Boundary vibration control for a two-link rigid–flexible manipulator with quantized input

Abstract

The aim of this paper is to investigate the control problem for a two-link rigid–flexible manipulator with input quantization. Abundant studies of the quantized control problem are based on the lumped parameter system, which is expressed using ordinary differential equations. In this paper, the dynamic model for a two-link rigid–flexible manipulator is represented using partial differential equations. The controller design and analysis with input quantization are based on the partial differential equation model. By means of an auxiliary system, tracking errors and boundary vibrations can be eliminated. Simulations are given to verify the effectiveness of the proposed controller with quantized input.

Keywords

Two-link rigid–flexible manipulator boundary vibration control input quantization distributed parameter system partial differential equation model

1. Introduction

In recent years, modeling and controlling for flexible mechanical systems have been studied intensively by many researchers. Typical applications of flexible systems include flexible robotic manipulators (Heidari et al. 2013, 2017; He et al., 2019), flexible wings (He and Zhang, 2017), and axially moving systems (Zhao et al., 2018). The motivation behind the research is a desire for lighter weight, smaller size, and more workspace of the flexible mechanical systems. However, there is greater control complexity in flexible systems than in bulky rigid systems (Xin, 2011). Owing to the structural flexibility, more difficulty will occur in the design and analysis of the flexible system control.

As a consequence, many control schemes have been proposed to settle the control problem. Mahamood and Pedro (2011) designed a hybrid collocated proportional– derivative (PD) and a noncollocated proportional–integral–derivative (PID) controller for input tracking and vibration control of a two-link flexible manipulator. Xing and Liu (2017) develop a controller using linear matrix inequalities to reduce the vibration of a flexible string with external disturbance. A second-order PID terminal sliding mode control is proposed by Lochan et al. (2017), to achieve fast suppression of tip deflection for flexible manipulators.

More recently, the improvement of control performance where there are engineering problems has become a hot topic. For example, disturbances are considered by He et al. (2018a) and Zhao et al. (2019) and corresponding control schemes are designed for flexible mechanical systems. Liu et al. (2017), Meng et al. (2018), and He et al. (2018b) involve an input limitation problem in the control design. As for actuator faults, fault-tolerant control laws are proposed for a flexible manipulator by Cao and Liu (2018a) and a flexible wind turbine blade in Gao and Liu (2018).

With the development of networks, the design of networked control has drawn lots of attention. Quantization is a key issue in the high precision of networked control systems, since there will be bad effects on the system performance caused by quantization errors (Yun et al., 2010). Several approaches have been proposed to solve the problem in the last few years. Liberzon (2003) is concerned with the global asymptotic stabilization of continuous-time systems subject to quantization. The H_∞ state-feedback controller design for networked control systems is studied in Tian et al. (2007), with uncertain network-induced delay and data packet dropout. Liu et al. (2012) present a novel recursive design approach for output-feedback control of a class of nonlinear systems with actuator quantization. A quantized system with finite-level quantized input computed from quantized measurements is considered by Xia et al. (2013). A general class of strict feedback nonlinear systems is studied by Zhou et al. (2014); in this class, the input signal takes quantized values. Li and Lin (2017) aim to address the problem of global adaptive stabilization by output feedback for a class of nonlinear systems, in which both the input and output are quantized.

In this paper, the problem of input quantization for a two-link rigid–flexible manipulator via an auxiliary system is considered. In the cited literature, the analysis is based on the lumped parameter system, which can be represented using ordinary differential equations. By contrast, the two-link rigid–flexible manipulator is a distributed parameter system and should be described by partial differential equations (PDEs). Many control strategies for flexible systems are based on truncated models represented by ordinary differential equations. Most ordinary differential equation models are derived using finite-element methods or assumed modes methods from original PDE models. On the one hand, this strategy is convenient for controller design and analysis. On the other hand, it may lead to system instability problems, since it neglects higher modes. Recently, there are more and more studies adopting PDE models directly for flexible systems. Many control methods have subsequently been proposed for PDE model systems. For example, optimal trajectory control is studied by Zhang and Liu (2012) for a flexible two-link manipulator based on the PDE dynamic model. A state observer for PDE model-based control of a flexible manipulator is designed by Jiang et al. (2015). Boundary control for flexible string systems is proposed by Liu et al. (2019) in the presence of disturbances, on the basis of a PDE model. A restrained adaptive boundary iterative learning control law is proposed by He et al. (2018c) for an Euler–Bernoulli beam system based on a time-weighted Lyapunov–Krasovskii-like composite energy function. Inspired by these studies, the model in this paper for the two-link rigid–flexible manipulator is a PDE model derived using Hamilton's principle.

The main difference from our previous work on the two-link rigid–flexible manipulator (Cao and Liu 2017, 2018a, 2018b) is that the control problem is different. In Cao and Liu (2018b), the output constraints problem is studied. In Cao and Liu (2017), to perform repetitive tasks, an adaptive boundary iterative learning control scheme is proposed. In Cao and Liu (2018a), the actuator fault problem is studied in the presence of a boundary disturbance. These works did not refer to the input quantization problem. Input quantization is an inevitable problem in network control.

In this paper, the control problem for a two-link rigid–flexible manipulator is studied in the presence of input quantization. There are three contributions of this paper.

The PDE model of a two-link rigid–flexible manipulator is given in the presence of input quantization.

A quantized feedback controller is proposed to achieve both joint angle control and boundary vibration suppression.

The controller design and analysis with input quantization are based on the PDE model, by means of an auxiliary system.

This paper is organized as follows. In Section 2, the PDE dynamic model for a two-link rigid–flexible manipulator is presented. In Section 3, a boundary controller is designed via an auxiliary system, in the presence of quantized input signals. Numerical simulation results are shown in Section 4 and conclusions are made in Section 5.

2. System description

In this paper, the two-link rigid–flexible manipulator, which is shown in Figure 1, is considered.

Referring to Figure 1, $XOY$ and $xoy$ represent the global inertial coordinate system and the body-fixed coordinate system attached to the second link, respectively. x gives the location of the undeformed point on the flexible link. R denotes the position vector of an arbitrary point on the second link relative to the inertial frame. According to Figure 1, we give the following definition

\begin{matrix} R = [\begin{matrix} R_{x} \\ R_{y} \end{matrix}] = [\begin{matrix} L_{1} cos θ_{1} + x cos (θ_{1} + θ_{2}) - w (x, t) sin (θ_{1} + θ_{2}) \\ L_{1} sin θ_{1} + x sin (θ_{1} + θ_{2}) + w (x, t) cos (θ_{1} + θ_{2}) \end{matrix}] \end{matrix}

(1)

Hamilton's principle is applied to derive the dynamic model, which is expressed by

\int_{t_{1}}^{t_{2}} (δ E_{k} - δ E_{p} + δ W) d t = 0

(2)

where

δ (\cdot)

represents the variation of

(\cdot)

t_{1}

and

t_{2}

are two time constants, and

t_{1} < t < t_{2}

is the operating time.

