Adaptive fuzzy practical tracking control for flexible-joint robots via command filter design

Abstract

This paper investigates the issue of finite-time tracking control for flexible-joint robots. In the design scheme, the unknown continuous function is identified by a fuzzy system. By introducing the command filter technique, “explosion of complexity” problem which arises from repeated differentiation of virtual controllers is avoided. Meanwhile, errors resulting from the first-order filters can be reduced with the introduced compensation signal. Besides, the proposed method ensures that the tracking performance could be achieved within a limited time. Eventually, the simulation is given to demonstrate the effectiveness of the proposed scheme.

Keywords

Flexible-joint (FJ) robots adaptive tracking control command filter backstepping finite-time control

Introduction

Compared to the rigid-joint robots, flexible-joint (FJ) robots have many advantages of high performance, such as light mass, small size, and low energy consumption. They have been widely studied in the past decades.^1–5 Therefore, the research on tracking control of FJ robots also is of great significance.^6–8 For example, based on the singular perturbation method, Kim and Croft⁸ realize the full-state tracking control of FJ robots. With the aid of the tan-type barrier Lyapunov function, Sun et al.⁹ propose an adaptive tracking controller for FJ robot systems with full-state constraints. In fact, the FJ robot system is a typical under-actuated system, and the research about this kind of system can refer to the wheeled inverted pendulum system,¹⁰ the crane system,^11,12 and so on.

The backstepping method is used in many of the above papers; however, one of its disadvantages is that it requires repeated derivatives, which can result in “explosion of complexity” problem and increase the complexity of controllers, especially for the system with a higher dimension. Although the above troubles were handled by the dynamic surface control (DSC) method in previous works,^13–19 errors arising from the filters are not solved and the quality of the controller is also greatly reduced in this way. Another way is to apply the command filter technique to the backstepping design, by which the first problem can be successfully avoided. By introducing the compensated signal, the drawback of the DSC can be overcome (see the work by Farrell et al.,²⁰ Hu and Zhang,²¹ and Niu et al.²²). Considering the uncertain nonlinear systems with actuator faults, Li²³ developed a fault-tolerant control scheme by the aid of command filter design. For the switched nonlinear systems in Hou and Tong,²⁴ the issue of output feedback control is addressed with the command filter backstepping technique. For nonlinear systems with saturation input, the finite-time tracking control problem with command filter is investigated in Yu et al.²⁵

As we know, due to the ability to deal with structural uncertainty, the adaptive control method is widely employed to address uncertain nonlinear systems. With the quality of approximating unknown function, fuzzy logic systems (FLS) play a crucial part in handling the unknown items needed in control design. Therefore, the successful application of FLS in adaptive control can properly avoid burdensome computations and significantly improve the control performance of systems; many results have been obtained.^26–30 With the help of Nussbaum-type function in Sun et al.,²⁷ an adaptive fuzzy control method is proposed for the nonlinear systems with unknown control directions. For the high-order stochastic nonlinear systems, Sun et al.²⁸ consider the issue of reduced adaptive fuzzy control. Similarly, the unknown and uncertainty problems in this paper are also addressed by the adaptive fuzzy control scheme.

Inspired by the above works, this paper studies the problem of finite-time tracking control for FJ robots and develops an adaptive fuzzy control algorithm with the help of the command filter technique. The main contributions are summarized as follows:

Compared with the design in Sun et al.,⁹ the explosion of complexity problem is avoided by applying the command filter technique to the backstepping design. Thus, the computing burden is also reduced to some extent. With the aid of compensated signals, the errors resulting from the utilization of DSC in Liu and Wu¹⁵ can be removed.

Different from the existing schemes that can only guarantee infinite-time stability, this paper considers the convergence rate of tracking error and makes full use of the finite-time stability criterion to design an adaptive fuzzy controller, which ensures that the tracking error can achieve practical finite stable.

System description and preliminaries

The dynamic model of an n-link FJ robot can be expressed as

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) + G (q) + F (\overset{\cdot}{q}) + Kq = K q_{m}

(1)

J {\overset{\cdot\cdot}{q}}_{m} + B {\overset{\cdot}{q}}_{m} + K (q_{m} - q) = u

(2)

in which $q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q} \in R^{n}$ represent the link position, velocity, and acceleration vectors, respectively. $M (q) \in R^{n \times n}$ stands for the inertia matrix that is symmetric and positive definite, $C (q, \overset{\cdot}{q}) \in R^{n}$ is the Coriolis and centripetal forces, $G (q) \in R^{n}$ represents the gravity vector, and $F (\overset{\cdot}{q}) \in R^{n}$ denotes the friction term. $q_{m}, {\overset{\cdot}{q}}_{m}, {\overset{\cdot\cdot}{q}}_{m} \in R^{n}$ represent the rotor angular position, velocity, and acceleration vectors, respectively. $K, J, B \in R^{n \times n}$ are constant positive definite diagonal matrices and denote the joint flexibility, the actuator inertia, and the natural damping term, respectively. $u \in R^{n}$ is the torque input at each actuator.

