]^r=k_1\textrm{sgn}(\xi _1)|\xi _1|\textrm{sgn} (x_2^*)|x_2^*|^r=-(c_{11}+l_{111})|x_1|^{r+1}\), it follows from (13) that

$$\begin{aligned} \mathscr {L}V_1(x_1)\le -c_{11}|\xi _1|^{r+1}+k_1\xi _1\big ([x_2]^r-[x_2^*]^r\big ). \end{aligned}$$
(14)

Generally, for the \((x_1,x_2,\ldots ,x_{i-1})\)-subsystem of system (1), assume that we can find \(V_{i-1}(x_1,x_2,\ldots ,x_i)\) and two sets of coordinate transformations

$$\begin{aligned} \xi _j=x_j-x_j^*,\quad \xi _{jd}=x_{jd}-x_{jd}^*,\quad j\in {\mathcal {N}}_i, \end{aligned}$$
(15)

where \(x_j^*=-\beta _{j-1}\xi _{j-1}, x_{jd}^*=-\beta _{j-1}\xi _{j-1,d}\) with \(\beta _0=0\) and \(\beta _j, c_{i-1,j}>0, j\in {\mathcal {N}}_{i-1}\), such that

$$\begin{aligned}&\mathscr {L}V_{i-1}(x_1,x_2,\ldots ,x_{i-1})\nonumber \\&\le -\sum _{j=1}^{i-1}c_{i-1,j}|\xi _j|^{r+1}+k_{i-1}\xi _{i-1}([x_i]^r-[x_i^*]^{r}). \end{aligned}$$
(16)

Next, we will prove that formula (16) still holds for the \((x_1,x_2,\ldots ,x_i)\)-subsystem. In fact, choose a Lyapunov–Krasovskii functional for the subsystem

$$\begin{aligned}&V_i(x_1,x_2,\ldots ,x_i) \nonumber \\&=\!V_{i\!-\!1}\!\!+\!\!\frac{k_i}{2}\xi _i^2\!\!+\!\!K \sum _{j\!=\!1}^{i\!-\!1}\!\!\int _{t-d_j}^{t}\!\!\!|\xi _j|^{r\!+\!1}(\!s\!)ds\\&\!\!+\!\!\frac{a_i}{1\!-\!v_i}\!\!\int _{t\!-\!d_i}^{t}\!\!|\xi _i|^{r\!+\!1}(\!s\!)ds \end{aligned}$$

with arbitrary positive constants \(k_i\) and K. Based on coordinate transformation (15), a simple calculation obtains that

$$\begin{aligned}&\mathscr {L}V_i(x_1,x_2,\ldots ,x_i)\nonumber \\&=\mathscr {L}V_{i-1}+k_i\xi _i\big ([x_{i+1}]^r+f_i+\sum _{j=1}^{i-1}B_{ij}([x_{j+1}]^r+f_j)\big )\nonumber \\&\quad \!+\!\!\frac{k_i}{2}||g_i\!\!+\!\!\sum _{j=1}^{i\!-\!1}\!B_{ij}g_j||^2\!\!+\!\!K\!\sum _{j=1}^{i-1}|\xi _j|^{r+1} \!\!-\!\!K\!\sum _{j\!+\!1}^{i-1}\!|\xi _{jd}|^{r+1} (1\!\!-\!\!\dot{d}_j)\nonumber \\&\quad +\frac{a_i}{1-v_i}|\xi _i|^{r+1}-\frac{a_i}{1-v_i}|\xi _{id}|^{r+1}(1-\dot{d}_i)\nonumber \\&\le \!-\!\!\sum _{j\!=\!1}^{i\!-\!1}\!\!c_{i\!-\!1,j}\!|\xi _j|^{r\!+\!1}\!\! +\!\!k_{i\!-\!1}\xi _{i\!-\!1}\big (\![x_i]^r\!\!\!-\!\![x_i^*]^r\!\big ) \!\!+\!\!k_i\xi _i\big (\![x_{i\!+\!1}]^r\!\!\!-\!\! [x_{i\!+\!1}^*]^r\!\big )\nonumber \\&\quad +k_i\xi _i[x_{i+1}^*]^r+k_i\xi _i \big (f_i+\sum _{j=1}^{i}B_{ij}([x_{j+1}]^r+f_j)\big )\nonumber \\&\quad +\frac{k_i}{2}||g_i+\sum _{j=1}^{i-1}B_{ij}g_j||^2+K\sum _{j=1}^{i-1}|\xi _j|^{r+1}\nonumber \\&\quad -\!K\sum _{j\!=\!1}^{i\!-\!1}(1\!-\!v_j)|\xi _{jd}|^{r\!+\!1}\!+\!\frac{a_i}{1\!-\!v_i}\!|\xi _i|^{r\!+\!1}\!-\!a_i|\xi _{id}|^{r\!+\!1}, \end{aligned}$$
(17)

where \(B_{ij}=\beta _{i-1}\beta _{i-2}\cdots \beta _j,j\in {\mathcal {N}}_{i-1}\).

We will investigate the terms on the right of inequality (17). According to coordinate transformation (15), Assumption 1, Lemmas 5 and 6, it follows that

$$\begin{aligned}&\frac{k_i}{2}||g_i+\sum _{j=1}^{i-1}B_{ij}g_j||^2\nonumber \\&\le k_i||g_i||^2+k_i(i-1)\sum _{j=1}^{i-1}B_{ij}||g_j||^2\nonumber \\&\le 2ik_ib_i^2\sum _{j=1}^{i}(|x_j|^{r+1}+|x_{jd}|^{r+1})\nonumber \\&\quad +2(i-1)k_i\sum _{j=1}^{i-1}jb_j^2B_{ij}^2\sum _{m=1}^{j}(|x_m|^{r+1}+|x_{md}|^{r+1})\nonumber \\&\le 2^rk_i\sum _{j=1}^{i-1}(\bar{b}_{ij}+\beta _j^{r+1}\bar{b}_{i,j+1})(|\xi _{j}|^{r+1}+|\xi _{jd}|^{r+1})\nonumber \\&\quad +2^rk_i\bar{b}_{ii}(|\xi _{i}|^{r+1}+|\xi _{id}|^{r+1}). \end{aligned}$$
(18)

where \(\bar{b}_{ij}=2ik_ia_i^2+2(i-1)\sum _{m=j}^{i-1}mB_{im}^2b_m^2,\ j\in {\mathcal {N}}_i\). Substituting (18) into (17) yields

$$\begin{aligned}&\mathscr {L}V_i(x_1,x_2,\ldots ,x_i)\nonumber \\&\le -\sum _{j=1}^{i-1}c_{i-1,j}|\xi _j|^{r+1}+k_{i-1}\xi _{i-1}([x_i]^r-[x_i^*]^r)\nonumber \\&\quad +k_i\xi _i([x_{i+1}]^r-[x_{i+1}^*]^r)+k_i\xi _i[x_{i+1}^*]^r \nonumber \\&\quad +k_i\xi _i\big (f_i+\sum _{j=1}^{i}B_{ij}([x_{j+1}]^r+f_j)\big )\nonumber \\&\quad +\!\sum _{j=1}^{i\!-\!1}\!l_{ij1}\xi _j^{r\!+\!1}\!+\!h_{i1}\xi _i^{r\!+\!1}\!-\!\sum _{j=1}^{i\!-\!1}P_{ij}\xi _{jd}^{r+1}\!-\!P_{ii}\xi _{id}^{r\!+\!1}, \end{aligned}$$
(19)

