Adaptive output feedback asymptotic tracking for uncertain nonlinear systems with dead-zone input

Abstract

In this paper, the adaptive output feedback asymptotic tracking problem is investigated for nonlinear systems with dead-zone input and unknown nonlinearities, where only system output is available. First, an observer is constructed to estimate unavailable states. Then, by the observer and barrier Lyapunov function, an adaptive output feedback controller is designed, which compensates the influence of dead-zone input and unknown nonlinearities. Furthermore, based on the proposed controller and tuning function, the global boundedness of all signals in closed-loop system is achieved, and the tracking error can asymptotically converge to zero. Particularly, it should be noted that there is no need to calculate the derivatives of virtual control laws in controller design, thus reducing the computational complexity. Finally, two numerical examples are taken to verify the effectiveness of the designed controller.

Keywords

Nonlinear systems asymptotic tracking dead-zone input barrier Lyapunov function output feedback controller

Introduction

With the continuous improvement of system performance requirements in modern industry, the nonlinear systems have received extensive attention (Batista et al., 2022; Gao et al., 2022; Liu et al., 2022a; Lu and Li, 2021; Wang et al., 2020; Xu et al., 2022). Many important control methods have been proposed to investigate nonlinear systems. By barrier Lyapunov function (BLF) and Nussbaum gain technique, the full state constraints problem was solved in Liu and Tong (2017). In Chen et al. (2018), the dual-domination approach was utilized to address the output feedback stabilization problem. With the help of the time-varying high-gain approach, the state feedback and output feedback regulation problems were studied in Chen et al. (2020). By the adaptive backstepping control scheme, the asymptotic tracking problem was investigated in Li (2020). However, it should be emphasized that the controllers proposed above were based on the known prior knowledge of system nonlinearities.

For nonlinear systems with unknown nonlinearities, fuzzy logic systems and neural networks have been widely used in controller design due to their approximation property (Hua et al., 2019; Li et al., 2022; Parsa et al., 2021; Zhang and Yang, 2020). In fact, the action scope of approximation methods is limited to compact set, so that the relevant research only obtain semi-global results. In order to obtain global results, scholars have also made a lot of effort (Li et al., 2022; Liu and Li, 2018; Ma et al., 2015). Instead of utilizing the approximation methods, the switching universal control approach was proposed in Ma et al. (2015) to achieve global stabilization for power integrator triangular systems. The system nonlinearities in Liu and Li (2018) and Liu et al. (2022b) were regarded as bounded “disturbance-like” terms and were compensated adaptively at each step. Nevertheless, the dead-zone input was not discussed in above references.

Due to the physical constraints of actuators and components aging, the dead-zone input inevitably exists in many actual systems, and it has negative effects on system performance (Shen et al., 2021; Wan et al., 2022). Therefore, scholars were committed to dealing with the influence of dead-zone input on control systems. The backstepping technique has been extensively applied to study the nonlinear systems with dead-zone input and known nonlinearities (Li et al., 2019; Su et al., 2018). An adaptive finite-time tracking control scheme was proposed in Li et al. (2019) to achieve bounded tracking. In Su et al. (2018), the event-triggered state feedback controller was designed to implement asymptotic tracking. Furthermore, many excellent results have been obtained for the nonlinear systems with unknown nonlinearities and dead-zone input. By an adaptive neural network tracking control method, the tracking error in Ni and Shi (2021) converged to the predefined accuracy within a predefined time. The backstepping technique was combined with fuzzy logic systems or neural networks in Lan et al. (2021), Zhou et al. (2021) and Zong et al. (2022), such that the semi-global boundedness of all signals in closed-loop system and the bounded tracking were obtained. With the aid of the dynamic gain control technique, an output feedback control scheme was proposed in Jia et al. (2019) to achieve the global practical tracking.

It should be mentioned that although the above references have achieved control objectives successfully, there are still some problems to be further explored. On one hand, it is unrealistic to obtain all state information in actual systems. Therefore, we consider the case where only system output is available. On the other hand, most of the existing results considering dead-zone input guaranteed the semi-global boundedness of the closed-loop system, or realized the bounded tracking. For nonlinear systems with dead-zone input and unknown nonlinearities, it is challenging to design a controller to achieve global boundedness and asymptotic tracking, which motivates the research of this paper. By applying tuning function and Barbalat lemma, it is concluded that under the proposed controller, all signals of the closed-loop system are globally bounded and the tracking error asymptotically converge to zero. We present the main contribution of the paper as follows:

Compared with the work in Liu and Li (2018), we consider a more practical situation that only system output is available, and an observer is constructed to estimate unavailable states. Particularly, based on the observer and BLF, we design an adaptive output feedback controller without the derivative of virtual control laws, which has lower design complexity than that in Ni and Shi (2021).

Different from the results in Lan et al. (2021), the asymptotic tracking is achieved despite the presence of dead-zone input and unknown nonlinearities. Specifically, we compensate the influence of dead-zone input with the help of BLF and the proposed controller. Furthermore, the unknown nonlinearities are tackled by the designed adaptive law instead of approximation methods.

It is worth pointing out that a tuning function is introduced into the coordinate transformations to eliminate the limitations on system initial values. Thereby, the global boundedness of all signals in closed-loop system is guaranteed by the proposed controller and tuning function, which improves the semi-global boundedness results in Zhou et al. (2021) and Zong et al. (2022).

The rest of the paper is structured as follows. The system model is presented in section “Problem statement.” In section “Main result,” an adaptive output feedback controller is designed, and its effectiveness is further verified by two examples in section “Simulation examples.” Finally, section “Conclusion” summarizes the paper.