The goal of design is to construct the adaptive tracking controller which can guarantee that the link position $q$ tracks the target signal $q_{d}$ in a finite time and all signals in the closed-loop system remain bounded, where $q_{d} \in C^{1}$ , $q_{d}$ and ${\overset{\cdot}{q}}_{d}$ are bounded.

Let $x_{1} = q$ , $x_{2} = \overset{\cdot}{q}$ , $x_{3} = q_{m}$ , and $x_{4} = {\overset{\cdot}{q}}_{m}$ , then equations (1) and (2) can be converted into

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = M^{- 1} (x_{1}) {Kx}_{3} + M^{- 1} (x_{1}) \\ [- C (x_{1}, x_{2}) - G (x_{1}) - F (x_{2}) - {Kx}_{1}] \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = J^{- 1} u + J^{- 1} [- {Bx}_{4} - K (x_{3} - x_{1})] \end{matrix}

(3)

In what follows, for simplicity, we note

{\begin{matrix} g_{1} = I \\ g_{2} = M^{- 1} (x_{1}) K \\ g_{3} = I \\ g_{4} = J^{- 1} \\ f_{2} = M^{- 1} (x_{1}) [- C (x_{1}, x_{2}) - G (x_{1}) - F (x_{2}) - K x_{1}] \\ f_{4} = J^{- 1} [- B x_{4} - K (x_{3} - x_{1})] \end{matrix}

(4)

Hence, equation (3) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = g_{2} x_{3} + f_{2} \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = g_{4} u + f_{4} \end{matrix}

(5)

Next, we will introduce some definitions and lemmas that are useful to achieve control objective.

Definition 1

For the nonlinear system $\overset{\cdot}{x} = f (x (t))$ , the equilibrium $x = 0$ is practical finite-time stable, if for any initial condition $x (0) \in x_{0}$ there exists a constant $ε > 0$ and the settling time $T (ε, x_{0}) < \infty$ such that³¹

‖ x (t) ‖ < ε, \forall t > T

Lemma 1

For any real numbers $α, β > 0$ , $0 < p, μ < 1$ , $0 < γ < \infty$ , if there exists the function $V (x)$ satisfying²³

\overset{\cdot}{V} (x) \leq - α V (x) - β V^{p} (x) + γ

then the trajectory of $\overset{\cdot}{x} = f (x (t))$ is practical finite-time stale, and the settling time $T$ satisfies

T \leq \frac{1}{α (1 - p)} \ln \frac{α V^{1 - p} x (0) + μ β}{μ β}

Lemma 2

For any constant $b_{i} > 0 (i = 1, 2, 3)$ and real variables $x, y$ , one has³²

{| x |}^{b_{1}} {| y |}^{b_{2}} \leq \frac{b_{1}}{b_{1} + b_{2}} b_{3} {| x |}^{b_{1} + b_{2}} + \frac{b_{2}}{b_{1} + b_{2}} b_{3}^{- \frac{b_{1}}{b_{2}}} {| y |}^{b_{1} + b_{2}}

FLS will be used to estimate the unknown continuous function in the subsequent design process.

IF-THEN Rules: $R_{i}$ : If $x_{1}$ is $F_{i}^{1}$ and ... and $x_{n}$ is $F_{i}^{n}$ , then $y$ is $G^{i}$ , $i = 1, . . ., n$ .

FLS can be expressed as

y (x) = \frac{\sum_{i = 1}^{N} Φ_{i} Π_{j = 1}^{n} μ_{F_{j}^{i}} (x_{j})}{\sum_{i = 1}^{N} [Π_{j = 1}^{n} μ_{F_{j}^{i}} (x_{j})]}

(6)

Let

w_{i} (x) = \frac{Π_{j = 1}^{n} μ_{F_{j}^{i}} (x_{j})}{\sum_{i = 1}^{N} [Π_{j = 1}^{n} μ_{F_{j}^{i}} (x_{j})]}

W (x) = [w_{1} (x), w_{2} (x), \dots, w_{N} (x)]^{T}, Φ = [Φ_{1}, Φ_{2}, \dots, Φ_{N}]^{T}

then we have

y (x) = Φ^{T} W (x)

(7)

Lemma 3

For any $ϵ > 0$ and a continuous function $h (x)$ defined on a compact set $U$ , there is an FLS $Φ^{T} W (x)$ satisfying³³

sup_{x \in U} | h (x) - Φ^{T} W (x) | \leq ϵ

(8)

Finite-time controller

This section is devoted to the design procedure of the command filtered controller.

Define the tracking error as

z_{1} = x_{1} - q_{d}

(9)

z_{i} = x_{i} - {\bar{α}}_{i}

(10)

where ${\bar{α}}_{i} \in R^{n}$ is the output of command filter with respect to $α_{i}$ , which is defined as

ε_{i} \overset{\cdot}{\bar{α_{i}}} + {\bar{α}}_{i} = α_{i}, {\bar{α}}_{i} (0) = α_{i} (0), i = 2, 3, 4

with $ε_{i} > 0$ being a designed constant.