The total kinetic energy $E_{k}$ can be calculated by

\begin{matrix} E_{k} = \frac{1}{2} J_{1} {\overset{\cdot}{θ}}_{1}^{2} + \frac{1}{2} [M_{h} L_{1}^{2} {\overset{\cdot}{θ}}_{1}^{2} + J_{2} ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) 2] \\ + \frac{1}{2} ρ \int_{0}^{L_{2}} {\overset{\cdot}{R}}^{T} \overset{\cdot}{R} d x + \frac{1}{2} M_{p} {\overset{\cdot}{R}}_{L_{2}}^{T} {\overset{\cdot}{R}}_{L_{2}} \end{matrix}

(3)

where

R_{L_{2}} = R | x = L_{2}

. In the following equations throughout this paper, a dot denotes the derivative with respect to time, namely,

(\overset{\cdot}{*}) = \partial (*) / \partial (*) \partial t \partial t

and

(\overset{··}{*}) = \partial 2 (*) / \partial 2 (*) \partial t^{2} \partial t^{2}

The potential energy $E_{p}$ is given by

E_{p} = \frac{EI}{2} \int_{0}^{L_{2}} {[w_{xx} (x, t)]}^{2} d x

(4)

For clarity, the following notation is introduced: $w_{x} = \partial w / \partial w \partial x \partial x$ , $w_{xx} = \partial 2 w / \partial 2 w \partial x^{2} \partial x^{2}$ , $w_{xxx} = \partial 3 w / \partial 3 w \partial x^{3} \partial x^{3}$ , $w_{xxxx} = \partial 4 w / \partial 4 w \partial x^{4} \partial x^{4}$ .

The nonconservative work W can be written as

W = τ_{1} θ_{1} + τ_{2} θ_{2} + uw (L_{2}, t)

(5)

Substitution of equations (3) to (5) into equation (2) yields the following dynamic equations

\begin{matrix} (\begin{matrix} J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3} + ρ L_{1} L_{2}^{2} cos θ_{2} \\ + ρ \int_{0}^{L_{2}} w^{2} (x, t) d x - 2 ρ L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{1}^{2} + L_{2}^{2} + w^{2} (L_{2}, t) + 2 L_{1} L_{2} cos θ_{2} - 2 L_{1} w (L_{2}, t) sin θ_{2}) \end{matrix}) {\overset{··}{θ}}_{1} \\ + (\begin{matrix} J_{2} + ρ \frac{L_{2}^{3}}{3} + ρ L_{1} \frac{L_{2}^{2}}{2} cos θ_{2} - ρ L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{2}^{2} + w^{2} (L_{2}, t) + L_{1} L_{2} cos θ_{2} - L_{1} w (L_{2}, t) sin θ_{2}) \end{matrix}) {\overset{··}{θ}}_{2} \\ - (\begin{matrix} ρ L_{1} L_{2}^{2} sin θ_{2} + 2 ρ L_{1} \int_{0}^{L_{2}} (w (x, t) cos θ_{2}) d x \\ + 2 M_{p} L_{1} L_{2} sin θ_{2} + 2 M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{1} {\overset{\cdot}{θ}}_{2} \\ - (\begin{matrix} ρ L_{1} \frac{L_{2}^{2}}{2} sin θ_{2} + ρ L_{1} \int_{0}^{L_{2}} (w (x, t) cos θ_{2}) d x \\ + M_{p} L_{1} L_{2} sin θ_{2} + M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{2}^{2} \\ - (\begin{matrix} 2 ρ L_{1} \int_{0}^{L_{2}} (\overset{\cdot}{w} (x, t) sin θ_{2}) d x - 2 ρ \int_{0}^{L_{2}} w (x, t) \overset{\cdot}{w} (x, t) d x \\ + 2 M_{p} L_{1} \overset{\cdot}{w} (L_{2}, t) sin θ_{2} - 2 M_{p} w (L_{2}, t) \overset{\cdot}{w} (L_{2}, t) \end{matrix}) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) \\ + ρ L_{1} \int_{0}^{L_{2}} (\overset{··}{w} (x, t) cos θ_{2}) d x + ρ \int_{0}^{L_{2}} x \overset{··}{w} (x, t) d x \\ + M_{p} L_{1} (\overset{··}{w} (L_{2}, t) cos θ_{2}) + M_{p} L_{2} \overset{··}{w} (L_{2}, t) = τ_{1} \end{matrix}

(6)

\begin{matrix} (\begin{matrix} J_{2} + ρ \int_{0}^{L_{2}} w (x, t) 2 d x + ρ L_{1} \frac{L_{2}^{2}}{2} cos θ_{2} \\ + ρ \frac{L_{2}^{3}}{3} - ρ L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{2}^{2} + L_{1} L_{2} cos θ_{2} + w (L_{2}, t) 2 - L_{1} sin θ_{2} w (L_{2}, t)) \end{matrix}) {\overset{··}{θ}}_{1} \\ + (J_{2} + ρ \frac{L_{2}^{3}}{3} + ρ \int_{0}^{L_{2}} w (x, t) 2 d x + M_{p} (L_{2}^{2} + w (L_{2}, t) 2)) {\overset{··}{θ}}_{2} \\ + (2 ρ \int_{0}^{L_{2}} w (x, t) \overset{\cdot}{w} (x, t) d x + 2 M_{p} w (L_{2}, t) \overset{\cdot}{w} (L_{2}, t)) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) \\ + (\begin{matrix} ρ L_{1} \frac{L_{2}^{2}}{2} sin θ_{2} + ρ L_{1} \int_{0}^{L_{2}} w (x, t) cos θ_{2} d x \\ + M_{p} L_{1} L_{2} sin θ_{2} + M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{1}^{2} \\ + ρ \int_{0}^{L_{2}} x \overset{··}{w} (x, t) d x - M_{p} L_{1} ({\overset{··}{θ}}_{1} sin θ_{2} w (L_{2}, t)) \\ + M_{p} L_{2} \overset{··}{w} (L_{2}, t) = τ_{2} \end{matrix}

