Neural network adaptive command filtered control of robotic manipulators with input saturation

Abstract

This paper investigates finite-time control of uncertain robotic manipulators with external disturbances by means of neural network control and backstepping technique. To solve the “explosion of terms” in traditional backstepping control, a second-order command filter is designed, and the virtual input and its first-order derivative can be obtained accurately in a finite time. The parameters of the neural network are updated by using the tracking error signals. The proposed controller can guarantee that the tracking error converges to a small region of the origin in some finite time. Finally, we give a simulation study to show the effectiveness of the proposed method.

Keywords

Adaptive neural network control robotic manipulator finite-time stability command filtered control

Introduction

As we all know, robots have a wider range of applications, such as deep-sea rescue, medical care, education, and complex environments. However, system uncertainties usually exist in a robot system, and will result in unpleasant control performance, or even the instability of the robot. Thus, it is meaningful to study the control of uncertain robotic manipulators. Recently, control algorithms represented by fuzzy systems and neural networks (NNs) have been used extensively in the control of robotic manipulators, because these systems can be well used to eliminate the system uncertainties.^1

–9 In the study by Yang et al.,² a NN control method was proposed for robotic manipulators based on an observer, where the proposed method is very effective for canceling the effect of external disturbance and has very good robustness. To enhance the control performance for robotic manipulators, Deng¹⁰ proposed an adaptive fuzzy control method by using backstepping technique, and in which the finite-time control was also considered. In the study by Yen et al.,¹¹ a novel control method named recurrent fuzzy wavelet NN sliding mode control was proposed. In the study by Muñoz-Vázquez et al.,¹² fractional-order control together with fuzzy control was proposed for robotic manipulators. It should be mentioned that although the above methods are effective for control uncertain robots, the saturation problem that often exists in robotic manipulators is not considered.

In real-world systems, the system variables are often constrained by some conditions, especially in the robot systems because the driving force of the robot is limited. When input saturation takes place, it is a challenging work to design adaptive controllers for the controlled systems. Thus, it makes sense to study the control of robotic manipulators considering the input saturation. In this field, some prior works have been done.^13
–15 On the other hand, backstepping control methods are often used to control nonlinear systems with triangular form.^16

–19 In this method, some medium variables are recursively treated as virtual control inputs. Integer-order backstepping controller needs to remove the nonlinearities of system. However, when the system with unknown parameters or model is uncertain, this method is often unable to achieve the expected goal. In fact, robotic manipulators can be written as strict-feedback systems, and thus, backstepping control methods can also be used. For example, in the study by Nikdel et al.,²⁰ a fractional fuzzy backstepping controller was implemented for robot; in Liu et al.,²¹ finite-time $H_{\infty}$ control of robotic manipulators was investigated by means of backstepping control. However, there is a problem that usually occurs in backstepping control: the “explosion of terms”. To solve this problem, the command filtered control (CFC) is proposed.^17,22,23 In the CFC method, the virtual control input is driven to pass through a command filter, and thus its derivative can be estimated by the filter, and the estimation error can be arbitrarily small. As far as we know, the CFC for robotic manipulators with input saturation has rarely been investigated up to now.

There are two main reasons that drive us to do this work. Firstly, it has been shown that in above literature, finite-time control of robotic manipulators has rarely been investigated by using NNs. Secondly, in existing literature that considers backstepping control of robotic manipulators, the “explosion of terms” problem has not been well solved. Thus, in this article, we will address the finite-time control of robotic manipulators by using NN control. The main contributions of this work are concluded as follows: (1) A second-order finite-time filter is designed to obtain the finite-time estimation of the virtual input and its derivative, and thus the “explosion of terms” problem can be solved; (2) compensated signals are designed to tackle the estimation errors of the virtual inputs; (3) input saturation problem is also considered in the finite-time NN controller design, and the uncertain term of the saturation as well as the system uncertainties are approximated by NNs.

This work is arranged as follows: The second section presents some basic results about the NN and the finite-time stability of nonlinear systems. Finite-time controller design by using a backstepping procedure is given in the third section. Simulation studies are presented in the fourth section. Finally, the main conclusions of this work are summarized in the fifth section.

Preliminaries

Description of a NN

A multi-layer NN is effective for modeling nonlinear functions due to its excellent capabilities in function approximation, which is given by

y_{k} (s, w_{k}) = \sum_{j = 1}^{h} ω_{k j} φ_{k j} (\sum_{i = 1}^{n} v_{j i} s_{i} + θ_{j}) = w_{k}^{T} ψ_{k} (\cdot)

with $n, h, m \in ℕ_{+}$ respectively being the amount of neurons of input, hidden layer as well as output of the NN, $w_{k} = [\begin{matrix} ω_{k 1} \\ ⋮ \\ ω_{k h} \end{matrix}]$ , $ψ_{k} = [\begin{matrix} φ_{k 1} (\sum_{i = 1}^{n} v_{1 i} s_{i} + θ_{1}) \\ ⋮ \\ φ_{k h} (\sum_{i = 1}^{n} v_{h i} s_{i} + θ_{h}) \end{matrix}]$ . $v_{j i}$ is a weight with $j = 1, \dots, h, i = 1, \dots, n$ which can be chosen on $[- 1, 1]$ randomly. Usually, $φ (\cdot)$ is defined by

