Finite-time command-filtered backstepping control for nonlinear systems with input delay and time-varying full-state constraints

Abstract

This paper examines a finite-time command-filtered backstepping control proposed for a class of strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. The existing control method either ignores the effect of input delay or converges asymptotically in infinite time. A command-filtered backstepping design is used to decrease computational burden. A first-order Levant differentiator is employed to replace the command filter for estimating the virtual control signals in finite time. The time-varying asymmetric barrier Lyapunov function (TVABLF) and command-filtered backstepping design are applied to alleviate the significant challenges caused by the backstepping approach and full-state constraints. A novel finite-time delay compensation mechanism is proposed to remove the impact of input delay. The closed-loop signals are proved to be practical bounded. Simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Keywords

Command-filtered backstepping barrier Lyapunov functions finite time input-delay systems

Introduction

Backstepping control, which has been widely used in practical nonlinear dynamic processes, is an effective way to design a controller (Arefi et al., 2015; Li et al., 2017; Liu et al., 2017a; Yu et al., 2018b; Zhao et al., 2016). In classical backstepping, virtual control signals are repeated differentiations of the control law design, which leads to a heavy computational burden called “the explosion of complexity” (Liu et al., 2014). To deal with this challenge, dynamic surface control (DSC) is investigated by introducing first-order low-pass filters, which estimate the virtual control laws in the backstepping design (Li et al., 2010; Tong et al., 2011; Zhang and Ge, 2008). However, DSC does not consider any issues of error compensation caused by the filters (Yu et al., 2018a).

The command-filtered backstepping approach, which introduced command filters to approximate the derivative of the virtual control signals, was first proposed by Farrell et al. (2009). Dong et al. (2012) extended it to the adaptive case for strict-feedback systems. Unlike DSC, error compensation signals are used in command-filtered backstepping to reduce the error caused by the command filters (Yu et al., 2020). Based on this approach, Yu et al. (2015) considered adaptive fuzzy output feedback control with observers for uncertain nonlinear systems. In Wang et al. (2021a, 2021b), an adaptive neural network controller was developed using command-filtered backstepping for a class of nonlinear systems with input nonlinearities and asymmetric time-varying output constraints. It should be noted that the above approaches cannot handle the situation when state constraints are required.

Barrier Lyapunov function (BLF; Tee et al., 2009), extremum-seeking control (DeHaan and Guay, 2005), and model predictive control (Mayne et al., 2000) are the methods that deal with nonlinear systems with state constraints. Log-type BLF and log-type integral barrier Lyapunov function (IBLF) are combined with adaptive control to deal with state constraints (Cui et al., 2020; Liu et al., 2018, 2021b; Wang et al., 2021a). Li et al. (2019b) constructed a neural network–based controller for a class of strict-feedback nonlinear systems with state constraints and input delay. An observer-based adaptive controller was designed for a nontriangular nonlinear system with full-state constraints by Zhang et al. (2021). However, the methods mentioned above were proposed for nonlinear systems with constant constraints. Nonlinear systems with time-varying asymmetric state constraints are dealt with effectively by the time-varying asymmetric barrier Lyapunov function (TVABLF) (He et al., 2017; Liu et al., 2017b; Mishra et al., 2021; Yang et al., 2020). It is worth noting that the nonlinear mapping method (Guo and Wu, 2014; Yan et al., 2022; Zhang et al., 2017; Zhao et al., 2020), which transfers the original nonlinear system with constraints to an equivalent one without constraints, is also an efficient way to handle nonlinear systems with time-varying asymmetric state constraints. Li et al. (2019a) designed an adaptive neural controller for nonlinear systems with time-varying full-state constraints.

Time delay is a classic control problem that often exists in practical engineering (Kim et al., 2016; Li et al., 2021). It can lead to poor system performance and system instability. The control problem of time delay has attracted the attention of many researchers (Li et al., 2019b; Liu et al., 2014; Wang et al., 2015; Xia et al., 2019, 2020, 2021a; Zhao et al., 2016). Xia et al. (2021a, 2021b) designed an event-triggered filter for delayed discrete-time Markovian jump systems, which utilized the Lyapunov–Krasovskii function to eliminate the effect of time delay. The effect caused by input delay was transferred to an intermediate variable by employing the Pade approximation in Li et al. (2019b). Zhou et al. (2022) introduced a novel auxiliary system to remove the impact of input delay. As far as we know, the problem of the finite-time command-filtered backstepping tracking control of strict-feedback nonlinear systems with input delay and time-varying full-state constraints has not been addressed well.

Following the above observations, we aim to tackle the issue of finite-time command-filtered control for strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. To the best of the authors’ knowledge, information about finite-time control for nonlinear systems with input delay is limited. This may be due to the inherent difficulties of designing a finite-time input delay compensated mechanism. In this paper, a finite-time input delay compensated mechanism is constructed and combined with the finite-time command-filtered backstepping technique, which ensures that the tracking error converges to a small neighborhood of the zero in finite time. The main contributions are summarized as follows:

A novel finite-time delay compensation mechanism is proposed to reduce the impact of input delay.

Wang et al. (2021a) and Cui et al. (2020) have previously only focused on constant symmetric state constraints, but this paper considers the influence of time-varying asymmetric state constraints in the design process.

By utilizing the finite-time command-filtered technique, the explosion of complexity problem is avoided, and the tracking error can converge to a small neighborhood of the zero in finite time.

This paper has been organized as follows. Section “System description and preliminaries” will address the formulation of the problem and preliminaries. In section “Controller design,” the design and analysis of control systems are presented. The simulation result is provided to demonstrate the approach in section “Simulation results,” while section “Conclusion” contains the conclusions.

System description and preliminaries

Consider the following class of nth nonlinear systems with input delay and saturation

{\begin{matrix} {\dot{x}}_{i} (t) = f_{i} ({\bar{x}}_{i} (t)) + g_{i} ({\bar{x}}_{i} (t)) x_{i + 1} (t) \\ {\dot{x}}_{n} (t) = f_{n} (x (t)) + g_{n} (x (t)) u (w (t - τ)) \\ y (t) = x_{1} (t) \end{matrix}

(1)

where $x = [x_{1}, x_{2}, \dots, x_{n}]^{T} \in R^{n}$ is the system states vector with $x (0) = x_{0}$ . $y \in R$ is the system output; ${\bar{x}}_{i} = [x_{1}, x_{2}, \dots, x_{i}]^{T}$ ; $f_{i} (\cdot)$ and $g_{i} (\cdot)$ stand for the known bounded smooth functions; $τ > 0$ presents the input delay. The saturated input $u (w (t))$ is defined as

u (w (t)) = sat (w (t)) = {\begin{matrix} sign (w (t)) u_{M}, & | w (t) | \geq u_{M} \\ w (t), & | w (t) | < u_{M} \end{matrix}

(2)

where $u_{M} > 0$ is the bound of actual control input $u (t)$ .

Remark 1. Many practical physical systems can be described as or simplified to system equation (1), such as manipulators (He et al., 2017; Zhao and Song, 2020), suspension systems (Liu et al., 2021c), and airships (Wei et al., 2019; Wu et al., 2020).

