Robust adaptive finite-time tracking control of uncertain mechanical systems with input saturation and deadzone

Abstract

This paper addresses the problem of robust adaptive finite-time tracking control for a class of mechanical systems in the presence of model uncertainties, unknown external disturbances, and input nonlinearities containing saturation and deadzone. Without imposing any conditions on the model uncertainties, radial basis function neural networks are used to approximate unknown nonlinear continuous functions, and an adaptive tracking control scheme is proposed by exploiting the recursive design method. It is shown that the input saturation and deadzone model can be expressed as a simple linear system with a time-varying gain and bounded disturbance. An adaptive compensation term for the upper bound of the lumped disturbance is introduced. The semi-global finite-time uniform ultimate boundedness of the corresponding closed-loop tracking error system is proved with the help of the finite-time Lyapunov stability theory. Finally, an example is given to demonstrate the effectiveness of the proposed method.

Keywords

Finite-time control adaptive control mechanical systems input saturation and deadzone neural networks

Introduction

It is well known that a large number of mechanical systems can be modelled by Euler–Lagrange equations, for example robotic manipulators (Yu and Fei, 2014), biped robots (Ge et al., 2012), exoskeleton robots (Lu et al., 2014), underwater vehicles (Cui et al., 2010), high-speed trains (Liu et al., 2009), and so on. In recent years, adaptive control and robust adaptive control for systems with uncertainties or disturbances has received much attention (Chen et al., 2010; Rafique et al., 2015; Siddique and Rehan, 2016). In the community of adaptive control for mechanical systems, fuzzy logic systems or neural networks (NNs) are always used to approximate uncertain continuous functions (see Liu and Zhang, 2013, 2014) due to their inherent approximation ability (Zhao et al., 2014).

In the past two decades, finite-time control for nonlinear systems has attracted more and more attention. It has been proved that finite-time-stable systems have fast convergence and good performance when it comes to robustness and disturbance rejection (Bhat and Bernstein, 1997, 2000). In recent years, finite-time control for mechanical systems has been studied. In Hong et al. (2002), the finite-time control of robot systems was studied by using state feedback and dynamic output feedback control. The global finite-time tracking control of robot manipulators was addressed in Su (2009). In Galicki (2015), a continuous finite-time controller for robotic manipulators was proposed by using the non-singular terminal sliding mode control method. Recently, the stabilization of simple mechanical systems in finite time with continuous state feedback was considered by Sanyal and Bohn (2015). It should be pointed out that all the aforementioned works concerned with finite-time control for mechanical systems are based on the assumption that the dynamic model is exactly known. In order to relax such an assumption, some of the latest research has taken the uncertainties into consideration when investigating finite-time control for mechanical systems. Based on the approximation capability of NNs, Liu and Zhang considered the adaptive finite-time control of uncertain robotic manipulators (Liu and Zhang, 2013, 2014). In Hu et al. (2015), the finite-time tracking control of uncertain Euler–Lagrange systems was studied by constructing a non-singular terminal sliding mode controller.

It is interesting to observe that all the aforementioned finite-time control results for mechanical systems with or without uncertainties are obtained under the presupposition that the actuators are able to provide any requested torque. However, due to physical constraints, the actuators of mechanical systems usually have saturation and deadzone properties, for example underactuated cranes with actuator saturation (Sun et al., 2013, 2017a), and robot manipulators with actuator deadzones (Ma et al., 2015; Park and Han, 2011). These undesirable properties are unavoidable and neglecting their effects may lead to bad performance such as large steady-state error, poor transient response, and large overshoot. Many researchers have developed a large quantity of adaptive control methods for nonlinear systems with saturation and deadzones (Hussain and Rehan, 2016; Rehan et al., 2016). In Tao and Kokotović (1994) and Tong and Li (2012), by constructing the inverse of the deadzone and using an adaptive control method to estimate unknown parameters in the inverse model, the problem of the unknown deadzone was handled. Another method without constructing the inverse of the deadzone was employed in Wang et al. (2004) and Zhang and Ge (2009), in which the input deadzone model was transformed into a linear system with an unknown gain and bounded disturbance. Under some appropriate assumptions, a robust adaptive control strategy for a class of multi input multi output (MIMO) nonlinear systems with both saturation and deadzones was proposed in Chen et al. (2010), where the input nonlinearity was denoted by a linear system with unknown gain whose bound was known. Some research has been done regarding mechanical systems with input saturation or deadzones. In Santibañez et al. (2005), the set-point control of robot manipulators with friction and input saturation was addressed. By using recurrent fuzzy NN, tracking control for robot manipulators with asymmetric deadzones and dynamic friction was dealt with in Park and Han (2011). In Chen et al. (2017), fuzzy control of uncertain MIMO mechanical systems with both deadzones and saturation input nonlinearities was considered.

More recently, finite-time control for mechanical systems with input deadzones or saturation was investigated. In Su and Swevers (2014), a finite-time tracking control scheme was proposed for robot manipulators in the presence of actuator saturation by replacing the linear error in the proportional–derivative (PD) plus dynamics compensation scheme with saturated non-smooth but continuous exponential-like ones. A more recent finite-time continuous stabilization scheme for mechanical systems was proposed in Zavala-Río and Zamora-Gómez (2017), based on the theoretical framework of local homogeneity. Zavala-Río and Zamora-Gómez proposed a bounded continuous control design scheme characterized by a saturating-PD-type term for constrained-input mechanical systems, guaranteeing global stabilization with either finite-time or (local) exponential convergence. Nevertheless, the model uncertainties were not considered in the above two bounded control approaches. In Ma et al. (2015), adaptive finite-time tracking control for uncertain robotic manipulators with unknown deadzones was considered, and it has been proved by using finite-time Lyapunov stability theory that the tracking error system is finite-time stable when the parameters of the controller are suitably chosen.

After reviewing the existing results, there are some points needing further improvement.

In Hu et al. (2015), the uncertainties are formulated as an unmodelled dynamic term which, together with the external disturbance and other terms, is defined as the lumped uncertainty. The lumped uncertainty is required to be differentiable, and the bound of its time derivative is to be known, which is really restricted. In Liu and Zhang (2014), only the external disturbance is unknown. Besides, in Liu and Zhang (2013, 2014) and Ma et al. (2015), a control parameter that needs to be larger than an unknown term is introduced, and it is obtained by a trial and error method.

In Su and Swevers (2014), Ma et al. (2015), and Zavala-Río and Zamora-Gómez (2017), only input saturation or deadzones were considered. When the considered mechanical systems possess both input saturation and deadzone constraints, how to design the controller needs to be further investigated.

Based on the above discussion, finite-time tracking control for uncertain mechanical systems with both saturation and deadzone input nonlinearities is still an open area and needs to be further studied, which motivates our work in this paper.

In this paper, we will investigate the problem of finite-time control for uncertain mechanical systems in the presence of input saturation and deadzones. Due to the existence of uncertainties, radial basis function neural networks (RBFNNs) are used to approximate unknown continuous functions. A time-varying state feedback controller and adaptive laws are developed based on the recursive design method. In addition, stability analysis is provided to show that the proposed control scheme can achieve semi-globally finite-time uniformly ultimately bounded (SGFTUUB) stabilization. The main contributions of this paper are twofold.

Compared with Liu and Zhang (2013, 2014) and Hu et al. (2015), more general uncertainties are considered. Besides, we introduce a time-varying function to eliminate the unknown term appearing in the control design procedure instead of using a trial and error method.

The considered mechanical systems possess both input saturation and deadzone constraints. Under some assumptions, it is shown that the input saturation and deadzone model can be expressed by a time-varying linear system with an unknown gain and bounded disturbance. An adaptive compensation term for the upper bound of the disturbance is introduced.

The remainder of this paper is organized as follows. In ‘Problem formulation and preliminary results’, the problem formulation and some preliminary results are given. The robust adaptive finite-time control scheme is derived in ‘Main results’. ‘Numerical example’ provides an example to illustrate the proposed results, and concluding remarks are given in the final section.