To remove the filtering errors arising from the command filters, we employ the compensating signal $ξ_{i}$

{\begin{matrix} \dot{ξ_{1}} = - k_{1} ξ_{1} + ({\bar{α}}_{2} - α_{2}) + ξ_{2} - L_{1} sgn (ξ_{1}) \\ \dot{ξ_{2}} = - k_{2} ξ_{2} + g_{2} ({\bar{α}}_{3} - α_{3}) - ξ_{1} + g_{2} ξ_{3} - L_{2} sgn (ξ_{2}) \\ \dot{ξ_{3}} = - k_{3} ξ_{3} + ({\bar{α}}_{4} - α_{4}) - g_{2}^{T} ξ_{2} + ξ_{4} - L_{3} sgn (ξ_{3}) \\ \dot{ξ_{4}} = - k_{4} ξ_{4} + g_{4} ({\bar{α}}_{5} - α_{5}) - ξ_{3} - L_{4} sgn (ξ_{4}) \end{matrix}

(11)

where $k_{i} > 0$ , $ξ_{i} \in R^{n}$ , $ξ_{i} (0) = 0$ , $L_{i} = diag [l_{i 1}, l_{i 2}, . . ., l_{in}] \in R^{n \times n}$ , $l_{i} = [l_{i 1}, l_{i 2}, . . ., l_{in}]^{T} \in R^{n}$ , $l_{ij} > 0$ , $i = 1, . . ., 4$ , $j = 1, . . ., n$ , and $sgn (ξ_{i}) = [sgn (ξ_{i 1}), sgn (ξ_{i 2}), . . ., sgn (ξ_{in})]^{T} \in R^{n}$ . Furthermore, define the compensating tracking error as

χ_{i} = z_{i} - ξ_{i}, i = 1, . . ., 4

(12)

Before providing the following detailed design, we need to define a constant

θ = m a x {‖ Φ_{i} ‖^{2}}, i = 1, 2

and $\hat{θ}$ is the estimation of $θ$ .

Step 1. Taking the derivative of $χ_{1}$ yields

\begin{matrix} {\overset{\cdot}{χ}}_{1} = {\overset{\cdot}{z}}_{1} - {\overset{\cdot}{ξ}}_{1} \\ = x_{2} - {\overset{\cdot}{q}}_{d} - {\overset{\cdot}{ξ}}_{1} \\ = z_{2} + ({\bar{α}}_{2} - α_{2}) + α_{2} - {\overset{\cdot}{q}}_{d} - {\overset{\cdot}{ξ}}_{1} \\ = χ_{2} + α_{2} - {\overset{\cdot}{q}}_{d} + k_{1} z_{1} - k_{1} χ_{1} + L_{1} sgn (ξ_{1}) \end{matrix}

(13)

Define

V_{1} = \frac{1}{2} χ_{1}^{T} χ_{1}

(14)

then we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = χ_{1}^{T} [χ_{2} + α_{2} - {\overset{\cdot}{q}}_{d} + k_{1} z_{1} - k_{1} χ_{1} + L_{1} sgn (ξ_{1})] \\ = - k_{1} χ_{1}^{T} χ_{1} + χ_{1}^{T} χ_{2} + χ_{1}^{T} (α_{2} - {\overset{\cdot}{q}}_{d} + k_{1} z_{1}) \\ + χ_{1}^{T} L_{1} sgn (ξ_{1}) \end{matrix}

(15)

By the aid of Young’s inequality, one has

χ_{1}^{T} L_{1} sgn (ξ_{1}) \leq \frac{1}{2} χ_{1}^{T} χ_{1} + \frac{1}{2} l_{1}^{T} l_{1}

(16)

The virtual controller $α_{2}$ is designed as

α_{2} = {\overset{\cdot}{q}}_{d} - k_{1} z_{1} - c_{1} χ_{1} (χ_{1}^{T} χ_{1})^{p - 1}

(17)

with a known constant $c_{1} > 0$ . By plugging equations (16) and (17) into equation (15), we have

{\overset{\cdot}{V}}_{1} \leq - (k_{1} - \frac{1}{2}) χ_{1}^{T} χ_{1} - c_{1} {(χ_{1}^{T} χ_{1})}^{p} + χ_{1}^{T} χ_{2} + \frac{1}{2} l_{1}^{T} l_{1}

(18)

Step 2. The time derivative of $χ_{2}$ is

\begin{matrix} {\overset{\cdot}{χ}}_{2} = {\overset{\cdot}{z}}_{2} - {\overset{\cdot}{ξ}}_{2} \\ = g_{2} x_{3} + f_{2} - \overset{\cdot}{\bar{α_{2}}} - {\overset{\cdot}{ξ}}_{2} \\ = g_{2} z_{3} + g_{2} ({\bar{α}}_{3} - α_{3}) + g_{2} α_{3} + f_{2} - \overset{\cdot}{\bar{α_{2}}} - {\overset{\cdot}{ξ}}_{2} \\ = g_{2} χ_{3} + g_{2} α_{3} + f_{2} - \overset{\cdot}{\bar{α_{2}}} + k_{2} z_{2} - k_{2} χ_{2} \\ + ξ_{1} + L_{2} sgn (ξ_{2}) \end{matrix}

(19)

Choose

V_{2} = V_{1} + \frac{1}{2} χ_{2}^{T} χ_{2} + \frac{1}{2 r} {\tilde{θ}}^{2}

(20)

where $\tilde{θ} = θ - \hat{θ}$ , and $r > 0$ is a known constant. Then, we could get