Remark 2. Compared with previous works (Liu et al., 2021a, 2021b; Mishra et al., 2021; Zhao and Song, 2020), which considered time-varying state constrained systems, the effect of input delay is not taken into account. Therefore, the system studied is more general and relevant to the practical engineering context. The effect of input delay is transferred to an intermediate variable (Xing et al., 2021; Zhou et al., 2022) or estimated by an auxiliary system (Li et al., 2019b; You et al., 2019), but it cannot converge in finite time.

The saturation is nonlinear and unsmooth because of the sharp corner at $| w (t) | = u_{M}$ . To solve this problem, smooth function $ρ (w)$ is applied to approximate the nonlinear saturation function (Wen et al., 2011; Zhou et al., 2017)

ρ (w) = u_{M} \tanh (\frac{w}{u_{M}}) = u_{M} \frac{e^{w / u_{M}} - e^{- w / u_{M}}}{e^{w / u_{M}} + e^{- w / u_{M}}} .

(3)

Then, $sat (w)$ in equation (2) can be rewritten as

sat (w) = ρ (w) + d

(4)

where $| d | = sat (w) - ρ (w) \leq u_{M} (1 - \tanh (1)) = \bar{d}$ .

According to intermediate value theorem, we can obtain

ρ (w) = ρ (w_{0}) + \frac{\partial ρ (w)}{\partial w} |_{w = w_{σ}} (w - w_{0})

(5)

where $w_{σ} = σ w + (1 - σ) w_{0}$ and $0 < σ < 1$ .

By choosing $w_{0} = 0$ , equation (2) can be described as

u (w) = ρ_{w_{σ}} w + d

(6)

where $ρ_{w_{σ}} = (\partial ρ (w) / \partial w) |_{w = w_{σ}}$ .

The control objective of this paper is to design a finite-time controller $w$ for nonlinear systems equation (1), and the desired tracking signal $y_{d}$ is tracked pre-eminently by system output $y$ . Meanwhile, the time-varying asymmetric state constraints are not violated, that is, $- {\underline{k}}_{c_{i}} (t) < x_{i} (t) < {\bar{k}}_{c_{i}} (t) (i = 1, 2, \dots, n)$ , where ${\underline{k}}_{c_{i}} (t)$ and ${\bar{k}}_{c_{i}} (t)$ are the time-varying functions.

The control objective can be implemented in the following assumptions and lemmas.

Assumption 1. (Yu et al., 2018a). The origin and initial condition $x_{0}$ are included in an open set $Ω_{d} \subset R^{n}$ . In system (1), $f_{i}^{(p)} (\cdot)$ , $g_{i}^{(p)} (\cdot)$ , $f_{n} (\cdot)$ , $g_{n} (\cdot)$ , ${\overset{\cdot}{f}}_{n} (\cdot)$ , and ${\overset{\cdot}{g}}_{n} (\cdot)$ are bounded in the closed set ${\bar{Ω}}_{d}$ , where $i = 1, 2, \dots, n - 1$ and $p = 1, 2, \dots, n - i$ , and $\underline{g} \leq | g_{i} | \leq \bar{g}$ with $\underline{g}$ and $\bar{g}$ are the known positive constants.

Assumption 2. (Liu et al., 2017b). Constants ${\bar{K}}_{c_{i}} > 0$ and ${\underline{K}}_{c_{i}} > 0$ $i = 1, \dots, n$ satisfy ${\bar{k}}_{c_{i}} (t) \leq {\bar{K}}_{c_{i}}$ and ${\underline{k}}_{c_{i}} (t) \geq {\underline{K}}_{c_{i}}$ , and the time-varying functions ${\underline{k}}_{c_{i}} (t)$ and ${\bar{k}}_{c_{i}} (t)$ , $i = 1, 2, \dots, n$ and their first and second derivatives are bounded and smooth.

Assumption 3. The desired tracking signal $y_{d}$ and its first derivative ${\overset{\cdot}{y}}_{d}$ are the bounded, smooth, and known functions, and two functions satisfy ${\underline{k}}_{c_{1}} (t) < {\underline{y}}_{0} (t) \leq y_{d} (t) \leq {\bar{y}}_{0} (t) < {\bar{k}}_{c_{1}} (t)$ and $| {\overset{\cdot}{y}}_{d} | \leq {\bar{y}}_{1}$ , where ${\bar{y}}_{1}$ is a positive constant.

Remark 3. Assumption 1 is a common assumption in a tracking control problem (Li et al., 2019b; Ren et al., 2010; Yu et al., 2018a). Assumptions 2 and 3 are sufficient conditions to prove the effectiveness of the control approach.

Remark 4. Classical backstepping requires the time derivative of the desired tracking signal in each step to compute the time derivative of the virtual control signal. It means the desired tracking signal and its nth derivative should be available and bounded. Here, the virtual control signals and their first derivative are estimated by the first-order Levant differentiator, which will be introduced later, and availability has been relaxed.

Lemma 1. (Ren et al., 2010). For $| ξ | < 1$ , $\log 1 / 1 - ξ^{2} < ξ^{2} / 1 - ξ^{2}$ holds.

Lemma 2. (Huang et al., 2005). For $x_{i} \in R (i = 1, 2, \dots, n)$ and $0 < p \leq 1$

{(\sum_{i = 1}^{n} | x_{i} |)}^{p} \leq \sum_{i = 1}^{n} | x_{i} |^{p} \leq n^{1 - p} {(\sum_{i = 1}^{n} | x_{i} |)}^{p} .

(7)

Lemma 3. (Yu et al., 2005). For any real numbers $ϵ_{1} > 0$ , $ϵ_{2} > 0$ , and $γ > 0$ , an extended Lyapunov condition of finite-time stability can be given as $\overset{\cdot}{V} (x) + ϵ_{1} V (x) + ϵ_{2} V^{γ} (x) \leq 0$ , and the settling time can be given as

Tr \leq t_{0} + \frac{1}{ϵ_{1} (1 - γ)} \ln \frac{ϵ_{1} V^{1 - γ} (t_{0}) + ϵ_{2}}{ϵ_{2}} .