Notation

Throughout this paper, the superscript ‘T’ stands for matrix transposition; $A^{- 1}$ indicates the inverse of any square matrix $A$ ; $R^{n}$ denotes the $n$ -dimensional real space; $R^{n \times n}$ represents the space of $n \times n$ matrices with real entries; $R_{odd}^{+} ≜ {q \in R : q > 0 and q is a ratio of odd integers}$ ; $C^{i}$ denotes the set of all functions with continuous $i$ th partial derivatives; $| a |$ denotes the absolute value of scalar $a$ ; $‖ x ‖$ ( $‖ A ‖$ ) denotes the Euclidean norm of a vector $x$ (matrix $A$ ); $sign (\cdot)$ denotes the standard signum function; and $diag {\cdot}$ stands for a block-diagonal matrix.

Problem formulation and preliminary results

Consider the uncertain mechanical systems with non-symmetric input saturation and deadzone described by

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + D (t, q, \overset{\cdot}{q}) = τ

(1)

where $q$ , $\overset{\cdot}{q}$ and $\overset{\cdot\cdot}{q} \in R^{n}$ denote the vectors of position, velocity and acceleration, respectively. $M (q) \in R^{n \times n}$ is a bounded symmetric positive-definite inertia matrix, $C (q, \overset{\cdot}{q}) \in R^{n \times n}$ is the centripetal and Coriolis matrix, $G (q) \in R^{n}$ is the gravity term, and $D (t, q, \overset{\cdot}{q}) \in R^{n}$ denotes the continuous and bounded external disturbance. Due to the existence of uncertainties, the matrices $M (q)$ and $C (q, \overset{\cdot}{q})$ and the vectors $G (q)$ and $D (t, q, \overset{\cdot}{q})$ are unknown. Further, $τ = {[τ_{1}, τ_{2}, \dots, τ_{n}]}^{T} \in R^{n}$ is the actual control input, which is the output of input nonlinearity and can be expressed as

τ_{i} = Π_{i} (τ_{c i}) = {\begin{matrix} τ_{l i m}, & if & τ_{c i} \leq h_{l i 1} \\ π_{l i} (τ_{c i}), & if & h_{l i 1} < τ_{c i} \leq h_{l i 0} \\ 0, & if & h_{l i 0} < τ_{c i} \leq h_{r i 0} \\ π_{r i} (τ_{c i}), & if & h_{r i 0} < τ_{c i} \leq h_{r i 1} \\ τ_{r i m}, & if & τ_{c i} > h_{r i 1} \end{matrix} i \in N = {1, 2, \dots, n}

(2)

where $τ_{ci} \in R$ denotes the $i$ th input of the input nonlinearity (the control signal to be designed), $π_{li} (τ_{ci})$ and $π_{ri} (τ_{ci})$ are unknown nonlinear smooth functions, $τ_{\lim} < 0$ and $τ_{rim} > 0$ are the unknown saturation values, and $h_{li 1} < h_{li 0} < 0$ , $h_{ri 1} > h_{ri 0} > 0$ are unknown constants and denote the input nonlinearity parameters. Figure 1 shows the input nonlinearity with non-symmetrical saturation and deadzone.

Remark 1. One example of a physical system that can be described by system (1) is robotic manipulators. As we know, for robotic manipulators the parameters in model (1) are usually the functions of some physical parameters, such as link mass, link length and inertial moment. In practice, it is very difficult to obtain the precise values of these parameters because of the existence of measurement errors, payload variation and external disturbances. Therefore, it is of practical significance to investigate the control problem for uncertain mechanical systems with $M (q)$ , $C (q, \overset{\cdot}{q})$ and $G (q)$ being unknown.

Throughout the paper, the following assumptions are imposed on system (1) and input nonlinearity (2), respectively.

Assumption 1. (Wen et al., 2011). System (1) is input-to-state stable.

Assumption 2. There exist unknown constants $π_{li 0} > 0$ , $π_{li 1} > 0$ , $π_{ri 0} > 0$ and $π_{ri 1} > 0$ , $i \in N$ , such that

\begin{matrix} 0 < π_{li 0} \leq π'_{li} (τ_{ci}) \leq π_{li 1}, \forall τ_{ci} \in [h_{li 1}, h_{li 0}] \\ 0 < π_{ri 0} \leq π'_{ri} (τ_{ci}) \leq π_{ri 1}, \forall τ_{ci} \in [h_{ri 0}, h_{ri 1}] \end{matrix}

where $π_{l i}^{'} (τ_{c i}) = {(d π_{l i} (z) / d z) |}_{z = τ_{c i}}$ and $π_{r i}^{'} (τ_{c i}) = {(d π_{r i} (z) / d z) |}_{z = τ_{c i}}$ .

Remark 2. It has been pointed out in Wen et al. (2011) that Assumption 1 is reasonable. It can be shown that an unstable plant cannot be globally stabilized in the presence of input saturation by using the following example given in Wen et al. (2011). Consider a simple system $\overset{\cdot}{x} = ax + u (v)$ where $x \in R$ is a state variable and $u (v) \in R$ denotes the saturated input satisfying $| u (v) | \leq u_{m}$ . When $a > 0$ and the initial condition $x (0) > u_{m} / a$ , there does not exist any control that satisfies the saturation constraint to stabilize the system.

According to the differential mean value theorem, there exist $υ_{li} (τ_{ci}) \in (h_{li 1}, h_{li 0})$ and $υ_{ri} (τ_{ci}) \in (h_{ri 0}, h_{ri 1})$ such that

\begin{matrix} π_{li} (τ_{ci}) = π_{li} (τ_{ci}) - π_{li} (h_{li 0}) = π'_{li} (υ_{li} (τ_{ci})) (τ_{ci} - h_{li 0}), \\ \forall τ_{ci} \in [h_{li 1}, h_{li 0}] \\ π_{ri} (τ_{ci}) = π_{ri} (τ_{ci}) - π_{ri} (h_{ri 0}) = π'_{ri} (υ_{ri} (τ_{ci})) (τ_{ci} - h_{ri 0}), \\ \forall τ_{ci} \in [h_{ri 0}, h_{ri 1}] \end{matrix}

(3)

Combining (2) and (3), the control input $τ_{i}$ can be represented as follows:

τ_{i} = φ_{i} (τ_{ci}) τ_{ci} + β_{i} (τ_{ci}), i \in N

(4)

where

φ_{i} (τ_{ci}) = {\begin{matrix} τ_{\lim} / τ_{ci}, & if τ_{ci} \leq h_{li 1} \\ π'_{li} (υ_{li} (τ_{ci})), & if h_{li 1} < τ_{ci} \leq 0 \\ π'_{ri} (υ_{ri} (τ_{ci})), & if 0 < τ_{ci} \leq h_{ri 1} \\ τ_{rim} / τ_{ci}, & if τ_{ci} > h_{ri 1} \end{matrix}

(5)

and

β_{i} (τ_{ci}) = {\begin{matrix} 0, & if τ_{ci} \leq h_{li 1} \\ - π'_{li} (υ_{li} (τ_{ci})) h_{li 0}, & if h_{li 1} < τ_{ci} \leq h_{li 0} \\ - π'_{li} (υ_{li} (τ_{ci})) τ_{ci}, & if h_{li 0} < τ_{ci} \leq 0 \\ - π'_{ri} (υ_{ri} (τ_{ci})) τ_{ci}, & if 0 < τ_{ci} \leq h_{ri 0} \\ - π'_{ri} (υ_{ri} (τ_{ci})) h_{ri 0}, & if h_{ri 0} < τ_{ci} \leq h_{ri 1} \\ 0, & if τ_{ci} > h_{ri 1} \end{matrix}

(6)

From (2), (5) and (6), we know that $φ_{i} (τ_{ci})$ and $β_{i} (τ_{ci})$ are unknown functions. According to Assumption 1 and (5), it is easy to find that there exist constants ${\underline{τ}}_{li}$ and ${\underline{τ}}_{ri}$ such that the inequalities $τ_{\lim} / h_{li 1} \geq τ_{\lim} / τ_{ci} > {\underline{τ}}_{li} > 0$ (when $τ_{ci} \leq h_{li 1}$ ) and $τ_{rim} / h_{ri 1} \geq τ_{rim} / τ_{ci} > {\underline{τ}}_{ri} > 0$ (when $τ_{ci} \geq h_{ri 1}$ ) are satisfied, $i \in N$ . Then, combining with Assumption 2, it is apparent that $φ_{i} (τ_{ci})$ and $β_{i} (τ_{ci})$ satisfy $0 < φ_{i 0} \leq φ_{i} (τ_{ci}) \leq φ_{i 1}$ and $| β_{i} (τ_{ci}) | < β_{im}$ with $φ_{i 0} = min {{\underline{τ}}_{li}, {\underline{τ}}_{ri}, π_{li 0}, π_{ri 0}}$ , $φ_{i 1} = max {π_{li 1}, π_{ri 1}}$ , and $β_{im} = max {π_{li 1}, π_{ri 1}} \cdot max {- h_{li 0}, h_{ri 0}}$ , that is, $φ_{i} (τ_{ci})$ and $β_{i} (τ_{ci})$ are bounded functions, $i \in N$ .