φ (ζ) = \frac{e^{ζ} - e^{- ζ}}{e^{ζ} + e^{- ζ}}

Thus, a NN can be given by

y = W^{T} ψ (ζ)

where $W = [\begin{matrix} w_{1}^{T} \\ w_{2}^{T} \\ ⋮ \\ w_{m}^{T} \end{matrix}]$ , $ψ (ζ) = [\begin{matrix} ψ_{1} (ζ) & 0 & \dots & 0 \\ 0 & ψ_{2} (ζ) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & ψ_{m} (ζ) \end{matrix}]$ . To approximate an unknown function $f (ζ), ζ \in ℝ^{n}$ , NN (3) can be used as

f (x) = W^{* T} ψ (ζ) + ε (ζ)

with $ε (ζ)$ being the optimal approximation error, and the optimal weight matrix $W^{*}$ is

W^{*} = arg min_{W} [sup | \hat{f} (ζ) - f (ζ) |]

with $\hat{f} (ζ) = W^{T} ψ (ζ)$ .

Basic lemmas

The following lemmas are presented to facilitate the subsequent analysis.

Lemma 1

Suppose that $V (ζ) \in ℂ^{1} \cap ℝ^{+}$ satisfies $\dot{V} (ζ) + π_{1} V^{π_{2}} (ζ) \leq 0$ with $ζ \in Ω_{1} \subset ℝ^{n}$ with $Ω_{1}$ being an open set and $π_{1} \in ℝ^{+}$ , $0 < π_{2} < 1$ . Thus, there exists $Ω_{2} \subset ℝ^{n}$ such that any $V (ζ)$ began within $Ω_{2}$ can arrive at $V (ζ) \equiv 0$ in some finite time $T^{*}$ which is estimated by

T^{*} \leq \frac{V^{1 - π_{2}} (ζ (0))}{π_{1} (1 - π_{2})}

Lemma 2

Let $g_{1}, g_{2} \in ℝ^{+}$ and $g_{3} \in (0, 1)$ , and

\dot{V} (ζ) + g_{1} V (ζ) + g_{2} V^{g_{3}} (ζ) \leq 0

with $V (ζ)$ being a Lyapunov function, $ζ \in ℝ^{n}$ . The stability time $T^{*}$ can be estimated by

T^{*} \leq \frac{1}{[ln [g_{1} V^{1 - g_{3}} (0) + g_{2}] - ln g_{2}] g_{1} (1 - g_{3})}

Lemma 3

Consider

{\begin{array}{l} {\dot{λ}}_{1} & = F \\ F & = - g_{4} Λ sign (λ_{1} - ζ) + λ_{2} \\ {\dot{λ}}_{2} & = - g_{5} sign (λ_{2} - F) \end{array}

where $ζ \subset ℝ^{n}$ , $g_{4}, g_{5} \in ℝ^{+}$ , and $Λ \in ℝ^{n \times n}, Λ > 0$ .^24,25 The following equality holds within some infinite time that is related to the design parameters $g_{4}, g_{5}$

λ_{1} = ζ, Ϝ = \dot{ζ}

Control design

An n-link robotic manipulator can be described by²⁶

M (x) \ddot{x} + C (x, \dot{x}) \dot{x} + G (x) + F (\dot{x}) = sat (u (t)) + d (t)

with $x = [x_{1}, x_{2}, \dots, x_{n}]^{T} \in ℝ^{n}$ being the position vector, $\dot{x}$ and $\ddot{x}$ being velocity and acceleration vectors, $M (x) \in ℝ^{n \times n}$ being an inertia matrix, $C (x, \dot{x})$ being the Coriolis matrix, $G (x) \in ℝ^{n \times n}$ representing the gravity force term, $F (\dot{x})$ being the friction term, $sat (u (t)) \in ℝ^{n}$ denoting the input with saturation, and $d (t) \in ℝ$ denoting a disturbance term.

In robotic manipulators control, one hopes that the output can change evenly with the input. However, it is well known that system constraints usually exist in robotic manipulators, that is, if the input arrives at a certain value, the output cannot increase anymore. A mapping from $ℝ$ to $ℝ$ $sat : u \to sat (u)$ is called a saturator, which is described by

sat (u) = {\begin{array}{l} u_{r}, & u \geq u_{r} \\ u, & u_{l} < u < u_{r} \\ u_{l}, & u \leq u_{l} \end{array}

where $u_{r} = [u_{r 1}, \dots, u_{r n}]^{T}, u_{l} = [u_{l 1}, \dots, u_{l n}]^{T}$ with $u_{r i} > 0, u_{l i} < 0, i = 1, \dots, n$ are two constant vectors. The saturation phenomenon of the input $u_{i} (t)$ is depicted in Figure 1.