Notations

In this paper, $a (t)$ is denoted as $a$ , $x^{0} = x (0)$ represents the initial condition, and $r^{m}$ is called the $m th$ power of $r$ . The Euclidean norm is represented by $∥ \cdot ∥$ . $R$ denotes the real field, $R^{+}$ shows the nonnegative real field, and $R^{n}$ expresses the $n$ -dimensional vector space. $I_{n}$ shows the $n$ -order identity matrix, $0_{n}$ indicates the $n$ -dimensional zero vector, and $X^{T}$ stands for the transpose of matrix $X$ or vector $X$ .

Problem statement

Consider the following nonlinear system

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = x_{i + 1} (t) + g_{i} (t, y (t)) + β_{i} (t), i = 1, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} (t) = v (t) + g_{n} (t, y (t)) + β_{n} (t) \\ y (t) = x_{1} (t) \end{matrix}

(1)

where $x (t) = (x_{1} (t), \dots, x_{n} (t))^{T} \in R^{n}$ is system state, $y (t) \in R$ is system output; $g_{i} (\cdot) : R^{+} \times R \to R$ is an unknown function, piecewise continuous in $t$ and locally Lipschitz in $y (t)$ ; $β_{i} (t)$ is an unknown bounded external disturbance; and $v (t) = D (u (t))$ is actuator dead-zone described by Jia et al. (2019) and Li et al. (2019)

v (t) = D (u (t)) = {\begin{matrix} ϑ_{1} (u (t) - μ_{1}), u (t) \geq μ_{1} \\ 0, - μ_{2} < u (t) < μ_{1} \\ ϑ_{2} (u (t) + μ_{2}), u (t) \leq - μ_{2} \end{matrix}

(2)

where $ϑ_{i} > 0 (i = 1, 2)$ is the slope of dead-zone, $μ_{i} > 0 (i = 1, 2)$ shows the break point of dead-zone, $ϑ_{i}$ and $μ_{i}$ are unknown constants, and $v (t)$ is available. Without causing confusion, the explicit dependency of the symbol on $t$ will be omitted in the following. Then, equation (2) can be rewritten as

D (u) = ϑ (u) u + ρ (u)

(3)

with

\begin{matrix} ϑ (u) = {\begin{matrix} ϑ_{1}, u \geq μ_{1} \\ \frac{ϑ_{1} + ϑ_{2}}{2}, - μ_{2} < u < μ_{1} \\ ϑ_{2}, u \leq - μ_{2} \end{matrix}, \\ ρ (u) = {\begin{matrix} - ϑ_{1} μ_{1}, u \geq μ_{1} \\ - \frac{(ϑ_{1} + ϑ_{2}) u}{2}, - μ_{2} < u < μ_{1} \\ ϑ_{2} μ_{2}, u \leq - μ_{2} \end{matrix} \end{matrix}

Thus, $\underline{ϑ} = min {ϑ_{1}, ϑ_{2}} \leq ϑ (u) \leq \bar{ϑ} = max {ϑ_{1}, ϑ_{2}}$ and $| ρ (u) | \leq max {\bar{ϑ} μ_{1}, \bar{ϑ} μ_{2}} = \bar{ρ}$ .

Some assumptions and lemmas are given to prepare for the proof of main result.

Assumption 1. For system nonlinearity $g_{i} (t, y)$ , there exist a smooth function ${\bar{g}}_{i} (y)$ and an unknown constant $c > 0$ , such that $| g_{i} (t, y) | \leq c {\bar{g}}_{i} (y), i = 1, \dots, n$ .

Assumption 2. The external disturbance $β_{i} (t)$ satisfies $| β_{i} (t) | < {\bar{β}}_{i}$ , where ${\bar{β}}_{i} > 0 (i = 1, \dots, n)$ is an unknown constant.

Assumption 3. The reference signal $y_{0} (t)$ and its derivative ${\overset{\cdot}{y}}_{0} (t)$ are continuous and bounded.

Remark 1. It is obvious from Assumption 1 that the system nonlinearities satisfy the growth of arbitrary output function with unknown constant rate, which is more general than that in Ilchmann et al. (2007) and (Ye, 1999). In practice, some engineering systems can be modeled as such systems after a series of transformations, such as Josephson junction circuit system and robot arm system (Cheng and Chen, 2020; Wang and Liu, 2021). As seen in Su et al. (2018), the reference signal $y_{0} (t)$ and its $i$ th (i = 1, …, n) order derivatives were continuous and bounded. It should be stated that the condition of reference signal is relaxed in Assumption 3. To be specific, only the reference signal and its first order derivative are continuous and bounded.

Lemma 1. For matrix $A_{1} = (\begin{matrix} 0_{n - 1} & I_{n - 1} \\ 0 & 0_{n - 1}^{T} \end{matrix}) \in R^{n \times n}$ , vectors $K = (k_{1}, \dots, k_{n})^{T} \in R^{n}$ and $C = (1, 0, \dots, 0)^{T} \in R^{n}$ , and positive definite matrix $P$ , there holds $A^{T} P + PA \leq - I_{n}$ , where $A = A_{1} - K C^{T}$ , and $k_{i}$ is coefficient of the Hurwitz polynomial $h (s) = s^{n} + k_{1} s^{n - 1} + \dots + k_{n - 1} s + k_{n}$ (Chen et al., 2020).

Lemma 2. For any $ϱ > 0$ and $z \in R$ , there holds $0 \leq | z | - \frac{z^{2}}{\sqrt{z^{2} + ϱ^{2}}} < ϱ$ (Liu and Li, 2018).

Lemma 3. (Barbalat lemma) Consider the function $x (t) : R^{+} \to R$ , if $x (t)$ is square-integrable function, that is $lim_{t \to + \infty} \int_{0}^{t} x^{2} (s) ds < + \infty$ , and $\overset{\cdot}{x} (t)$ is bounded, then $lim_{t \to + \infty} x (t) = 0$ (Li et al., 2022).