Lemma 4. (Yu et al., 2018a). Consider the system $\overset{\cdot}{x} = f (x)$ . If a continuous function $V (x)$ exists, which satisfies $\overset{\cdot}{V} (x) \leq - ϵ_{1} V (x) - ϵ_{2} V^{γ} (x) + χ$ with {\epsilon_1} \gt 0, $ϵ_{2} > 0$ , $0 < γ < 1$ , and $0 < χ < \infty$ , then the trajectory of the system $\overset{\cdot}{x} = f (x)$ is practical finite-time stable, and the residual set of the solution of system $\overset{\cdot}{x} = f (x)$ is given by

{lim_{t \to T_{r}} x | V (x) \leq min {\frac{χ}{(1 - θ_{0}) ϵ_{1}}, {(\frac{χ}{(1 - θ_{0}) ϵ_{2}})}^{\frac{1}{γ}}}},

(8)

where $0 < θ_{0} < 1$ . The convergence time is bounded as

\begin{matrix} T_{r} \leq max {t_{0} + \frac{1}{θ_{0} ϵ_{1} (1 - γ)} \ln \frac{θ_{0} ϵ_{1} V^{1 - γ} (t_{0}) + ϵ_{2}}{ϵ_{2}} \\ t_{0} + \frac{1}{ϵ_{1} (1 - γ)} \ln \frac{ϵ_{1} V^{1 - γ} (t_{0}) + θ_{0} ϵ_{2}}{θ_{0} ϵ_{2}}} . \end{matrix}

(9)

The first-order Levant differentiator (Levant, 1998, 2003) is introduced as follows

{\begin{matrix} {\overset{\cdot}{φ}}_{1} = ν_{1} \\ ν_{1} = - r_{1} | φ_{1} - α_{r} |^{\frac{1}{2}} sign (φ_{1} - α_{r}) + φ_{2} \\ {\overset{\cdot}{φ}}_{2} = - r_{2} sign (φ_{2} - ν_{1}) \end{matrix}

(10)

where $α_{r}$ is the input signal, and $r_{1}$ and $r_{2}$ are the design parameters. The following lemma satisfies if $r_{1}$ and $r_{2}$ are chosen properly.

Lemma 5. (Levant, 1998, 2003). If $r_{1}$ and $r_{2}$ are chosen properly and the input of differentiator (10) does not have input noise, the following equalities hold

\begin{matrix} ν_{1} = {\overset{\cdot}{α}}_{r 0} \\ φ_{1} = α_{r 0} \end{matrix}

(11)

and the solutions of the differentiator have finite-time convergence.

Lemma 6. (Levant, 1998, 2003). If the input noise satisfies $| α_{r} - α_{r 0} | \leq μ$ , the following equalities satisfy in finite time if $r_{1}$ and $r_{2}$ are chosen properly

\begin{matrix} | φ_{1} - α_{r 0} | \leq ι_{1} μ = ψ_{1} \\ | ν_{1} - {\overset{\cdot}{α}}_{r 0} | \leq ι_{2} μ^{\frac{1}{2}} = ψ_{2} \end{matrix}

(12)

where $μ$ , $ι_{1}$ , and $ι_{2}$ are the positive constants dependent on $r_{1}$ and $r_{2}$ in differentiator (10).

Remark 5. According to Assumption 1, functions $f_{i}$ and $g_{i}$ on ${\bar{Ω}}_{d}$ are Lipschitz. Lemmas 5 and 6 describe the different conditions when the differentiator has input noise or not, respectively. It should be noted that the scheme for choosing parameters $r_{1}$ and $r_{2}$ was given in Levant (2003).

A novel finite-time input delay compensation mechanism is proposed to remove the impact of the input delay, which is defined as follows

{\begin{matrix} {\overset{\cdot}{λ}}_{i} = - q_{i} λ_{i} + g_{i} λ_{i + 1} - l_{i} sign (λ_{i}) \\ {\overset{\cdot}{λ}}_{n} = - q_{n} λ_{n} + g_{n} u (w (t - τ)) - g_{n} u (w (t)) - l_{n} sign (λ_{n}) \end{matrix}

(13)

where $i = 1, 2, \dots, n - 1$ and $λ (0) = 0$ . $q_{i}$ and $l_{i}$ are the positive designed parameters that will be described later.

Lemma 7. The input delay compensation mechanism in equation (13) can guarantee that all states are practical finite-time stable. The settle time is bounded as

T_{λ} \leq max {\frac{2}{θ_{λ} a_{λ}} \ln \frac{θ_{λ} a_{λ} V_{λ}^{1 / 2} (0) + b_{λ}}{b_{λ}}, \frac{2}{a_{λ}} \ln \frac{a_{λ} V_{λ}^{1 / 2} (0) + θ_{λ} b_{λ}}{θ_{λ} b_{λ}}}

(14)

where $a_{λ} = 2 min {(q_{1} - (1 / 2) \bar{g}), (q_{i} - \bar{g}), (q_{n} - (3 / 2) \bar{g})}$ and $b_{λ} = \sqrt{2} min {l_{i}}$ .

Proof. Choose the Lyapunov function as

V_{λ} = \frac{1}{2} \sum_{i = 1}^{n} λ_{i}^{2} .

(15)

The derivative of $V_{λ}$ is

\begin{array}{l} {\dot{V}}_{λ} = λ_{1} {\dot{λ}}_{1} + λ_{2} {\dot{λ}}_{2} + \dots + λ_{n} {\dot{λ}}_{n} \\ = - \sum_{i = 1}^{n} q_{i} λ_{i}^{2} + \sum_{i = 1}^{n - 1} g_{i} λ_{i} λ_{i + 1} - \sum_{i = 1}^{n} l_{i} | λ_{i} | \\ + g_{n} λ_{n} (u (w (t - τ)) - u (w (t))) . \end{array}

(16)

According to Assumption 1, Lemma 2, and Young’s inequality, equation (16) can be obtained

\begin{matrix} {\overset{\cdot}{V}}_{λ} \leq - (q_{1} - \frac{1}{2} \bar{g}) λ_{1}^{2} - \sum_{i = 2}^{n - 1} (q_{i} - \bar{g}) λ_{i}^{2} - (q_{n} - \frac{3}{2} \bar{g}) λ_{n}^{2} \\ - \sum_{i = 1}^{n} l_{i} | λ_{i} | + \bar{g} u_{M}^{2} \\ \leq - a_{λ} V_{λ} - b_{λ} V_{λ}^{\frac{1}{2}} + c_{λ} \end{matrix}

(17)

Then, if we choose suitable parameters such that $a_{λ} > 0$ and $b_{λ} > 0$ , states $λ_{i}$ will achieve practical finite-time stability from Lemma 4. $λ_{i}$ will converge to the region $| λ_{i} | \leq E_{λ} = min {\sqrt{2 c_{λ} / (1 - θ_{λ}) a_{λ}}, \sqrt{2} c_{λ} / (1 - θ_{λ}) b_{λ}}$ in finite time $T_{r λ} \leq max {2 / θ_{λ} a_{λ} \ln θ_{λ} a_{λ} V_{λ}^{1 / 2} (0) + b_{λ} / b_{λ}, 2 / a_{λ} \ln a_{λ} V_{λ}^{1 / 2} (0) + θ_{λ} b_{λ} / θ_{λ} b_{λ}}$ , $0 < θ_{λ} < 1$ .

Remark 6. Consider system (1) and input delay compensation mechanism (13) and the term $z_{n} = x_{n} - {\bar{α}}_{n} - λ_{n}$ , the actual control signal $w$ will be employed to eliminate the impacts of the input delay $λ_{n}$ . It should be emphasized that the auxiliary system for input delay given in Zhou et al. (2022) can only guarantee that the system is bounded. The input delay compensation mechanism in equation (13) can guarantee that the signals in the system are finite-time stable.

Controller design

In this section, the finite-time command-filtered backstepping control method will be proposed for system equation (1) with time-varying asymmetric full-state constraints.