Figure 1.

The saturation phenomenon.

Denoting the term that exceeds the saturation limiter as $α (t)$

α = {\begin{matrix} u_{r} - u, & u \geq u_{r} \\ 0, & u_{l} < u < u_{r} \\ u_{l} - u, & u \leq u_{l} \end{matrix}

For system (6), we have the following assumption.

Assumption 1

$M (x)$ is positive definite and satisfies $a_{1} I \leq M (x) \leq a_{2} I$ with $a_{1}, a_{2} \in ℝ^{+}$ , and $\dot{M} (x) - 2 C (x, \dot{x})$ satisfies $ξ^{T} [\dot{M} (x) - 2 C (x, \dot{x})] ξ = 0$ with $ξ \in ℝ^{n}$ .

Remark 1

It should be pointed out that assumption 1 is not strict because most robotic manipulators satisfy this condition. In a robotic manipulator, $M (x)$ is an inertia matrix that is designed to be positive definite, and thus the robotic manipulator will have good nonlinear dynamics and is easy to be controlled. In fact, assumption 1 can be seen in most related literature.^6,27
–29 In addition, this assumption is very important for analyzing the stability of the controlled robotic manipulators.

For facilitating the controller design, let us rearrange system (6) into the following triangular form by denoting $y_{1} = x, y_{2} = \dot{x}$

{\begin{array}{l} {\dot{y}}_{1} (t) & = y_{2} (t) \\ {\dot{y}}_{2} (t) & = M^{- 1} (y) sat (u (t)) + M^{- 1} (y) d (t) + f (y) \end{array}

in which $y = [y_{1}^{T}, y_{2}^{T}]^{T} \in ℝ^{2 n}$ , $f (y) = M^{- 1} (y) [- C (y_{1}, y_{2}) y_{2} - G (y_{1}) - F (y_{2})] \in ℝ^{n}$ .

The control objective is to drive the position vector y tracking a smooth referenced signal $y_{d} \in ℝ^{n}$ within an infinite time. To design the controller, we will use the backstepping method. Firstly, denoting the tracking error as $ϕ_{1} (t) = y_{1} (t) - y_{d} (t)$ , and constructing a virtual input $π (t) \in ℝ^{n}$ , then we have

\begin{array}{l} {\dot{ϕ}}_{1} & = y_{2} - {\dot{y}}_{d} = π + \hat{π} - π + y_{2} - \hat{π} - {\dot{y}}_{d} \\ = π + \hat{π} - π + ϕ_{2} - {\dot{y}}_{d} \end{array}

where $ϕ_{2} = y_{2} - {\hat{π}}_{1}$ , and $\hat{π} \in ℝ^{n}$ is the estimation of $π$ . Different from the traditional CFC method,^30
–32 in this article, we will use an error compensation term to tackle the estimation error of the virtual input $π$ and its derivative. Define

{\dot{ϖ}}_{1} = - k_{1} ϖ_{1} + \hat{π} - π + ϖ_{2} - c_{1} sign (ϖ_{1})

with $ϖ_{2}$ being also a compensation signal, $k_{1}, c_{1} \in ℝ^{+}$ are constants, and $ϖ (0) = 0$ . Noting that the derivative of the virtual input $π$ is very complicated, we will obtain the estimation of $π$ and $\dot{π}$ by means of lemma 3

{\begin{array}{l} {\dot{λ}}_{1} & = Ϝ \\ Ϝ & = - b_{1} Λ sign (λ_{1} - π) + λ_{2} \\ {\dot{λ}}_{2} & = - b_{2} sign (λ_{2} - Ϝ) \end{array}

in which $Λ = diag (\sqrt{| λ_{11} - π_{1} |}, \sqrt{| λ_{12} - π_{2} |}, \dots, \sqrt{| λ_{1 n} - π_{n} |})$ . It follows from (12) and lemma 3 that $\hat{π} = λ_{1}$ and $\dot{\hat{π}} = Ϝ$ after some infinite time $T^{*} \in R^{+}$ . Compensated tracking errors are defined by

{\begin{array}{l} {\tilde{ϕ}}_{1} & = ϕ_{1} - ϖ_{1} \\ {\tilde{ϕ}}_{2} & = ϕ_{2} - ϖ_{2} \end{array}

with $ϕ_{2} = y_{2} - \hat{π}$ . Thus, we can use the following virtual control signal