where \(l_{ij1}=K+2^rk_i\sum _{}^{}(\bar{b}_{ij}+\beta _{j}^{r+1}\bar{b}_{i,j+1}), h_{i1}=\frac{a_i}{1-v_i}+2^rk_i\bar{b}_{ii},\) \(P_{ij}=K(1-v_i)-2^rk_i(\bar{b}_{ij}+\beta _{j}^{r+1}\bar{b}_{i,j+1}), j\in {\mathcal {N}}_{i-1}\), and \(P_{ii}=a_i-2^rk_i\bar{b}_{ii}\). For the second term, by using (6) and (8), we can get

$$\begin{aligned}&k_{i-1}\xi _{i-1}([x_i]^r-[x_i^*]^r)\nonumber \\&\le 2^{1\!-\!r}k_{i\!-\!1}|\xi _{i\!-\!1}||\xi _{i}|^r\le l_{i,i-1,2}|\xi _{i\!-\!1}|^{r+1}\!+\!h_{i2}|\xi _i|^{r\!+\!1}, \end{aligned}$$
(20)

where \(l_{i,i-1,2}, h_{i2}>0\) are two constants. Similarly, it follows from Assumption 1 and Lemma 4 that

$$\begin{aligned}&k_i\xi _i\big (f_i+\sum _{j=1}^{i}B_{ij}([x_{j+1}]^r+f_j)\big )\nonumber \\&\le k_i|\xi _i|\Big (a_i\sum _{j=1}^{i}|x_j|^r+|x_{jd}|^r)+\sum _{j=1}^{i-1}B_{ij}\big ([x_{j+1}]^r\nonumber \\&\quad +a_j\sum _{m=1}^{j}(|x_m|^r+|x_{md}|^r)\big )\Big )\nonumber \\&\le \!k_i|\xi _i|\!\sum _{j\!=\!1}^{i}\!\!\Big (\!\tilde{b}_{ij}(|\xi _j|^r\!\!+\!\!\beta _{j\!-\!1}^r|\xi _{j\!-\!1}|^r)\nonumber \\&\quad \!\!+\!\!\hat{b}_{ij}(|\xi _{jd}|^r\!\!+\!\!\beta _{j\!-\!1}^r|\xi _{j\!-\!1,d}|^r )\!\!\Big )\nonumber \\&\le \!\!\sum _{j\!=\!1}^{i\!-\!1}\!l_{ij3}|\xi _j|^{r\!+\!1}\!\!+\!\!h_{i3}|\xi _i|^{r\!+\!1}\!\!+\!\!\!\sum _{j\!=\!1}^{i\!-\!1}\!\!P_{ij}|\xi _{jd}|^{r\!+\!1} \!\!+\!\!P_{ii}|\xi _{id}|^{r\!+\!1}\!\!, \end{aligned}$$
(21)

where for any \(j\in {\mathcal {N}}_{i-1}\), \(\hat{b}_{ij}=a_i+\sum _{m=j}^{i-1}a_mB_{im}\), \(\hat{b}_{ii}=a_i\), \(\tilde{b}_{ij}=a_i+B_{i,j-1}+\sum _{m=j}^{i-1}a_mB_{im}\), \(\tilde{b}_{ii}=a_i+B_{i,i-1}\), and \(l_{ij3}\), \(h_{i3} >0\) are constants. Then, substituting (20) and (21) into (19) results in

$$\begin{aligned}&\mathscr {L}V_i(x_1,x_2,\dots ,x_i)\nonumber \\&\le -\sum _{j=1}^{i-1}c_{ij}|\xi _j|^{r+1}+k_i\xi _i([x_{i+1}]^r-[x_{i+1}^*]^{r})\nonumber \\&\quad +(h_{i1}+h_{i2}+h_{i3})|\xi _i|^{r+1}+k_i\xi _i[x_{i+1}^*]^r, \end{aligned}$$
(22)

where \(c_{ij}=c_{i-1,j}-l_{ij1}-l_{ij3}>0, \ j\in {\mathcal {N}}_{i-2}\), and \(c_{i,i-1}=c_{i-1,i-1}-l_{i,i-1,1}-l_{i,i-1,2}-l_{i,i-1,3}>0\). By designing \(x_{i+1}^*=-\beta _i\xi _i\) with \(\beta _i=(\frac{c_{ii}+h_{i1}+h_{i2}+h_{i3}}{k_i})^{\frac{1}{r}}>0\) and the arbitrary positive constant \(c_{ii}\), it follows that

$$\begin{aligned}&\mathscr {L}V_i(x_1,x_2,\dots ,x_i)\nonumber \\&\le -\sum _{j=1}^{i}c_{ij}|\xi _j|^{r+1}+k_i\xi _i\big ([x_{i+1}]^r-[x_{i+1}^*]^r\big ). \end{aligned}$$
(23)

The induction proof above shows that (16) holds for every \(i\in {\mathcal {N}}_n\). Thus, for system (1), we can choose

$$\begin{aligned}&V_n(x_1,x_2,\ldots ,x_n)\nonumber \\&=V_{n-1}+\frac{k_n}{2}\xi _n^2+K\sum _{j=1}^{n-1}\int _{t-d_j}^{t}|\xi _j(s)|^{r+1}ds\nonumber \\&\quad +\frac{a_n}{1-v_n}\int _{t-d_n}^{t}|\xi _n(s)|^{r+1}ds \end{aligned}$$

with \(k_n>0\), and design \(x^*_{n+1}=-\beta _n\xi _n\) with \(\beta _n=(\frac{c_{nn}+h_{n1}+h_{n2}+h_{n3}}{k_n})^{\frac{1}{r}}>0\) and the arbitrary positive constant \(c_{nn}\), such that

$$\begin{aligned}&\mathscr {L}V_n(x_1,x_2,\ldots ,x_n)\nonumber \\&\le -\sum _{j=1}^{n}c_{nj}|\xi _j|^{r+1}+k_n\xi _n([u]^{r}-[x_{n+1}^*]^r). \end{aligned}$$
(24)

Remark 5

It is noted that the multiple time-varying delays \(d_i(t)\) need to satisfy \(\dot{d}(t)\le v_i<1\) with the positive constant numbers \(v_i, i\in {\mathcal {N}}_n\). This constraint condition on time delays is necessary in our controller design, which has wide applications in the study of time-delay systems, such as [21, 30]. It means that the change of the time delays is relatively slow. From the design process above, we can see that \(v_i, i\in {\mathcal {N}}_n\) greatly effect the performance of the controller and cannot be too close to one in practice.

3.2 Reduced-order observer design

In this part, a reduced-order state observer will be introduced to estimate the unmeasurable system states \(x_i,i\in {\mathcal {I}}_n\). Let \(\hat{z}_i=\hat{x}_i-s_{i-1}\hat{x}_{i-1}, i\in {\mathcal {I}}_n\), where \(\hat{x}_1=x_1\) and \(s_j>1,j\in {\mathcal {N}}_{n-1}\) are dynamic gains to be determined later. Construct the following dynamic observer equation to generate \(\hat{z}_i, i\in {\mathcal {I}}_n\)

$$\begin{aligned} \left\{ \begin{array}{lll} \dot{\hat{z}}_i=[\hat{x}_{i+1}]^r-s_{i-1}[\hat{x}_i]^r,\ \ i\in {\mathcal {I}}_{n-1},\\ \dot{\hat{z}}_n=[u]^r-s_{n-1}[\hat{x}_n]^r. \end{array} \right. \end{aligned}$$
(25)