Define the tracking error as

e_{1} = x_{1} - y_{d}, e_{i} = x_{i} - {\bar{α}}_{i}, i = 2, \dots, n,

(18)

where ${\bar{α}}_{i + 1} = φ_{i, 1}$ is the output of the first-order Levant differentiator, with the virtual controller $α_{i}$ as input. The formulation of the first-order Levant differentiator has been introduced as follows

{\begin{matrix} {\overset{\cdot}{φ}}_{i, 1} = ν_{i, 1} \\ ν_{i, 1} = - r_{i, 1} | φ_{i, 1} - α_{i} |^{\frac{1}{2}} sign (φ_{i, 1} - α_{i}) + φ_{i, 2} \\ {\overset{\cdot}{φ}}_{i, 2} = - r_{i, 2} sign (φ_{i, 2} - ν_{i, 1}) \end{matrix}

(19)

Furthermore, the compensated tracking error signals are constructed as

v_{i} = z_{i} - ζ_{i}, i = 1, 2, \dots, n,

(20)

where $z_{i} = e_{i} - λ_{i}$ and $ζ_{i}$ is the error compensation signal with $ζ (0) = 0$ .

Consider the TVABLF

V_{i} (t) = \frac{h_{i} (t)}{2 κ} \log \frac{k_{b_{i}}^{2 κ} (t)}{k_{b_{i}}^{2 κ} (t) - v_{i} {(t)}^{2 κ}} + \frac{1 - h_{i} (t)}{2 κ} \log \frac{k_{a_{i}}^{2 κ} (t)}{k_{a_{i}}^{2 κ} (t) - v_{i} {(t)}^{2 κ}},

(21)

where $2 κ \geq n$ is a positive constant to ensure that the virtual control signals $α_{i}$ and their nth derivative is available. $k_{a_{i}} (t)$ and $k_{b_{i}} (t)$ will be described later along with $k_{a_{i}} (t) < v_{i} (t) < k_{b_{i}} (t)$ . The $h_{i} (t)$ is defined as

h_{i} (t) = {\begin{matrix} 0, v_{i} (t) \leq 0 \\ 1, v_{i} (t) > 0 \end{matrix} i = 1, 2, \dots, n .

(22)

With $ξ_{i} = h_{i} (v_{i} / k_{b_{i}}) + (1 - h_{i}) (v_{i} / k_{a_{i}})$ , equation (21) can be rewritten as

V_{i} = \frac{1}{2 κ} \log \frac{1}{1 - ξ_{i}^{2 κ}} .

(23)

Remark 7. In classical TVABLF, $2 κ \geq n$ is required to ensure the virtual control signals $α_{i}$ and their nth derivative are available. In this paper, availability has been relaxed using the first-order command filter to estimate the virtual control signals. It also reduces the computational burden.

Step 1. Based on $v_{1} = z_{1} - ζ_{1}$ in equation (20), the derivative of $v_{1}$ is

\begin{matrix} {\overset{\cdot}{v}}_{1} = {\overset{\cdot}{x}}_{1} - {\overset{\cdot}{y}}_{d} - {\overset{\cdot}{λ}}_{1} - {\overset{\cdot}{ζ}}_{1} \\ = f_{1} + g_{1} (z_{2} + {\bar{α}}_{2}) + q_{1} λ_{1} + l_{1} sign (λ_{1}) - {\overset{\cdot}{y}}_{d} - {\overset{\cdot}{ζ}}_{1} . \end{matrix}

(24)

Construct the TVABLF as

V_{1} = \frac{h_{1}}{2} \log \frac{k_{b_{1}}^{2}}{k_{b_{1}}^{2} - v_{1}^{2}} + \frac{1 - h_{1}}{2} \log \frac{k_{a_{1}}^{2}}{k_{a_{1}}^{2} - v_{1}^{2} .}

(25)

Then, the derivative of $V_{1}$ is

\begin{matrix} {\overset{\cdot}{V}}_{1} = ϕ_{1} v_{1} ({\overset{\cdot}{v}}_{1} - v_{1} h_{1} \frac{{\overset{\cdot}{k}}_{b_{1}}}{k_{b_{1}}} - v_{1} (1 - h_{1}) \frac{{\overset{\cdot}{k}}_{a_{1}}}{k_{a_{1}}}) \\ = ϕ_{1} v_{1} (f_{1} + g_{1} (z_{2} + {\bar{α}}_{2}) + q_{1} λ_{1} - {\overset{\cdot}{y}}_{d} - {\overset{\cdot}{ζ}}_{1} - v_{1} h_{1} \frac{{\overset{\cdot}{k}}_{b_{1}}}{k_{b_{1}}} - v_{1} (1 - h_{1}) \frac{{\overset{\cdot}{k}}_{a_{1}}}{k_{a_{1}}}) \end{matrix}

(26)

where $ϕ_{1} = h_{1} / (k_{b_{1}}^{2} - v_{1}^{2}) + (1 - h_{1}) / (k_{a_{1}}^{2} - v_{1}^{2})$ .

Construct a virtual control signal $α_{1}$ and a compensating signal $ζ_{1}$ as

α_{1} = \frac{1}{g_{1}} (- (k_{1} + {\bar{k}}_{1} (t)) z_{1} - f_{1} + {\overset{\cdot}{y}}_{d} - q_{1} λ_{1} - \frac{p_{1} + l_{1}}{2} ϕ_{1} v_{1} - s_{1} ϕ_{1}^{η - 1} v_{1}^{2 η - 1})

(27)

{\overset{\cdot}{ζ}}_{1} = - (k_{1} + {\bar{k}}_{1} (t)) ζ_{1} + g_{1} ({\bar{α}}_{2} - α_{1}) + g_{1} ζ_{2} - p_{1} sign (ζ_{1})

(28)

where ${\bar{k}}_{1} (t) = \sqrt{{({\overset{\cdot}{k}}_{b_{1}} / k_{b_{1}})}^{2} + {({\overset{\cdot}{k}}_{a_{1}} / k_{a_{1}})}^{2} + β_{1}}$ is a time-varying gain. $k_{1}$ , $β_{1}$ , $s_{1}$ , and $p_{1}$ are the positive design parameters to be constructed. $0 \leq η = η_{1} / η_{2} < 1$ , where $η_{1}$ and $η_{2}$ are the positive odd integers. By submitting equations (27) and (28) into equation (26), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = ϕ_{1} v_{1} (- k_{1} v_{1} - ({\bar{k}}_{1} (t) + h_{1} \frac{{\overset{\cdot}{k}}_{b_{1}}}{k_{b_{1}}} + (1 - h_{1}) \frac{{\overset{\cdot}{k}}_{a_{1}}}{k_{a_{1}}}) v_{1} - s_{1} ϕ_{1}^{η - 1} v_{1}^{2 η - 1} \\ - \frac{p_{1} + l_{1}}{2} ϕ_{1} v_{1} + p_{1} sign (ζ_{1}) + l_{1} sign (λ_{1}) + g_{1} v_{2}) . \end{matrix}

(29)

Using Young’s inequality, the following inequalities hold

p_{1} ϕ_{1} v_{1} sign (ζ_{1}) \leq \frac{p_{1}}{2} ϕ_{1}^{2} v_{1}^{2} + \frac{p_{1}}{2}

(30)

l_{1} ϕ_{1} v_{1} sign (λ_{1}) \leq \frac{l_{1}}{2} ϕ_{1}^{2} v_{1}^{2} + \frac{l_{1}}{2} .