Let $M_{d} (q) = diag {M_{d 1} (q), M_{d 2} (q), \dots, M_{dn} (q)} \in R^{n \times n}$ be a known bounded diagonal matrix and $M_{di} (q) \neq 0$ , $i \in N$ . Then, the unknown inertia matrix $M (q)$ can be divided into two parts, that is,

M (q) = M_{d} (q) + M_{Δ} (q)

(7)

where $M_{Δ} (q) \in R^{n \times n}$ is an unknown matrix. It is obvious that $M_{Δ} (q)$ is bounded because $M (q)$ and $M_{d} (q)$ are bounded, and $‖ M_{Δ} (q) ‖ = ‖ M (q) - M_{d} (q) ‖ \leq ‖ M (q) ‖ + ‖ M_{d} (q) ‖$ .

By letting $v = \overset{\cdot}{q} = {[v_{1}, v_{2}, \dots, v_{n}]}^{T} \in R^{n}$ and

φ (τ_{c}) = diag {φ_{1} (τ_{c 1}), φ_{2} (τ_{c 2}), \dots, φ_{n} (τ_{cn})} \in R^{n \times n}

β (τ_{c}) = {[β_{1} (τ_{c 1}), β_{2} (τ_{c 2}), \dots, β_{n} (τ_{cn})]}^{T}

and noting the fact that $M_{d}^{- 1} (q) φ (τ_{c}) = φ (τ_{c}) M_{d}^{- 1} (q)$ because both $M_{d} (q)$ and $φ (τ_{c})$ are diagonal matrices, we can transform system (1) with input nonlinearity (4) into the following form by combining with (7):

\overset{\cdot}{q} = v, \overset{\cdot}{v} = φ (τ_{c}) u_{c} + ϕ (q, v) + ψ (t, q, v, τ, τ_{c})

(8)

where $u_{c} = M_{d}^{- 1} (q) τ_{c}$ , $ϕ (q, v) = - M^{- 1} (q) (C (q, v) v + G (q))$ , and

\begin{array}{l} ψ (t, q, v, τ, τ_{c}) = M_{d}^{- 1} (q) β (τ_{c}) - M_{d}^{- 1} (q) M_{Δ} (q) M^{- 1} (q) τ \\ - M^{- 1} (q) D (t, q, v) \end{array}

It is obvious that $ϕ (q, v)$ and $ψ (t, q, v, τ, τ_{c})$ are unknown nonlinear function vectors because they contain unknown terms $M_{Δ} (q)$ , $M (q)$ , $C (q, v)$ , $G (q)$ and $β (τ_{c})$ . It should be pointed out that $M_{d}^{- 1} (q) = diag {M_{d 1}^{- 1} (q), M_{d 2}^{- 1} (q), \dots, M_{dn}^{- 1} (q)}$ and $M^{- 1} (q)$ are bounded because $M_{di} (q) \neq 0$ and $M (q)$ is bounded and positive-definite. Besides, the external disturbance $D (t, q, v)$ is bounded, and $τ$ is obviously bounded due to the existence of input saturation. Therefore, we have that $ψ (t, q, v, τ, τ_{c})$ is also bounded.

Let $q_{d} \in R^{n}$ and $v_{d} \in R^{n}$ be the desired position and velocity signals, which satisfy

{\overset{\cdot}{q}}_{d} = v_{d}, {\overset{\cdot}{v}}_{d} = f (t, q_{d}, v_{d})

(9)

where $f (t, q_{d}, v_{d}) \in R^{n}$ is a continuous function vector. Define $e_{1} = q - q_{d}$ and $e_{2} = v - v_{d}$ as the tracking errors of position and velocity, and it follows from (8) and (9) that the dynamics of the error system is

{\overset{\cdot}{e}}_{1} = e_{2}, {\overset{\cdot}{e}}_{2} = φ (τ_{c}) u_{c} + ϕ (q, v) + ψ (t, q, v, τ, τ_{c}) - f (t, q_{d}, v_{d})

(10)

Before introducing the purpose of this paper, and referring to Definition 1 in Ni et al. (2016), we give the following definition.

Definition 1. Consider the following nonlinear system:

\overset{\cdot}{x} = f (x), x (0) = x_{0} \in U_{0}

(11)

where $x \in R^{n}$ is the state vector, $x_{0}$ is the initial condition, and $U_{0} \subset R^{n}$ is a compact set. The state of system (11) is said to be SGFTUUB if there exist $ε > 0$ and $T (ε, x_{0}) < \infty$ such that $‖ x ‖ < ε$ holds for all $t \geq T$ .

Lemma 1. (Zhu et al., 2015) Consider nonlinear system $\overset{\cdot}{x} = f (x)$ , $x (0) = x_{0} \in U_{0} \subset R^{n}$ . Suppose that there exist continuous differentiable function $V (x)$ and scalars $λ > 0$ , $0 < α < 1$ and $0 < η < \infty$ such that

\overset{\cdot}{V} (x) \leq - λ V^{α} (x) + η

Then the trajectory of system $\overset{\cdot}{x} = f (x)$ is SGFTUUB.

Remark 3. It should be pointed out that the state of the system is also SGFTUUB when $α = 1$ . If $η = 0$ and $0 < α < 1$ , the system is finite-time stable. If $η = 0$ and $α = 1$ , the system is asymptotically stable. The system with $0 < α < 1$ possesses better disturbance rejection and robustness properties. Therefore, in this paper we only consider the case where $0 < α < 1$ .

The control objective of this paper is to design a robust adaptive control law for system (1) in the presence of input nonlinearity and uncertainties to make the position and velocity of the system track the desired position and velocity given by system (9) in finite time with satisfactory accuracy, that is, to make the state of tracking error system (10) be SGFTUUB.

Due to the existence of uncertainties, RBFNNs will be used to approximate unknown continuous nonlinear functions because of their inherent approximation capabilities (Ge et al., 2002). An unknown continuous nonlinear function $H (x) : R^{n} \to R$ can be modelled by a RBFNN as follows:

H (x) = θ^{* T} W (x) + δ (x), \forall x \in Ξ \subset R^{n}

where $x$ is the input vector of the RBFNN, $W (x) = {[W_{1} (x), W_{2} (x), \dots, W_{l} (x)]}^{T}$ is the basis function with the NN node number $l > 1$ , $δ (x)$ is the approximation error, $θ^{*}$ is an unknown constant parameter vector and $Ξ$ is a compact region. The RBFNN uses Gaussian functions as the basis functions, that is,

W_{i} (x) = \exp [- \frac{{(x - c_{i})}^{T} (x - c_{i})}{μ_{i}^{2}}], i = 1, 2, \dots, l

(12)

where $c_{i}$ is the centre and $μ_{i}$ is the width of $W_{i} (x)$ . The optimal parameter vector $θ^{*}$ is selected as the value of $θ$ that minimizes $δ (x)$ for all $x \in Ξ$ , that is,

θ^{*} : = \underset{θ \in R^{l}}{\arg min} {sup_{x \in Ξ} | H (x) - θ^{T} W (x) |}

The following assumption is made regarding the approximation error.