π = - k_{1} ϕ_{1} + {\dot{y}}_{d} - a_{1} {\tilde{ϕ}}_{1}^{ν}

with $a_{1} \in ℝ^{+}$ , $ν \in (0, 1)$ being a constant, and ${\tilde{ϕ}}_{1}^{ν} = [{\tilde{ϕ}}_{11}^{ν}, {\tilde{ϕ}}_{12}^{ν}, \dots, {\tilde{ϕ}}_{1 n}^{ν}]^{T} \in ℝ^{n}$ . Define $V_{1} = \frac{1}{2} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1}$ . The derivative of V ₁ is given by

\begin{array}{l} {\dot{V}}_{1} & = {\tilde{ϕ}}_{1}^{T} {\dot{\tilde{ϕ}}}_{1} (t) \\ = {\tilde{ϕ}}_{1}^{T} [π + \hat{π} - π + ϕ_{2} - {\dot{y}}_{d} + k_{1} ϖ_{1} - \hat{π} + π - ϖ_{2} + c_{1} sign (ϖ_{1})] \\ = {\tilde{ϕ}}_{1}^{T} [- k_{1} ϕ_{1} - a_{1} {\tilde{ϕ}}_{1}^{ν} + ϕ_{2} + k_{1} ϖ_{1} - ϖ_{2} + c_{1} sign (ϖ_{1})] \\ = - k_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1} - a_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1}^{ν} + {\tilde{ϕ}}_{1}^{T} ϕ_{2} - {\tilde{ϕ}}_{1}^{T} z_{2} + c_{1} {\tilde{ϕ}}_{1}^{T} sign (ϖ_{1}) \\ = - k_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1} - a_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1}^{ν} + {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{2} + c_{1} {\tilde{ϕ}}_{1}^{T} sign (ϖ_{1}) \end{array}

On the other hand, (7), (8), (9), and (13) imply

\begin{array}{l} {\dot{\tilde{ϕ}}}_{2} & = M^{- 1} (y) sat (u (t)) + M^{- 1} (y) d (t) + f (y) - {\dot{\hat{π}}}_{1} - {\dot{ϖ}}_{2} \\ = M^{- 1} (y) (u + α) + M^{- 1} (y) d (t) + f (y) - {\dot{\hat{π}}}_{1} - {\dot{ϖ}}_{2} \\ = M^{- 1} (y) u + Θ (y) - {\dot{\hat{π}}}_{1} - {\dot{ϖ}}_{2} \end{array}

in which $Θ (y) = M^{- 1} (y) d (t) + f (y) + M^{- 1} (y) α = [Θ_{1} (y), Θ_{2} (y), \dots, Θ_{n} (y {)]}^{T} \in ℝ^{n}$ , $\dot{\hat{π}}$ is the output of (12). Since we do not know the function $Θ (y)$ , we can estimate its value by using NN (3)

{\hat{Θ}}_{i} = W_{i}^{T} ψ_{i} (y)

Define the ideal parameter of NN as $W_{i}^{*} = arg min_{W_{i}} [sup_{y} | {\hat{Θ}}_{i} - Θ_{i} |]$ . Define the ideal estimation error of ${\hat{Θ}}_{i}$ and the ideal parameter estimation error as ${\tilde{W}}_{i} = W_{i} - W_{i}^{*}$ and $ε_{i} = {\hat{Θ}}_{i}^{*} - Θ_{i}$ , respectively. It is rational to suppose that $| ε_{i} | \leq {\bar{ε}}_{i},$ with ${\bar{ε}}_{i} \in ℝ^{+}$ . As a result

\begin{array}{l} {\hat{Θ}}_{i} - Θ_{i} & = W_{i}^{T} ψ_{i} (y) - W_{i}^{* T} ψ_{i} (y) + W_{i}^{* T} ψ_{i} (y) - Θ_{i} \\ = {\tilde{W}}_{i}^{T} ψ_{i} (y) - ε_{i} \end{array}

For convenience, denote $ε = [ε_{1}, ε_{2}, \dots, ε_{n}]^{T} \in ℝ^{n}$ , $W (t) = [W_{1}^{T} (t), \dots, W_{n}^{T} (t {)]}^{T} \in ℝ^{N \times n}$ , $W^{T} ψ (y) = [W_{1}^{T} ψ_{1} (y), \dots, W_{n}^{T} ψ_{n} (y {)]}^{T} \in ℝ^{n \times n}$ , and ${\tilde{W}}^{T} ψ (y) = [{\tilde{W}}_{1}^{T} ψ_{1} (y), \dots, {\tilde{W}}_{n}^{T} ψ_{n} (y {)]}^{T} \in ℝ^{n \times n}$ . Thus, it follows from (18) that