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + χ_{2}^{T} [g_{2} χ_{3} + g_{2} α_{3} + f_{2} - \overset{\cdot}{\bar{α_{2}}} + k_{2} z_{2} - k_{2} χ_{2} \\ + ξ_{1} + L_{2} sgn (ξ_{2})] - \frac{1}{r} \tilde{θ} \overset{\cdot}{\hat{θ}} \\ \leq - (k_{1} - \frac{1}{2}) χ_{1}^{T} χ_{1} - c_{1} {(χ_{1}^{T} χ_{1})}^{p} + χ_{2}^{T} g_{2} χ_{3} \\ + \frac{1}{2} l_{1}^{T} l_{1} - \frac{1}{r} \tilde{θ} \overset{\cdot}{\hat{θ}} \\ + χ_{2}^{T} (z_{1} + g_{2} α_{3} - \overset{\cdot}{\bar{α_{2}}} + k_{2} z_{2}) + ‖ χ_{2}^{T} ‖ ‖ f_{2} ‖ \\ - k_{2} χ_{2}^{T} χ_{2} + χ_{2}^{T} L_{2} sgn (ξ_{2}) \end{matrix}

(21)

The following inequality is similar to equation (16)

χ_{2}^{T} L_{2} sgn (ξ_{2}) \leq \frac{1}{2} χ_{2}^{T} χ_{2} + \frac{1}{2} l_{2}^{T} l_{2}

(22)

In view of Lemma 3, $| | f_{2} | |$ is estimated by following FLS. For any $ϵ_{1} > 0$

\begin{matrix} ‖ f_{2} ‖ = Φ_{1}^{T} W_{1} (X_{1}) + δ_{1} (X_{1}), \\ | δ_{1} (X_{1}) | \leq ϵ_{1}, \end{matrix}

where $δ_{1} (X_{1})$ is the approximation error. With the completion of squares, it is obtained that

\begin{matrix} ‖ χ_{2}^{T} ‖ ‖ f_{2} ‖ = ‖ χ_{2}^{T} ‖ Φ_{1}^{T} W_{1} (X_{1}) + ‖ χ_{2}^{T} ‖ δ_{1} (X_{1}) \\ \leq \frac{θ χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} + \frac{χ_{2}^{T} χ_{2}}{2} + \frac{a_{1}^{2}}{2} + \frac{ϵ_{1}^{2}}{2} \end{matrix}

(23)

Substituting equations (22) and (23) into equation (21) produces

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - \sum_{i = 1}^{2} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - c_{1} {(χ_{1}^{T} χ_{1})}^{p} + χ_{2}^{T} g_{2} χ_{3} \\ + \sum_{i = 1}^{2} \frac{1}{2} l_{i}^{T} l_{i} - \frac{1}{r} \tilde{θ} \overset{\cdot}{\hat{θ}} \\ + χ_{2}^{T} (z_{1} + g_{2} α_{3} - \overset{\cdot}{\bar{α_{2}}} + k_{2} z_{2}) + \frac{θ χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} \\ + \frac{χ_{2}^{T} χ_{2}}{2} + \frac{a_{1}^{2}}{2} + \frac{ϵ_{1}^{2}}{2} \end{matrix}

(24)

The virtual controller $α_{3}$ is designed as

\begin{matrix} α_{3} = g_{2}^{- 1} \\ [- z_{1} + \overset{\cdot}{\bar{α_{2}}} - k_{2} z_{2} - c_{2} χ_{2} {(χ_{2}^{T} χ_{2})}^{p - 1} - \frac{\hat{θ} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} - \frac{χ_{2}}{2}] \end{matrix}

(25)

with $c_{2} > 0$ being a known constant. By plugging equation (25) into equation (24), one can get

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - \sum_{i = 1}^{2} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{2} {c_{i} (χ_{i}^{T} χ_{i})}^{p} + χ_{2}^{T} g_{2} χ_{3} \\ + \sum_{i = 1}^{2} \frac{1}{2} l_{i}^{T} l_{i} + \frac{a_{1}^{2}}{2} + \frac{ϵ_{1}^{2}}{2} + \tilde{θ} (\frac{χ_{2}^{T} χ_{2} P_{1}^{T} P_{1}}{2 a_{1}^{2}} - \frac{1}{r} \overset{\cdot}{\hat{θ}}) \end{matrix}

(26)

Step 3. From equation (12), the derivative of $χ_{3}$ gives

\begin{matrix} {\overset{\cdot}{χ}}_{3} = {\overset{\cdot}{z}}_{3} - {\overset{\cdot}{ξ}}_{3} \\ = x_{4} - \overset{\cdot}{\bar{α_{3}}} - {\overset{\cdot}{ξ}}_{3} \\ = z_{4} + ({\bar{α}}_{4} - α_{4}) + α_{4} - \overset{\cdot}{\bar{α_{3}}} - {\overset{\cdot}{ξ}}_{3} \\ = χ_{4} + α_{4} - \overset{\cdot}{\bar{α_{3}}} + k_{3} z_{3} - k_{3} χ_{3} + g_{2}^{T} ξ_{2} + L_{3} sgn (ξ_{3}) \end{matrix}