Main result

Observer design

In this paper, we consider the case where only system output is available. To estimate the unavailable states in system (1), an observer is constructed as

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i} = {\hat{x}}_{i + 1} + r^{i} k_{i} (x_{1} - {\hat{x}}_{1}), i = 1, \dots, n - 1 \\ {\overset{\cdot}{\hat{x}}}_{n} = D (u) + r^{n} k_{n} (x_{1} - {\hat{x}}_{1}) \end{matrix}

(4)

where $r \geq 1$ is a constant to be determined, ${\hat{x}}_{i}$ is the estimation of $x_{i}$ , and $k_{i}$ is Hurwitz coefficient mentioned in Lemma 1.

The observer error between original system state and observer state is defined as $χ_{i} = x_{i} - {\hat{x}}_{i}$ . Furthermore, the following transformations are introduced

η_{i} = r^{1 - i} χ_{i}, i = 1, \dots, n

(5)

By virtue of equations (1) and (4), the derivative of equation (5) is

{\begin{matrix} {\overset{\cdot}{η}}_{i} = r η_{i + 1} - r k_{i} η_{1} + r^{1 - i} (g_{i} + β_{i}), i = 1, \dots, n - 1 \\ {\overset{\cdot}{η}}_{n} = - r k_{n} η_{1} + r^{1 - n} (g_{n} + β_{n}) \end{matrix}

(6)

Letting $η = (η_{1}, \dots, η_{n})^{T}$ , we rewrite equation (6) as

\overset{\cdot}{η} = rA η + G_{1} + G_{2}

(7)

where $G_{1} = (g_{1}, \dots, r^{1 - n} g_{n})^{T}$ , $G_{2} = (β_{1}, \dots, r^{1 - n} β_{n})^{T}$ , $A$ is defined in Lemma 1, and the boundedness of $η$ is proved later.

Output feedback control design

In order to deal with the problem of arbitrary initial values, we introduce a tuning function $ω (t)$ into the following coordinate transformations

{\begin{matrix} ξ_{1} = x_{1} - y_{0} - ω (t) (x_{1}^{0} - y_{0}^{0}) \\ ξ_{i} = {\hat{x}}_{i} - α_{i - 1} - ω (t) {\hat{x}}_{i}^{0}, i = 2, \dots, n \end{matrix}

(8)

where $ξ_{i} (i = 1, \dots, n)$ is defined as state error; $α_{i - 1}$ is virtual control law to be designed later; $x_{1}^{0}$ , $y_{0}^{0}$ , and ${\hat{x}}_{i}^{0}$ are the initial condition of $x_{1}$ , $y_{0}$ , and ${\hat{x}}_{i}$ , respectively; and $ω (t)$ meets the following condition

ω (t) = {\begin{matrix} \cos (t), t < \frac{π}{2} \\ 0, t \geq \frac{π}{2} \end{matrix}

(9)

Obviously, $ω (t)$ is bounded and $\overset{\cdot}{ω} (t)$ exists on intervals $(0, \frac{π}{2})$ and $(\frac{π}{2}, + \infty)$ . For convenience, $ω$ is used to indicate $ω (t)$ in subsequent description.

To better understand and express the following content, we define

φ_{i} = \frac{ξ_{i}}{τ_{i} (t)}, ϕ_{i} = \frac{φ_{i}}{1 - φ_{i}^{2}}, i = 1, \dots, n

(10)

with $τ_{i} (t) : R^{+} \to R^{+} - {0}$ , ${\bar{τ}}_{1} \leq | τ_{i} (t) | \leq {\bar{τ}}_{2}$ , $| {\overset{\cdot}{τ}}_{i} (t) | \leq {\bar{τ}}_{3}$ , where ${\bar{τ}}_{i} > 0$ , $i = 1, 2, 3$ are known constants, and $τ_{i} (t)$ is denoted by $τ_{i}$ for the latter use.

Next, the virtual control law $α_{i} (i = 1, \dots, n - 1)$ , the designed controller $u$ and the corresponding compensator ${\hat{ε}}_{i} (i = 1, \dots, n)$ are expressed as

α_{i} = - ϕ_{i} (d_{i} + \frac{σ_{i} {\hat{ε}}_{i}}{\sqrt{ϕ_{i}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{i} = \frac{b_{i} ϕ_{i}^{2}}{\sqrt{ϕ_{i}^{2} + κ^{2} (t)}}, {\hat{ε}}_{i}^{0} \geq 0

(11)

and

u = - ϕ_{n} (d_{n} + \frac{σ_{n} {\hat{ε}}_{n}}{\sqrt{ϕ_{n}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{n} = \frac{b_{n} ϕ_{n}^{2}}{\sqrt{ϕ_{n}^{2} + κ^{2} (t)}}, {\hat{ε}}_{n}^{0} \geq 0

(12)

where $d_{i}$ , $σ_{i}$ , and $b_{i}$ are positive design parameters arbitrarily; ${\hat{ε}}_{i}$ is the estimation of $ε_{i}$ , $ε_{i}$ is an unknown constant to be determined; $ϕ_{i}$ is defined in equation (10); and $κ (t)$ is a positive continuous function satisfying $\int_{0}^{t} κ (s) ds \leq κ_{1} < + \infty$ and $| \overset{\cdot}{κ} (t) | \leq κ_{2}$ , where $κ_{1} > 0$ and $κ_{2} > 0$ are constants. For convenience, $κ (t)$ is represented by $κ$ in following content.