Assumption 3. The approximation error satisfies

| δ (x) | \leq δ_{N}, \forall x \in Ξ

where $δ_{N} > 0$ is an unknown bound.

In what follows, we review two lemmas that will play a key role in the control design procedure.

Lemma 2. (Qian and Lin, 2001) For $x \in R$ , $y \in R$ and $q \geq 1$ a constant, the following inequalities hold:

{| x + y |}^{q} \leq 2^{q - 1} | x^{q} + y^{q} |, (| x | + | y |)^{1 / q} \leq {| x |}^{1 / q} + {| y |}^{1 / q}

Moreover, when $q \geq 1$ is an odd integer or a ratio of two odd integers,

{| x - y |}^{q} \leq 2^{q - 1} | x^{q} - y^{q} |

or | x^{1 / q} - y^{1 / q} | \leq 2^{1 - (1 / q)} {| x - y |}^{1 / q}

Lemma 3. (Qian and Li, 2006) For any positive real numbers $c$ , $d$ , and any real-valued function $υ (x, y) > 0$ ,

{| x |}^{c} {| y |}^{d} \leq \frac{c}{c + d} υ (x, y) {| x |}^{c + d} + \frac{d}{c + d} υ^{- c / d} (x, y) {| y |}^{c + d}

Main results

In this section, the main result will be derived in two steps. In the first step, a robust adaptive finite-time control design scheme is developed for the tracking error system (10) in the framework of the recursive design method. In the second step, based on Lemma 1, the state of the resulting closed-loop system is proved to be SGFTUUB.

Adaptive control law design

The control design procedure contains two steps based on the recursive design method. In Step 1, a virtual control signal $e_{2}^{*}$ for $e_{2}$ is designed such that the position error converges to zero in finite time. In Step 2, an adaptive controller and parameter update laws are designed such that $e_{2}$ converges to a small neighbourhood of $e_{2}^{*}$ in finite time.

Step 1

Consider ${\overset{\cdot}{e}}_{1} = e_{2}$ and choose a Lyapunov function candidate as follows:

V_{1} = \frac{1}{2} e_{1}^{T} e_{1} = \frac{1}{2} \sum_{i = 1}^{n} e_{1 i}^{2}

(13)

Its time derivative is

{\overset{\cdot}{V}}_{1} = \sum_{i = 1}^{n} e_{1 i} {\overset{\cdot}{e}}_{1 i} = \sum_{i = 1}^{n} e_{1 i} (e_{2 i} - e_{2 i}^{*}) + \sum_{i = 1}^{n} e_{1 i} e_{2 i}^{*}

(14)

where $e_{2 i}^{*}$ is the virtual control signal to be designed, and $i \in N$ .

Choose $e_{2 i}^{*}$ as follows:

e_{2 i}^{*} = - k_{1} e_{1 i}^{α}

(15)

where $k_{1} > 0$ is a design parameter, $1 / 2 < α < 1$ , and $α \in R_{odd}^{+}$ is a constant.

Substituting (15) into (14) yields

{\overset{\cdot}{V}}_{1} = - k_{1} \sum_{i = 1}^{n} e_{1 i}^{1 + α} + \sum_{i = 1}^{n} e_{1 i} (e_{2 i} - e_{2 i}^{*})

(16)

This completes Step 1.

Step 2

Because $ψ (t, q, v, τ, τ_{c})$ is bounded, there exists an unknown positive constant $ρ$ such that $‖ ψ (t, q, v, τ, τ_{c}) ‖ \leq ρ$ . Since $ϕ (q, v)$ is an unknown continuous function vector, RBFNNs are used to approximate every element of it. Suppose $ϕ_{i} (q, v)$ is the $i$ th element of $ϕ (q, v)$ , $i \in N$ . It can be modelled by a RBFNN on a compact set $Ω$ as follows:

ϕ_{i} (q, v) = θ_{i}^{* T} ω_{i} (q, v) + δ_{i} (q, v), {[q^{T}, v^{T}]}^{T} \in Ω

(17)

where $θ_{i}^{*}$ , $ω_{i} (q, v) \in R^{l_{Ni}}$ are the optimal parameter vector and basis function vector, respectively, $δ_{i} (q, v) \in R$ is the approximation error satisfying $| δ_{i} (q, v) | \leq δ_{iN}$ from Assumption 3, and $l_{Ni} > 1$ is the NN node number.

Noting the fact that the basis function vector $ω_{i} (q, v)$ satisfies $0 < ω_{i}^{T} (q, v) ω_{i} (q, v) \leq l_{Ni}$ and letting $ϑ_{i}^{*} = [θ_{i}^{* T}, δ_{iN}]^{T}$ and $ϖ_{i} (q, v) = [ω_{i}^{T} (q, v), 1]^{T}$ , then from (17) and $| δ_{i} (q, v) | \leq δ_{iN}$ we have

\begin{array}{l} ϕ_{i} (q, v) \leq ϑ_{i}^{* T} ϖ_{i} (q, v) \leq | ϑ_{i}^{* T} ϖ_{i} (q, v) | \leq ‖ ϑ_{i}^{*} ‖ ‖ ϖ_{i} (q, v) ‖ \\ \leq \sqrt{l_{N i} + 1} ‖ ϑ_{i}^{*} ‖ \end{array}

(18)

Obviously, $‖ ϑ_{i}^{*} ‖ \in R$ is unknown.

Consider ${\overset{\cdot}{e}}_{2} = φ (τ_{c}) u_{c} + ϕ (q, v) + ψ (t, q, v, τ, τ_{c}) - f (t, q_{d}, v_{d})$ and choose the following Lyapunov function candidate:

\begin{array}{l} V_{2} = V_{1} + \sum_{i = 1}^{n} U_{i} + \frac{1}{2 φ_{0}} \sum_{i = 1}^{n} {\tilde{Θ}}_{i}^{2} + \frac{1}{2 φ_{0}} {\tilde{ρ}}^{2}, \\ U_{i} = \int_{e_{2 i}^{*}}^{e_{2 i}} {(s^{1 / α} - e_{2 i}^{* 1 / α})}^{2 - α} d s \end{array}

(19)

where $φ_{0} = min_{i \in N} {φ_{i 0}} > 0$ is an unknown parameter, ${\tilde{Θ}}_{i} = Θ_{i}^{*} - φ_{0} {\hat{Θ}}_{i}$ and $\tilde{ρ} = ρ^{*} - φ_{0} \hat{ρ}$ are parameter estimation errors, $Θ_{i}^{*} = ‖ ϑ_{i}^{*} ‖^{(1 + α) / (2 - α)}$ and $ρ^{*} = ρ^{(1 + α) / (2 - α)}$ , and ${\hat{Θ}}_{i}$ and $\hat{ρ}$ are the estimations. It is easy to verify that (19) works as a $C^{1}$ Lyapunov function with reference to the proof of Proposition B.1 in Polendo and Qian (2008).

