Control Design for Flexible Manipulator Model with Nonlinear Input and State Constraints Based on Symmetric Barrier Lyapunov Function

Abstract

Flexible manipulators are widely applied in many fields. Here, the control design for a simplified flexible manipulator model with nonlinear inputs and state constraints is studied. The impact of two inputs and disturbances on the system was considered. One torque input comes from the joint motor, and the other input force comes from the linkage actuator tip. The input constraints of a dead zone are applied to both inputs to the manipulator. To offset the effect of the nonlinear input, we first linearize the dead zone and convert it into a linear-input characteristic and a finite error value. Then, the adaptive rate is designed to compensate for the effects of the nonlinear input. For the state constraints, an adaptive controller is proposed based on a symmetric tangent-type barrier Lyapunov function which can operate under closer constraint conditions, and parameter tunability offers flexibility in balancing the constraints’ tightness and performance. The stability proof ensures that all states are within the given constraint range. The provided simulation results indicate that the system is not sensitive to the initial values, and when the initial values are taken to be between open intervals (−0.4, 0.34), this ensures the stability of the system and does not violate the constraint bounds.

1. Introduction

Flexible structures have important applications in daily life, such as flexible beams [1], flexible hoisting systems [2], conveyor belts [3], flexible marine risers [4], etc. Flexible systems are ubiquitous in real life. Almost all mechanical structures can be considered flexible systems. Flexible structures are lightweight, so the number of components can be reduced. And miniaturization, design, and process integration are easy. This makes them suitable for many applications where space is limited. These advantages have made flexible systems increasingly popular.

In recent times, numerous domestic and international studies have proposed various solutions for suppressing flexible vibration issues, primarily categorized into passive and active control. Among these, reducing vibrations in flexible mechanical arms by installing mechanical devices that can dampen the system’s vibrations is called passive control. While this approach yields noticeable effects, it often comes with drawbacks such as a high mass and cost. Boundary control, as an effective strategy for suppressing vibrations in flexible materials, has garnered increasing attention and has become a cutting-edge research problem.

System models for flexible manipulators are more complex. In the research on the control for flexible manipulators, attention should be paid to the impact of nonlinear factors in their mechanical structural components on the system performance. Common nonlinear inputs are the dead zone [5,6], backlash [7,8], and saturation [9,10,11]. Dead-zone input constraints refer to phenomena where the input variables of a flexible manipulator do not affect the system within a specified range. This means that as the input signal changes, the output signal remains zero, and after a certain range is exceeded, the output signal changes nonlinearly. In [12], the authors studied the control stability problem with a dead zone using the backstepping technique. In [13], a state observer was constructed to study control methods for fuzzy dead zones. In [3,5], the nonlinear dead-zone characteristic is converted into a linear-input characteristic and a finite error value. For other studies with constraints, see [14,15,16,17,18,19,20,21]. Reference [14] proposes an adaptive controller based on a tangent-type barrier Lyapunov function (BLF) for output tracking. A piecewise BLF for reducing the boundary displacement in an asymmetric open set was introduced in the literature [15]. In [22], the researchers developed a novel flexible manipulator with a compact design capable of accessing confined spaces, making it suitable for applications such as minimally invasive medical surgery. For cases considering the state constraints for flexible manipulator systems, in [23], a backstepping tracking control was studied. Reference [24] investigated boundary-event-based control of flexible manipulator agents under input–output constraints. In [25,26], the authors provided an overview of the background and categories of methods in the field of robotics. For more research related to control over flexible link manipulators and flexible joint manipulators, we refer the reader to [27,28,29,30,31,32].

However, the literature mentioned above has only involved studies from one aspect. Given the widespread application of flexible manipulators in industry, aerospace, navigation, and other fields, it is usually necessary to overcome both a nonlinear input and distributed parameter constraints simultaneously to ensure the effectiveness and safety of the system. A natural question is what kind of controller should be designed to achieve system stability when considering problems with both dead zones and constraints. As far as we know, currently, there is no existing research simultaneously studying dead-zone inputs and state constraints in flexible manipulator systems. This paper focuses on the adaptive control for a flexible manipulator with dead-zone input constraints and state constraints. A controller that simultaneously constrains two inputs is designed based on the symmetric tangent-type BLF method, taking into account the influence of external disturbances. Compared with the highly conservative logarithmic-type BLF, the tangent-type BLF can operate under closer constraint conditions, and parameter tunability offers flexibility in balancing between the constraints’ tightness and performance. It can simplify the controller design and reduce complexity. The contribution of this manuscript is as follows:

• An adaptive control method based on the tangent-type BLF is proposed. It handles both nonlinear input and state constrains.
• A compensation mechanism that integrates external disturbances and dead-zone errors is established;
• A theoretical basis for future research on control problems in such PDE systems is provided.