(7)

\begin{matrix} ρ (x ({\overset{··}{θ}}_{1} + {\overset{··}{θ}}_{2}) + \overset{··}{w} (x, t)) + EI w_{xxxx} (x, t) + ρ L_{1} {\overset{··}{θ}}_{1} cos θ_{2} \\ + ρ L_{1} {\overset{\cdot}{θ}}_{1}^{2} sin θ_{2} - ρ w (x, t) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})^{2} = 0 \end{matrix}

(8)

\begin{matrix} (M_{p} \overset{··}{w} (L_{2}, t) + M_{p} L_{2} ({\overset{··}{θ}}_{1} + {\overset{··}{θ}}_{2})) + M_{p} L_{1} {\overset{··}{θ}}_{1} cos θ_{2} \\ + M_{p} L_{1} {\overset{\cdot}{θ}}_{1}^{2} sin θ_{2} - M_{p} w (L_{2}, t) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})^{2} \\ - EI w_{xxx} (L_{2}, t) = u \end{matrix}

(9)

w_{xx} (L_{2}, t) = 0

(10)

The natural boundary condition of the flexible beam is

w (0, t) = 0, w_{x} (0, t) = 0

(11)

In this paper, input quantization is considered. This is an unavoidable problem in the network control, owing to the limit of the communication bandwidth and word length. The feedback system in the presence of input quantization is shown in Figure 2.

In Figure 2, the quantizer could be viewed as the information coder. It can transform the analog signal to the finite-length signal.

For the system described by equations (6) to (11), the quantized inputs are defined as

τ_{1} = q (v_{1}) = δ_{1} round (\frac{v_{1}}{δ_{1}})

(12)

τ_{2} = q (v_{2}) = δ_{2} round (\frac{v_{2}}{δ_{2}})

(13)

u = q (v_{3}) = δ_{3} round (\frac{v_{3}}{δ_{3}})

(14)

$q (\cdot)$ is the quantization operator defined by a function round $(\cdot)$ that rounds a number to the nearest integer. $δ_{i} (i = 1, 2, 3)$ is called a quantizing level. Every quantizer has a δ, which represents the sensitivity of the quantizer. A smaller value of $δ_{i}$ indicates a higher sensitivity. If $δ_{i}$ is a constant, the quantizer is a static quantizer. If $δ_{i}$ is time-varying, the quantizer is a dynamic quantizer. In this paper, the quantizer is considered as a static quantizer, since this is convenient in actual engineering.

Lemma 1

For $q (\cdot)$ , we have (Fu and Xie, 2005)

| q (v_{i}) - v_{i} | \leq | v_{i} | (i = 1, 2, 3)

(15)

The proof is given in Appendix 2.

3. Controller design with input quantization

In this section, a boundary controller for the two-link rigid–flexible manipulator with input quantization is proposed. The final aim of the system is $θ_{1} \to θ_{1 d}$ , ${\overset{\cdot}{θ}}_{1} \to {\overset{\cdot}{θ}}_{1 d}$ , $θ_{2} \to θ_{2 d}$ , ${\overset{\cdot}{θ}}_{2} \to {\overset{\cdot}{θ}}_{2 d}$ , $w (L_{2}, t) \to 0$ and $\overset{\cdot}{w} (L_{2}, t) \to 0$ when $t \to \infty$ . $θ_{1 d}$ and $θ_{2 d}$ are desired angular positions.

Assumption 1

Considering that $θ_{1 d}$ and $θ_{2 d}$ are constants, we have ${\overset{\cdot}{θ}}_{1} - {\overset{\cdot}{θ}}_{1 d} = {\overset{\cdot}{θ}}_{1}$ and ${\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{2 d} = {\overset{\cdot}{θ}}_{2}$ .

First, the state space variables are defined as

\begin{matrix} {\begin{matrix} x_{1} = θ_{1} - θ_{1 d} \\ x_{2} = {\overset{\cdot}{θ}}_{1} - {\overset{\cdot}{θ}}_{1 d} \\ x_{3} = θ_{2} - θ_{2 d} \\ x_{4} = {\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{2 d} \\ x_{5} = w (L_{2}, t) \\ x_{6} = \overset{\cdot}{w} (L_{2}, t) \\ x_{7} = w (x, t) \\ x_{8} = \overset{\cdot}{w} (x, t) \end{matrix} \end{matrix}

(16)

In consideration of Assumption 1, we have

\begin{matrix} {\begin{matrix} x_{1} = θ_{1} - θ_{1 d} \\ x_{2} = {\overset{\cdot}{θ}}_{1} \\ x_{3} = θ_{2} - θ_{2 d} \\ x_{4} = {\overset{\cdot}{θ}}_{2} \\ x_{5} = w (L_{2}, t) \\ x_{6} = \overset{\cdot}{w} (L_{2}, t) \\ x_{7} = w (x, t) \\ x_{8} = \overset{\cdot}{w} (x, t) \end{matrix} \end{matrix}

(17)

Then, the system described by equations (6) to (11) can be rewritten as

\begin{matrix} {\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \frac{1}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} (τ_{1} + ♦_{1}) \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = \frac{1}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} (τ_{2} + ♦_{2}) \\ {\overset{\cdot}{x}}_{5} = x_{6} \\ {\overset{\cdot}{x}}_{6} = \frac{1}{M_{p}} (u + ♦_{3}) \\ {\overset{\cdot}{x}}_{7} = x_{8} \\ {\overset{\cdot}{x}}_{8} = \frac{1}{ρ} ♦_{4} \end{matrix} \end{matrix}

(18)

where

\begin{matrix} ♦_{1} = - (\begin{matrix} ρ L_{1} L_{2}^{2} cos θ_{2} + ρ \int_{0}^{L_{2}} w^{2} (x, t) d x - 2 L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{1}^{2} + L_{2}^{2} + w^{2} (L_{2}, t) + 2 L_{1} L_{2} cos θ_{2} \\ - 2 L_{1} w (L_{2}, t) sin θ_{2} \end{matrix})) {\overset{··}{θ}}_{1} \\ - (\begin{matrix} J_{2} + ρ \frac{L_{2}^{3}}{3} + ρ L_{1} \frac{L_{2}^{2}}{2} cos θ_{2} - ρ L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{2}^{2} + w^{2} (L_{2}, t) + L_{1} L_{2} cos θ_{2} - L_{1} w (L_{2}, t) sin θ_{2}) \end{matrix}) {\overset{··}{θ}}_{2} \\ + (\begin{matrix} ρ L_{1} L_{2}^{2} sin θ_{2} + 2 ρ L_{1} \int_{0}^{L_{2}} (w (x, t) cos θ_{2}) d x \\ + 2 M_{p} L_{1} L_{2} sin θ_{2} + 2 M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{1} {\overset{\cdot}{θ}}_{2} \\ + (\begin{matrix} ρ L_{1} \frac{L_{2}^{2}}{2} sin θ_{2} + ρ L_{1} \int_{0}^{L_{2}} (w (x, t) cos θ_{2}) d x \\ + M_{p} L_{1} L_{2} sin θ_{2} + M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{2}^{2} \end{matrix}