For any \(i\in {\mathcal {I}}_n\), the estimator \(\hat{x}_i\) of \(x_i\) can be obtained by \(\hat{x}_i=\hat{z}_i+s_{i-1}\hat{x}_{i-1}\). Meanwhile, let \(z_i=x_i-s_{i-1}x_{i-1}\), and define an error variable \(e_i=z_i-\hat{z}_i\). According to dynamic systems (1) and (25), we can get that

$$\begin{aligned} \left\{ \begin{array}{lll} \!de_i\!=\!\big (([x_{i+1}]^r\!-\![\hat{x}_{i+1}]^r)\! +\!(f_i\!-\!s_{i-1}f_{i-1})\!-\!s_{i\!-\!1}([x_i]^r\\ \quad \ \ \ \ -[\hat{x}_i]^r) \big )dt\!+\!\big (g_i-s_{i\!-\!1} g_{i\!-\!1}\big )^T\!dw,\ i\in {\mathcal {I}}_{n\!-\!1},\\ \!de_n\!=\!\big (\!(f_n\!-\!s_{n-1}f_{n\!-\!1})\!-\!s_{n-1} ([x_n]^r-[\hat{x}_n]^r) \big )dt\\ \quad \ \ \ \ +\big (g_n\!-\!s_{n\!-\!1}g_{n\!-\!1}\!\big )^Tdw. \end{array} \right. \end{aligned}$$
(26)

For error system (26), construct a Lyapunov–Krasovskii functional

$$\begin{aligned} U(e_2,e_3,\ldots ,e_n)&\!=&\!\sum _{i=2}^{n}\!\frac{m_i}{2}e_i^2\!\\{} & {} \quad +\!\sum _{i\!=\!1}^{n}\frac{w_i}{1\!-\!v_i}\!\int _{t-d_i}^{t}\!|\xi _i(s)|^{r+1}ds \end{aligned}$$

with arbitrary positive constants \(m_i\) and \(w_i\). A simple calculation yields

$$\begin{aligned}&\mathscr {L}U(e_2,e_3,\ldots ,e_n)\nonumber \\&=\!\sum _{i=2}^{n-1}m_ie_i\big ([(x_{i+1}]^r\!\!-\![\hat{x}_{i\!+\!1}]^r\big )\nonumber \\&\quad \!-\!\!\sum _{i=2}^{n}m_ie_is_{i-1}\big ([x_i]^r\!\!-\![\hat{x}_i]^r\big )\nonumber \\&\quad +\!\sum _{i=2}^{n}m_ie_i\big (f_i-s_{i-1}f_{i-1}\big )\nonumber \\&\quad \!+\!\frac{1}{2}\sum _{i=2}^{n}m_i||g_i-s_{i-1}g_{i-1}||^2\nonumber \\&\quad +\sum _{i=1}^{n}\frac{w_i}{1-v_i} \big (|\xi _i|^{r+1}-|\xi _{id}|^{r+1}(1-\dot{d}_i)\big ). \end{aligned}$$
(27)

Consider the first term of (27). According to inequality (8) we obtain that

$$\begin{aligned}&\sum _{i=2}^{n-1}m_ie_i\big ([(x_{i+1}]^r-[\hat{x}_{i+1}]^r\big )\nonumber \\&\ \le 2^{1-r}\sum _{i=2}^{n-1}m_i|e_i|\ |x_{i+1}-\hat{x}_{i+1}|^r. \end{aligned}$$
(28)

For the second term of (27), we rewrite it as the following form

$$\begin{aligned}&-\sum _{i=2}^{n}m_ie_is_{i-1}\big ([x_i]^r-[\hat{x}_i]^r\big )\nonumber \\&=-\sum _{i=2}^{n}m_ie_is_{i-1}\big ([x_i]^r-[x_i-e_i]^r)\big )\nonumber \\&\quad -\sum _{i=2}^{n}m_ie_is_{i-1}\big ([x_i-e_i]^r-[\hat{x}_i]^r\big ). \end{aligned}$$
(29)

For any \(i\in {\mathcal {I}}_n\), assuming that \(e_i\ne 0\), let \(v=\frac{x_i}{e_i}, \varepsilon =s_{i-1}^{-\frac{1}{r+1}}\) and \(s=r\). Then, according to Lemma 7, it follows that

$$\begin{aligned}&-m_ie_is_{i-1}\big ([x_i]^r-[x_i-e_i]^r)\nonumber \\&\le m_is_{i-1}^{\frac{r-1}{r+1}}|x_i|^{r+1}-(2^r-1)m_i s_{i-1}^{\frac{2r}{r+1}}|e_i|^{r+1}. \end{aligned}$$
(30)

Clearly, inequality (30) holds when \(e_i=0\). In addition, with the help of \(\hat{x}_i=x_i-e_i-s_{i-1}(x_{i-1}-\hat{x}_{i-1})\) and Lemma 5, we can get that

$$\begin{aligned}&-m_ie_is_{i-1}\big ([x_i-e_i]^r-[\hat{x}_i]^r\big )\nonumber \\&\le 2^{1-r}m_is_{i-1}^{r+1}|e_i|\ |x_{i-1}-\hat{x}_{i-1}|^r. \end{aligned}$$
(31)

Substituting (30) and (31) into (29) gives rise to

$$\begin{aligned}&-\sum _{i=2}^{n}m_ie_is_{i-1}\big ([x_i]^r-[\hat{x}_i]^r\big )\nonumber \\&\le \sum _{i=2}^{n}m_is_{i-1}^{\frac{r-1}{r+1}}|x_i|^{r+1}-\sum _{i=2}^{n}m_iH(s_{i-1})|e_i|^{r+1}\nonumber \\&\quad +2^{1-r}\sum _{i=2}^{n}m_is_{i-1}^{r+1}|e_i|\ |x_{i-1}-\hat{x}_{i-1}|^r, \end{aligned}$$
(32)

where \(H(s_{i-1})=(2^r-1)s_{i-1}^{\frac{2r}{r+1}}\). For the forth term of (27), by using Assumption 1, Lemma 6 and coordinate transformation (15), we have the following estimation

$$\begin{aligned}&\frac{1}{2}\sum _{i=2}^{n}m_i||g_i-s_{i-1}g_{i-1}||^2\nonumber \\&\le \sum _{i=2}^{n}m_i\big (||g_i||^2+s_{i-1}^2||g_{i-1}||^2\big )\nonumber \\&\le \sum _{i=2}^{n}m_i\Big (2ib_i^2\sum _{j=1}^{i}(|x_j|^{\frac{r+1}{2}}+|x_{jd}|^{\frac{r+1}{2}})^2\nonumber \\&\quad +2(i-1)b_{i-1}^2s_{i-1}^2\sum _{j=1}^{i-1}(|x_j|^{\frac{r+1}{2}}+|x_{jd}|^{\frac{r+1}{2}})^2\Big )\nonumber \\&\le \sum _{i=2}^{n}\Big (\sum _{j=1}^{i-1}R_{ij}(s_{i-1})|\xi _j|^{r+1}+R_{ii}|\xi _i|^{r+1}\nonumber \\&\quad +\sum _{j=1}^{i-1}R_{ij}(s_{i-1})|\xi _{jd}|^{r+1}+R_{ii}|\xi _{id}|^{r+1} \Big ), \end{aligned}$$
(33)

where \(R_{ij}(s_{i-1})=2^{r+1}m_i\big (ib_i^2+(i-1)s_{i-1}^2b_{i-1}^2\big )\big (1+\beta _j\big )^{r+1}, j\in {\mathcal {N}}_{i-2}\), \(R_{i,i-1}(s_{i-1})=2^{r+1}m_i\big (ib_i^2+(i-1)s_{i-1}^2b_{i-1}^2+ib_i^2\beta _{i-1}^{r+1}\big )\) and \(R_{ii}=2^{r+1}m_iib_i^2\). Let \(\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})=R_{ii}+R_{i+1,i}(s_i)+R_{i+2,i}(s_{i+1})+\cdots +R_{ni}(s_{n-1}), i\in {\mathcal {N}}_{n-1}\) with \(R_{11}=0\) and \(\lambda _{n1}=R_{nn}\). Then, formula (33) can be further expressed as