(31)

It is clear that ${\bar{k}}_{1} + h_{1} ({\overset{\cdot}{k}}_{b_{1}} / k_{b_{1}}) + (1 - h_{1}) ({\overset{\cdot}{k}}_{a_{1}} / k_{a_{1}}) \geq 0$ , so we can obtain

{\overset{\cdot}{V}}_{1} \leq - k_{1} \frac{ξ_{1}^{2}}{1 - ξ_{1}^{2}} - s_{1} {(\frac{ξ_{1}^{2}}{1 - ξ_{1}^{2}})}^{η} + \frac{p_{1}}{2} + \frac{l_{1}}{2} + ϕ_{1} g_{1} v_{1} v_{2} .

(32)

Step i $(i = 2, 3, \dots, n - 2)$ : construct the Lyapunov function as

V_{i} = V_{i - 1} + \frac{h_{i}}{2} \log \frac{k_{b_{i}}^{2}}{k_{b_{i}}^{2} - v_{i}^{2}} + \frac{1 - h_{i}}{2} \log \frac{k_{a_{i}}^{2}}{k_{a_{i}}^{2} - v_{i}^{2}} .

(33)

Similar to equation (26), the derivative of $V_{i}$ is

\begin{array}{l} {\dot{V}}_{i} = {\dot{V}}_{i - 1} + ϕ_{i} v_{i} (f_{i} + g_{i} (z_{i + 1} + {\bar{α}}_{i + 1}) + q_{i} λ_{i} - {\dot{\bar{α}}}_{i} - {\dot{ζ}}_{i} \\ - v_{i} h_{i} \frac{{\dot{k}}_{b_{i}}}{k_{b_{i}}} - v_{i} (1 - h_{i}) \frac{{\dot{k}}_{a_{i}}}{k_{a_{i}}}) \end{array}

(34)

where $ϕ_{i} = h_{i} / (k_{b_{i}}^{2} - v_{i}^{2}) + (1 - h_{i}) / (k_{a_{i}}^{2} - v_{i}^{2})$ .

The virtual control signal $α_{i}$ and compensation signal $ζ_{i}$ are constructed as

\begin{array}{l} α_{i} = \frac{1}{g_{i}} (- (k_{i} + {\bar{k}}_{i} (t)) z_{i} - f_{i} + {\dot{\bar{α}}}_{i} - q_{i} λ_{i} - \frac{p_{i} + l_{i}}{2} ϕ_{1} v_{1} ϕ_{i} v_{i} . \\ - g_{i - 1} ζ_{i - 1} - \frac{ϕ_{i - 1}}{ϕ_{i}} g_{i - 1} v_{i - 1} - s_{i} ϕ_{i}^{η - 1} v_{i}^{2 η - 1}) \end{array}

(35)

\begin{matrix} {\overset{\cdot}{ζ}}_{i} = - (k_{i} + {\bar{k}}_{i} (t)) ζ_{i} + g_{i} ({\bar{α}}_{i + 1} - α_{i}) - g_{i - 1} ζ_{i - 1} \\ + g_{i} ζ_{i + 1} - p_{i} sign (ζ_{i}) \end{matrix}

(36)

where $k_{i}$ , $s_{i}$ , and $p_{i}$ are the positive parameters to design, and $ϕ_{i} = h_{i} / (k_{b_{i}}^{2} - v_{i}^{2}) + (1 - h_{i}) / (k_{a_{i}}^{2} - v_{i}^{2})$ . By substituting equations (35) and (36) into equation (34), we have

\begin{array}{l} {\dot{V}}_{i} = {\dot{V}}_{i - 1} + ϕ_{i} v (_{i} - k_{i} v_{i} - ({\bar{k}}_{i} (t) + h_{i} \frac{{\dot{k}}_{b_{i}}}{k_{b_{i}}} + (1 - h_{i}) \frac{{\dot{k}}_{a_{i}}}{k_{a_{i}}}) v_{i} - \frac{p_{i} + l_{i}}{2} ϕ_{i} v_{i} . \\ - s_{i} ϕ_{i}^{η - 1} v_{i}^{2 η - 1} + p_{i} sign (ζ_{i}) + l_{i} sign (λ_{i}) - \frac{ϕ_{i - 1}}{ϕ_{i}} g_{i - 1} v_{i - 1} + g_{i} v_{i + 1}) \end{array}

(37)

It is clear that ${\bar{k}}_{i} + h_{i} ({\overset{\cdot}{k}}_{b_{i}} / k_{b_{i}}) + (1 - h_{i}) ({\overset{\cdot}{k}}_{a_{i}} / k_{a_{i}}) \geq 0$ , so we obtain

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq {\overset{\cdot}{V}}_{i - 1} - k_{i} \frac{ξ_{i}^{2}}{1 - ξ_{i}^{2}} - s_{i} {(\frac{ξ_{i}^{2}}{1 - ξ_{i}^{2}})}^{η} + \frac{p_{i}}{2} + \frac{l_{i}}{2} \\ - ϕ_{i - 1} g_{i - 1} v_{i - 1} v_{i} + ϕ_{i} g_{i} v_{i} v_{i + 1} \end{matrix}

(38)

where $ϕ_{1} = h_{1} / (k_{b_{1}}^{2} - v_{1}^{2}) + (1 - h_{1}) / (k_{a_{1}}^{2} - v_{1}^{2})$ .

By mathematical induction, it follows that

\begin{matrix} {\overset{\cdot}{V}}_{i - 1} \leq \sum_{j = 1}^{i - 1} [- \frac{k_{j} ξ_{j}^{2}}{1 - ξ_{j}^{2}} - s_{j} {(\frac{ξ_{j}^{2}}{1 - ξ_{j}^{2}})}^{η} + \frac{p_{j}}{2} + \frac{l_{j}}{2}] + ϕ_{i - 1} g_{i - 1} v_{i - 1} v_{i} . \end{matrix}

(39)

Combining equations (38) and (39), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq \sum_{j = 1}^{i} [- \frac{k_{j} ξ_{j}^{2}}{1 - ξ_{j}^{2}} - s_{j} {(\frac{ξ_{j}^{2}}{1 - ξ_{j}^{2}})}^{η} + \frac{p_{j}}{2} + \frac{l_{j}}{2}] + ϕ_{i} g_{i} v_{i} v_{i + 1} . \end{matrix}

(40)

Step n: the derivative of $v_{n}$ is

\begin{array}{l} {\dot{v}}_{n} = {\dot{x}}_{n} - {\dot{\bar{α}}}_{n} - {\dot{λ}}_{n} - {\dot{ζ}}_{n} \\ = f_{n} + g_{n} (ρ_{w_{σ}} w + d) - {\dot{\bar{α}}}_{n} + q_{n} λ_{n} + l_{n} sign (λ_{n}) - {\dot{ζ}}_{n} . \end{array}

(41)

Choose the Lyapunov function as

V_{n} = V_{n - 1} + \frac{h_{n}}{2} \log \frac{k_{b_{n}}^{2}}{k_{b_{n}}^{2} - v_{n}^{2}} + \frac{1 - h_{n}}{2} \log \frac{k_{a_{n}}^{2}}{k_{a_{n}}^{2} - v_{n}^{2}} .