Remark 2. The tuning function $ω (t)$ in equation (8) is introduced to eliminate the restrictions on initial values. Specifically, equation (9) reveals $ω (0) = 1$ and it follows from equation (8) that $ξ_{1} (0) = 0 < τ_{1} (0)$ holds for any $x_{1}^{0}$ and $y_{0}^{0}$ . Then, as seen from equation (10), we obtain $φ_{1} (0) = 0$ and $ϕ_{1} (0) = 0$ . Furthermore, $α_{1} (0) = 0$ is also deduced by equation (11), which means $ξ_{2} (0) = 0 < τ_{2} (0)$ for any ${\hat{x}}_{2}^{0}$ . Repeating above process, it is not difficult to obtain $ξ_{i} (0) = 0 < τ_{i} (0)$ for any ${\hat{x}}_{i}^{0} (i = 3, \dots, n)$ .

According to the above analysis, we obtain the following theorem.

Theorem 1. Consider the closed-loop system consisting of equations (1), (4), and (12). Under Assumptions 1 to 3, the following statements hold:

All signals of the closed-loop system are globally bounded;

The state error $ξ_{i} (t)$ in equation (8) satisfies $| ξ_{i} (t) | < τ_{i} (t) (i = 1, \dots, n)$ ; and

The tracking error satisfies $lim_{t \to \infty} (y (t) - y_{0} (t)) = 0$ .

where the bound can converge to a small neighborhood of the origin by adjusting the parameters appropriately.

Proof. The BLF candidates $V_{i}$ are chosen as

{\begin{matrix} V_{i} = \frac{1}{2} \log \frac{1}{1 - φ_{i}^{2}} + \frac{σ_{i} θ_{i}}{2 b_{i}} {\tilde{ε}}_{i}^{2}, i = 1, \dots n - 1 \\ V_{n} = \frac{1}{2 \underline{ϑ}} \log \frac{1}{1 - φ_{n}^{2}} + \frac{σ_{n} θ_{n}}{2 b_{n}} {\tilde{ε}}_{n}^{2} \end{matrix}

(13)

where ${\tilde{ε}}_{i} = ε_{i} - {\hat{ε}}_{i}$ , $ε_{i}$ and $θ_{i}$ are unknown constants to be determined. Following the steps in Liu and Tong (2017) and (Li, 2020), we define a compact set $Ω_{φ_{i}} = {φ_{i} : | φ_{i} | < 1} (i = 1, \dots, n)$ , and $V_{i}$ is a continuous function in the set $Ω_{φ_{i}}$ .

Step 1. By aid of equations (1), (5), (8), and (10), the derivative of $φ_{1}$ is represented as

\begin{matrix} {\overset{\cdot}{φ}}_{1} = \frac{1}{τ_{1}} (r η_{2} + ξ_{2} + α_{1} + ω {\hat{x}}_{2}^{0} + g_{1} + β_{1} \\ - {\overset{\cdot}{y}}_{0} - \overset{\cdot}{ω} (x_{1}^{0} - y_{0}^{0}) - \frac{{\overset{\cdot}{τ}}_{1}}{τ_{1}} ξ_{1}) \end{matrix}

(14)

Then, on the basis of Assumptions 1 and 2 and equation (11), the derivative of $V_{1}$ is

\begin{matrix} {\overset{\cdot}{V}}_{1} = \frac{φ_{1}}{1 - φ_{1}^{2}} {\overset{\cdot}{φ}}_{1} - \frac{σ_{1} θ_{1} {\tilde{ε}}_{1}}{b_{1}} {\overset{\cdot}{\hat{ε}}}_{1} \\ = - \frac{ϕ_{1}^{2}}{τ_{1}} (d_{1} + \frac{σ_{1} {\hat{ε}}_{1}}{\sqrt{ϕ_{1}^{2} + κ^{2}}}) - \frac{σ_{1} θ_{1} {\tilde{ε}}_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} \\ + \frac{ϕ_{1}}{τ_{1}} (r η_{2} + ξ_{2} + ω {\hat{x}}_{2}^{0} + g_{1} + β_{1} - y_{0} - \overset{\cdot}{ω} (x_{1}^{0} - y_{0}^{0}) - \frac{{\overset{\cdot}{τ}}_{1}}{τ_{1}} ξ_{1}) \\ \leq - d_{1} δ_{1} ϕ_{1}^{2} - \frac{δ_{1} σ_{1} {\hat{ε}}_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} + | ϕ_{1} | ϒ_{1} - \frac{σ_{1} θ_{1} {\tilde{ε}}_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} \end{matrix}

(15)

where $Υ_{1} = δ_{1} (| r η_{2} | + | ξ_{2} | + | ω {\hat{x}}_{2}^{0} | + | c {\bar{g}}_{1} | + \bar{β} + | {\dot{y}}_{0} | + | \dot{ω} (x_{1}^{0} - y_{0}^{0}) | + | δ_{1} ξ_{1} {\dot{τ}}_{1} |)$ and $δ_{1} = 1 / τ_{1}$ . As observed from (13) that $| φ_{i} | < 1$ , which together with the bounded $τ_{i}$ , gives the boundedness of $ξ_{i}$ in equation (10). Furthermore, $y$ is also bounded from equation (8), and ${\bar{g}}_{i} (y)$ in Assumption 1 is a smooth function, thereby, the boundedness of ${\bar{g}}_{i} (y)$ is obtained due to bounded $y$ .