\hat{Θ} - Θ = {\tilde{W}}^{T} ψ (y) - ε

Based on the above discussion, we can use the following compensated signal

{\dot{ϖ}}_{2} = - k_{2} ϖ_{2} - z_{1} - c_{2} sign (ϖ_{2})

with $k_{2}, c_{2} \in ℝ^{+}$ . Consequently, we implement the following controller

u = M (y) [- k_{2} ϕ_{2} + \dot{\hat{π}} - {\hat{W}}^{T} ψ (y) - σ sign ({\tilde{ϕ}}_{2}) - a_{2} {\tilde{ϕ}}_{2}^{ν}]

with $σ, a_{2} \in ℝ^{+}$ satisfying $σ \geq max {{\bar{ε}}_{1}, {\bar{ε}}_{2}, \dots, {\bar{ε}}_{n}}$ . Substituting (19), (20), and (21) into (16) yields

\begin{array}{l} {\dot{\tilde{ϕ}}}_{2} & = - k_{2} ϕ_{2} - {\hat{W}}^{T} ψ (y) - σ sign ({\tilde{ϕ}}_{2}) - a_{2} {\tilde{ϕ}}_{2}^{ν} + Θ (y) - {\dot{ϖ}}_{2} \\ = - k_{2} ϕ_{2} - {\tilde{W}}^{T} ψ (y) + ε - σ sign ({\tilde{ϕ}}_{2}) - a_{2} {\tilde{ϕ}}_{2}^{ν} + k_{2} ϖ_{2} - {\tilde{ϕ}}_{1} + c_{2} sign (ϖ_{2}) \\ = - k_{2} {\tilde{ϕ}}_{2} - {\tilde{W}}^{T} ψ (y) + ε - σ sign ({\tilde{ϕ}}_{2}) - a_{2} {\tilde{ϕ}}_{2}^{ν} - {\tilde{ϕ}}_{1} + c_{2} sign (ϖ_{2}) \end{array}

Multiplying ${\tilde{ϕ}}_{2}^{T}$ to both sides of (22) yields

\begin{array}{l} {\tilde{ϕ}}_{2}^{T} {\dot{\tilde{ϕ}}}_{2} & = - k_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2} - {\tilde{ϕ}}_{2}^{T} {\tilde{W}}^{T} ψ (y) + {\tilde{ϕ}}_{2}^{T} ε - σ {\tilde{ϕ}}_{2}^{T} sign ({\tilde{ϕ}}_{2}) - a_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2}^{ν} - {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{1} + c_{2} {\tilde{ϕ}}_{2}^{T} sign (ϖ_{2}) \\ = - k_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2} - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) + \sum_{i = 1}^{n} {\tilde{e}}_{2 i} ε_{i} - σ \sum_{i = 1}^{n} | {\tilde{e}}_{2 i} | - a_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2}^{ν} - {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{1} + c_{2} {\tilde{ϕ}}_{2}^{T} sign (ϖ_{2}) \\ \leq - k_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2} - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) + \sum_{i = 1}^{n} | {\tilde{e}}_{2 i} | {\bar{ε}}_{i} - σ \sum_{i = 1}^{n} | {\tilde{e}}_{2 i} | - a_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2}^{ν} - {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{1} + c_{2} {\tilde{ϕ}}_{2}^{T} sign (ϖ_{2}) \\ \leq - k_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2} - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) - a_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2}^{ν} - {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{1} + c_{2} {\tilde{ϕ}}_{2}^{T} sign (ϖ_{2}) \end{array}

Let the Lyapunov function be

V_{2} = V_{1} + \frac{1}{2} {\tilde{ϕ}}_{2} T {\tilde{ϕ}}_{2}

Then, (15), (23), and (24) imply

\begin{array}{l} {\dot{V}}_{2} & = - k_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1} - a_{1} {\tilde{ϕ}}_{1}^{T} {\tilde{ϕ}}_{1}^{ν} + c_{1} {\tilde{ϕ}}_{1}^{T} sign (ϖ_{1}) - k_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2} - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) \\ - a_{2} {\tilde{ϕ}}_{2}^{T} {\tilde{ϕ}}_{2}^{ν} + c_{2} {\tilde{ϕ}}_{2}^{T} sign (ϖ_{2}) \\ = - \sum_{j = 1}^{2} k_{j} {\tilde{ϕ}}_{j}^{T} {\tilde{ϕ}}_{j} - \sum_{j = 1}^{2} a_{j} {\tilde{ϕ}}_{j}^{T} {\tilde{ϕ}}_{j}^{ν} + \sum_{j = 1}^{2} c_{j} {\tilde{ϕ}}_{j}^{T} sign (ϖ_{j}) - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) \\ = \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- k_{j} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1} + c_{j} {\tilde{ϕ}}_{j i} sign (ϖ_{j})] - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) \end{array}

We can use the following adaptation law

{\dot{W}}_{i} = β_{1 i} e_{2 i} ψ_{i} (y) - β_{1 i} β_{2 i} W_{i}

with $β_{1 i}, β_{2 i} \in ℝ^{+}$ .

Thus, we have the following theorem:

Theorem 1

Consider system (9) under assumption 1. The virtual input in the first step is given by (14), which together with its derivative is estimated by the command filter (12). To cancel the estimation error, the compensated signals are given by (11) and (20). Thus, the implemented controller (21) and the adaptation law (26) guarantee that tracking error $ϕ_{1}$ converges to an arbitrary small region in finite time and all signals keep bounded.