(42)

Taking the derivative of $V_{n}$ yields

\begin{array}{l} {\dot{V}}_{n} = {\dot{V}}_{n - 1} + ϕ_{n} v_{n} (f_{n} + g_{n} (ρ_{w_{σ}} w + d) - {\dot{\bar{α}}}_{n} + q_{n} λ_{n} + l_{n} sign (λ_{n}) \\ - {\dot{ζ}}_{n} - v_{n} h_{n} \frac{{\dot{k}}_{b_{n}}}{k_{b_{n}}} - v_{n} (1 - h_{n}) \frac{{\dot{k}}_{a_{n}}}{k_{a_{n}}}) . \end{array}

(43)

The actual control signal $w$ and compensation signal $ζ_{n}$ are given as follows

\begin{matrix} w = \frac{1}{g_{n}} (- (k_{n} + {\bar{k}}_{n} (t)) z_{n} - f_{n} + {\overset{\cdot}{\bar{α}}}_{n} - q_{n} λ_{n} - g_{n - 1} ζ_{n - 1} \\ - \frac{p_{n} + l_{n} + 1}{2} ϕ_{n} v_{n} - \frac{ϕ_{n - 1}}{ϕ_{n}} g_{n - 1} v_{n - 1} - s_{n} ϕ_{n}^{η - 1} v_{n}^{2 η - 1}) \end{matrix}

(44)

{\overset{\cdot}{ζ}}_{n} = - (k_{n} + {\bar{k}}_{n} (t)) ζ_{n} - g_{n - 1} ζ_{n - 1} - p_{n} sign (ζ_{n}) .

(45)

Substituting equations (45) and (44) into equation (43) yields

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq {\overset{\cdot}{V}}_{n - 1} + ϕ_{n} v_{n} (- k_{n} v_{n} - ({\bar{k}}_{n} (t) + h_{n} \frac{{\overset{\cdot}{k}}_{b_{n}}}{k_{b_{n}}} + (1 - h_{n}) \frac{{\overset{\cdot}{k}}_{a_{n}}}{k_{a_{n}}}) v_{n} \\ - s_{n} ϕ_{n}^{η - 1} v_{n}^{2 η - 1} + g_{n} d - \frac{p_{n} + l_{n} + 1}{2} ϕ_{n} v_{n} + p_{n} sign (ζ_{n}) + l_{n} sign (λ_{n}) \\ - \frac{ϕ_{n - 1}}{ϕ_{n}} g_{n - 1} v_{n - 1}) . \end{matrix}

(46)

Theorem 1. Consider the closed-loop nonlinear system equation (1) under Assumptions 1–3. The controller (equation (44)), the virtual controllers (equations (27) and (35)), error compensation signals (equations (28), (36), and (45)), and input delay compensation mechanism (equation (13)) have the following results:

1. All internal signals in the closed-loop system are semi-globally uniformly ultimately bounded and converge to a small neighborhood of the zero in finite time.

2. The asymmetric time-varying states constraints are never violated.

Proof. Similar to equation (40), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq \sum_{j = 1}^{n} [- \frac{k_{j} ξ_{j}^{2}}{1 - ξ_{j}^{2}} - s_{j} {(\frac{ξ_{j}^{2}}{1 - ξ_{j}^{2}})}^{η} + \frac{p_{j}}{2} + \frac{l_{j}}{2}] + \frac{1}{2} {\bar{g}}^{2} {\bar{d}}^{2} \end{matrix} .

(47)

From Lemma 1, we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq \sum_{j = 1}^{n} [- k_{j} \log \frac{1}{1 - ξ_{j}^{2}} - s_{j} {(\log \frac{1}{1 - ξ_{j}^{2}})}^{η} + \frac{p_{j}}{2} + \frac{l_{j}}{2}] + \frac{1}{2} {\bar{g}}^{2} {\bar{d}}^{2} . \end{matrix}

(48)

Let $V = V_{n}$ , $a_{V} = 2 \min {k_{i}}$ , $b_{V} = 2^{η} \min {s_{i}}$ , and $c_{V} = \sum_{j = 1}^{n} [p_{j} / 2 + l_{j} / 2] + 1 / 2 {\bar{g}}^{2} {\bar{d}}^{2}$ , then equation (48) can be rewritten as

\overset{\cdot}{V} \leq - a_{V} V - b_{V} V^{η} + c_{V} .

(49)

On the basis of Lemma 4, $v_{i}$ will converge to the region

- k_{a_{i}} E_{1} \leq v_{i} \leq k_{b_{i}} E_{1}

(50)

where $E_{1} = \sqrt{1 - e^{- 2 E_{2}}}$ and $E_{2} = min {c_{V} / (1 - θ_{0}) a_{V}, (c_{V} / (1 - θ_{0}) b_{V})^{\frac{1}{η}}}$ . The setting time is

\begin{matrix} T_{r 1} \leq max {\frac{1}{θ_{0} a_{V} (1 - η)} \ln \frac{θ_{0} a_{V} V^{1 - η} (0) + b_{V}}{b_{V}} \\ \frac{1}{a_{V} (1 - η)} \ln \frac{a_{V} V^{1 - η} (0) + θ_{0} b_{V}}{θ_{0} b_{V}}} . \end{matrix}

(51)

According to equation (42), we can conclude that

- k_{a_{i}} < v_{i} < k_{b_{i}} .

(52)

Since $v_{i} = z_{i} - ζ_{i}$ , it can be proved that $ζ_{i}$ is bounded in finite time.

Choose the Lyapunov function of the error compensation signal as follows

V_{ζ} = \frac{1}{2} \sum_{i = 1}^{n} ζ_{i}^{2} .