(27)

Choose

V_{3} = V_{2} + \frac{1}{2} χ_{3}^{T} χ_{3}

(28)

The derivation of equation (28) is presented as

\begin{matrix} {\overset{\cdot}{V}}_{3} = {\overset{\cdot}{V}}_{2} + χ_{3}^{T} [χ_{4} + α_{4} - \overset{\cdot}{\bar{α_{3}}} + k_{3} z_{3} - k_{3} χ_{3} \\ + g_{2}^{T} ξ_{2} + L_{3} sgn (ξ_{3})] \\ \leq - \sum_{i = 1}^{2} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{2} {c_{i} (χ_{i}^{T} χ_{i})}^{p} + χ_{3}^{T} χ_{4} \\ + \sum_{i = 1}^{2} \frac{1}{2} l_{i}^{T} l_{i} + \frac{a_{1}^{2}}{2} + \frac{ϵ_{1}^{2}}{2} + \tilde{θ} (\frac{χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} - \frac{1}{r} \overset{\cdot}{\hat{θ}}) \\ + χ_{3}^{T} [g_{2}^{T} z_{2} + α_{4} - \overset{\cdot}{\bar{α_{3}}} + k_{3} z_{3}] \\ - k_{3} χ_{3}^{T} χ_{3} + χ_{3}^{T} L_{3} sgn (ξ_{3}) \end{matrix}

(29)

Similar to equation (16), we obtain

χ_{3}^{T} L_{3} sgn (ξ_{3}) \leq \frac{1}{2} χ_{3}^{T} χ_{3} + \frac{1}{2} l_{3}^{T} l_{3}

(30)

Design the virtual controller $α_{4}$ as

α_{4} = - g_{2}^{T} z_{2} + \overset{\cdot}{\bar{α_{3}}} - k_{3} z_{3} - c_{3} χ_{3} (χ_{3}^{T} χ_{3})^{p - 1}

(31)

where $c_{3} > 0$ is a known constant. By substituting equations (30) and (31) into equation (29), we have

\begin{matrix} {\overset{\cdot}{V}}_{3} \leq - \sum_{i = 1}^{3} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{3} {c_{i} (χ_{i}^{T} χ_{i})}^{p} + χ_{3}^{T} χ_{4} \\ + \sum_{i = 1}^{3} \frac{1}{2} l_{i}^{T} l_{i} + \frac{a_{1}^{2}}{2} + \frac{ε_{1}^{2}}{2} + \tilde{θ} (\frac{χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} - \frac{1}{r} \overset{\cdot}{\hat{θ}}) \end{matrix}

(32)

Step 4. From equation (12), we have

\begin{matrix} {\overset{\cdot}{χ}}_{4} = {\overset{\cdot}{z}}_{4} - {\overset{\cdot}{ξ}}_{4} \\ = {\overset{\cdot}{x}}_{4} - \overset{\cdot}{\bar{α_{4}}} - {\overset{\cdot}{ξ}}_{4} \\ = g_{4} α_{5} + f_{4} - \overset{\cdot}{\bar{α_{4}}} + k_{4} z_{4} - k_{4} χ_{4} + ξ_{3} + L_{4} sgn (ξ_{4}) \end{matrix}

(33)

Choose the Lyapunov function as

V_{4} = V_{3} + \frac{1}{2} χ_{4}^{T} χ_{4}

(34)

It can be concluded that

\begin{matrix} {\overset{\cdot}{V}}_{4} = {\overset{\cdot}{V}}_{3} + χ_{4}^{T} [g_{4} α_{5} + f_{4} - \overset{\cdot}{\bar{α_{4}}} + k_{4} z_{4} - k_{4} χ_{4} + ξ_{3} \\ + L_{4} sgn (ξ_{4})] \\ \leq - \sum_{i = 1}^{3} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{3} {c_{i} (χ_{i}^{T} χ_{i})}^{p} \\ + \sum_{i = 1}^{3} \frac{1}{2} l_{i}^{T} l_{i} + \frac{a_{1}^{2}}{2} + \frac{ϵ_{1}^{2}}{2} \\ + \tilde{θ} (\frac{χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} - \frac{1}{r} \overset{\cdot}{\hat{θ}}) + χ_{4}^{T} (z_{3} + g_{4} α_{5} - \overset{\cdot}{\bar{α_{4}}} + k_{4} z_{4}) \\ + ‖ χ_{4}^{T} ‖ ‖ f_{4} ‖ - k_{4} χ_{4}^{T} χ_{4} + χ_{4}^{T} L_{4} sgn (ξ_{4}) \end{matrix}

(35)

By the aid of equation (8), $| | f_{4} | |$ can be approximated by following FLS

\begin{matrix} ‖ f_{4} ‖ = Φ_{2}^{T} W_{2} (X_{2}) + δ_{2} (X_{2}) \\ | δ_{2} (X_{2}) | \leq ϵ_{2} \end{matrix}

where $ϵ_{2}$ is an arbitrary positive constant and $δ_{2} (X_{2})$ is the approximation error. Based on the completion of squares, it is concluded that

\begin{matrix} ‖ χ_{4}^{T} ‖ ‖ f_{4} ‖ = ‖ χ_{4}^{T} ‖ Φ_{2}^{T} W_{2} (X_{2}) + ‖ χ_{4}^{T} ‖ δ_{2} (X_{2}) \\ \leq \frac{θ χ_{4}^{T} χ_{4} W_{2}^{T} W_{2}}{2 a_{2}^{2}} + \frac{χ_{4}^{T} χ_{4}}{2} + \frac{a_{2}^{2}}{2} + \frac{ϵ_{2}^{2}}{2} \end{matrix}