$$\begin{aligned}&\frac{1}{2}\sum _{i=2}^{n}m_i||g_i-s_{i-1}g_{i-1}||^2\nonumber \\&\le \sum _{i=1}^{n}\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})\big (|\xi _i|^{r+1} +|\xi _{id}|^{r+1}\big ), \end{aligned}$$
(34)

where \(\lambda _{i1}(s_i,s_{i-1},\ldots ,s_{n-1})\) are nonnegative functions of \(s_i,s_{i-1},\ldots ,s_{n-1}\), \(i\in {\mathcal {N}}_{n-1}\) and \(\lambda _{n1}\) is a positive constant. For the last term of (27), it follows from \(\dot{d}_i(t)\le v_i<1, i\in {\mathcal {N}}_{n}\) that

$$\begin{aligned}&\sum _{i=1}^{n}\frac{w_i}{1-v_i}\big (|\xi _i|^{r+1} -|\xi _{id}|^{r+1}(1-\dot{d}_i)\big )\nonumber \\&\le \sum _{i=1}^{n}\lambda _{i2}|\xi _i|^{r+1} -\sum _{i=1}^{n}w_i|\xi _{id}|^{r+1}, \end{aligned}$$
(35)

where \(\lambda _{i2}=\frac{w_i}{1-v_i}>0, i\in {\mathcal {N}}_n\). Substituting (28), (32), (34) and (35) into (27), we have

$$\begin{aligned}&\mathscr {L}U(e_2,e_3,\ldots ,e_n)\nonumber \\&\le \sum _{i=2}^{n}m_is_{i-1}^{\frac{r-1}{r+1}}|x_i|^{r+1}-\sum _{i=2}^{n}m_iH(s_{i-1})|e_i|^{r+1}\nonumber \\&\quad +\!2^{1\!-\!r}\!\Big (\!\sum _{i\!=\!2}^{n\!-\!1}\!m_i|e_i||x_{i\!+\!1}\!\!-\!\!\hat{x}_{i\!+\!1}|^r \!\!+\!\!\sum _{i\!=\!2}^{n}\!m_is_{i\!-\!1}^{r\!+\!1}|e_i||x_{i\!-\!1}\!\!-\!\!\hat{x}_{i\!-\!1}|^r \!\Big )\nonumber \\&\quad +\sum _{i=1}^{n}\big (\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})+\lambda _{i2}\big )|\xi _i|^{r+1}\nonumber \\&\quad -\sum _{i=1}^{n}Q_i|\xi _{id}|^{r+1}+\sum _{i=2}^{n}m_ie_i\big (f_i-s_{i-1}f_{i-1}\big ), \end{aligned}$$
(36)

where \(Q_i=w_i-\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1}), i\in {\mathcal {N}}_n\).

For the first term of (36), according to (7) and (15) it follows that

$$\begin{aligned}&\sum _{i=2}^{n}m_is_{i-1}^{\frac{r-1}{r+1}}|x_i|^{r+1}\nonumber \\&\le 2^r\sum _{i=2}^{n}m_is_{i-1}^{\frac{r-1}{r+1}}\big (|\xi _i|^{r+1}+\beta _{i-1}^{r+1}|\xi _{i-1}|^{r+1} \big )\nonumber \\&=\sum _{i=1}^{n}\lambda _{i3}(s_{i-1},s_i)|\xi _i|^{r+1}, \end{aligned}$$
(37)

where \(\lambda _{i3}(s_{i-1},s_{i})=2^r\big (m_is_{i-1}^{\frac{r-1}{r+1}} +m_{i+1}s_i^{\frac{r-1}{r+1}}\beta _i^{r+1}\big ), i\in {\mathcal {N}}_n\) with \(s_0=s_n=0\). By using the variable transformations \(x_i=z_i+s_{i-1}x_{i-1}\), \(\hat{x}_i=\hat{z}_i+\hat{s}_{i-1}\hat{x}_{i-1}\) and \(e_i=z_i-\hat{z}_i\), for any \(i\in {\mathcal {I}}_n\), we have

$$\begin{aligned} x_i\!-\!\hat{x}_i \!=\!e_i\!+\!s_{i\!-\!1}e_{i\!-\!1}\!+\!s_{i\!-\!1}s_{i\!-\!2}e_{i\!-\!2}\!+\!s_{i\!-\!1}s_{i\!-\!2}\!\cdots \!s_{2}e_2. \end{aligned}$$
(38)

This means that for any \(i\in {\mathcal {I}}_n\), \(x_i-\hat{x}_i\) can be expressed as the linear combination of \(e_2,e_3,\ldots ,e_i\). Based on (6), (9) and (38), for the third term of (36), we have

$$\begin{aligned}&2^{1\!-\!r}\!\Big (\sum _{i=2}^{n\!-\!1}\!m_i|e_i||x_{i\!+\!1}\! -\!\hat{x}_{i\!+\!1}|^r \!+\!\sum _{i\!=\!2}^{n}\!m_is_{i-1}^{r+1}|e_i||x_{i\!-\!1}\! -\!\hat{x}_{i\!-\!1}|^r \Big )\nonumber \\&=2^{1-r}m_2|e_2||e_3+s_2e_2|^r+2^{1-r}\sum _{i=3}^{n-1} m_i|e_i|\big (|e_{i+1}+s_ie_i\nonumber \\&\quad +s_is_{i-1}e_{i-1}+\cdots +s_is_{i-1}\cdots s_2e_2|^r +s_{i-1}^{r+1}|e_{i-1}\nonumber \\&\quad +s_{i-2}e_{i-2}+s_{i-2}s_{i-3}e_{i-3} +\cdots +s_{i-2}s_{i-3}\cdots s_2e_2|^r\big )\nonumber \\&\quad +2^{1-r}m_ns_{n-1}^{r+1}|e_n||e_{n-1} +s_{n-2}e_{n-2}+s_{n-2}s_{n-3}e_{n-3}\nonumber \\&\quad +\cdots +s_{n-2}s_{n-3}\cdots s_2e_2|^r \nonumber \\&\le \sum _{i=2}^{n-1}\sigma _{i1}(s_i,s_{i+1}, \ldots ,s_{n-1})|e_i|^{r+1}+\sigma _{n1}|e_n|^{r+1}\nonumber \\&\triangleq \sum _{i=2}^{n}\sigma _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})|e_i|^{r+1}, \end{aligned}$$
(39)

where \(\sigma _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})\) are nonnegative functions of the gains \(s_i,s_{i+1},\ldots ,s_{n-1}, i\in {\mathcal {N}}_{n-1}\) and \(\sigma _{n1}\) is a positive constant. Next, we concentrate on the last term of (36). In fact, by using Assumption 1, coordinate transform (15) and Lemma 5, it follows that

$$\begin{aligned}&\sum _{i=2}^{n}m_ie_i(f_i-s_{i-1}f_{i-1})\nonumber \\&\le \!\sum _{i=2}^{n}\!m_i|e_i|\!\Big (\!\!a_i\!\!\sum _{j=1}^{i}\big (\!|x_j|^r\!\!+\!\!|x_{jd}|^r \!\big ) \!\!+\!\!s_{i\!-\!1}a_{i\!-\!1}\!\!\sum _{j=1}^{i-1} \big (\!|x_j|^r\!\!+\!\!|x_{jd}|^r\!\big )\!\!\Big )\!\nonumber \\&\le \!\sum _{i=2}^{n}\!m_i|e_i|\!\Big (\! \sum _{j=1}^{i-1}\!A_{ij}(s_{i-1})\big (\!|\xi _j|^r\!+\!|\xi _{jd}|^r\!\big ) \!\!+\!\!A_{ii}\big (\!|\xi _i|^r\!\!+\!\!|\xi _{id}|^r\!\big )\!\Big )\!, \end{aligned}$$
(40)