(19)

\begin{matrix} + (\begin{matrix} 2 ρ L_{1} \int_{0}^{L_{2}} (\overset{\cdot}{w} (x, t) sin θ_{2}) d x - 2 ρ \int_{0}^{L_{2}} w (x, t) \overset{\cdot}{w} (x, t) d x \\ + 2 M_{p} L_{1} \overset{\cdot}{w} (L_{2}, t) sin θ_{2} - 2 M_{p} w (L_{2}, t) \overset{\cdot}{w} (L_{2}, t) \end{matrix}) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) \\ - ρ L_{1} \int_{0}^{L_{2}} (\overset{··}{w} (x, t) cos θ_{2}) d x + ρ \int_{0}^{L_{2}} x \overset{··}{w} (x, t) d x \\ - M_{p} L_{1} (\overset{··}{w} (L_{2}, t) cos θ_{2}) + M_{p} L_{2} \overset{··}{w} (L_{2}, t) \end{matrix}

\begin{matrix} ♦_{2} = - (\begin{matrix} J_{2} + ρ \int_{0}^{L_{2}} w (x, t) 2 d x + ρ L_{1} \frac{L_{2}^{2}}{2} cos θ_{2} \\ + ρ \frac{L_{2}^{3}}{3} - ρ L_{1} \int_{0}^{L_{2}} sin θ_{2} w (x, t) d x \\ + M_{p} (L_{2}^{2} + L_{1} L_{2} cos θ_{2} + w (L_{2}, t) 2 \\ - L_{1} sin θ_{2} w (L_{2}, t) \end{matrix})) {\overset{··}{θ}}_{1} \\ - (ρ \int_{0}^{L_{2}} w (x, t) 2 d x + M_{p} w (L_{2}, t) 2) {\overset{··}{θ}}_{2} \\ - (2 ρ \int_{0}^{L_{2}} w (x, t) \overset{\cdot}{w} (x, t) d x + 2 M_{p} w (L_{2}, t) \overset{\cdot}{w} (L_{2}, t)) \\ \times ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) \\ - (\begin{matrix} ρ L_{1} \frac{L_{2}^{2}}{2} sin θ_{2} + ρ L_{1} \int_{0}^{L_{2}} w (x, t) cos θ_{2} d x \\ + M_{p} L_{1} L_{2} sin θ_{2} + M_{p} L_{1} w (L_{2}, t) cos θ_{2} \end{matrix}) {\overset{\cdot}{θ}}_{1}^{2} \\ - ρ \int_{0}^{L_{2}} x \overset{··}{w} (x, t) d x + M_{p} L_{1} ({\overset{··}{θ}}_{1} sin θ_{2} w (L_{2}, t)) \\ - M_{p} L_{2} \overset{··}{w} (L_{2}, t) \end{matrix}

(20)

\begin{matrix} ♦_{3} = - M_{p} L_{2} ({\overset{··}{θ}}_{1} + {\overset{··}{θ}}_{2}) - M_{p} L_{1} {\overset{··}{θ}}_{1} cos θ_{2} \\ - M_{p} L_{1} {\overset{\cdot}{θ}}_{1}^{2} sin θ_{2} + M_{p} w (L_{2}, t) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})^{2} \\ + EI w_{xxx} (L_{2}, t) \end{matrix}

(21)

\begin{matrix} ♦_{4} = - ρ x ({\overset{··}{θ}}_{1} + {\overset{··}{θ}}_{2}) - EI w_{xxxx} (x, t) - ρ L_{1} {\overset{··}{θ}}_{1} cos θ_{2} \\ - ρ L_{1} {\overset{\cdot}{θ}}_{1}^{2} sin θ_{2} + ρ w (x, t) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})^{2} \end{matrix}

(22)

We consider the following auxiliary system

\begin{matrix} {\begin{matrix} {\overset{\cdot}{\overset{\land}{}} x}_{1} = {\overset{\land}{x}}_{2} + l_{1} (x_{1} - {\overset{\land}{x}}_{1}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{2} = \frac{1}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} (v_{1} + ♦_{1}) + l_{2} (x_{1} - {\overset{\land}{x}}_{1}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{3} = {\overset{\land}{x}}_{4} + l_{3} (x_{3} - {\overset{\land}{x}}_{3}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{4} = \frac{1}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} (v_{2} + ♦_{2}) + l_{4} (x_{3} - {\overset{\land}{x}}_{3}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{5} = {\overset{\land}{x}}_{6} + l_{5} (x_{5} - {\overset{\land}{x}}_{5}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{6} = \frac{1}{M_{p}} (v_{3} + ♦_{3}) + l_{6} (x_{5} - {\overset{\land}{x}}_{5}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{7} = {\overset{\land}{x}}_{8} + l_{7} (x_{7} - {\overset{\land}{x}}_{7}) \\ {\overset{\cdot}{\overset{\land}{}} x}_{8} = \frac{1}{ρ} ♦_{4} + l_{8} (x_{7} - {\overset{\land}{x}}_{7}) \end{matrix} \end{matrix}

(23)

To achieve system stability according to the Lyapunov functions (equations (25) and (27)) and counteract the negative effect of quantized inputs, the controller is designed as

\begin{matrix} {\begin{matrix} v_{1} = J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3} (- k_{1} {\overset{\land}{x}}_{1} - k_{2} {\overset{\land}{x}}_{2} - ♦_{1}) \\ τ_{1} = q (v_{1}) \\ v_{2} = J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2} (- k_{1} {\overset{\land}{x}}_{3} - k_{2} {\overset{\land}{x}}_{4} - ♦_{2}) \\ τ_{2} = q (v_{2}) \\ v_{3} = M_{p} (- k_{1} {\overset{\land}{x}}_{5} - k_{2} {\overset{\land}{x}}_{6} - ♦_{3}) \\ u = q (v_{3}) \end{matrix} \end{matrix}

(24)

We define $e_{i} = x_{i} - {\overset{\land}{x}}_{i} (i = 1, 2, 3)$ . Next, we choose two Lyapunov functions to demonstrate $e_{i} \to 0$ and $x_{i} \to 0 (i = 1, 2, 3, 4, 5, 6)$ .