Subsequently, we need to verify that $η$ is bounded with the aid of bounded ${\bar{g}}_{i} (y)$ . Define the scalar function $V_{η} = η^{T} P η$ , where $P$ is defined in Lemma 1. By choosing $r = 2 ‖ P ‖ + 1$ and $\bar{β} = max {{\bar{β}}_{1}, \dots, {\bar{β}}_{n}}$ , the derivative of $V_{η}$ with respect to equation (7) is

\begin{matrix} {\overset{\cdot}{V}}_{η} \leq - r {‖ η ‖}^{2} + 2 ‖ P ‖ {‖ η ‖}^{2} + ‖ P ‖ ({‖ G_{1} ‖}^{2} + {‖ G_{2} ‖}^{2}) \\ \leq - {‖ η ‖}^{2} + ‖ P ‖ {‖ G_{1} ‖}^{2} + \sqrt{n} ‖ P ‖ \bar{β} \end{matrix}

(16)

in which $G_{1} = (g_{1}, \dots, r^{1 - n} g_{n})$ . Based on Assumptions 1 and the boundedness of ${\bar{g}}_{i} (y)$ , we know that $‖ G_{1} ‖$ is bounded. Naturally, equation (16) is converted to ${\overset{\cdot}{V}}_{η} \leq - K_{1} V_{η} + K_{2}$ , where $K_{1} = \frac{1}{λ_{\max} (P)}$ and $K_{2} = ‖ P ‖ ‖ G_{1} ‖^{2} + \sqrt{n} ‖ P ‖ \bar{β}$ . Consequently, the boundedness of $η$ is guaranteed. Moreover, ${\overset{\cdot}{y}}_{0}$ , $\overset{\cdot}{ω} (t)$ and ${\overset{\cdot}{τ}}_{1}$ are also bounded, thus there exist unknown constants $δ_{11}$ , $δ_{12}$ , $ϒ_{11}$ , and $ϒ_{12}$ such that $0 < δ_{11} \leq δ_{1} \leq δ_{12}$ and $0 < ϒ_{11} \leq ϒ_{1} \leq ϒ_{12}$ . With the help of Lemma 2, equation (15) is expressed as

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - d_{1} δ_{11} ϕ_{1}^{2} + | ϕ_{1} | ϒ_{1} - \frac{δ_{11} σ_{1} {\hat{ε}}_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} - \frac{θ_{1} σ_{1} {\tilde{ε}}_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} \\ \leq - d_{1} δ_{11} ϕ_{1}^{2} - \frac{δ_{11} σ_{1} ε_{1} ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}} + | ϕ_{1} | ϒ_{12} \\ = - d_{1} δ_{11} ϕ_{1}^{2} + ϒ_{12} (| ϕ_{1} | - \frac{ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2}}}) \\ \leq - d_{1} δ_{11} ϕ_{1}^{2} + ϒ_{12} κ \end{matrix}

(17)

where $θ_{1} = δ_{11}$ and $ε_{1} = \frac{ϒ_{12}}{δ_{11} σ_{1}}$ . Integrating equation (17) from $0$ to $t$ , we get

V_{1} (t) + d_{1} δ_{11} \int_{0}^{t} ϕ_{1}^{2} (s) ds \leq V_{1} (0) + ϒ_{12} κ_{1}

Combined with equation (13), $\frac{1}{2} \log \frac{1}{1 - φ_{1}^{2}} \leq V_{1} \leq ϵ_{1}$ and $\frac{σ_{1} θ_{1}}{2 b_{1}} {\tilde{ε}}_{1}^{2} \leq V_{1} \leq ϵ_{1}$ are derived, where $ϵ_{1} = V_{1} (0) + ϒ_{12} κ_{1}$ . Thereby, the following inequalities hold

| φ_{1} | \leq \sqrt{1 - e^{- 2 ϵ_{1}}} < 1, | {\hat{ε}}_{1} | \leq \sqrt{\frac{2 b_{1} ϵ_{1}}{σ_{1} θ_{1}}} + ε_{1}

(18)

which implies that $α_{1}$ in equation (11) is bounded, and the boundedness of ${\hat{x}}_{2}$ is further obtained from equation (8). By means of the boundedness of $η_{2}$ and ${\hat{x}}_{2}$ , we deduce that $x_{2}$ is bounded. It follows from the boundedness analysis above that ${\overset{\cdot}{α}}_{1}$ is bounded.

Step $i$ $(i = 2, \dots, n - 1)$ : Through equations (4), (5), (8), and (10), the derivative of $φ_{i}$ is

{\overset{\cdot}{φ}}_{i} = \frac{1}{τ_{i}} (ξ_{i + 1} + α_{i} + ω {\hat{x}}_{i + 1}^{0} - {\overset{\cdot}{α}}_{i - 1} + r^{i} k_{i} η_{1} - \overset{\cdot}{ω} {\hat{x}}_{i}^{0} - \frac{{\overset{\cdot}{τ}}_{i}}{τ_{i}} ξ_{i})

(19)

From equations (11), (13), and (19), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - d_{i} δ_{i} ϕ_{i}^{2} - \frac{δ_{i} σ_{i} {\hat{ε}}_{i} ϕ_{i}^{2}}{\sqrt{ϕ_{i}^{2} + κ^{2}}} + | ϕ_{i} | ϒ_{i} - \frac{σ_{i} θ_{i} {\tilde{ε}}_{i} ϕ_{i}^{2}}{\sqrt{ϕ_{i}^{2} + κ^{2}}} \end{matrix}

(20)

where $Υ_{i} = δ_{i} (| r^{i} k_{i} η_{1} | + | ξ_{i + 1} | + | ω {\hat{x}}_{i + 1}^{0} | + | {\dot{α}}_{i - 1} | + | \dot{ω} {\hat{x}}_{i}^{0} | + | δ_{i} ξ_{i} {\dot{τ}}_{i} |)$ and $δ_{i} = 1 / τ_{i}$ . Note that $δ_{i}$ is bounded due to bounded $τ_{i}$ , and from step 1, $η_{1}$ , $ξ_{i}$ , $ξ_{i + 1}$ and ${\overset{\cdot}{α}}_{i - 1}$ are also bounded. Thus, the unknown constants $δ_{i 1}$ , $δ_{i 2}$ , $ϒ_{i 1}$ , and $ϒ_{i 2}$ are found such that $0 < δ_{i 1} \leq δ_{i} \leq δ_{i 2}$ and $0 < ϒ_{i 1} \leq ϒ_{i} \leq ϒ_{i 2}$ . Similar to equation (17) in Step 1, on the basis of Lemma 2, equation (20) is reconstructed as