Proof

Define

V = V_{2} + \sum_{i = 1}^{n} \frac{1}{2 β_{1 i}} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i}

Then, based on (25), (26), and (27), we obtain

\begin{array}{l} \dot{V} & \leq \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- k_{j} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1} + c_{j} {\tilde{ϕ}}_{j i} sign (ϖ_{j})] - \sum_{i = 1}^{n} e_{2 i} {\tilde{W}}_{i}^{T} ψ_{i} (y) + \sum_{i = 1}^{n} \frac{1}{β_{1 i}} {\tilde{W}}_{i}^{T} {\dot{W}}_{i} \\ = \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- k_{j} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1} + c_{j} {\tilde{ϕ}}_{j i} sign (ϖ_{j})] - \sum_{i = 1}^{n} β_{2 i} {\tilde{W}}_{i}^{T} W_{i} \\ \leq \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} β_{2 i} {\tilde{W}}_{i}^{T} W_{i} + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} \\ = \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} β_{2 i} {\tilde{W}}_{i}^{T} ({\tilde{W}}_{i} + W_{i}^{*}) + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} \\ = \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} β_{2 i} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} + \sum_{i = 1}^{n} β_{2 i} {\tilde{W}}_{i}^{T} W_{i}^{*} \\ \leq \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} \frac{3 β_{2 i}}{4} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} + \sum_{i = 1}^{n} β_{2 i} W_{i}^{* T} W_{i}^{*} \end{array}

Then, (28) implies

\begin{array}{l} \dot{V} & \leq \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} {(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} - \sum_{i = 1}^{n} \frac{3 β_{2 i}}{4} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} \\ + \sum_{i = 1}^{n} β_{2 i} W_{i}^{* T} W_{i}^{*} + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} + \sum_{i = 1}^{n} {(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} \end{array}

When ${(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} \geq 1$ , we have

\begin{array}{l} {(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} - \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + β_{2 i} W_{i}^{* T} W_{i}^{*} & \leq \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} - \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + β_{2 i} W_{i}^{* T} W_{i}^{*} = β_{2 i} W_{i}^{* T} W_{i}^{*} \end{array}

Else, when ${(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} < 1$ , we have

{(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} - \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + β_{2 i} W_{i}^{* T} W_{i}^{*} < 1 - \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + β_{2 i} W_{i}^{* T} W_{i}^{*} \leq 1 + β_{2 i} W_{i}^{* T} W_{i}^{*}

As a result, (30) and (31) imply

\begin{array}{l} {(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} - \frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + β_{2 i} W_{i}^{* T} W_{i}^{*} \leq 1 + β_{2 i} W_{i}^{* T} W_{i}^{*} \end{array}

Substituting (32) into (29) yields

\begin{array}{l} \dot{V} & \leq \sum_{j = 1}^{2} \sum_{i = 1}^{n} [- \frac{2 k_{j} - c_{j}}{2} {\tilde{ϕ}}_{j i}^{2} - a_{j} {\tilde{ϕ}}_{j i}^{ν + 1}] - \sum_{i = 1}^{n} {(\frac{β_{2 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2} (ν + 1)} - \sum_{i = 1}^{n} \frac{β_{2 i}}{4} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} \\ + \sum_{i = 1}^{n} [1 + β_{2 i} W_{i}^{* T} W_{i}^{*}] + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2} \\ \leq - q_{1} V - q_{2} V^{\frac{ν + 1}{2}} + q_{3} \end{array}

with $q_{1} = min {\frac{2 k_{1} - c_{1}}{2}, \frac{2 k_{2} - c_{2}}{2}, \dots, \frac{2 k_{n} - c_{n}}{2}, \frac{β_{min}}{2}},$ $q_{2} = min {2^{\frac{ν + 1}{2}} a_{1} {,2}^{\frac{ν + 1}{2}} a_{2}, \dots {,2}^{\frac{ν + 1}{2}} a_{n}, β_{min}^{\frac{ν + 1}{2}}},$ and $q_{3} = \sum_{i = 1}^{n} [1 + β_{2 i} W_{i}^{* T} W_{i}^{*}] + \sum_{j = 1}^{2} \sum_{i = 1}^{n} \frac{c_{j}}{2}$ being three constants, and $β_{min} = min {β_{1}, β_{2}, \dots, β_{n}}$ . Thus, we can rewrite (33) as

\dot{V} \leq - (q_{1} - \frac{q_{3}}{2 V}) V - (q_{2} - \frac{q_{3}}{2 V^{\frac{ν + 1}{2}}}) V^{\frac{ν + 1}{2}}

Based on (34) and lemma 2, if $k_{i} > \frac{1}{2} c_{i}$ , we know that $\tilde{ϕ} (t)$ tends to the region