(36)

Similar to equation (16), we obtain

χ_{4}^{T} L_{4} sgn (ξ_{4}) \leq \frac{1}{2} χ_{4}^{T} χ_{4} + \frac{1}{2} l_{4}^{T} l_{4}

(37)

Substituting equations (36) and (37) into equation (35) yields

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{3} c_{i} {(χ_{i}^{T} χ_{i})}^{p} + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} \\ + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) + \tilde{θ} (\frac{χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} - \frac{1}{r} \overset{\cdot}{\hat{θ}}) \\ + χ_{4}^{T} (z_{3} + g_{4} α_{5} - \overset{\cdot}{\bar{α_{4}}} + k_{4} z_{4}) + \frac{θ χ_{4}^{T} χ_{4} W_{2}^{T} W_{2}}{2 a_{2}^{2}} + \frac{χ_{4}^{T} χ_{4}}{2} \end{matrix}

(38)

Design the actual controller $u$ and the adaptive law $\overset{\cdot}{\hat{θ}}$ as

\begin{matrix} u = α_{5} = g_{4}^{- 1} \\ [- z_{3} + \overset{\cdot}{\bar{α_{4}}} - k_{4} z_{4} - c_{4} χ_{4} {(χ_{4}^{T} χ_{4})}^{p - 1} - \frac{\hat{θ} χ_{4} W_{2}^{T} W_{2}}{2 a_{2}^{2}} - \frac{χ_{4}}{2}] \end{matrix}

(39)

\overset{\cdot}{\hat{θ}} = r (\frac{χ_{2}^{T} χ_{2} W_{1}^{T} W_{1}}{2 a_{1}^{2}} + \frac{χ_{4}^{T} χ_{4} W_{2}^{T} W_{2}}{2 a_{2}^{2}}) - ϱ \hat{θ}

(40)

where $c_{4} > 0$ and $ϱ > 0$ are known constants.

Substituting equations (39) and (40) into equation (38), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{4} {c_{i} (χ_{i}^{T} χ_{i})}^{p} \\ + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) + \frac{ϱ}{r} \tilde{θ} \hat{θ} \end{matrix}

(41)

Stability analysis

Theorem 1

Consider the nonlinear system (equation (3)), under the virtual controller (equations (17), (25), and (31)), the actual controller (equation (39)), and the adaptive law (equation (40)), the tracking error is practical finite stable and all signals in the resulting system are bounded.

Proof

According to $\tilde{θ} = θ - \hat{θ}$ , one has

\frac{ϱ}{r} \tilde{θ} \hat{θ} \leq - \frac{ϱ}{2 r} {\tilde{θ}}^{2} + \frac{ϱ}{2 r} θ^{2}

(42)

Substituting equation (42) into equation (41) produces

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{4} c_{i} {(χ_{i}^{T} χ_{i})}^{p} + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} \\ + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) - \frac{ϱ}{2 r} {\tilde{θ}}^{2} + \frac{ϱ}{2 r} θ^{2} \\ = - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{4} c_{i} {(χ_{i}^{T} χ_{i})}^{p} + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} \\ + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) - \frac{ϱ}{2 r} {\tilde{θ}}^{2} + \frac{ϱ}{2 r} θ^{2} - ϱ {(\frac{{\tilde{θ}}^{2}}{2 r})}^{p} \\ + ϱ {(\frac{{\tilde{θ}}^{2}}{2 r})}^{p} \end{matrix}

(43)

Using Lemma 2 to the term $ϱ ({\tilde{θ}}^{2} / 2 r)^{p}$ , the following inequality holds

ϱ {(\frac{{\tilde{θ}}^{2}}{2 r})}^{p} \leq q_{1} \frac{ϱ}{2 r} {\tilde{θ}}^{2} + ϱ (1 - p) {(\frac{p}{q_{1}})}^{\frac{p}{1 - p}}

(44)

with $0 < q_{1} < 1$ . By substituting equation (44) into equation (43), it can be proved that

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - \sum_{i = 1}^{4} c_{i} {(χ_{i}^{T} χ_{i})}^{p} + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} \\ + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) - \frac{ϱ}{2 r} {\tilde{θ}}^{2} + \frac{ϱ}{2 r} θ^{2} \\ - ϱ {(\frac{{\tilde{θ}}^{2}}{2 r})}^{p} + q_{1} \frac{ϱ}{2 r} {\tilde{θ}}^{2} + ϱ (1 - p) {(\frac{p}{q_{1}})}^{\frac{p}{1 - p}} \\ = - \sum_{i = 1}^{4} (k_{i} - \frac{1}{2}) χ_{i}^{T} χ_{i} - (1 - q_{1}) \frac{ϱ}{2 r} {\tilde{θ}}^{2} \\ - \sum_{i = 1}^{4} c_{i} {(χ_{i}^{T} χ_{i})}^{p} - ϱ {(\frac{{\tilde{θ}}^{2}}{2 r})}^{p} \\ + \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) + \frac{ϱ}{2 r} θ^{2} \\ + ϱ (1 - p) {(\frac{p}{q_{1}})}^{\frac{p}{1 - p}} \\ \leq - α V_{4} - β V_{4}^{p} + γ \end{matrix}