{\overset{\cdot}{V}}_{i} \leq - d_{i} δ_{i 1} ϕ_{i}^{2} + ϒ_{i 2} κ

(21)

with $θ_{i} = δ_{i 1}$ and $ε_{i} = \frac{ϒ_{i 2}}{δ_{i 1} σ_{i}}$ . Integrating equation (21) from $0$ to $t$ , we have $\frac{1}{2} \log \frac{1}{1 - φ_{i}^{2}} \leq V_{i} \leq ϵ_{i}, \frac{σ_{i} θ_{i}}{2 b_{i}} {\tilde{ε}}_{i}^{2} \leq V_{i} \leq ϵ_{i}$ , where $ϵ_{i} = V_{i} (0) + ϒ_{i 2} κ_{1}$ . After that, the following inequalities are established

| φ_{i} | \leq \sqrt{1 - e^{- 2 ϵ_{i}}} < 1, | {\hat{ε}}_{i} | \leq \sqrt{\frac{2 b_{i} ϵ_{i}}{σ_{i} θ_{i}}} + ε_{i}

(22)

which means that $α_{i}$ in equation (11) is bounded. Furthermore, the boundedness of ${\hat{x}}_{i + 1}$ and $x_{i + 1}$ are deduced by equations (5) and (8), respectively. Moreover, ${\overset{\cdot}{α}}_{i}$ is also bounded.

Step $n$ : Based on equations (3)–(5), (8), and (10), the derivative of $φ_{n}$ is

\begin{matrix} {\overset{\cdot}{φ}}_{n} = \frac{1}{τ_{n}} (ϑ (u) u + ρ (u) - {\overset{\cdot}{α}}_{n - 1} + r^{n} k_{n} η_{1} - \overset{\cdot}{ω} {\hat{x}}_{n}^{0} - \frac{{\overset{\cdot}{τ}}_{n}}{τ_{n}} ξ_{n}) \end{matrix}

(23)

Similar to the process in step $i$ , in the light of equations (12), (13), and (23), the derivative of $V_{n}$ is constructed as

{\overset{\cdot}{V}}_{n} \leq - d_{n} δ_{n} ϕ_{n}^{2} - \frac{δ_{n} σ_{n} {\hat{ε}}_{n} ϕ_{n}^{2}}{\sqrt{ϕ_{n}^{2} + κ^{2}}} + | ϕ_{n} | ϒ_{n} - \frac{σ_{n} θ_{n} {\tilde{ε}}_{n} ϕ_{n}^{2}}{\sqrt{ϕ_{n}^{2} + κ^{2}}}

(24)

where $ϒ_{n} = \frac{δ_{n}}{\underline{ϑ}} (\bar{ρ} + | r^{n} k_{n} η_{1} | + | {\overset{\cdot}{α}}_{n - 1} | + | \overset{\cdot}{ω} {\hat{x}}_{n}^{0} | + | δ_{n} ξ_{n} {\overset{\cdot}{τ}}_{n} |)$ and $δ_{n} = 1 / τ_{n}$ .

By virtue of the boundedness of $τ_{n}$ , ${\overset{\cdot}{τ}}_{n}$ , $η_{1}$ , ${\overset{\cdot}{α}}_{n - 1}$ , $\overset{\cdot}{ω} (t)$ , and $ξ_{n}$ , we know that there are unknown constants $δ_{n 1}$ , $δ_{n 2}$ , $ϒ_{n 1}$ , and $ϒ_{n 2}$ such that $0 < δ_{n 1} \leq δ_{n} \leq δ_{n 2}$ and $0 < ϒ_{n 1} \leq ϒ_{n} \leq ϒ_{n 2}$ . In the following, combined with Lemma 2, equation (24) is shown as

{\overset{\cdot}{V}}_{n} \leq - d_{n} δ_{n 1} ϕ_{n}^{2} + ϒ_{n 2} κ

(25)

where $θ_{n} = δ_{n 1}$ and $ε_{n} = \frac{ϒ_{n 2}}{δ_{n 1} σ_{n}}$ . Integrating equation (25) from $0$ to $t$ , we obtain $\frac{1}{2} \log \frac{1}{1 - φ_{n}^{2}} \leq V_{n} \leq ϵ_{n}, \frac{σ_{n} θ_{n}}{2 b_{n}} {\tilde{ε}}_{n}^{2} \leq V_{n} \leq ϵ_{n}$ , where $ϵ_{n} = V_{n} (0) + ϒ_{n 2} κ_{1}$ . Furthermore, we have

| φ_{n} | \leq \sqrt{1 - e^{- 2 ϵ_{n}}} < 1, | {\hat{ε}}_{n} | \leq \sqrt{\frac{2 b_{n} ϵ_{n}}{σ_{n} θ_{n}}} + ε_{n} .

(26)

From equations (12) and (26), $u$ in is bounded.

In summary, all signals of the closed-loop system are globally bounded, and state errors satisfy $| ξ_{i} | < τ_{i}, i = 1, \dots, n$ . In addition, integrating (17) from 0 to $t$ , we have $\int_{0}^{t} d_{1} δ_{11} ϕ_{1}^{2} (s) ds \leq ϵ_{1}$ . According to the boundedness analysis in above process, we conclude that ${\overset{\cdot}{ϕ}}_{1}$ is bounded. In view of Lemma 3, one has $lim_{t \to + \infty} ϕ_{1} (t) = 0$ , which means $lim_{t \to + \infty} ξ_{1} (t) = 0$ . Therefore

lim_{t \to + \infty} (y (t) - y_{0} (t)) = 0

(27)

and it indicates that the asymptotic tracking is achieved. The proof is completed.