∥ \tilde{ϕ} (t) ∥ \leq max {\sqrt{\frac{q_{3}}{q_{1}}}, \sqrt{2 {(\frac{q_{3}}{2 q_{2}})}^{\frac{ν + 1}{2}}}}

in a finite time. Note $ϕ = \tilde{ϕ} + ϖ$ with $ϖ = [ϖ_{1}, ϖ_{2}]^{T} \in ℝ^{2 n}$ , to prove the boundedness of all signals, we only need to show that $ϖ (t)$ is bounded. Define $V_{3} (t) = \frac{1}{2} ϖ^{T} z$ . It follows from (11) and (20) that

\begin{array}{l} {\dot{V}}_{3} & = ϖ_{1}^{T} {\dot{z}}_{1} + ϖ_{2}^{T} {\dot{z}}_{2} \\ = - k_{1} ϖ_{1}^{T} ϖ_{1} + ϖ_{1}^{T} (\hat{π} - π) + ϖ_{1}^{T} ϖ_{2} - c_{1} ϖ_{1}^{T} sign (ϖ_{1}) - k_{2} ϖ_{2}^{T} ϖ_{2} - ϖ_{2}^{T} z_{1} - c_{2} ϖ_{2}^{T} sign (ϖ_{2}) \\ \leq - \sum_{j = 1}^{2} \sum_{i = 1}^{n} k_{j} z_{j i}^{2} - \sum_{j = 1}^{2} \sum_{i = 1}^{n} c_{j} z_{j i} sign (z_{j i}) + ϖ_{1}^{T} (\hat{π} - π) \\ \leq - \sum_{j = 1}^{2} \sum_{i = 1}^{n} k_{j} z_{j i}^{2} - \sum_{j = 1}^{2} \sum_{i = 1}^{n} c_{j} | z_{j i} | + \sum_{j = 1}^{2} \sum_{i = 1}^{n} ρ_{j} z_{j i} {\tilde{π}}_{j i} \end{array}

with $ρ_{1} = 1, ρ_{2} = 0$ , ${\tilde{π}}_{1 i} = {\hat{π}}_{1 i} - π_{1 i}$ , ${\tilde{π}}_{2 i} = 0$ for $i = 1, 2, \dots, n$ .

Thus, (12) and lemma 3 imply that ${\tilde{π}}_{1 i}$ will be bounded in finite time. Then, it is easy to know that

\begin{array}{l} {\dot{V}}_{3} & \leq - \sum_{j = 1}^{2} \sum_{i = 1}^{n} k_{j} z_{j i}^{2} - \sum_{j = 1}^{2} \sum_{i = 1}^{n} c_{j} | z_{j i} | + \sum_{j = 1}^{2} \sum_{i = 1}^{n} ρ_{j} δ_{i} | z_{j i} | \\ \leq \underline{k} V_{3} - \underline{c} \sqrt{V_{3}} + \bar{ρ} \bar{δ} \sqrt{V_{3}} \\ = \underline{k} V_{3} - (\underline{c} - \bar{ρ} \bar{δ}) \sqrt{V_{3}} \end{array}

where $\underline{k} = 2 min {k_{i}}$ , $\underline{c} = min {c_{i}}$ , $\bar{ρ} = max {ρ_{1}, ρ_{2}}$ , and $\bar{δ} = max {δ_{i}}$ . Thus, the finite-time stability of $\tilde{ϕ}$ is achieved.

Remark 2

To prove theorem 1, a conclusion, that is, $| {\tilde{π}}_{1 i} | \leq δ_{i}$ is used. In fact, this conclusion is true because of the following reason: Based on lemma 3, we know that $| {\tilde{π}}_{1 i} |$ converges to zero in some finite time. However, if there the system is affected by external disturbances, for example, the disturbances given in the study by Levant²⁴ and Yu et al.,²⁵ we cannot guarantee that $| {\tilde{π}}_{1 i} |$ tends to zero in finite time. But in this case, we know that this signal will converge to a very small region near the origin in finite time.

Remark 3

In the controller design, some design parameters are given, for example, b ₁ and b ₂ in (12), which determine the converge speed of the filters, and they should be chosen largely to get rapid convergence; $k_{1}, k_{2}$ in the virtual input and the controller can be chosen randomly on $(0, + \infty)$ , but the greater value, the greater burden is put on the control system (in the simulation, they are set small to show the effectiveness of the control method); c ₁ and c ₂ in the compensated signal which can be chosen randomly on $(0, + \infty)$ ; $β_{1 i}, β_{2 i}$ in (26), small value of $β_{2 i}$ can be chosen to guarantee the boundedness of the signal W .

Remark 4

It can be concluded from (36) that all signals in the closed-loop system are bounded, and so does the signal u. Thus, from (8), since u_r is a bounded constant, thus $α = u - u_{r}$ is also a bounded signal.

Remark 5

It is well known that conventional backstepping control method has a very nerve-racking problem, that is, “explosion of terms,” which is produced by differentiating the virtual control input again and again. To solve this problem, in this article, we give a second-order command filter (12). Thus, we do not need to solve its derivative in the second step, and its derivative is also estimated by (12). In addition, the proposed filter (12) can estimate $π$ and its derivative in some finite time which is determined by design parameters. However, some related backstepping method, for example, the control method proposed by Farrell et al.,²² Dong et al.,³¹ and Pan and Yu,³³ has no such ability.