Define a new variable $η_{i} = e_{2 i}^{1 / α} - e_{2 i}^{* 1 / α}$ . From the definition of $U_{i}$ and after simple computation, we can obtain

\frac{\partial U_{i}}{\partial e_{2 i}} = η_{i}^{2 - α}

(20)

and

\frac{\partial U_{i}}{\partial e_{1 i}} = - (2 - α) \frac{\partial e_{2 i}^{* 1 / α}}{\partial e_{1 i}} \int_{e_{2 i}^{*}}^{e_{2 i}} (s^{1 / α} - e_{2 i}^{* 1 / α})^{1 - α} ds

(21)

From Lemma 2, we have

e_{2 i} - e_{2 i}^{*} \leq | e_{2 i} - e_{2 i}^{*} | \leq 2^{1 - α} {| η_{i} |}^{α}

(22)

Combining (16) and (22), and using Lemma 3, we get

\begin{array}{l} {\dot{V}}_{1} \leq - k_{1} \sum_{i = 1}^{n} e_{1 i}^{1 + α} + 2^{1 - α} \sum_{i = 1}^{n} | e_{1 i} | {| η_{i} |}^{α} \\ \leq - (k_{1} - \frac{2^{1 - α}}{1 + α}) \sum_{i = 1}^{n} e_{1 i}^{1 + α} + \frac{2^{1 - α} α}{1 + α} \sum_{i = 1}^{n} η_{i}^{1 + α} \end{array}

(23)

Taking the time derivative of $V_{2}$ and combining with (23), we have

\begin{array}{l} {\dot{V}}_{2} \leq - (k_{1} - \frac{2^{1 - α}}{1 + α}) \sum_{i = 1}^{n} e_{1 i}^{1 + α} + \frac{2^{1 - α} α}{1 + α} \sum_{i = 1}^{n} η_{i}^{1 + α} \\ + \sum_{i = 1}^{n} \frac{\partial U_{i}}{\partial e_{1 i}} {\dot{e}}_{1 i} \\ + \sum_{i = 1}^{n} \frac{\partial U_{i}}{\partial e_{2 i}} {\dot{e}}_{2 i} - \sum_{i = 1}^{n} {\tilde{Θ}}_{i} {\dot{\hat{Θ}}}_{i} - \tilde{ρ} \dot{\hat{ρ}} \end{array}

(24)

From (21), it can be obtained by using Lemma 2 that

\begin{array}{l} \frac{\partial U_{i}}{\partial e_{1 i}} {\dot{e}}_{1 i} \leq (2 - α) | e_{2 i} - e_{2 i}^{*} | {| η_{i} |}^{1 - α} | \frac{\partial e_{2 i}^{* 1 / α}}{\partial e_{1 i}} {\dot{e}}_{1 i} | \\ \leq (2 - α) 2^{1 - α} | η_{i} | | \frac{\partial e_{2 i}^{* 1 / α}}{\partial e_{1 i}} {\dot{e}}_{1 i} | \end{array}

(25)

From (15) and $η_{i} = e_{2 i}^{1 / α} - e_{2 i}^{* 1 / α}$ , it can be calculated with the help of Lemma 2 that

| \frac{\partial e_{2 i}^{* 1 / α}}{\partial e_{1 i}} {\overset{\cdot}{e}}_{1 i} | \leq k_{1}^{1 / α} | e_{2 i} | \leq k_{1}^{1 / α} {| η_{i} + e_{2 i}^{* 1 / α} |}^{α} \leq k_{1}^{1 / α} (k_{1} | e_{1 i}^{α} | + | η_{i}^{α} |)

(26)

Substituting (26) into (25), and from Lemma 3, we have

\begin{array}{l} \frac{\partial U_{i}}{\partial e_{1 i}} {\dot{e}}_{1 i} \leq (2 - α) 2^{1 - α} | η_{i} | k_{1}^{1 / α} (k_{1} | e_{1 i}^{α} | + | η_{i}^{α} |) \\ \leq \frac{2^{1 - α} (2 - α) α}{1 + α} e_{1 i}^{1 + α} + 2^{1 - α} (2 - α) \\ (\frac{k_{1}^{1 / α} + k_{1}^{{(1 + α)}^{2} / α}}{1 + α}) η_{i}^{1 + α} \end{array}

(27)

From (10) and (20), we get

\frac{\partial U_{i}}{\partial e_{2 i}} {\overset{\cdot}{e}}_{2 i} = η_{i}^{2 - α} (φ_{i} (τ_{ci}) u_{ci} + ϕ_{i} (q, v) + ψ_{i} (t, q, v, τ, τ_{c}) - f_{i} (t, q_{d}, v_{d}))

(28)

From (18) and Lemma 3, we have

\begin{array}{l} η_{i}^{2 - α} ϕ_{i} (q, v) \leq | η_{i}^{2 - α} | \sqrt{l_{i} + 1} ‖ ϑ_{i}^{*} ‖ \\ \leq \frac{2 - α}{1 + α} ς_{i} {(l_{i} + 1)}^{\frac{1 + α}{2 (2 - α)}} η_{i}^{1 + α} Θ_{i}^{*} + \frac{2 α - 1}{1 + α} ς_{i}^{(α - 2) / (2 α - 1)} \end{array}

(29)

where $ς_{i} > 0$ is a constant.

Since $‖ ψ (t, q, v, τ, τ_{c}) ‖ < ρ$ ,

ψ_{i} (t, q, v, τ, τ_{c}) \leq | ψ_{i} (t, q, v, τ, τ_{c}) | < ‖ ψ (t, q, v, τ, τ_{c}) ‖ < ρ

(30)

Then from Lemma 3 we obtain

\begin{array}{l} η_{i}^{2 - α} ψ_{i} (t, q, v, τ, τ_{c}) \leq | η_{i}^{2 - α} | ρ \leq \frac{2 - α}{1 + α} ζ η_{i}^{1 + α} ρ^{*} \\ + \frac{2 α - 1}{1 + α} ζ^{(α - 2) / (2 α - 1)} \end{array}

(31)

where $ζ > 0$ is a constant.

By using Lemma 3, one has

\begin{array}{l} η_{i}^{2 - α} f_{i} (t, q_{d}, v_{d}) \leq \frac{2 - α}{1 + α} φ_{0} χ_{i} η_{i}^{1 + α} F_{i} (t, q_{d}, v_{d}) \\ + \frac{2 α - 1}{1 + α} {(φ_{0} χ_{i})}^{(α - 2) / (2 α - 1)} \end{array}

(32)

where $F_{i} (t, q_{d}, v_{d}) = {| f_{i} (t, q_{d}, v_{d}) |}^{(1 + α) / (2 - α)}$ and $χ_{i} > 0$ is a constant.

Choose the control input in the following form:

u_{ci} (t, X) = - η_{i}^{2 α - 1} μ_{i} (t, X), i \in N

(33)

where $μ_{i} (t, X) > 0$ is a continuous function to be designed, $X = [e_{1}^{T}, e_{2}^{T}, {\hat{Θ}}^{T}, \hat{ρ}]^{T}$ and $\hat{Θ} = [{\hat{Θ}}_{1}, {\hat{Θ}}_{2}, \dots, {\hat{Θ}}_{n}]^{T}$ . Since $α \in R_{odd}^{+}$ , we have $η_{i}^{1 + α} \geq 0$ , and thus the following inequality holds:

\begin{array}{l} η_{i}^{2 - α} φ_{i} (τ_{c}) u_{c i} (t, X) = - η_{i}^{1 + α} φ_{i} (τ_{c}) μ_{i} (t, X) \\ \leq - φ_{i 0} η_{i}^{1 + α} μ_{i} (t, X) \leq - φ_{0} η_{i}^{1 + α} μ_{i} (t, X) \end{array}

(34)

Substituting (27), (29), (31), (32) and (34) into (24), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - λ_{1} \sum_{i = 1}^{n} e_{1 i}^{1 + α} + λ_{2} \sum_{i = 1}^{n} η_{i}^{1 + α} - φ_{0} \sum_{i = 1}^{n} η_{i}^{1 + α} μ_{i} (t, X) \\ + \frac{2 - α}{1 + α} \sum_{i = 1}^{n} ς_{i} {(l_{Ni} + 1)}^{\frac{1 + α}{2 (2 - α)}} η_{i}^{1 + α} Θ_{i}^{*} + \frac{2 - α}{1 + α} ζ ρ^{*} \sum_{i = 1}^{n} η_{i}^{1 + α} \\ + \frac{2 - α}{1 + α} φ_{0} \sum_{i = 1}^{n} χ_{i} η_{i}^{1 + α} F_{i} (t, q_{d}, v_{d}) - \sum_{i = 1}^{n} {\tilde{Θ}}_{i} {\overset{\cdot}{\hat{Θ}}}_{i} - \tilde{ρ} \overset{\cdot}{\hat{ρ}} + ε \end{matrix}

(35)

where

\begin{array}{l} λ_{1} = k_{1} - \frac{2^{1 - α}}{1 + α} - \frac{2^{1 - α} (2 - α) α}{1 + α} > 0, \\ λ_{2} = \frac{2^{1 - α} α}{1 + α} + 2^{1 - α} (2 - α) (k_{1}^{1 / α} + \frac{k_{1}^{{(1 + α)}^{2} / α}}{1 + α}) \end{array}

and

\begin{array}{l} ε = \frac{2 α - 1}{1 + α} {(φ_{0} χ_{i})}^{(α - 2) / (2 α - 1)} + \frac{2 α - 1}{1 + α} ζ^{(α - 2) / (2 α - 1)} \\ + \frac{2 α - 1}{1 + α} ς_{i}^{(α - 2) / (2 α - 1)} \end{array}

are constants.