where \(A_{ij}(s_{i-1})=(a_i+s_{i-1}a_{i-1})(1+\beta _j^r), j\in {\mathcal {N}}_{i-2}\), \(A_{i,i-1}(s_{i-1})=s_{i-1}a_i+a_i(1+\beta _{i-1}^r)\) and \(A_{ii}=a_i\). According to Lemma 4, inequality (40) can be further calculated as

$$\begin{aligned}&\sum _{i=2}^{n}m_ie_i(f_i-s_{i-1}f_{i-1})\nonumber \\&\le \sum _{i=2}^{n}\Big (\sum _{j=1}^{i-1}\!\big (\bar{A}_{ij}(s_{i\!-\!1})|\xi _j|^{r\!+\!1} \tilde{A}_{ij}(s_{i\!-\!1})|\xi _{jd}|^{r\!+\!1}\big )\nonumber \\&\quad \!+\!\bar{A}_{ii}|\xi _i|^{r\!+\!1} +\tilde{A}_{ii}|\xi _{id}|^{r+1}+\sum _{j=1}^{i}\hat{A}_{ij}|e_i|^{r+1}\Big ), \end{aligned}$$
(41)

where \(\bar{A}_{ij}(s_{i-1}),\tilde{A}_{ij}(s_{i-1}),j\in {\mathcal {N}}_{i-1}\) are appropriate nonnegative functions of \(s_{i-1}\), and \(\bar{A}_{ii},\tilde{A}_{ii}, \hat{A}_{ij},j\in {\mathcal {N}}_i\) are appropriate positive constants. For any \(i\in {\mathcal {N}}_{n}\), let \(\lambda _{i4}(s_i,s_{i+1},\ldots ,s_{n-1})=\sum _{j=i+1}^{n}\bar{A}_{ji}(s_{j-1}) +\bar{A}_{ii}\) with \(\bar{A}_{11}=0\) and \(\lambda _{n4}=\bar{A}_{nn}\), and choose appropriate \(\tilde{A}_{ij}(s_{i-1})\) and \(\tilde{A}_{ii}\) such that \(Q_i=\sum _{j=i+1}^{n}\tilde{A}_{ji}(s_{j-1}) +\tilde{A}_{ii}\) with \(\tilde{A}_{11}=0\) and \(Q_n=\tilde{A}_{nn}\). Then, it follows that

$$\begin{aligned}&\sum _{i=2}^{n}m_ie_i(f_i-s_{i-1}f_{i-1})\nonumber \\&\le \sum _{i=1}^{n}\lambda _{i4}(s_i,s_{i+1},\ldots ,s_{n-1})|\xi _i|^{r+1}+\sum _{i=1}^{n}Q_i|\xi _{id}|^{r+1}\nonumber \\&\quad +\sum _{i=2}^{n}\sigma _{i2}|e_i|^{r+1}, \end{aligned}$$
(42)

where \(\sigma _{i2}=\sum _{j=1}^{i}\hat{A}_{ij},i\in {\mathcal {I}}_n\). Substituting (37), (39) and (42) into (36) yields

$$\begin{aligned}&\mathscr {L}U(e_2,e_3,\ldots ,e_n) \nonumber \\&\le \sum _{i=1}^{n}\big (\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})+\lambda _{i2}+\lambda _{i3}(s_{i-1},s_i) \nonumber \\&\quad +\lambda _{i4}(s_i,s_{i+1},\ldots ,s_{n-1})\big )\!|\xi _i|^{r+1}+\sum _{i=2}^{n}\big (\!-\!m_iH(s_{i\!-\!1}) \nonumber \\&\quad +\sigma _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})+\sigma _{i2} \big )|e_i|^{r+1}. \end{aligned}$$
(43)

3.3 Main results

Based on the design obtained in Sects. 3.1 and 3.2 above, we give the main stabilization results for system (1) in the third part. Output feedback stabilizer, stability analysis and the procedure of selecting dynamic gains \(s_i,i\in {\mathcal {N}}_{n-1}\) will be provided here.

Theorem 1

Consider system (1) under Assumption 1. There exists an output feedback control law \(u=u(y,\hat{x}_2,\ldots ,\hat{x}_n)\) with dynamic observer (25), such that the closed-loop system is GSASP at \(x=0\).

Proof

By using the system output y and the state estimations \(\hat{x}_i, i\in {\mathcal {I}}_n\) generated by (25), we construct the following controller for system (1)

$$\begin{aligned} u&=-\beta _n\xi _n(y,\hat{x}_2,\ldots ,\hat{x}_n) \nonumber \\&=-(\beta _n\hat{x}_n+\beta _n\beta _{n-1}\hat{x}_{n-1} +\cdots +\beta _n\beta _{n-1}\cdots \beta _2\hat{x}_2\nonumber \\&\quad +\beta _n\beta _{n-1}\cdots \beta _1y) \end{aligned}$$
(44)

To investigate the stochastic stability of (1)–(44), the Lyapunov–Krasovskii functional \(V=V_n(x_1,x_2,\cdots ,x_n)+U(e_1,e_2,\ldots ,e_n)\) is chosen for the system. According to (24) and (43), the differential of V is given as

$$\begin{aligned} \mathscr {L}V&=\mathscr {L}V_n(x_1,x_2,\ldots ,x_n)+\mathscr {L}U(e_1,e_2,\ldots ,e_n)\nonumber \\&\le \sum _{i=1}^{n}\big (-c_{ni}+\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})+\lambda _{i2}\nonumber \\&\quad +\lambda _{i3}(s_{i-1},s_i)\lambda _{i4}(s_i,s_{i+1},\ldots ,s_{n-1})\big )|\xi _i|^{r+1}\nonumber \\&\quad +\sum _{i=2}^{n}\big (-m_iH(s_{i-1})+\sigma _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})\nonumber \\&\quad +\sigma _{i2} \big )|e_i|^{r+1}+k_n\xi _n([u]^{r}-[x_{n+1}^*]^r). \end{aligned}$$
(45)

From the state feedback controller design, we can see that \(x_{n+1}^*=-\beta _n\xi _n=-(\beta _nx_n+\beta _n\beta _{n-1}x_{n-1}+\cdots +\beta _n\beta _{n-1}\cdots \beta _2x_2 +\beta _n\beta _{n-1}\cdots \beta _1y\). Thus, according to Lemma 5, (44) and (38), we have

$$\begin{aligned}&k_n\xi _n([u]^{r}-[x_{n+1}^*]^r)\nonumber \\&\le 2^{1-r}k_n|\xi _n|\ |u-x_{n+1}^*|^r\nonumber \\&=2^{1-r}k_n|\xi _n|\ |\beta _n(x_n-\hat{x}_n)+\beta _{n}\beta _{n-1}(x_{n-1}-\hat{x}_{n-1})\nonumber \\&\quad +\cdots +\beta _n\beta _{n-1}\cdots \beta _2(x_2-\hat{x}_2)|^r\nonumber \\&=2^{1\!-\!r}k_n|\xi _n|\Big |\sum _{i=2}^{n\!-\!1}\!L_i (s_i,s_{i\!+\!1},\ldots ,s_{n\!-\!1})e_i\!+\!L_ne_n\!\Big |^r, \end{aligned}$$
(46)

where \(L_i(s_i,s_{i+1},\ldots ,s_{n\!-\!1})\!=\!\beta _ns_{n-1}s_{n-2}\cdots s_i+\beta _n\beta _{n\!-\!1}s_{n-2}\cdots s_i+\cdots +\beta _n\beta _{n-1}\beta _{n-2}\cdots \beta _i, i\in {\mathcal {I}}_{n-1}\) and \(L_n=\beta _n\). It follows from Lemmas 4 and 6 that inequality (46) is further rewritten as