Simulation results

To show the control performance of the proposed method, in this part, simulation studies will be carried out by using a two-link robotic manipulator, which can be seen in Figure 2.

Figure 2.

A two-link robotic manipulator.

The details of system model (6) are given as follows: $F (\dot{x}) = [0.4 {\dot{x}}_{1},0.4 {\dot{x}}_{2}]^{T}$ , $G (x) = {[\begin{matrix} (\frac{1}{2} m_{1} + m_{2}) g l_{1} cos x_{2} + \frac{1}{2} m_{2} l_{2} g cos (x_{1} + x_{2}), \frac{1}{2} m_{2} g l_{2} cos (x_{1} + x_{2}) \end{matrix}]}^{T}$ , $M (x) = [\begin{matrix} m_{11} & m_{12} \\ m_{12} & m_{22} \end{matrix}],$ $C (x, \dot{x}) = {[\begin{matrix} - \frac{1}{2} m_{2} l_{1} l_{2} (2 {\dot{x}}_{1} {\dot{x}}_{2} + {\dot{x}}_{2}^{2}) sin x_{2}, \frac{1}{2} m_{2} l_{1} l_{2} {\dot{x}}_{1}^{2} sin x_{2} \end{matrix}]}^{T}$ , with $m_{11} = (\frac{1}{4} m_{1} + m_{2}) l_{1}^{2} + \frac{1}{4} m_{2} l_{2}^{2} + m_{2} l_{1} l_{2} cos x_{2}$ , $m_{12} = \frac{1}{4} m_{2} l_{2}^{2} + \frac{1}{2} m_{2} l_{1} l_{2} cos x_{2}$ , $m_{22} = \frac{1}{4} m_{2} l_{2}^{2}$ , $m_{1}, m_{2}$ being the mass, $l_{1}, l_{2}$ denoting the length. In this example, let $m_{1} = 4.0, m_{2} = 2.0, l_{1} = 2.0, l_{2} = 1.0$ .

According to the theoretical analysis, denoting $y = [y_{1}^{T}, y_{2}^{T}]^{T} \in ℝ^{4}$ with $y_{1} (t) = [x_{1}, x_{2}]^{T}, y_{2} (t) = [{\dot{x}}_{1}, {\dot{x}}_{2}]^{T}$ . Let desired signal be $y_{d} = {[0.5 sin (t),0.5 cos (t)]}^{T}$ . Let the parameters of the controller be $b_{1} = b_{2} = 0.7, k_{1} = 1.3, k_{2} = 1, β_{11} = β_{12} = 4, β_{21} = β_{22} = 0.01, c_{1} = c_{2} = 1.5$ . The initial condition is $y (0) = [1.5, 2, 0.2, - {0.5]}^{T}$ . The NNs use $x_{1}, x_{2}, {\dot{x}}_{1}$ , and ${\dot{x}}_{2}$ as its inputs. The initial condition for W is $W (0) = 0 \in ℝ^{81}$ . The saturation parameters are chosen as $u_{l 1} = u_{u 2} = - 5, u_{r 1} = u_{r 2} = 5$ .

Simulation results are indicated in Figures 3 and 4. In Figure 3(a) and (b), it is shown that the tracking errors and their derivatives converge to zero in short time, and the tracking performance is good. The control inputs are given in Figure 3(c), it is smooth because in the simulation, we replace the sign function with $arctan (\cdot)$ . The saturation phenomenon is shown in Figure 3(d). The tracking performance of x ₁ and x ₂ is shown in Figure 4(a) and (b), respectively. The simulation results of the proposed filter (12) is shown in Figure 4(c) and (d), where we can see that the tracking performance is very good.

Figure 3.

Control performance in (a) tracking errors, (b) derivatives of tracking errors, (c) control inputs, and (d) input saturation.

Figure 4.

Simulation results in (a) tracking performance of x ₁, (b) tracking performance of x ₂, (c) tracking performance of $π_{1}$ , and (d) tracking performance of $π_{2}$ .

Conclusion

In this article, we mainly investigate the finite-time tracking control for robotic manipulators with system uncertainties and disturbances. Firstly, we give a second-order command filter that can estimate the virtual input and its derivative in finite time, and the estimation error can be eliminated by a compensated signal. By doing this, we do not need to calculate the derivative of the virtual input repeatedly, and thus the “explosion of terms” problem will not occur. It is shown that the proposed method can drive the tracking error to the origin in finite time and keep all the signals bounded. It has also been shown that the proposed control method has good robustness. How to implement controller for robotic manipulators with experiment studies is one of our future research directions.

Footnotes

Data availability

The data used to support the findings of this study are included within the article.

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 Natural Science Research Key Project for Universities of Anhui Province (KJ2019A0108).

ORCID iD

Lin Wang

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