To achieve $e_{i} \to 0 (i = 1, 2, 3, 4, 5, 6)$ , the first Lyapunov function is chosen as

V_{1} = \frac{1}{2} e_{1}^{2} + \frac{1}{2} e_{2}^{2} + \frac{1}{2} e_{3}^{2} + \frac{1}{2} e_{4}^{2} + \frac{1}{2} e_{5}^{2} + \frac{1}{2} e_{6}^{2}

(25)

Taking the derivative of equation (25), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} {\overset{\cdot}{e}}_{1} + e_{2} {\overset{\cdot}{e}}_{2} + e_{3} {\overset{\cdot}{e}}_{3} + e_{4} {\overset{\cdot}{e}}_{4} + e_{5} {\overset{\cdot}{e}}_{5} + e_{6} {\overset{\cdot}{e}}_{6} \\ = e_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{\overset{\land}{}} x}_{1}) + e_{2} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{\overset{\land}{}} x}_{2}) \\ + e_{3} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{\overset{\land}{}} x}_{3}) + e_{4} ({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{\overset{\land}{}} x}_{4}) \\ + e_{5} ({\overset{\cdot}{x}}_{5} - {\overset{\cdot}{\overset{\land}{}} x}_{5}) + e_{6} ({\overset{\cdot}{x}}_{6} - {\overset{\cdot}{\overset{\land}{}} x}_{6}) \\ = e_{1} (x_{2} - {\overset{\land}{x}}_{2} - l_{1} (x_{1} - {\overset{\land}{x}}_{1})) \\ + e_{2} (\frac{τ_{1} + ♦_{1} - v_{1} - ♦_{1}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - l_{2} (x_{1} - {\overset{\land}{x}}_{1})) \\ + e_{3} (x_{4} - {\overset{\land}{x}}_{4} - l_{3} (x_{3} - {\overset{\land}{x}}_{3})) \\ + e_{4} (\frac{τ_{2} + ♦_{2} - v_{2} - ♦_{2}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - l_{4} (x_{3} - {\overset{\land}{x}}_{3})) \\ + e_{5} (x_{6} - {\overset{\land}{x}}_{6} - l_{5} (x_{5} - {\overset{\land}{x}}_{5})) \\ + e_{6} (\frac{u + ♦_{3} - v_{3} - ♦_{3}}{M_{p}} - l_{6} (x_{5} - {\overset{\land}{x}}_{5})) \\ = e_{1} (e_{2} - l_{1} e_{1}) \\ + e_{2} (\frac{q (v_{1}) - v_{1}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - l_{2} e_{1}) \\ + e_{3} (e_{4} - l_{3} e_{3}) \\ + e_{4} (\frac{q (v_{2}) - v_{2}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - l_{4} e_{3}) \end{matrix}

(26)

\begin{matrix} + e_{5} (e_{6} - l_{5} e_{5}) \\ + e_{6} (\frac{q (v_{3}) - v_{3}}{M_{p}} - l_{6} e_{5}) \\ = - l_{1} e_{1}^{2} + (1 - l_{2}) e_{1} e_{2} + \frac{e_{2} (q (v_{1}) - v_{1})}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} \\ - l_{3} e_{3}^{2} + (1 - l_{4}) e_{3} e_{4} + \frac{e_{4} (q (v_{2}) - v_{2})}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} \\ - l_{5} e_{5}^{2} + (1 - l_{6}) e_{5} e_{6} + \frac{1}{M_{p}} e_{6} (q (v_{3}) - v_{3}) \\ \leq - λ_{1} (e_{1}^{2} + e_{2}^{2}) \\ + \frac{\sqrt{k_{1}^{2} + k_{2}^{2}} ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2}) + \sqrt{k_{1}^{2} + k_{2}^{2}} (e_{1}^{2} + e_{2}^{2})}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} \\ - λ_{3} (e_{3}^{2} + e_{4}^{2}) \\ + \frac{\sqrt{k_{3}^{2} + k_{4}^{2}} ({\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2}) + \sqrt{k_{3}^{2} + k_{4}^{2}} (e_{3}^{2} + e_{4}^{2})}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} \\ - λ_{5} (e_{5}^{2} + e_{6}^{2}) + \frac{1}{M_{p}} \\ \times (\sqrt{k_{5}^{2} + k_{6}^{2}} ({\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) + \sqrt{k_{5}^{2} + k_{6}^{2}} (e_{5}^{2} + e_{6}^{2})) \end{matrix}

To obtain $x_{i} \to 0 (i = 1, 2, 3, 4, 5, 6)$ , we choose the second Lyapunov function as

V_{2} = \frac{1}{2} {\overset{\land}{x}}_{1}^{2} + \frac{1}{2} {\overset{\land}{x}}_{2}^{2} + \frac{1}{2} {\overset{\land}{x}}_{3}^{2} + \frac{1}{2} {\overset{\land}{x}}_{4}^{2} + \frac{1}{2} {\overset{\land}{x}}_{5}^{2} + \frac{1}{2} {\overset{\land}{x}}_{6}^{2}

(27)