(45)

where

α = \min {(k_{i} - \frac{1}{2}), (1 - q_{1}) ϱ}, β = \min {2^{p} c_{i}, ϱ}, and

γ = \sum_{i = 1}^{4} \frac{1}{2} l_{i}^{T} l_{i} + \sum_{i = 1}^{2} (\frac{a_{i}^{2}}{2} + \frac{ϵ_{i}^{2}}{2}) + \frac{ϱ}{2 r} θ^{2} + ϱ (1 - p) {(\frac{p}{q_{1}})}^{\frac{p}{1 - p}}

Applying Lemma 1, we can draw the following conclusion

‖ χ_{i} ‖ \leq \sqrt{2 {(\frac{γ}{(1 - μ) β})}^{\frac{1}{p}}}, 0 < μ < 1

and $‖ χ_{i} ‖$ is bounded in a fixed time $T_{1}$ . Because $z_{i} = χ_{i} + ξ_{i}$ , we can conclude that $z_{i}$ is convergent in a fixed time if $ξ_{i}$ is bounded. Next, we will deal with this problem. Construct the Lyapunov function

V_{5} = \sum_{i = 1}^{4} \frac{1}{2} ξ_{i}^{T} ξ_{i}

(46)

It is concluded that

\begin{matrix} {\overset{\cdot}{V}}_{5} = - k_{1} ξ_{1}^{T} ξ_{1} + ξ_{1}^{T} ({\bar{α}}_{2} - α_{2}) + ξ_{1}^{T} ξ_{2} - ξ_{1}^{T} L_{1} sgn (ξ_{1}) \\ - k_{2} ξ_{2}^{T} ξ_{2} + ξ_{2}^{T} g_{2} ({\bar{α}}_{3} - α_{3}) - ξ_{2}^{T} ξ_{1} + ξ_{2}^{T} g_{2} ξ_{3} - ξ_{2}^{T} L_{2} sgn (ξ_{2}) \\ - k_{3} ξ_{3}^{T} ξ_{3} + ξ_{3}^{T} ({\bar{α}}_{4} - α_{4}) - ξ_{3}^{T} g_{2}^{T} ξ_{2} + ξ_{3}^{T} ξ_{4} - ξ_{3}^{T} L_{3} sgn (ξ_{3}) \\ - k_{4} ξ_{4}^{T} ξ_{4} + ξ_{4}^{T} g_{4} ({\bar{α}}_{5} - α_{5}) - ξ_{4}^{T} ξ_{3} - ξ_{4}^{T} L_{4} sgn (ξ_{4}) \\ = - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} + \sum_{i = 1}^{4} ξ_{i}^{T} g_{i} ({\bar{α}}_{i + 1} - α_{i + 1}) \\ - \sum_{i = 1}^{4} ξ_{i}^{T} L_{i} sgn (ξ_{i}) \\ = - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} + \sum_{i = 1}^{4} ξ_{i}^{T} [g_{i} ({\bar{α}}_{i + 1} - α_{i + 1}) - L_{i} sgn (ξ_{i})] \end{matrix}

\begin{matrix} \leq - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} + \sum_{i = 1}^{4} ‖ ξ_{i}^{T} ‖ ‖ g_{i} ({\bar{α}}_{i + 1} - α_{i + 1}) - L_{i} sgn (ξ_{i}) ‖ \\ = - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} - \sum_{i = 1}^{4} - ‖ ξ_{i}^{T} ‖ ‖ g_{i} ({\bar{α}}_{i + 1} - α_{i + 1}) - L_{i} sgn (ξ_{i}) ‖ \\ \leq - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} - \sum_{i = 1}^{4} ‖ ξ_{i}^{T} ‖ (‖ L_{i} sgn (ξ_{i}) ‖ - ‖ g_{i} ({\bar{α}}_{i + 1} - α_{i + 1}) ‖) \end{matrix}

(47)

According to the lemma in Farrell et al.,²⁰ $| | {\bar{α}}_{i + 1} - α_{i + 1} | | \leq η_{i}$ can be obtained in the fixed time $T_{2}$ with a known constant $η_{i}$ . Hence, we obtain the following conclusion by choosing a suitable matrix $L_{i}$

\begin{matrix} {\overset{\cdot}{V}}_{5} \leq - \sum_{i = 1}^{4} k_{i} ξ_{i}^{T} ξ_{i} - \sum_{i = 1}^{4} ‖ ξ_{i}^{T} ‖ (‖ L_{i} sgn (ξ_{i}) ‖ - ‖ g_{i} ‖ η_{i}) \\ \leq - k_{m} V_{5} - k_{n} V_{5}^{\frac{1}{2}} \end{matrix}

(48)

where $k_{m} = 2 \min {k_{i}}$ and $k_{n} = \sqrt{2} \min {| | L_{i} sgn (ξ_{i}) | | - | | g_{i} | | η_{i}}$ . Based on Lemma 1, we know that $ξ_{i}$ can converge to the origin in a finite time $T_{3}$ . It can be concluded that $z_{i}$ is practical finite stable within the fixed time $T = T_{1} + T_{2} + T_{3}$ .