$$\begin{aligned}&k_n\xi _n([u]^{r}-[x_{n+1}^*]^r)\nonumber \\&\le \lambda _{n5}|\xi _n|^{r+1}+\sum _{i=2}^{n}\sigma _{i3}(s_i,s_{i+1},\ldots ,s_{n-1})|e_i|^{r+1}, \end{aligned}$$
(47)

where \(\lambda _{n5}\), \(\sigma _{n3}\) are two appropriate positive constants and \(\sigma _{i3}(s_i,s_{i+1},\ldots ,s_{n-1}), i\in {\mathcal {I}}_{n-1}\) are nonnegative functions of \(s_i,s_{i+1},\ldots ,s_{n-1}\). Let \(\lambda _{i5}=0, i\in {\mathcal {N}}_{n-1}\). Then, substituting (47) into (45) yields

$$\begin{aligned} \mathscr {L}V&\le \sum _{i=1}^{n}\big (-c_{ni}+\lambda _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})+\lambda _{i2}\nonumber \\&\quad +\lambda _{i3}(s_{i\!-\!1},s_i)\!+\!\lambda _{i4}(s_i,s_{i\!+\!1},\! \cdots \!,s_{n\!-\!1})\!+\!\lambda _{i5}\big )|\xi _i|^{r\!+\!1}\nonumber \\&\quad +\sum _{i=2}^{n}\big (-m_iH(s_{i-1})+\sigma _{i1}(s_i,s_{i+1},\ldots ,s_{n-1})\nonumber \\&\quad +\sigma _{i2}+\sigma _{i3}(s_i,s_{i+1},\ldots ,s_{n-1}) \big )|e_i|^{r+1}. \end{aligned}$$
(48)

Now, by using inequality (48), we recursively select the dynamic gains \(s_i,i\in {\mathcal {N}}_{n-1}\) such that \(\mathscr {L}W<0\) holds. First, for numbers \(\mu _{n}>0\) and \(\rho _{n}>0\) we select appropriate \(s_{n-1}\) such that the following condition is to be satisfied

$$\begin{aligned} \left\{ \begin{array}{lll} -c_{nn}+\lambda _{n1}+\lambda _{n2}+\lambda _{n3}(s_{n-1})+\lambda _{n4}+\lambda _{n5}\le -\mu _{n},\\ -m_nH(s_{n-1})+\sigma _{n1}+\sigma _{n2}+\sigma _{n3}\le -\rho _{n}. \end{array} \right. \end{aligned}$$

Based on \(s_{n-1}\) fixed above, taking two real numbers \(\mu _{n-1}>0\) and \(\rho _{n-1}>0\) we select appropriate \(s_{n-2}\) such that

$$\begin{aligned} \left\{ \begin{array}{lll} -c_{n,n-1}+\lambda _{n-1,1}(s_{n-1})\!+\!\lambda _{n-1,2}\!+\!\lambda _{n-1,3}(s_{n-2},s_{n-1})\\ \quad \quad \quad \ \ +\lambda _{n-1,4}(s_{n-1})\le -\mu _{n-1},\\ -m_{n\!-\!1}\!H\!(s_{n\!-\!2})\!\!+\!\sigma _{n\!-\!1,1}(\!s_{n\!-\!1}\!)\!+\!\sigma _{n2}\!\!+\!\sigma _{n\!-\!1,3}\!(\!s_{n\!-\!1}\!)\!\le \!-\rho _{n\!-\!1}. \end{array} \right. \end{aligned}$$

Following this process until step \(n-2\), we have fixed \(n-2\) dynamic gains \(s_{n-1},s_{n-2},\ldots ,s_2\). Now, we take three real numbers \(\mu _1>0,\mu _2>0\) and \(\sigma _2>0\), and select appropriate \(s_1\) such that the following three conditions hold

$$\begin{aligned} \left\{ \begin{array}{lll} -c_{n2}+\lambda _{21}(s_2,s_3\ldots ,s_{n-1})+\lambda _{22}+\lambda _{23}(s_1,s_2)\\ \quad \quad \ +\lambda _{24}(s_2,s_3\ldots ,s_{n-1})\le -\mu _2,\\ -m_2H(s_{1})+ \sigma _{21}(s_2,s_3,\ldots ,s_{n-1})+\sigma _{22}\\ \quad \quad \ + \sigma _{23}(s_2,s_3,\ldots ,s_{n-1})\le -\rho _2,\\ -c_{n1}+\lambda _{11}(s_1,s_2\ldots ,s_{n-1})+\lambda _{12}+\lambda _{13}(s_1)\\ \quad \quad \ +\lambda _{14}(s_1,s_2\ldots ,s_{n-1})\le -\mu _1.\\ \end{array} \right. \end{aligned}$$

Based on the selection of the gains above, (48) becomes

$$\begin{aligned} \mathscr {L}V\le -\sum _{i=1}^{n}\mu _i|\xi _i|^{r+1}-\sum _{i=2}^{n}\rho _j|e_i|^{r+1}<0. \end{aligned}$$
(49)

Let \(W=\sum _{i=1}^{n}\mu _i|\xi _i|^{r+1}+\sum _{i=2}^{n}\rho _j|e_i|^{r+1}\). Clearly, W is nonnegative, continuous and positive definite. So, from formula (49) and Lemma 1 we can see that closed-loop system (1)–(44) is GSASP by selecting the dynamic gains appropriately.

Table 2 List of system variables and adjustable parameters
Full size table

Remark 6

For system (1), Sect. 3 provides a systemical design method of the output feedback controller and a recursive selection procedure of the dynamic observer gains. According to the proof of Theorem 1, the \(n-1\) pending gains \(s_i, i\in {\mathcal {N}}_{n-1}\) can be effectively selected, under which the stability of the resulting system will be achieved. It should be noted that the \(4n-2\) adjustable parameters \(m_i, w_i, \mu _i\) and \(\sigma _i\) can make the selection of the gains more flexible. Furthermore, for the sake of increasing the flexibility and convenience of the controller, the \(2n+1\) adjustable parameters \(K, k_{i}\) and \(c_{ii}\) are introduced in the construction of LKF candidates \(V_i, i\in {\mathcal {N}}_n\). From the construct process, we can see that a better performance of the proposed control strategy may be obtained by properly selecting them. The system variables and these parameters are described in Table 2. In addition, it follows from Sect. 3 that the information of the bounds \(a_i\) and \(b_i\) in Assumption 1 is only required rather than the functions \(f_i(\cdot )\) and \(g_i(\cdot ), i\in {\mathcal {N}}_n\) in the design of our controller (44). This indicates that even if the functions \(f_i(\cdot )\) and \(g_i(\cdot )\) have some uncertainty, it will not affect the design process of the controller. That is to say, the proposed control scheme in our paper has a certain degree of robustness to uncertainties of the system.

Remark 7

When the system parameters are determined, the proposed control algorithm can be divided into three main steps. First, the estimators \(\hat{x}_i(k)\) of unmeasurable system states are computed by the equation \(\hat{x}_i(k)=\hat{z}_i(k)+s_{i-1}\hat{x}_{i-1}(k), i\in {\mathcal {I}}_n\). Second, by using the measured values y(k) of the system output and the estimators \(\hat{x}_i(k), i\in {\mathcal {I}}_n\), the control input u(k) can be computed from (44). Third, according to Eq. (25), the values of the state observer \(\hat{z}_i(k+1), i\in {\mathcal {I}}_n\), at time \(k+1\) are calculated in regard to u(k), y(k) and \(\hat{x}_i(k), i\in {\mathcal {I}}_n\). Meanwhile, the states of the original system at time \(k+1\) are updated by using the control input u(k). From the calculation process above, it can be seen that the computational cost of the proposed method in our paper is low. To illustrate this point, the function tic-toc in MATLAB is used to measure the computation time in simulation examples of Sect. 4.