Taking the derivative of equation (27), we get

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\land}{x}}_{1} {\overset{\cdot}{\overset{\land}{}} x}_{1} + {\overset{\land}{x}}_{2} {\overset{\cdot}{\overset{\land}{}} x}_{2} + {\overset{\land}{x}}_{3} {\overset{\cdot}{\overset{\land}{}} x}_{3} + {\overset{\land}{x}}_{4} {\overset{\cdot}{\overset{\land}{}} x}_{4} + {\overset{\land}{x}}_{5} {\overset{\cdot}{\overset{\land}{}} x}_{5} + {\overset{\land}{x}}_{6} {\overset{\cdot}{\overset{\land}{}} x}_{6} \\ = {\overset{\land}{x}}_{1} ({\overset{\land}{x}}_{2} + l_{1} (x_{1} - {\overset{\land}{x}}_{1})) + {\overset{\land}{x}}_{2} \\ \times (\frac{1}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} (v_{1} + ♦_{1}) + l_{2} (x_{1} - {\overset{\land}{x}}_{1})) \\ + {\overset{\land}{x}}_{3} ({\overset{\land}{x}}_{4} + l_{3} (x_{3} - {\overset{\land}{x}}_{3})) + {\overset{\land}{x}}_{4} \\ \times (\frac{1}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} (v_{2} + ♦_{2}) + l_{4} (x_{3} - {\overset{\land}{x}}_{3})) \\ + {\overset{\land}{x}}_{5} ({\overset{\land}{x}}_{6} + l_{5} (x_{5} - {\overset{\land}{x}}_{5})) + {\overset{\land}{x}}_{6} \\ \times (\frac{1}{M_{p}} (v_{3} + ♦_{3}) + l_{6} (x_{5} - {\overset{\land}{x}}_{5})) \\ = {\overset{\land}{x}}_{1} ({\overset{\land}{x}}_{2} + l_{1} e_{1}) + {\overset{\land}{x}}_{2} (- k_{1} {\overset{\land}{x}}_{1} - k_{2} {\overset{\land}{x}}_{2} + ♦_{1} + l_{2} e_{1}) \\ + {\overset{\land}{x}}_{3} ({\overset{\land}{x}}_{4} + l_{2} e_{3}) + {\overset{\land}{x}}_{4} (- k_{3} {\overset{\land}{x}}_{3} - k_{4} {\overset{\land}{x}}_{4} + ♦_{2} + l_{4} e_{3}) \\ + {\overset{\land}{x}}_{5} ({\overset{\land}{x}}_{6} + l_{1} e_{5}) + {\overset{\land}{x}}_{6} (- k_{5} {\overset{\land}{x}}_{5} - k_{6} {\overset{\land}{x}}_{6} + ♦_{3} + l_{6} e_{5}) \\ = (1 - k_{1}) {\overset{\land}{x}}_{1} {\overset{\land}{x}}_{2} - k_{2} {\overset{\land}{x}}_{2}^{2} + (l_{1} {\overset{\land}{x}}_{1} + l_{2} {\overset{\land}{x}}_{2}) e_{1} \\ + (1 - k_{3}) {\overset{\land}{x}}_{3} {\overset{\land}{x}}_{4} - k_{4} {\overset{\land}{x}}_{4}^{2} + (l_{3} {\overset{\land}{x}}_{3} + l_{4} {\overset{\land}{x}}_{4}) e_{3} \\ + (1 - k_{5}) {\overset{\land}{x}}_{5} {\overset{\land}{x}}_{6} - k_{6} {\overset{\land}{x}}_{6}^{2} + (l_{5} {\overset{\land}{x}}_{5} + l_{6} {\overset{\land}{x}}_{6}) e_{5} \end{matrix}

(28)

\begin{matrix} \leq - λ_{2} ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2}) + ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2}) + (l_{1}^{2} + l_{2}^{2}) (e_{1}^{2} + e_{2}^{2}) \\ - λ_{4} ({\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2}) + ({\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2}) + (l_{3}^{2} + l_{4}^{2}) (e_{3}^{2} + e_{4}^{2}) \\ - λ_{6} ({\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) + ({\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) + (l_{5}^{2} + l_{6}^{2}) (e_{5}^{2} + e_{6}^{2}) \end{matrix}

Therefore

\begin{matrix} {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2} \\ \leq - (λ_{1} - \frac{\sqrt{k_{1}^{2} + k_{2}^{2}}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - (l_{1}^{2} + l_{2}^{2})) \\ \times (e_{1}^{2} + e_{2}^{2}) \\ - (λ_{2} - \frac{\sqrt{k_{1}^{2} + k_{2}^{2}}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - 1) ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2}) \\ - (λ_{3} - \frac{\sqrt{k_{3}^{2} + k_{4}^{2}}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - (l_{3}^{2} + l_{4}^{2})) (e_{3}^{2} + e_{4}^{2}) \\ - (λ_{4} - \frac{\sqrt{k_{3}^{2} + k_{4}^{2}}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - 1) ({\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2}) \\ - (λ_{5} - \frac{\sqrt{k_{5}^{2} + k_{6}^{2}}}{M_{p}} - (l_{5}^{2} + l_{6}^{2})) (e_{5}^{2} + e_{6}^{2}) \\ - (λ_{6} - \frac{\sqrt{k_{5}^{2} + k_{6}^{2}}}{M_{p}} - 1) ({\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) \end{matrix}

(29)

where

λ_{1} = max l_{1} + \sqrt{l_{1}^{2} + (l_{2} - 1) 2}, l_{1} - \sqrt{l_{1}^{2} + (l_{2} - 1) 2}

λ_{2} = max k_{2} + \sqrt{k_{2}^{2} + (k_{1} - 1) 2}, k_{2} - \sqrt{k_{2}^{2} + (k_{1} - 1) 2}

λ_{3} = max l_{3} + \sqrt{l_{3}^{2} + (l_{4} - 1) 2}, l_{3} - \sqrt{l_{3}^{2} + (l_{4} - 1) 2}

λ_{4} = max k_{4} + \sqrt{k_{4}^{2} + (k_{3} - 1) 2}, k_{4} - \sqrt{k_{4}^{2} + (k_{3} - 1) 2}

λ_{5} = max l_{5} + \sqrt{l_{5}^{2} + (l_{6} - 1) 2}, l_{5} - \sqrt{l_{5}^{2} + (l_{6} - 1) 2}

, and

λ_{6} = max k_{6} + \sqrt{k_{6}^{2} + (k_{5} - 1) 2}, k_{6} - \sqrt{k_{6}^{2} + (k_{5} - 1) 2}

We choose

\begin{matrix} λ_{1} - \frac{\sqrt{k_{1}^{2} + k_{2}^{2}}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - (l_{1}^{2} + l_{2}^{2}) > 0, \\ λ_{2} - \frac{\sqrt{k_{1}^{2} + k_{2}^{2}}}{J_{1} + M_{h} L_{1}^{2} + J_{2} + ρ L_{1}^{2} L_{2} + ρ \frac{L_{2}^{3}}{3}} - 1 > 0, \end{matrix}