(53)

The derivative of $V_{ζ}$ is

\begin{matrix} {\overset{\cdot}{V}}_{ζ} = ζ_{1} {\overset{\cdot}{ζ}}_{1} + ζ_{2} {\overset{\cdot}{ζ}}_{2} + \dots + ζ_{n} {\overset{\cdot}{ζ}}_{n} \\ = - \sum_{i = 1}^{n} (k_{i} + {\bar{k}}_{i} (t)) ζ_{i}^{2} - \sum_{i = 1}^{n} p_{i} ζ_{i} sign (ζ_{i}) + \sum_{i = 1}^{n - 1} g_{i} ζ_{i} ({\bar{α}}_{i + 1} - α_{i}) \\ \leq - \sum_{i = 1}^{n} k_{i} ζ_{i}^{2} - \sum_{i = 1}^{n} p_{i} | ζ_{i} | + \sum_{i = 1}^{n - 1} g_{i} ζ_{i} ({\bar{α}}_{i + 1} - α_{i}) . \end{matrix}

(54)

From Lemmas 5 and 6, $| {\bar{α}}_{i + 1} - α_{i} | \leq ψ_{i}$ can be achieved in finite-time $T_{i 2}$ . According to $\underline{g} \leq | g_{i} | \leq \bar{g}$ , for $t \geq max {T_{rc}}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{ζ} \leq - \sum_{i = 1}^{n} k_{i} ζ_{i}^{2} - \sum_{i = 1}^{n} p_{i} | ζ_{i} | + \sum_{i = 1}^{n - 1} \bar{g} ψ_{i} | ζ_{i} | + \bar{g} ψ_{n} | ζ_{n} | \\ \leq - a_{ζ} V_{ζ} - p_{0} V_{ζ}^{\frac{1}{2}} + \sqrt{2 n} \bar{g} \bar{ψ} V_{ζ}^{\frac{1}{2}} \\ \leq - a_{ζ} V_{ζ} - b_{ζ} V_{ζ}^{\frac{1}{2}} \end{matrix}

(55)

where $p_{0} = \sqrt{2} min {p_{i}}$ , $\bar{ψ} = max {ψ_{i}}$ , $a_{ζ} = 2 \min {k_{i}}$ , and $b_{ζ} = p_{0} - \sqrt{2 n} \bar{g} \bar{ψ}$ . Choosing suitable $k_{i}$ and $p_{i}$ to make $a_{ζ} > 0$ and $b_{ζ}$ , $ζ_{i}$ will achieve finite-time stability based on Lemma 3. Then, we obtain $ζ_{1} = 0$ when $T_{r ζ} > max {T_{rc}} + 2 / a_{ζ} \ln a_{ζ} \sqrt{V (0)} + b_{ζ} / b_{ζ}$ . From equations (20) and (50), we have $- k_{a_{1}} E_{1} \leq e_{1} - ζ_{1} - λ_{1} \leq k_{b_{1}} E_{1}$ . For $t \geq max {T_{r 1}, T_{λ}, T_{r ζ}}$ , the tracking error $e_{1}$ will satisfy

- k_{a_{1}} E_{1} - E_{λ} \leq e_{1} \leq k_{b_{1}} E_{1} + E_{λ}

(56)

From equations (18), (20), and (52), $e_{i}$ and $z_{i}$ are bounded and $- k_{a_{1}} + λ_{1} + ξ_{1} + y_{d} \leq x \leq k_{b_{1}} + λ_{1} + ξ_{1} + y_{d}$ . Once ${\underline{k}}_{c_{1}} = λ_{1} + ξ_{1} + y_{d} - k_{a_{1}}$ and ${\bar{k}}_{c_{1}} = k_{b_{1}} + λ_{1} + ξ_{1} + y_{d}$ , we can obtain ${\underline{k}}_{c_{1}} \leq x_{1} \leq {\bar{k}}_{c_{1}}$ . Since $α_{1}$ is a function of $z_{1}$ , $v_{1}$ , ${\overset{\cdot}{y}}_{d}$ , ${\overset{\cdot}{k}}_{b_{1}}$ , $k_{b_{1}}$ , ${\overset{\cdot}{k}}_{a_{1}}$ , $k_{a_{1}}$ , and $λ_{1}$ , it can be concluded that $α_{1}$ is bounded, so a constant ${\bar{A}}_{2}$ satisfies $| {\bar{α}}_{2} | \leq {\bar{A}}_{2}$ . From $z_{2} = x_{2} - {\bar{α}}_{2} - λ_{2} - ζ_{2}$ , we can infer that ${\underline{k}}_{c_{2}} \leq x_{2} \leq {\bar{k}}_{c_{2}}$ with ${\underline{k}}_{c_{2}} = {\bar{A}}_{2} + λ_{2} + ζ_{2} - k_{a_{2}}$ and ${\bar{k}}_{c_{2}} = k_{b_{2}} + λ_{2} + ζ_{2} + {\bar{A}}_{2}$ . By mathematical induction, the boundedness of $α_{i}$ and ${\bar{α}}_{i + 1}$ can be obtained, and the boundedness of state $x_{i}$ can be proved that ${\underline{k}}_{c_{i}} \leq x_{i} \leq {\bar{k}}_{c_{i}}$ with ${\underline{k}}_{c_{i}} = {\bar{A}}_{i} + λ_{i} + ζ_{i} - k_{a_{i}}$ and ${\bar{k}}_{c_{i}} = k_{b_{i}} + λ_{i} + ζ_{i} + {\bar{A}}_{i}$ , where $| {\bar{α}}_{i} | \leq {\bar{A}}_{i}$ . Thus, all signals in the closed-loop system are bounded, the system output tracks the desired tracking signal pre-eminently, and the asymmetric time-varying system states constraints are not violated.

Construct the Lyapunov function candidate

\begin{matrix} V_{all} = V_{n} + V_{λ} + V_{ζ} . \end{matrix}

(57)

Taking the derivative of $V_{all}$ yields

\begin{matrix} {\overset{\cdot}{V}}_{all} = {\overset{\cdot}{V}}_{n} + {\overset{\cdot}{V}}_{λ} + {\overset{\cdot}{V}}_{ζ} \\ \leq - a_{V} V_{n} - b_{V} V_{n}^{η} + c_{V} - a_{λ} V_{λ} - b_{λ} V_{λ}^{\frac{1}{2}} + c_{λ} - a_{ζ} V_{ζ} - b_{ζ} V_{ζ}^{\frac{1}{2}} \end{matrix}

(58)

Let $a = max {a_{V}, a_{λ}, a_{ζ}}$ , $b = max {b_{V}, b_{λ}, b_{ζ}}$ , $ω = max {η, 1 / 2}$ , and $c = c_{V} + c_{λ}$ , then equation (58) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{all} & \leq - a V_{all} - {bV}_{all}^{ω} + c . \end{matrix}

(59)

Thus, all internal signals in the closed-loop system are semi-globally uniformly ultimately bounded and converge to a small neighborhood of the zero in finite time.

Remark 8. From equation (50), we can obtain some direction on how to pick the control parameters. We need to decrease $c_{V} / a_{V}$ and $c_{V} / b_{V}$ to reduce the radiuses of the region of tracking errors. Thus, the tracking errors can be made arbitrarily small. Moreover, from equation (55) and Lemma 7, it can be seen that the convergence rate of the error compensating system and the input delay compensating system can be guaranteed by larger $p_{i}$ and $l_{i}$ , respectively. However, large $p_{i}$ and $l_{i}$ will result in a larger $c_{V} / a_{V}$ . Therefore, more simulations are required for choosing the control parameters and balancing the control performance.

Remark 9. To the best of the authors’ knowledge, there is a drawback and a limitation of the proposed method in the paper. Higher system order requires more control parameters to be adjusted. Moreover, the proposed method requires that all system states should be measurable, which will be further improved in future work.