Remark 8

In Sect. 3, we developed a recursive design method that yields an output feedback stabilizer for low-order SNTDSs. This stabilizer is comprised of a state feedback controller and a reduced-order observer. It should be noted that the present paper is first to provide a solution to the stabilization problem of low-order SNTDSs. Compared with the recent works in [43, 44], where low-order stochastic nonlinear systems are considered and the corresponding stabilization problems in different scenarios have been solved, time-varying delays of the system states are taken into account in our work. This is one of main reasons for the difficulty in solving the problem. Although there are lots of research results on some special stochastic time-delay systems, such as [21,22,23] for strict-feedback SNTDSs, and [32,33,34,35] for high-order SNTDSs, it is still a difficult task to address the problem for low-order SNTDSs due to the intrinsic characteristics of such systems. For example, the backstepping-based design method commonly used in Refs. [21, 22] cannot be applied to our work because low-order SNTDSs are not feedback linearizable. Meanwhile, the AAPI-based design method used in Refs. [33, 35] also cannot be directly applied to our work because low-order SNTDS (1) is continuous but nondifferentiable. An appropriate LKF and a new reduced-order observer are developed to successfully overcome the obstacles in the present paper. In a sense, compared with the existing achievements mentioned above, the advantage of our work is to be able to solve the output feedback stabilization problem of low-order SNTDSs under some conditions.

4 Simulation example

In order to verify the effectiveness of the feedback control method developed in Sect. 3, we provide an illustrative example in this section.

Example 1

Consider a planar low-order SNTDS

$$\begin{aligned} \left\{ \begin{array}{lll} dx_1=[x_2]^{\frac{3}{4}}dt+\frac{1}{16}\cos (x_1x_{1d})dt+\frac{1}{8}[x_1]^\frac{7}{8}dw, \\ dx_2=[u]^{\frac{3}{4}}dt+\frac{1}{9}\sin (x_2)\cos (x_{2d})dt\\ \quad \quad \ \ +\frac{1}{10}\sin (x_1+x_{2d})dw,\\ y=x_1, \end{array} \right. \end{aligned}$$
(50)

where \(x_i=x_i(t)\) are system states, \(x_{id}=x_{i}\big (t-d_i\big )\) are delayed states, and \(d_i=d_i(t)\) are time-varying delays of the system, \(i=1,2\). Take \(d_1(t)=0.1(1+\sin (t))\) and \(d_2(t)=0.2(1-\cos (t))\). Clearly, system (50) satisfies Assumption 1 with \(r=\frac{3}{4},a_1=\frac{1}{16},b_1=\frac{1}{8},a_2=\frac{1}{9}\), \(b_2=\frac{1}{10}\), and the conditions \(\dot{d}_i(t)\le v_i<1, i=1,2\) with \(v_1=\frac{1}{10}\), \(v_2=\frac{1}{5}\). Assuming that only the output signal y can be measurable, we now apply the proposed control method to construct a stabilizer for system (50).

Table 3 Experimental data with the different parameter values
Full size table

For the \(x_1\)-subsystem of (50), choose \(V_1(x_1)=\frac{k_1}{2}x_1^2+\frac{5}{72}\int _{t}^{t-d_1}|x_1(s)|^{\frac{7}{4}}ds\) with \(k_1>0\). It follows that \(x_2^*=-\beta _1x_1\) with \(\beta _1=(\frac{c_{11}+l_{111}}{k_1})^{\frac{4}{3}}\) where \(c_{11}>0\) and \(l_{111}=\frac{5}{75} + \frac{5k_1}{64}+\frac{k_1}{28}(\frac{7}{3k_1}-\frac{7}{12})^{-\frac{3}{4}}\). For whole system (50), let \(\xi _2=x_2-x_2^*\), \(\xi _{2d}=x_{2d}-x_{2d}^*\), and choose \(V_2(x_1,x_2)=V_1(x_1)+\frac{k_2}{2}\xi _2^2+K\int _{t}^{t-d_1}|\xi _1(s)|^{\frac{7}{4}}ds +\frac{5}{36}\int _{t}^{t-d_2}|\xi _2(s)|^{\frac{7}{4}}ds\) with \(k_2>0, K>0\). We construct a state feedback controller \(u=u(x_1,x_2)=-\beta _2x_2-\beta _2\beta _1x_1\) with \(\beta _2=(\frac{c_{22}+h_{21}+h_{22}+h_{23}}{k_2})^{\frac{4}{3}}\) where \(c_{22}\) is an arbitrary positive constant, \(h_{21}=\frac{5}{36}+\frac{1}{25}2^{\frac{3}{4}}k_2\), \(h_{22}=\frac{3}{7}2^{\frac{9}{12}}k_1\) and \(h_{23}=\big (\frac{2}{63}(1+\beta _1^{\frac{3}{4}})+\frac{1}{9}+\beta _1+\frac{4}{7}\beta _1(1+\beta _1^{\frac{3}{4}}) \big )k_2+\frac{4}{63}(1+\beta _1^{\frac{3}{4}})k_2\Big (\frac{21}{2k_2(1+\beta _1^{\frac{3}{4}})} \big (\frac{9}{10}K-(\frac{1}{25}+\frac{1}{5}\beta _1^2 +\frac{1}{25}2^{\frac{3}{4}}\beta _1^{\frac{7}{4}})k_2\big )\Big )^{-\frac{3}{4}}+\frac{4}{63}k_2\big ( \frac{7}{3k_2}-\frac{21}{25}2^{\frac{3}{4}}\big )^{-\frac{3}{4}}+\frac{1}{28}k_1\beta _1\Big ( \frac{56}{3k_2\beta _1}\big (\frac{9}{10}K-(\frac{1}{25}+\frac{1}{5}\beta _1^2 +\frac{1}{25}2^{\frac{3}{4}}\beta _1^{\frac{7}{4}})k_2\big )\Big )^{-\frac{3}{4}}\).

Fig. 1
figure 1

The state trajectories of the closed-loop system

Full size image
Fig. 2
figure 2

The state trajectory of the observer system and the controller evolutions

Full size image
Fig. 3
figure 3

The Wiener process and the associated white Gaussian noise

Full size image

Next, we construct an observer for unmeasurable state \(x_2\). The symbol \(\hat{x}_2\) denotes the estimated value of \(x_2\). Let \(\hat{z}_2=\hat{x}_2-s_1y\), where \(s_1>1\) is the dynamic gain needing to be selected. Consider the following observer system

$$\begin{aligned} \dot{\hat{z}}_2=[u]^{\frac{3}{4}}+s_1[\hat{z}_2+s_1x_1]^{\frac{3}{4}}. \end{aligned}$$
(51)

By using the variable substitution \(\hat{x}_2=\hat{z}_2+s_1x_1\), dynamic equation (51) generates the estimated state \(\hat{x}_2\). Furthermore, substituted \(\hat{x}_2\) for \(x_2\) in \(u=u(x_1,x_2)\), we get the desired controller

$$\begin{aligned} u=u(y,\hat{x}_2)=-\beta _2\hat{x}_2-\beta _2\beta _1y. \end{aligned}$$
(52)

The numerical simulation results were obtained according to the Euler-Maruyama technique [59] in the MATLAB environment (MATLAB R2020b). The initial values of the original system and the observer system are set to be \((x_1(0),x_2(0))=(-0.6,0.7)\) and \(\hat{z}_2(0)=0.8\), respectively. In order to illustrate the influence of the adjustable parameters \(K, k_1, k_2, c_{11}\) and \(c_{22}\) on controller (52), the detailed experimental data with the different parameter values are described in Table 3. It can be seen from the table that the values of \(\beta _1, \beta _2\) and \(s_1\) can be changed by adjusting these parameters, which means that better performance of the system may be achieved by selecting proper parameter values.