(30)

\begin{matrix} λ_{3} - \frac{\sqrt{k_{3}^{2} + k_{4}^{2}}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - (l_{3}^{2} + l_{4}^{2}) > 0, \\ λ_{4} - \frac{\sqrt{k_{3}^{2} + k_{4}^{2}}}{J_{2} + ρ \frac{L_{2}^{3}}{3} + M_{p} L_{2}^{2}} - 1 > 0, \\ λ_{5} - \frac{\sqrt{k_{5}^{2} + k_{6}^{2}}}{M_{p}} - (l_{5}^{2} + l_{6}^{2}) > 0, \\ λ_{6} - \frac{\sqrt{k_{5}^{2} + k_{6}^{2}}}{M_{p}} - 1 > 0 \end{matrix}

It leads to

\begin{matrix} \overset{\cdot}{V} \leq - ɛ (e_{1}^{2} + e_{2}^{2} + e_{3}^{2} + e_{4}^{2} + e_{5}^{2} + e_{6}^{2}) \\ - ɛ ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2} + {\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2} + {\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) \leq 0 \end{matrix}

(31)

Since

\begin{matrix} V = V_{1} + V_{2} \\ = \frac{1}{2} e_{1}^{2} + \frac{1}{2} e_{2}^{2} + \frac{1}{2} e_{3}^{2} + \frac{1}{2} e_{4}^{2} + \frac{1}{2} e_{5}^{2} + \frac{1}{2} e_{6}^{2} \\ + \frac{1}{2} {\overset{\land}{x}}_{1}^{2} + \frac{1}{2} {\overset{\land}{x}}_{2}^{2} + \frac{1}{2} {\overset{\land}{x}}_{3}^{2} + \frac{1}{2} {\overset{\land}{x}}_{4}^{2} + \frac{1}{2} {\overset{\land}{x}}_{5}^{2} + \frac{1}{2} {\overset{\land}{x}}_{6}^{2} \\ \leq β (e_{1}^{2} + e_{2}^{2} + e_{3}^{2} + e_{4}^{2} + e_{5}^{2} + e_{6}^{2}) \\ + β ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2} + {\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2} + {\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) = β V \end{matrix}

(32)

where

β \geq 1 / 122

, we have

\begin{matrix} \overset{\cdot}{V} \leq - ɛ (e_{1}^{2} + e_{2}^{2} + e_{3}^{2} + e_{4}^{2} + e_{5}^{2} + e_{6}^{2}) \\ - ɛ ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2} + {\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2} + {\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) \\ = - \frac{ɛ}{β} β (e_{1}^{2} + e_{2}^{2} + e_{3}^{2} + e_{4}^{2} + e_{5}^{2} + e_{6}^{2}) \\ - \frac{ɛ}{β} β ({\overset{\land}{x}}_{1}^{2} + {\overset{\land}{x}}_{2}^{2} + {\overset{\land}{x}}_{3}^{2} + {\overset{\land}{x}}_{4}^{2} + {\overset{\land}{x}}_{5}^{2} + {\overset{\land}{x}}_{6}^{2}) \\ \leq - \frac{ɛ}{β} V \end{matrix}

(33)

Based on equation (29), we have

V (t) \leq e^{- \frac{ɛ}{β} t} V (0)

(34)

According to this analysis, when $t \to \infty$ , we have $x_{i} \to 0$ , that is, $θ_{1} \to θ_{1 d}$ , ${\overset{\cdot}{θ}}_{1} \to {\overset{\cdot}{θ}}_{1 d}$ , $θ_{2} \to θ_{2 d}$ , ${\overset{\cdot}{θ}}_{2} \to {\overset{\cdot}{θ}}_{2 d}$ , $w (L_{2}, t) \to 0$ , and $\overset{\cdot}{w} (L_{2}, t) \to 0$ .

4. Simulations

In this section, simulation technique is used to demonstrate the effectiveness of the proposed controller with input quantization. The dynamic PDE model of equations (6) to (11) constitutes a set of highly nonlinear and hybrid differential equations. The finite-difference approach is applied to solve the PDEs. The time step and the space step are, respectively, set as $Δ t = 2 \times 10^{- 4}$ s and $Δ x = 6 \times 10^{- 2}$ m. Without loss of generality, initial conditions of the system states are set to be zero. The parameters of the two-link rigid–flexible manipulator are: $L_{1} = 0.1$ m, $L_{2} = 0.6$ m, $J_{1} = 0.45$ kgm², $J_{2} = 0.08$ kgm², $M_{h} = 3.0$ kg, $M_{P} = 8.5$ kg, $EI = 9$ Nm², $ρ = 0.6$ kg/m. The control objectives are $θ_{1 d} = 0.2 rad$ , ${\overset{\cdot}{θ}}_{1 d} = 0$ , $θ_{2 d} = 0.25 rad$ , ${\overset{\cdot}{θ}}_{2 d} = 0$ , $w (L_{2}, t) = 0$ , $\overset{\cdot}{w} (L_{2}, t) = 0$ as $t \to \infty$ .

The parameters in the controller are listed in Table 1. The performance of the closed-loop system is provided through the simulation results shown in Figures 3 to 8. From Figure 3 we can see that the first desired angles can be tracked within $1.5$ s. Similarly, Figure 4 shows that the second desired angles can be tracked within $2$ s. From Figure 5, we know that boundary vibration can be eliminated. In conclusion, the following control objectives are achieved within 4 s: $θ_{1} - θ_{1 d} = 0$ , ${\overset{\cdot}{θ}}_{1} = 0$ , $θ_{2} - θ_{2 d} = 0$ , ${\overset{\cdot}{θ}}_{2} = 0$ , $w (L_{2}, t) = 0$ and $\overset{\cdot}{w} (L_{2}, t) = 0$ . Two joint angles and speeds can track the given values in a limit time. The boundary deflection and vibration is eliminated quickly with the proposed controller.

Table 1.

Parameters of control laws.

Parameter	Description	Value
$k_{1}$	Controller gains of control laws (23)	2
$k_{2}$	Controller gains of control laws (23)	0.5
$k_{3}$	Controller gains of control laws (24)	3
$k_{4}$	Controller gains of control laws (24)	0.8
$k_{5}$	Controller gains of control laws (25)	3
$k_{6}$	Controller gains of control laws (25)	1
$δ_{1}$	Quantizing level of the first input	0.05
$δ_{2}$	Quantizing level of the second input	0.4
$δ_{3}$	Quantizing level of boundary input	0.3

The relationships between actual inputs and designed inputs are demonstrated in Figures 6 to 8. The actual inputs take the quantized values of the designed inputs. In Figure 6, $δ_{1} = 0.05$ . In Figure 7, $δ_{2} = 0.3$ , and $δ_{3} = 0.4$ in Figure 8. The simulation results demonstrate that a smaller value of $δ_{i}$ indicates a higher sensitivity.