Simulation results

In this section, two simulation results are provided to show the effectiveness of the proposed approach. The proposed method in this paper will be compared with the method in Zhou et al. (2022). It should be noted that the method in Zhou et al. (2022) cannot handle the system with time-varying asymmetric state constraints.

Example 1. Denote $ψ$ and $r$ are the yaw angle and yaw angular rate of the airship, respectively. The simplified model of the stratospheric airship with input delay is given as follows (Miao et al., 2016)

\begin{matrix} \overset{\cdot}{ψ} (t) = r (t) \\ J \overset{\cdot}{r} (t) + Dr (t) = M (w (t - τ)) \end{matrix}

(60)

where $J$ denotes the airship inertia of z-axis moment, $D$ is the damping associated with the yaw rate, and $M$ represents the yaw moment control. The parameters’ values refer to Wang et al. (2010, 2018).

Denote $x_{1} = ψ$ , $x_{2} = r$ , and $M = u$ , then equation (60) can be rewritten as

\begin{matrix} {\overset{\cdot}{x}}_{1} (t) = x_{2} \\ {\overset{\cdot}{x}}_{2} (t) = f_{2} (x_{2}) + g_{2} u (w (t - τ)) \\ y = x_{1}, \end{matrix}

(61)

where $f_{2} (x_{2}) = (- D / J) x_{2}$ and $g_{2} = 1 / J$ . The desired tracking signal is given as $y_{d} = π (\sin (t^{2 / 3}) + \sin (t)) / 3$ . The system state variables are $x_{1}$ and $x_{2}$ limited in ${\underline{k}}_{c_{1}} (t) \leq x_{1} (t) \leq {\bar{k}}_{c_{1}} (t)$ and ${\underline{k}}_{c_{2}} (t) \leq x_{2} (t) \leq {\bar{k}}_{c_{2}} (t)$ , where ${\underline{k}}_{c_{1}} (t) = π (\sin (t^{2 / 3}) + \sin (t)) / 3 - π / 2$ , ${\bar{k}}_{c_{1}} (t) = π (\sin (t^{2 / 3}) + \sin (t)) / 3 + π / 2$ , ${\underline{k}}_{c_{2}} (t) = 0.4 (\sin (t^{2 / 3}) + \sin (t)) - 2$ and ${\bar{k}}_{c_{2}} (t) = 2 + 0.4 (\sin (t^{2 / 3}) + \sin (t))$ . The input delay is chosen as $τ = 0.01$ . The bound of the input saturation is set as $u_{M} = 10, 000$ .

For simulation, all design parameters are taken to be $k_{1} = 0.5$ , $k_{2} = 1$ , $β_{i} = 2$ , $p_{i} = 5$ , $q_{i} = 1$ , $l_{i} = 1$ , $s_{i} = 1$ , and $η = 7 / 9$ . $[x_{1} (0), x_{2} (0)]^{T} = [π / 5, 0]^{T}$ are chosen as the initial values of system states.

Note that Figure 1 shows the tracking trajectories under this method. It can be clearly seen that the method given in this paper can make the system output $y$ follow the desired tracking signal $y_{d}$ in finite time and satisfy the constraint. Figure 2 shows the tracking trajectories with the same parameters under the method given in Zhou et al. (2022). The tracking errors based on the two different methods are given in Figure 3. Compared with two methods, it can be seen that the method in this paper has faster convergence rate. Figure 4 shows the trajectories of the state $x_{2}$ and its constraint interval. The trajectories of the system input and controller signal are shown in Figure 5.

Example 2. Consider the following fourth-order nonlinear systems with input delay

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - 35.55 \sin x_{1} - 0.282 x_{2} + x_{3} \\ {\overset{\cdot}{x}}_{3} = - 0.93 x_{1} x_{2} - 5.19 x_{3} + x_{4} \\ {\overset{\cdot}{x}}_{4} = \sin x_{3} - x_{4} + u (w (t - τ)) \\ y = x_{1} . \end{matrix}

(62)

The initial conditions are chosen as $[3, 0, 0, 0]$ . The desired tracking signal is given as $y_{d} = 2 \sin (t) + 1.5 \cos (t)$ . The system states are constrained as ${\underline{k}}_{c_{i}} < x_{i} < {\bar{k}}_{c_{i}}, i = 1, \dots, 4$ with ${\underline{k}}_{c_{1}} = {\underline{k}}_{c_{2}} = \cos (2 t) - 5$ , ${\bar{k}}_{c_{1}} = {\bar{k}}_{c_{2}} = 2 \cos (2 t) + 5$ , ${\underline{k}}_{c_{3}} = 10 \cos (2 t) - 60$ , ${\bar{k}}_{c_{3}} = 10 \cos (2 t) + 60$ , ${\underline{k}}_{c_{4}} = 100 \cos (2 t) - 600$ , and ${\bar{k}}_{c_{4}} = 100 \sin (2 t) + 600$ . The input delay is chosen as $τ = 0.01$ . The bound of the input saturation is $u_{M} = 10, 000$ .

For simulation, the control parameters are taken to be $k_{1} = 2$ , $k_{2} = 4$ , $k_{3} = 5$ , $k_{4} = 10$ , $β_{1} = 3$ , $β_{2} = 3$ , $β_{3} = 3$ , $β_{4} = 5$ , $s_{1} = 2$ , $s_{2} = 2$ , $s_{3} = 5$ , $s_{4} = 3$ , and $η = 13 / 15$ for finite-time command-filtered controller. The other parameters are designed as $p_{i} = 1$ , $q_{i} = 1$ , and $l_{i} = 1$ .

The simulation results of this example are given in Figures 6 –12. The tracking trajectories under the method given in this paper are shown in Figure 6. It can be seen that the proposed method can make the system output $y$ follow the desired tracking signal $y_{d}$ in finite time, and the constraint is not violated. The tracking trajectories with the same parameters under the method given in Zhou et al. (2022) are shown in Figure 7, and the tracking errors based on the two different methods are given in Figure 8. It could be observed that the proposed method has a faster convergence time than the method in Zhou et al. (2022). Figures 9 –11 show the curves of the constrained system states $x_{2}$ , $x_{3}$ , and $x_{4}$ , respectively. It is obvious that the state constraints are not violated. The trajectories of the system input and controller signal are shown in Figure 12.

Remark 10. From Figures 3 and 8, we can conclude that the proposed method has a faster convergence rate. However, from Figures 5 and 12, the control signals are noisier, which can degrade tracking performances when the system has smaller input saturation.

Conclusion

This study has developed a finite-time command-filtered backstepping control for a class of strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. The TVABLF has been introduced to handle the issue of time-varying asymmetric full-state constraint. The command-filtered backstepping method is utilized to avoid “the explosion of complexity.” A new finite-time delay compensation mechanism is introduced to remove the impact of input delay. The proposed control method will be extended to nonlinear mapping and multi-input-multi-output nonlinear systems in the future.

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.

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, in part, by the National Natural Science Foundation of China (nos 51906141 and 62073216).

ORCID iD

Qiming Lin

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