Remark 3. Different from the work in Liu and Li (2018) and Ni and Shi (2021), a more practical situation is considered in this paper, where only the system output is available. Therefore, it is necessary to design the observer (4) to estimate unavailable states. Then, based on equation (4) and BLF, we design the adaptive output feedback controller (12) without the derivative of virtual control laws, which has simper structure than that in Lan et al. (2021) and Zhou et al. (2021). Particularly, instead of using the auxiliary system to handle the derivative of the virtual control laws as mentioned in Zong et al. (2022), we make adaptive compensation to them, thus further reducing the computational complexity.

Remark 4. It is worth pointing out that the asymptotic tracking is achieved despite the presence of dead-zone input and unknown nonlinearities. To be specific, by BLF and the controller (12), the influence of dead-zone input is compensated. Furthermore, the system nonlinearities are regarded as some bounded disturbances and are handled at each step by the adaptive law ${\hat{ε}}_{i}$ in equation (12), which avoids the limitation of compact domain caused by approximation methods in Hua et al. (2019) and Li et al. (2022).

Remark 5. As shown in the proof of Theorem 1, the state error $ξ_{i}$ remains inside the compact set $Ω_{φ_{i}}$ as time goes infinite for initial value satisfying $ξ_{i} (0) < τ_{i} (0)$ . It should be noted that the introduction of $ω (t)$ in equation (8) eliminates the restriction on initial value of signals in closed-loop system consisting of equations (1), (4), and (12). Thus, the global boundedness of the closed-loop system is guaranteed, which improves the semi-global boundedness results in Ni and Shi (2021) and Zong et al. (2022). Moreover, the size of the bound can be reduced by decreasing $b_{i}$ and increasing $d_{i}$ , $σ_{i}$ appropriately.

In sequel, we study the special case in Theorem 1, that is, the dead-zone parameters are known in system (1). Let $ϑ_{1} = ϑ_{2} = ϑ > 0$ , and equation (2) is constructed as $D (u) = ϑ u + ρ (u)$ , where $ρ (u) \leq max {ϑ μ_{1}, ϑ μ_{2}} = ρ$ . Then, inspired by equation (12), we construct the following controller $u$ and corresponding compensator ${\hat{ε}}_{n}$

u = - \frac{ϕ_{n}}{ϑ} (d_{n} + \frac{σ_{n} {\hat{ε}}_{n}}{\sqrt{ϕ_{n}^{2} + κ^{2}}}), {\overset{\cdot}{\hat{ε}}}_{n} = \frac{b_{n} ϕ_{n}^{2}}{\sqrt{ϕ_{n}^{2} + κ^{2}}}, {\hat{ε}}_{n}^{0} \geq 0

(28)

Next, we give the following corollary for system (1) with known dead-zone parameters.

Corollary 1. Under Assumptions 1 to 3, the adaptive output feedback controller (28) is proposed to realize the objective of Theorem 1.

Proof. Similar to equation (13) in the proof of Theorem 1, the BLFs candidates are designed as follows

V_{i} = \frac{1}{2} \log \frac{1}{1 - φ_{i}^{2}} + \frac{σ_{i} θ_{i}}{2 b_{i}} {\tilde{ε}}_{i}^{2}, i = 1, \dots, n

(29)

For simplicity, the detailed proof is omitted here.

Simulation examples

In this section, we use two simulation examples to demonstrate the effectiveness of the proposed control scheme.

Example 1. Consider the following Josephson junction circuit system

ϖ_{1} \overset{\cdot\cdot}{q} + ϖ_{2} \overset{\cdot}{q} + ϖ_{3} \sin (q) = D (u) + β (t)

(30)

where $ϖ_{i} > 0$ is known constant, $q$ is measurable, $D (u)$ is dead-zone input, and $β (t)$ is external disturbance. Letting $x_{1} = q$ , $x_{2} = ϖ_{1} \overset{\cdot}{q} + ϖ_{2} q$ , $ϖ_{1} = ϖ_{2} = ϖ_{3} = 1$ , we can rewrite system (30) as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + g_{1} (t, y), \\ {\overset{\cdot}{x}}_{2} = D (u) + g_{2} (t, y) + β (t) . \end{matrix}

(31)

where $g_{1} (t, y) = - x_{1}$ , $g_{2} (t, y) = - \sin (x_{1})$ , $β (t) = 0.05 + 0.1 e^{- t}$ , and some parameters in $D (u)$ are selected as $ϑ_{1} = 1$ , $ϑ_{2} = 0.9$ , $μ_{1} = 0.2$ , $μ_{2} = 0.1$ .

For simulation, we set $k_{1} = k_{2} = 1$ , $d_{1} = 0.8$ , $d_{2} = 0.1$ , $σ_{1} = 0.1$ , $σ_{2} = 0.2$ , $b_{1} = 1.2$ , $b_{2} = 0.5$ , $τ_{1} (t) = 0.2 + e^{- 2 t}$ , $τ_{2} (t) = 0.5 + e^{- t}$ , $κ (t) = \frac{1}{t^{2} + 0.1}$ , and $y_{0} (t) = 0.1 \sin (t)$ . In the light of previous data, the observer and the controller are constructed for system (31), respectively

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = {\hat{x}}_{2} + r (x_{1} - {\hat{x}}_{1}) \\ {\overset{\cdot}{\hat{x}}}_{2} = D (u) + r^{2} (x_{1} - {\hat{x}}_{1}) \end{matrix}

(32)

and

{\begin{matrix} α_{1} = - ϕ_{1} (0.8 + \frac{0.1 {\hat{ε}}_{1}}{\sqrt{ϕ_{1}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{1} = \frac{1.2 ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2} (t)}} \\ u = - ϕ_{2} (0.1 + \frac{0.2 {\hat{ε}}_{2}}{\sqrt{ϕ_{2}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{2} = \frac{0.5 ϕ_{2}^{2}}{\sqrt{ϕ_{2}^{2} + κ^{2} (t)}} \end{matrix}

(33)

where $r = 2.8090$ and the initial conditions are chosen as $x_{1} (0) = - 0.45$ , $x_{2} (0) = - 0.5$ , ${\hat{x}}_{1} (0) = {\hat{x}}_{2} (0) = 0$ , ${\hat{ε}}_{1} (0) = 0.7$ , and ${\hat{ε}}_{2} (0) = 0.1$ .