Choose the design function $μ_{i} (t, X)$ as follows:

\begin{matrix} μ_{i} (t, X) = k_{2} + h (t, X) + \frac{2 - α}{1 + α} ς_{i} (l_{Ni} + 1)^{\frac{1 + α}{2 (2 - α)}} \sqrt{1 + {\hat{Θ}}_{i}^{2}} \\ + \frac{2 - α}{1 + α} ζ \sqrt{1 + {\hat{ρ}}^{2}} + \frac{2 - α}{1 + α} χ_{i} F_{i} (t, q_{d}, v_{d}), i \in N \end{matrix}

(36)

where $k_{2} > 0$ is a constant and $h (t, X)$ is a time-varying continuous function that has the following properties:

lim_{t \to + \infty} inf_{X \in R^{3 n + 1}} h (t, X) = + \infty, lim_{‖ X ‖ \to + \infty} inf_{t \geq 0} h (t, X) = + \infty

(37)

Substituting (36) into (35), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - λ_{1} \sum_{i = 1}^{n} e_{1 i}^{1 + α} - φ_{0} k_{2} \sum_{i = 1}^{n} η_{i}^{1 + α} - (φ_{0} h (t, X) - λ_{2}) \sum_{i = 1}^{n} η_{i}^{1 + α} \\ + \frac{2 - α}{1 + α} \sum_{i = 1}^{n} ς_{i} {(l_{Ni} + 1)}^{\frac{1 + α}{2 (2 - α)}} η_{i}^{1 + α} {\tilde{Θ}}_{i} + \frac{2 - α}{1 + α} ζ \tilde{ρ} \sum_{i = 1}^{n} η_{i}^{1 + α} \\ - \sum_{i = 1}^{n} {\tilde{Θ}}_{i} {\overset{\cdot}{\hat{Θ}}}_{i} - \tilde{ρ} \overset{\cdot}{\hat{ρ}} + ε \end{matrix}

(38)

Choose the adaptive laws as follows:

{\overset{\cdot}{\hat{Θ}}}_{i} = \frac{2 - α}{1 + α} ς_{i} (l_{Ni} + 1)^{\frac{1 + α}{2 (2 - α)}} η_{i}^{1 + α} - γ_{i} {\hat{Θ}}_{i}, i \in N

(39)

\overset{\cdot}{\hat{ρ}} = \frac{2 - α}{1 + α} ζ \sum_{i = 1}^{n} η_{i}^{1 + α} - γ_{0} \hat{ρ}

(40)

where $γ_{i} > 0$ , $i \in N$ , and $γ_{0} > 0$ are constants. Hence (38) becomes

\begin{array}{l} {\dot{V}}_{2} \leq - λ_{1} \sum_{i = 1}^{n} q_{i}^{1 + α} - φ_{0} k_{2} \sum_{i = 1}^{n} η_{i}^{1 + α} - (φ_{0} h (t, X) - λ_{2}) \sum_{i = 1}^{n} η_{i}^{1 + α} \\ + \sum_{i = 1}^{n} γ_{i} {\tilde{Θ}}_{i} {\hat{Θ}}_{i} + γ_{0} \tilde{ρ} \hat{ρ} + ε \end{array}

(41)

This completes Step 2.

Remark 4. By using Lemma 3 we obtain inequalities (29) and (31), and define new parameters $Θ_{i}^{*} = ‖ ϑ_{i}^{*} ‖^{(1 + α) / (2 - α)}$ and $ρ^{*} = ρ^{(1 + α) / (2 - α)}$ , which are unknown constants. Thus the number of adaptation parameters is reduced from $l_{Ni} + 1$ of $ϑ_{i}^{*}$ to $1$ and the computational burden is relieved.

Stability analysis

So far, the robust adaptive control scheme has been established for uncertain tracking error system (10) possessing input saturation and deadzone. The following theorem summarizes the main result of this paper.

Theorem 4. Considering the tracking error system (10) with Assumptions 1 to 3, under the time-varying state-feedback controller (33) and the adaptive laws (39) and (40), the state of the corresponding closed-loop system is SGFTUUB.

Proof. From (19) and Lemma 2, we have

U_{i} \leq | e_{2 i} - e_{2 i}^{*} | {| η_{i} |}^{2 - α} \leq 2 {| η_{i} |}^{α} {| η_{i} |}^{2 - α} \leq 2 η_{i}^{2}

(42)

Thus

\begin{array}{l} V_{2}^{(1 + α) / 2} \leq 2 \sum_{i = 1}^{n} (e_{1 i}^{1 + α} + η_{i}^{1 + α}) + \sum_{i = 1}^{n} {(\frac{1}{2 φ_{0}} {\tilde{Θ}}_{i}^{2})}^{(1 + α) / 2} \\ + {(\frac{1}{2 φ_{0}} {\tilde{ρ}}^{2})}^{(1 + α) / 2} \end{array}

(43)

Combining (41) and (43), and letting $κ = min {λ_{1}, φ_{0} k_{2}}$ , from Lemma 2 we obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - \frac{κ}{2} V_{2}^{(1 + α) / 2} - (φ_{0} h (t, X) - λ_{2}) \sum_{i = 1}^{n} η_{i}^{1 + α} \\ + \frac{κ}{2} \sum_{i = 1}^{n} {(\frac{1}{2 φ_{0}} {\tilde{Θ}}_{i}^{2})}^{(1 + α) / 2} \\ + \frac{κ}{2} {(\frac{1}{2 φ_{0}} {\tilde{ρ}}^{2})}^{(1 + α) / 2} + \sum_{i = 1}^{n} γ_{i} {\tilde{Θ}}_{i} {\hat{Θ}}_{i} + γ_{0} \tilde{ρ} \hat{ρ} + ε \end{matrix}

(44)

By using Lemma 3, the following inequalities can be obtained:

\begin{array}{l} \frac{κ}{2} {(\frac{1}{2 φ_{0}} {\tilde{Θ}}_{i}^{2})}^{(1 + α) / 2} \leq {(\frac{κ}{2})}^{2 / (1 + α)} \frac{1}{2 φ_{0}} {\tilde{Θ}}_{i}^{2} \\ + \frac{1 - α}{2} {(\frac{1 + α}{2})}^{(1 + α) / (1 - α)}, i \in N \end{array}

(45)

\begin{array}{l} \frac{κ}{2} {(\frac{1}{2 φ_{0}} {\tilde{ρ}}^{2})}^{(1 + α) / 2} \leq {(\frac{κ}{2})}^{2 / (1 + α)} \frac{1}{2 φ_{0}} {\tilde{ρ}}^{2} \\ + \frac{1 - α}{2} {(\frac{1 + α}{2})}^{(1 + α) / (1 - α)} \end{array}

(46)

Noting the fact that ${\tilde{Θ}}_{i} = Θ_{i}^{*} - φ_{0} {\hat{Θ}}_{i}$ and $\tilde{ρ} = ρ^{*} - φ_{0} \hat{ρ}$ , by using Lemma 3 we have

γ_{i} {\tilde{Θ}}_{i} {\hat{Θ}}_{i} = \frac{γ_{i}}{φ_{0}} {\tilde{Θ}}_{i} (Θ_{i}^{*} - {\tilde{Θ}}_{i}) \leq - \frac{γ_{i}}{2 φ_{0}} {\tilde{Θ}}_{i}^{2} + \frac{γ_{i}}{2 φ_{0}} Θ_{i}^{* 2}, i \in N