Choose the Lyapunov function as

V = V_{4} = \sum_{i = 1}^{4} \frac{1}{2} χ_{i}^{T} χ_{i} + \frac{1}{2 r} {\tilde{θ}}^{2}

From equation (45), we obtain $\overset{\cdot}{V} (t) \leq - α V (t) + γ$ and $V (t) \leq (V (0) - (γ / α)) e^{- α t} + (γ / α)$ , which means that $\tilde{θ}$ and $\hat{θ}$ are bounded. $χ_{i} = z_{i} - ξ_{i}$ , then $z_{i}$ is bounded and $x_{i}$ must be bounded. $α_{i}$ is bounded since $α_{i}$ is the function of $z_{i}$ , $χ_{i}$ , and $\overset{\cdot}{\hat{θ}}$ . So all signals in the resulting system are bounded. The proof is completed.

Simulation example

To examine the efficiency of the proposed approach, we will carry out a simulation study for the single-link FJ manipulator

{\begin{matrix} m l^{2} \overset{\cdot\cdot}{q} + mgl \sin q + F (\overset{\cdot}{q}) + Kq = K q_{m} \\ J {\overset{\cdot\cdot}{q}}_{m} + B {\overset{\cdot}{q}}_{m} + K (q_{m} - q) = u \end{matrix}

(49)

where $q, q_{m} \in R$ . Let $m = 1 kg$ , $l = 1 m$ , $J = 1 kg \cdot m^{2}$ , $g = 10 m / s^{2}$ , $F (\overset{\cdot}{q}) = θ_{1} \cos \overset{\cdot}{q}$ , $K = 1$ , and $B = θ_{2}$ , where $θ_{1}$ and $θ_{2}$ are unknown parameters. $q_{d} = (1 / 2) \sin t$ is the expected trajectory. The following equations are the fuzzy membership functions required in the simulation

\begin{matrix} μ_{F_{i}^{1}} = e^{- 0.5 {(x + 1.5)}^{2}}, μ_{F_{i}^{2}} = e^{- 0.5 {(x + 1)}^{2}}, μ_{F_{i}^{3}} = e^{- 0.5 {(x + 0.5)}^{2}}, \\ μ_{F_{i}^{4}} = e^{- 0.5 {(x)}^{2}}, μ_{F_{i}^{5}} = e^{- 0.5 {(x - 0.5)}^{2}}, μ_{F_{i}^{6}} = e^{- 0.5 {(x - 1)}^{2}}, \\ μ_{F_{i}^{7}} = e^{- 0.5 {(x - 1.5)}^{2}} \end{matrix}

To achieve the control objective, we construct the virtual and actual controllers as equations (17), (25), (31), and (39) according to the method previously designed. Figures 1 –5 show the simulation results under the initial conditions $q (0) = 0.01$ , $\overset{\cdot}{q} (0) = 0$ , $q_{m} (0) = 0.01$ , and ${\overset{\cdot}{q}}_{m} (0) = 0$ . Figure 1 expresses the trajectories of the position state $q$ and reference signal $q_{d}$ . The trajectory of $q - q_{d}$ is shown in Figure 2, which indicates that $q - q_{d}$ can converge to a small neighborhood of zero in the finite time. That is, position state $q$ can follow target signal $q_{d}$ within a limited time. Figures 3 –5 are the trajectories of states $\overset{\cdot}{q}$ , $q_{m}$ , ${\overset{\cdot}{q}}_{m}$ , input $u$ , and adaptive law $\hat{θ}$ , respectively. It is found that states $\overset{\cdot}{q}$ , $q_{m}$ , ${\overset{\cdot}{q}}_{m}$ , input $u$ , and adaptive law $\hat{θ}$ are bounded. As a result, the proposed method can achieve the control objective.

Figure 1.

The trajectories of $q$ and $q_{d}$

Figure 2.

The trajectory of $q - q_{d} .$

Figure 3.

The trajectories of states $\overset{\cdot}{q}$ , $q_{m}$ , and ${\overset{\cdot}{q}}_{m}$ .

Figure 4.

The trajectory of input $u$ .

Figure 5.

The trajectory of adaptive law $\hat{θ}$ .

Conclusion

In this paper, the proposed scheme settles the issue of finite-time tracking control for FJ robots. With the help of the command filtered technology, both explosion of complexity and singularity problems in the standard backstepping design are avoided. By the aid of the finite-time control technique, the tracking error can achieve convergence quickly. The effectiveness of the proposed scheme is illustrated by simulation results. In this research direction, how to apply command filtered technology to under-actuated mechanical systems is meaningful work, such as the crane system.^34–36

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: The work was supported by the National Natural Science Foundation of China under Grants 61603170 and Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2019KJI010.

ORCID iD

Wei Sun

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