In the next step, the controller designed in equation (33) is applied to system (31), and the corresponding simulation results are shown in Figures 1 –5, which intuitively show that all signals of the closed-loop system consisting of equations (31), (32), and (33) are globally bounded. Particularly, it can be clearly seen in Figure 1 that the output $y$ achieves asymptotic tracking of the reference signal $y_{0}$ . The observer error $χ_{i} = x_{i} - {\hat{x}}_{i}$ is bounded from the analysis in section “Main result,” so there are some errors between the trajectory of original system state $x_{i}$ and observer state ${\hat{x}}_{i}$ in Figure 2. Moreover, Figure 3 shows that the state errors $ξ_{1}$ and $ξ_{2}$ satisfy $| ξ_{i} | < τ_{i}, i = 1, 2$ . In conclusion, the control objective of this paper is realized under the proposed controller.

Example 2. Consider the following nonlinear system

{\begin{matrix} \begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + \ln (1 + x_{1}^{2}) + β_{1} (t) \\ {\overset{\cdot}{x}}_{2} = D (u) + x_{1}^{2} + β_{2} (t) \end{matrix} \end{matrix}

(34)

where $β_{1} (t) = 0.015 \sin (t) + 0.1 e^{- 0.5 t}$ , $β_{2} (t) = 0.1 \cos (t)$ , and some parameters in $D (u)$ are set as $ϑ_{1} = 0.6$ , $ϑ_{2} = 0.9$ , and $μ_{1} = μ_{2} = 0.1$ .

Let $k_{1} = 3$ , $k_{2} = 3$ , $r = 2.1952$ , $d_{1} = 0.4$ , $d_{2} = 0.3$ , $σ_{1} = 0.2$ , $σ_{2} = 0.11$ , $b_{1} = 0.2$ , $b_{2} = 0.15$ , $τ_{1} (t) = 0.8 + e^{- 0.4 t}$ , $τ_{2} (t) = 0.5 + e^{- 2 t}$ , $κ (t) = \frac{2}{t^{2} + 1}$ , and $y_{0} (t) = 0.2 \cos (0.45 t) + 0.1 \sin (1.06 t)$ . Then, the observer and the controller are constructed for system (34) by aid of above data

{\begin{matrix} \begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = {\hat{x}}_{2} + 3 r (x_{1} - {\hat{x}}_{1}) \\ {\overset{\cdot}{\hat{x}}}_{2} = D (u) + 3 r^{2} (x_{1} - {\hat{x}}_{1}) \end{matrix} \end{matrix}

(35)

and

{\begin{matrix} α_{1} = - ϕ_{1} (0.4 + \frac{0.2 {\hat{ε}}_{1}}{\sqrt{ϕ_{1}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{1} = \frac{0.2 ϕ_{1}^{2}}{\sqrt{ϕ_{1}^{2} + κ^{2} (t)}} \\ u = - ϕ_{2} (0.3 + \frac{0.1 {\hat{ε}}_{2}}{\sqrt{ϕ_{2}^{2} + κ^{2} (t)}}), {\overset{\cdot}{\hat{ε}}}_{2} = \frac{0.15 ϕ_{2}^{2}}{\sqrt{ϕ_{2}^{2} + κ^{2} (t)}} \end{matrix}

(36)

The simulation results are displayed in Figures 6 –10 with initial conditions $x_{1} (0) = - 0.8$ , $x_{2} (0) = 1.1$ , ${\hat{x}}_{1} (0) = {\hat{x}}_{2} (0) = 0$ , ${\hat{ε}}_{1} (0) = 0.1$ , and ${\hat{ε}}_{2} (0) = 0.1$ . To be specific, the trajectories of output $y$ and reference signal $y_{0}$ are depicted in Figure 6, which means the asymptotic tracking is achieved. In Figure 7, we give the trajectory of original system state $x_{i}$ and observer state ${\hat{x}}_{i}$ , and we can observe in Figure 8 that $| ξ_{i} | < τ_{i}$ is satisfied. In addition, from Figures 6 –10, it is obvious that all signals of the closed-loop system consisting of equations (34), (35), and (36) are globally bounded. In summary, the control scheme proposed in section “Main result” is effective.

Conclusion

For the nonlinear systems with dead-zone input and unknown nonlinearities, the adaptive output feedback asymptotic tracking problem has been addressed in this paper. Under the premise that only system output was available, a dynamic observer has been introduced to estimate unavailable states. Then, drawing support from the observer and BLF, an adaptive output feedback controller has been constructed to compensate the influence of dead-zone input and unknown nonlinearities. By means of tuning function, the proposed controller has further ensured that all signals of the closed-loop system were globally bounded; meanwhile, the asymptotic tracking has been achieved. In particular, during the design process of the controller, it was unnecessary to calculate the derivative of virtual control laws, which reduced the complexity of calculation. In future study, a significant work is to extend the results to the event-triggered control for nonlinear systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethical approval

The work was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (61973188, 62192753, and 61973189); the Taishan Scholar Project of Shandong Province of China (ts20190905); and the Innovative Research Groups of National Natural Science Foundation of China (61821004).

ORCID iD

Zhiyu Duan

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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