(47)

γ_{0} \tilde{ρ} \hat{ρ} = \frac{γ_{0}}{φ_{0}} \tilde{ρ} (ρ^{*} - \tilde{ρ}) \leq - \frac{γ_{0}}{2 φ_{0}} {\tilde{ρ}}^{2} + \frac{γ_{0}}{2 φ_{0}} ρ^{* 2}

(48)

By choosing parameters $γ_{i}$ , $i \in N$ , and $γ_{0}$ such that $γ_{i} \geq (λ_{1} / 2)^{2 / (1 + α)}$ and $γ_{0} \geq (λ_{1} / 2)^{2 / (1 + α)}$ , we have $γ_{i} \geq (κ / 2)^{2 / (1 + α)}$ and $γ_{0} \geq (κ / 2)^{2 / (1 + α)}$ because $λ_{1} \geq κ$ . Substituting (45) to (48) into (44), we have

{\overset{\cdot}{V}}_{2} \leq - \frac{κ}{2} V_{2}^{(1 + α) / 2} - (φ_{0} h (t, X) - λ_{2}) \sum_{i = 1}^{n} η_{i}^{1 + α} + \bar{ε}

(49)

where $\bar{ε} = \sum_{i = 1}^{n} \frac{γ_{i}}{2 φ_{0}} Θ_{i}^{* 2} + \frac{γ_{0}}{2 φ_{0}} ρ^{* 2} + ε + (n + 1) \frac{1 - α}{2} {(\frac{1 + α}{2})}^{(1 + α) / (1 - α)}$ is a constant.

The remaining proof procedure contains two steps. Firstly, based on (49) and the properties (37) of the designed function $h (t, X)$ , it can be proved that the solutions of the closed-loop system are all defined on a time interval $[0, t_{f})$ with $t_{f} = + \infty$ for any initial condition. We can prove it by contradiction. Suppose that $t_{f}$ is finite. Then there exists at least one solution of the closed-loop system, denoted by $X (t, X_{0})$ with $X_{0}$ being the initial condition, which satisfies $lim_{t \to t_{f}} ‖ X (t, X_{0}) ‖ = + \infty$ . From the definition of $V_{2}$ , we have $lim_{t \to t_{f}} V_{2} = + \infty$ . With reference to the properties of $h (t, X)$ , it can be seen that there exists a time $t_{s} \in (0, t_{f})$ such that when $t \geq t_{s}$ we have

φ_{0} h (t, X) - λ_{2} \geq 0

(50)

Combining (49) and (50), it is not difficult to see that ${\overset{\cdot}{V}}_{2} \leq ν$ for some constant $ν \geq 0$ , $\forall t \in [0, t_{f})$ , which together with the continuity of all involved functions shows that $0 \leq V_{2} (t_{f}) \leq V_{2} (0) + ν t_{f} < + \infty$ , which obviously contradicts $lim_{t \to t_{f}} V_{2} = + \infty$ . Hence, $t_{f} = + \infty$ .

Secondly, it can be proved that the closed-loop system is SGFTUUB. Inspired by the above analysis, there exists a $t_{m} > 0$ such that

φ_{0} h (t, X) - λ_{2} \geq 0, \forall t \in [t_{m}, + \infty)

(51)

Accordingly, combining (49) and (51) gives

{\overset{\cdot}{V}}_{2} \leq - \frac{κ}{2} V_{2}^{(1 + α) / 2} + \bar{ε}

(52)

from which and according to Lemma 1 and the proof of Lemma 3.6 in Zhu et al. (2015), there exists a finite time $T$ satisfying $T \leq \frac{4 V_{2}^{(1 - α) / 2} (X (t_{m}))}{κ θ_{0} (1 - α)}$ such that, when $t \geq t_{m} + T$ , the trajectories of the closed-loop system are bounded as

Ξ = {X | V_{2} \leq {(\frac{2 \bar{ε}}{κ (1 - θ_{0})})}^{2 / (1 + α)}}

(53)

where $0 < θ_{0} < 1$ is a constant.

This completes the proof of Theorem 4.

Remark 5. From the proof of Theorem 4, we can see that the time-varying function $h (t, X)$ plays an important role in guaranteeing the semi-global finite-time uniform ultimate boundedness of the closed-loop system. Firstly, based on the property $lim_{‖ X ‖ \to + \infty} inf_{t \geq 0} h (t, X) = + \infty$ , it is proved that the solutions of the closed-loop system are all defined on a time interval $[0, t_{f})$ with $t_{f} = + \infty$ for any initial condition. Secondly, with the help of the property $lim_{t \to + \infty} inf_{X \in R^{3 n + 1}} h (t, X) = + \infty$ , we can obtain that there exists a $t_{m} > 0$ such that $φ_{0} h (t, X) - λ_{2} \geq 0$ , and thus the unknown term $- (φ_{0} h (t, X) - λ_{2}) \sum_{i = 1}^{n} η_{i}^{1 + α}$ in inequality (49) can be omitted and the SGFTUUB stabilization is achieved according to Lemma 1. Besides, it should be pointed out that function $h (t, X)$ can be easily constructed: for example, we can choose $h (t, X) = (1 + t) (1 + ‖ X ‖)$ , which obviously satisfies (37).

Remark 6. From the analysis of proof procedure, we can see that the convergence time $T$ of SGFTUUB stabilization is finite and dependent on the state value $X (t_{m})$ . In recent works (Ni et al., 2016, 2017a,b), a special finite-time control problem called ‘fixed-time control’ was investigated. The convergence time of fixed-time control is independent of initial conditions and bounded by a constant. However, in order to realize bounded time stabilization, the control laws of fixed-time control schemes are usually more complex than those of finite-time (non-fixed-time) control schemes. Unlike the results in Ni et al. (2016, 2017a,b), model uncertainties and input constraints are considered in this paper, and thus the proposed results cannot be directly extended to the case of fixed-time control. In the future, the fixed-time control of system (1) with uncertainties and input constraints will be an important research issue.

Remark 7. From $T \leq \frac{4 V_{2}^{(1 - α) / 2} (X (t_{m}))}{κ θ_{0} (1 - α)}$ it can be seen that the convergence time $T$ is adjustable by choosing suitable controller parameters and fractional power $α$ . In order to make the convergence time smaller, some improved sufficient conditions of finite-time stability were given in Shen and Xia (2008) and Shen and Huang (2012). However, the designed controller may be more complex (see equations (26) and (27) in Shen and Huang, 2012).

Numerical example

In this section, an example will be provided to demonstrate the effectiveness of the proposed results.

Consider a two-link manipulator, a schematic of which is shown in Figure 2. The model of a two-link manipulator can be expressed as (Spong et al., 2006)

\begin{array}{l} [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] [\begin{matrix} {\overset{\cdot\cdot}{q}}_{1} \\ {\overset{\cdot\cdot}{q}}_{2} \end{matrix}] + [\begin{matrix} h {\dot{q}}_{2} & h {\dot{q}}_{2} + h {\dot{q}}_{1} \\ - h {\dot{q}}_{1} & 0 \end{matrix}] [\begin{matrix} {\dot{q}}_{1} \\ {\dot{q}}_{2} \end{matrix}] + [\begin{matrix} c_{1} g \\ c_{2} g \end{matrix}] \\ + [\begin{matrix} D_{1} \\ D_{2} \end{matrix}] = [\begin{matrix} τ_{1} \\ τ_{2} \end{matrix}] \end{array}

(54)

where

a_{11} = m_{1} l_{c 1}^{2} + m_{2} (l_{1}^{2} + l_{c 2}^{2} + 2 l_{1} l_{c 2}^{2} + 2 l_{1} l_{c 2} \cos q_{2}) + I_{1} + I_{2}

a_{12} = a_{21} = m_{2} (l_{c 2}^{2} + l_{1} l_{c 2} \cos q_{2}) + I_{2}, a_{22} = m_{2} l_{c 2}^{2} + I_{2}

\begin{array}{l} c_{1} = (m_{1} l_{c 1} + m_{2} l_{1}) \cos q_{1} + m_{2} l_{c 2} \cos (q_{1} + q_{2}), \\ c_{2} = m_{2} l_{c 2} \cos (q_{1} + q_{2}) \end{array}

h = - m_{2} l_{1} l_{c 2} \sin q_{2}, D_{1} = 0.5 \sin t, D_{2} = \sin t

The desired position and velocity satisfy

{\overset{\cdot}{q}}_{d} = v_{d}, {\overset{\cdot}{v}}_{d} = - \sin (2 π t) v_{d}

In the simulation, the system’s parameters are chosen as shown in Table 1.