We choose the first set of parameter values for simulation display. Namely, take \(K=0.05,k_1=0.1,k_2=0.01,c_{11}=0.2\) and \(c_{22}=0.1\). Based on the selection procedure provided in Sect. 3.3, the dynamic gain is selected as \(s_1=20\). \(\beta _1\) and \(\beta _2\) can be calculated as 3.9 and 182.01, respectively. Figures 1, 2 and 3 demonstrate the simulation results. From the numerical results, it follows that system (50) is GSASP at \((x_1,x_2)=(0,0)\) under proposed control scheme (51)–(52). By using the function tic-tok in MATLAB, the computation time is measured as 0.024 s.

Example 2

Consider the liquid-level system with interaction (see [36]) shown in Fig. 4. Suppose that tank 1 and tank 2 have the same capacitances of c. The liquid levels of two tanks are \(H_1\) and \(H_2\), respectively, and their steady-state liquid levels are all \(\bar{H}\). Note that the difference of \(H_1\) and \(H_2\) at a certain moment will lead to the change of the flow rates \(Q_1\) and \(Q_2\), which can be described as

$$\begin{aligned} Q_1= \left\{ \begin{array}{lll} p_1|H_2-H_1|^{\frac{1}{2}},\quad H_2\le H_1\\ -p_1|H_2-H_1|^{\frac{1}{2}},\quad H_2 > H_1 \end{array} \right. \end{aligned}$$
(53)

and

$$\begin{aligned} Q_2=p_2H_2^{\frac{1}{2}}, \end{aligned}$$
(54)

where \(p_1\) and \(p_2\) are resistance coefficients of the corresponding valves. Assume that Q is inflow rate of the system. The goal is to adjust Q such that both \(H_1\) and \(H_2\) asymptotically converge to the steady state \(\bar{H}\). Let \(x_1=H_1-\bar{H}\), \(x_2=H_2-H_1\) and \([u]^{\frac{1}{2}}=\frac{1}{c} Q-\frac{1}{c} p_2\bar{H}^{\frac{1}{2}}\). When considering random disturbances and multiple time-varying delays, the dynamic of the liquid levels of the tanks 1 and 2 can be described as

$$\begin{aligned} \left\{ \begin{array}{lll} dx_1\!=\!\frac{1}{c}p_1[x_2]^{\frac{1}{2}}dt+\frac{1}{2}|x_{1d}|^{\frac{3}{4}}dw,\\ dx_2\!=\![u]^{\frac{1}{2}}dt+\varphi (\bar{x}_{2},\bar{x}_{2d})dt+\frac{1}{2} |x_{2d}|^{\frac{3}{4}}dw,\\ y = x_1, \end{array} \right. \end{aligned}$$
(55)

where \(\varphi (\bar{x}_{2},\bar{x}_{2d})=-\frac{2}{c} p_1[x_2]^{\frac{1}{2}}-\frac{1}{c} p_2(x_{1d}+x_{2d}+\bar{H})^{\frac{1}{2}}+\frac{1}{c} p_2\bar{H}^{\frac{1}{2}}\) and \(x_{1d}=x_1\big (t-d_1(t)\big )\), \(x_{2d}=x_2\big (t-d_2(t)\big )\), \(d_1(t)=0.1cos^2(t), d_2(t)=0.1sin^2(t)+0.2\). It is clear that system (55) satisfies Assumption 1 with \(r=\frac{1}{2}\), and the condition \(\dot{d}_i(t)\le v_i<1, i=1,2\) with \(v_1=\frac{1}{5}, v_2=\frac{1}{5}\). When only the output signal y is measurable, the proposed approach in the previous sections can be used to solve the stabilization problem of the system.

A state feedback controller is designed firstly for system (55) by using the proposed approach, i.e.,

$$\begin{aligned} u=u(x_1,x_2)=-\beta _2x_2-\beta _1\beta _2x_1, \end{aligned}$$
(56)

where \(\beta _1=(\frac{c}{k_1p_1})^{\frac{1}{r}}\big (c_{11}+k_1b_1^2+\frac{a_1}{1-v_1} +k_1a_1+\frac{k_1a_1}{1+r}(\frac{(1+r)(a_1-k_1b_1^2)}{k_1a_1r})^{-r}\big )^{\frac{1}{r}}\) and \(\beta _2=k_2^{-\frac{1}{r}}(c_{22}+h_{21}+2^{1-r}\frac{r\bar{k}_1}{1+r}+\frac{a_2}{1-v_2} )^{\frac{1}{r}}\) with \(h_{21}=\frac{k_2}{r+1}(a_2\beta _1^r+\frac{p_1}{c}\beta _1^{r+1}) +k_2(a_2+\frac{p_1}{c}\beta _1)+\frac{1}{r+1}k_2a_2(1+\beta _1^r)(\frac{r+1}{r})^{-r}\) \( (\frac{K(1-v_1)-2^rk_2\beta _1^{r+1}-k_2\beta _1^2}{k_2a_2(1+\beta _1^r)})^{-r} +\frac{k_2a_2}{r+1}(\frac{(a_2-2^rk_2)(r+1)}{rk_2a_2})^{-r}\). For the unmeasurable state \(x_2\), the estimated value can be taken as \(\hat{x}_2=\hat{z}_2+s_1x_1,\), where \(\hat{z}\) is generated by the dynamic

$$\begin{aligned} \dot{\hat{z}}_2=[u]^{\frac{1}{2}}+\frac{1}{c} p_1k_1s_1[\hat{z}_2+s_1x_1]^{\frac{1}{2}}. \end{aligned}$$
(57)

Using \(\hat{x}_2\) in place of \(x_2\) in (56), we get an output feedback stabilizer \(u=u(y,\hat{x}_2)=-\beta _2\hat{x}_2-\beta _1\beta _2y\). In the simulation, the initial conditions are given as \(x_1(0)=0.3, x_2(0)-0.4\) and \(\hat{z}_2(0)=0.6\). The dynamic gain is given as \(s_1=28\). The other simulation parameters are given as \(K=0.5, k_1=0.6, k_2=0.08, c_{11}=1.2, c_{22}=1.7\). The simulation results are shown in Figs. 5, 6 and 7. It can be seen that system (53) is stabilized by the proposed output feedback control scheme.

Fig. 4
figure 4

The liquid-level system with interaction

Full size image
Fig. 5
figure 5

The trajectory of system output y

Full size image
Fig. 6
figure 6

The trajectories of \(x_2\) and \(\hat{z}_2\)

Full size image
Fig. 7
figure 7

The trajectory of control input u

Full size image

5 Conclusions

In this technical paper, we dealt with the stabilization problem by using an output feedback approach for a class of low-order SNTDSs where the powers of integrators can be arbitrarily taken on the interval (0, 1). In order to overcome the difficulties arisen from the low-order nonlinearities, stochastic disturbers and multiple time-varying delays, we generalize the classical AAPI approach [27] and the observer design method introduced in [40] to construct an output feedback stabilizer. A novel LKF was skillfully chosen in the controller design, and the observer gains can be recursively selected to guarantee the GSASP of the resulting system at the trivial solution.

A meaningful work for further study is to extend the results obtained in this paper to a more general class of low-order SNTDSs with the different powers of chained integrator. In fact, the authors have attempted to solve the problem. But there are still some technical difficulties that cannot be tackled. Moreover, another work that deserves attention in the future is how to deal with the sampled-data stabilization problem of low-order SNTDSs. Clearly, it is a more difficult problem because some information and properties of the system will be lost due to sampling. A new design scheme based on the method proposed in our paper needs to be developed to solve the problem.