In conclusion, the simulation results verify the effectiveness of the proposed controller.

5. Conclusions

In this paper, the control problem for a two-link rigid–flexible manipulator with input quantization is studied. By use of Hamilton's principle, the PDE dynamic model for the flexible system is derived. With the help of an auxiliary system, the bad effect of input quantization can be removed. It is worth mentioning that all the analyses are based on the original PDE model. Simulation studies show that the tracking errors and boundary vibration are eliminated. Future work in this area will focus on the input delay. Many practical systems include input delays that, as is well-known, have negative effects on stability and performance. The controller design for the flexible manipulator in the presence of input delays will be studied in future.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 61873296) and the Academic Excellence Foundation of BUAA for PhD Students.

References

Cao

Liu

(2017) An adaptive iterative learning algorithm for boundary control of a coupled ODE–PDE two-link rigid–flexible manipulator. Journal of the Franklin Institute 354(1): 277–297.

Cao

Liu

(2018a) Adaptive actuator fault compensation control for a rigid-flexible manipulator with ODEs-PDEs model. International Journal of Systems Science 49(8): 1748–1759.

Cao

Liu

(2018b) Boundary control for a constrained two-link rigid-flexible manipulator with prescribed performance. International Journal of Control 91(5): 1091–1103.

Xie

(2005) The sector bound approach to quantized feedback control. IEEE Transactions on Automatic Control 50(11): 1698–1711.

Gao

Liu

(2018) Adaptive fault-tolerant vibration control of a wind turbine blade with actuator stuck. International Journal of Control. Epub ahead of print 16 Oct 2017. DOI: 10.1080/00207179.2018.1484572.

Zhang

(2017) Control design for nonlinear flexible wings of a robotic aircraft. IEEE Transactions on Control Systems Technology 25(1): 351–357.

Zou

, et al. (2019) PDE model based boundary control design for a flexible robotic manipulator with input backlash. IEEE Transactions on Control Systems Technology 27(2): 790–797.

Meng

, et al. (2018a) Iterative learning control for a flapping wing micro aerial vehicle under distributed disturbances. IEEE Transactions on Cybernetics 49(4): 1524–1535.

Meng

, et al. (2018b) Unified iterative learning control for flexible structures with input constraints. Automatica 86: 326–336.

10.

Meng

Huang

, et al. (2018c) Adaptive boundary iterative learning control for an Euler–Bernoulli beam system with input constraint. IEEE Transactions on Neural Networks and Learning Systems 29(5): 1539–1549.

11.

Heidari

Korayem

Haghpanahi

, et al. (2013) Optimal trajectory planning for flexible link manipulators with large deflection using a new displacements approach. Journal of Intelligent and Robotic Systems 72(3–4): 287–300.

12.

Heidari

Korayem

Haghpanahi

(2017) Optimal trajectory planning for increased stability of mobile flexible manipulators undergoing large deflection. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 231(1): 85–95.

13.

Jiang

Liu

(2015) A robust observer design for a flexible manipulator based on a PDE model. Journal of Vibration & Control 23(6): 871–882.

14.

Lin

(2017) Adaptive output feedback control for a class of nonlinear systems with quantised input and output. International Journal of Control 90(2): 239–248.

15.

Liberzon

(2003) Hybrid feedback stabilization of systems with quantized signals. Automatica 39(9): 1543–1554.

16.

Liu

Jiang

Hill

(2012) Small-gain based output-feedback controller design for a class of nonlinear systems with actuator dynamic quantization. IEEE Transactions on Automatic Control 57(5): 1326–1332.

17.

Liu

(2017) Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint. Automatica 77: 302–310.

18.

Liu

Zhao

Ahn

(2019) Boundary constrained control of flexible string systems subject to disturbances. IEEE Transactions on Circuits and Systems II. Epub ahead of print 25 February 2019. DOI: 10.1109/TCSII.2019.2901283.

19.

Lochan

Roy

Subudhi

(2017) Robust tip trajectory synchronisation between assumed modes modelled two-link flexible manipulators using second-order PID terminal SMC. Robotics and Autonomous Systems 97: 108–124.

20.

Mahamood R and Pedro JO (2011) Hybrid PD/PID controller design for two-link flexible manipulators. In: 8th Asian control conference, Kaohsiung, Taiwan, 15–18 May 2011, pp. 1358–1363. Piscataway, NJ: IEEE.

21.

Meng

Ren

Wen

(2018) Boundary adaptive robust control of a flexible riser system with input nonlinearities. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 49: 1971–1980.

22.

Tian

Yue

Zhao

(2007) Quantised control design for networked control systems. IET Control Theory & Applications 1(6): 1693–1699. .

23.

Xia

Yan

Shi

, et al. (2013) Stability analysis of discrete-time systems with quantized feedback and measurements. IEEE Transactions on Industrial Informatics 9(1): 313–324.

24.

Xin X (2011) Swing-up control for a two-link underactuated robot with a flexible elbow joint: New results beyond the passive elbow joint. In: 50th IEEE conference on decision and control and European control conference, Orlando, FL, USA, 12–15 December 2011, pp. 2481–2486. Piscataway, NJ: IEEE.

25.

Xing

Liu

(2017) LMI-based boundary and distributed control design for flexible string subject to disturbance. International Journal of Control 92(8): 1959–1969.

26.

Yun

Choi

Park

(2010) Dynamic output-feedback guaranteed cost control for linear systems with uniform input quantization. Nonlinear Dynamics 62(1–2): 95–104.

27.

Zhang L and Liu J (2012) Optimal trajectory control of flexible two-link manipulator based on PDE model. IEEE 51st annual conference on decision and control (CDC), Maui, HI, USA, 10–13 December 2012. Piscataway, NJ: IEEE.

28.

Zhao

Ren

, et al. (2018) Output feedback stabilization for an axially moving system. IEEE Transactions on Systems, Man, and Cybernetics: Systems 89(4): 2431–2440.

29.

Zhao

Lin

Zhu

, et al. (2019) Vibration control of a riser-vessel system subject to input backlash and extraneous disturbances. IEEE Transactions on Circuits and Systems II. Epub ahead of print 30 April 2019. DOI: 10.1109/TCSII.2019.2914061.

30.

Zhou

Wen

Yang

(2014) Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal. IEEE Transactions on Automatic Control 59(2): 460–464.