Table 1.

Parameters of two-link manipulator.

Symbols	Definition	Value
$m_{1}$	Mass of link 1	$1 kg$
$m_{2}$	Mass of link 2	$2 kg$
$l_{1}$	Length of link 1	$1 m$
$l_{c 1}$	Distance from the previous joint tothe centre of mass of link 1	$0.5 m$
$l_{c 2}$	Distance from the previous jointto the centre of mass of link 2	$0.6 m$
$I_{1}$	Moment of inertia of link 1	$0.12 kg \cdot m^{2}$
$I_{2}$	Moment of inertia of link 2	$0.25 kg \cdot m^{2}$

The parameters of the input nonlinearity are as follows:

h_{l 11} = - 50, h_{l 10} = - 2, h_{r 10} = 2, h_{r 11} = 300

h_{l 21} = - 50, h_{l 20} = - 2, h_{r 20} = 2, h_{r 21} = 50

π_{l 1} (τ_{c 1}) = 0.9 (τ_{c 1} - h_{l 10}), π_{r 1} (τ_{c 1}) = 1.1 (τ_{c 1} - h_{r 10})

π_{l 2} (τ_{c 2}) = 1.2 (τ_{c 2} - h_{l 20}), π_{r 2} (τ_{c 2}) = 1.1 (τ_{c 2} - h_{r 20})

where the saturation values are

\begin{array}{l} τ_{l 1 m} = 0.9 (h_{l 11} - h_{l 10}) = - 43.2 Nm, \\ τ_{r 1 m} = 1.1 (h_{r 11} - h_{r 10}) = 327.8 Nm \end{array}

\begin{array}{l} τ_{l 2 m} = 1.2 (h_{l 21} - h_{l 20}) = - 57.6 Nm, \\ τ_{r 2 m} = 1.1 (h_{r 21} - h_{r 20}) = 52.8 Nm \end{array}

First, the control performance by using a traditional PD control method is investigated, where the initial conditions are selected as shown in Table 2.

Table 2.

Initial conditions of PD control.

Position	Velocity
$q (0) = {[\begin{matrix} 0 0 \end{matrix}]}^{T}$	$v (0) = {[\begin{matrix} 0 0 \end{matrix}]}^{T}$
$q_{d} (0) = [\begin{matrix} 0.6 & - 0.5 \end{matrix}]^{T}$	$v_{d} (0) = {[\begin{matrix} - 0.3 0.5 \end{matrix}]}^{T}$

The simulation results for this case are shown in Figure 3. Figure 3(a) shows the trajectories of position and velocity under PD control, and Figure 3(b) depicts the tracking errors of position and velocity. It can be seen that PD control has large position and velocity tracking errors. The actual control input is shown in Figure 3(c).

Second, the proposed control method is adopted to solve the tracking control problem of system (54). Following the design procedure proposed in Section 3.1, we can obtain the controller and adaptive laws as follows:

\begin{matrix} u_{ci} = - η_{i}^{2 α - 1} μ_{i} (t, X) \\ {\overset{\cdot}{\hat{Θ}}}_{i} = \frac{2 - α}{1 + α} ς_{i} {(l_{Ni} + 1)}^{\frac{1 + α}{2 (2 - α)}} η_{i}^{1 + α} - γ_{i} {\hat{Θ}}_{i} \\ \overset{\cdot}{\hat{ρ}} = \frac{2 - α}{1 + α} ζ \sum_{i = 1}^{n} η_{i}^{1 + α} - γ_{0} \hat{ρ} \end{matrix}

(55)

where

\begin{matrix} μ_{i} (t, X) = k_{2} + h (t, X) + \frac{2 - α}{1 + α} ς_{i} {(l_{Ni} + 1)}^{\frac{1 + α}{2 (2 - α)}} \sqrt{1 + {\hat{Θ}}_{i}^{2}} \\ + \frac{2 - α}{1 + α} ζ \sqrt{1 + {\hat{ρ}}^{2}} + \frac{2 - α}{1 + α} χ_{i} F_{i} (t, q_{d}, v_{d}) \end{matrix}

\begin{matrix} h (t, X) = 0.1 (1 + t) (1 + ‖ {[q^{T}, v^{T}, q_{d}^{T}, v_{d}^{T}, {\hat{Θ}}^{T}, \hat{ρ}]}^{T} ‖) \\ η_{i} = e_{2 i}^{1 / α} - e_{2 i}^{* 1 / α}, e_{2 i}^{*} = - k_{1} e_{1 i}^{α}, i = 1, 2 \end{matrix}

Therefore, we can obtain the control input

τ_{c} = {[\begin{matrix} τ_{c 1} & τ_{c 2} \end{matrix}]}^{T} = M_{d} (q) {[\begin{matrix} u_{c 1} & u_{c 2} \end{matrix}]}^{T}

where

M_{d} (q) = [\begin{matrix} m_{1} l_{c 1}^{2} + m_{2} (l_{1}^{2} + l_{c 2}^{2} + 2 l_{1} l_{c 2}^{2} + 2 l_{1} l_{c 2} \cos q_{2}) & 0 \\ 0 & m_{2} l_{c 2}^{2} \end{matrix}]

is a known diagonal matrix.

Choose the design parameters in adaptive control law (55) as follows:

\begin{array}{l} α = 7 / 11, k_{1} = 2^{1 - α} + 2^{1 - α} (2 - α) α, k_{2} = 1, \\ ς_{1} = ς_{2} = χ_{1} = χ_{2} = ζ = 15 \end{array}

l_{N 1} = 8, l_{N 2} = 10, γ_{1} = γ_{2} = γ_{0} = (1 + (2 - α) α)^{2 / (1 + α)}

It is easy to verify that $λ_{1} > 0$ and $γ_{i} > (λ_{1} / 2)^{2 / (1 + α)}$ are satisfied, $i = 0, 1, 2$ .

The simulation results are shown in Figure 4, where the initial conditions are the same as those of PD control and $\hat{Θ} (0) = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T}, \hat{ρ} (0) = 0 .$ From Figure 4(a) it can be seen that fairly good tracking performance is obtained. Figure 4(b) indicates that the position and velocity tracking errors are smaller than those with PD control. Figure 4(c) gives the trajectories of the control input, which is obviously constrained. The trajectories of parameter estimations $\hat{Θ}$ and $\hat{ρ}$ are displayed in Figure 4(d). Figure 5 shows the norms of tracking errors based on PD control and the proposed control, from which we can see that the proposed control method has a faster convergence rate and higher-precision tracking performance. Therefore, the simulation results show the effectiveness of the proposed method.

Conclusions

This paper has investigated the problem of robust adaptive finite-time tracking control for a class of uncertain mechanical systems with input constraints. Based on the recursive design method and by using RBFNNs as approximators, a robust adaptive finite-time tracking scheme has been developed. In order to guarantee finite-time convergence of the closed-loop tracking error system, a time-varying designed function has been employed in the proposed control strategy. In addition, the application of the control scheme to a two-link manipulator has been provided to demonstrate its effectiveness. Simulation results have verified that the proposed control scheme has an advantage over the PD control method in the aspects of convergence rate and steady-state precision. In our further work, we will consider finite-time consensus tracking control for multiple uncertain mechanical systems, referring to some related works on the coordination control of Euler–Lagrange or mechanical systems (Sun et al., 2016, 2017b; Wang, 2014).

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 by the National Natural Science Foundation of China (grant numbers 61673214, 61673217).

ORCID iD

Baofang Wang

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