2. Materials and Methods

This section gives the system equations for a simplified flexible manipulator and the specific process of controller design.

2.1. System and Problem Description

A simplified flexible manipulator model that moves in one plane is investigated in this paper, as shown in Figure 1 [33].

A description of the symbols used is given in

Table 1. Parameters of flexible manipulator.

Symbol	Description	Symbol	Description
$EI$	Flexural stiffness	$I_h$	Rotational inertia of the hub
$\theta$	Angle of rotation of the manipulator	$\rho$	Mass of the flexible manipulator per degree of length
m	Mass of the end load	L	Length of the manipulator
$\tau \left(t\right)$	Control input generated by the engine	$F\left(t\right)$	External force at the connecting rod end
${\epsilon }_1\left(t\right)$	Unknown time-varying boundary disturbance generated by the engine	${\epsilon }_2\left(t\right)$	Unknown time-varying boundary disturbance generated by the end of the connecting rod

.

From reference [34], the system equation has the form

$$\rho \ddot{z}(x,t)=-EI{y}_{xxxx}(x,t)$$

(1)

with the boundary condition

$$\tau \left(t\right)+{\epsilon }_1\left(t\right)=I_{\mathrm{h}}\ddot{\theta }\left(t\right)-EI{y}_{xx}(0,t)$$

(2)

$$F\left(t\right)+{\epsilon }_2\left(t\right)=m\ddot{z}(L,t)-EI{y}_{xxx}(L,t)$$

(3)

$$y(0,t)=y_x(0,t)=y_{xx}(L,t)=0$$

(4)

where $z\left(x,t\right)=x\theta +y(x,t)$, $\ddot{z}\left(x,t\right)=x\ddot{\theta }+\ddot{y}(x,t)$, $\ddot{z}\left(L,t\right)=L\ddot{\theta }+\ddot{y}(L,t)$, $e_1=\theta -{\theta }_d$, $e_2=y(L,t)-y_d(L,t)$. Here, both $e_1$ and $e_2$ are constants. ${\theta }_d$ and $y_d(L,t)$ denote the desired angular position and the desired vibration. Because the energy of the external environmental perturbations to which the system is subjected is usually finite, ${\epsilon }_1\left(t\right)$ and ${\epsilon }_2\left(t\right)$ are bounded. Then, ${\dot{e}}_1$, ${\ddot{e}}_1$, ${\dot{e}}_2$, ${\ddot{e}}_2$ can be written as the following equation:

$${\dot{e}}_1=\dot{\theta }, {\ddot{e}}_1=\ddot{\theta }$$

(5)

$${\dot{e}}_2=\dot{y}(L,t), {\ddot{e}}_2=\ddot{y}(L,t)$$

(6)

For clarity, throughout this paper, simplified notations are used: $\left(\dot{\ast }\right)={\displaystyle \frac{\partial (\ast )}{\partial t}}, \left(\ddot{\ast }\right)={\displaystyle \frac{{\partial }^2(\ast )}{\partial t^2}}$, ${(\ast )}_x={\displaystyle \frac{\partial (\ast )}{\partial x}}, {(\ast )}_{xx}={\displaystyle \frac{{\partial }^2(\ast )}{\partial x^2}}, {(\ast )}_{xxx}={\displaystyle \frac{{\partial }^3(\ast )}{\partial x^3}}, {(\ast )}_{xxxx}={\displaystyle \frac{{\partial }^4(\ast )}{\partial x^4}}.$

2.2. Control Design

Firstly, the dead zone can be replaced by a piecewise function

$$\tau \left(t\right)=D_1\left(u_1\right)=\left\{\begin{array}{cc}{g}_{r1}\left(u_1\right),& u_1\ge b_{r1}\\ 0,& b_{r1}<u_1<b_{l1}\\ g_{l1}\left(u_1\right),& u_1\le b_{l1}\end{array}\right.$$

(7)

$$F\left(t\right)=D_2\left(u_2\right)=\left\{\begin{array}{cc}{g}_{r2}\left(u_2\right),& u_2\ge b_{r2}\\ 0,& b_{r2}<u_2<b_{l2}\\ g_{l2}\left(u_2\right),& u_2\le b_{l2}\end{array}\right.$$

(8)

where $\tau \left(t\right)$ and F are the control input of the engine and the external force of the manipulator, respectively, where $u_1\left(t\right)$, $u_2\left(t\right)$ are the designed control inputs. $b_{r1}$, $b_{r2}$, $b_{l1}$, and $b_{l2}$ are unknown bounded constants. $g_{r1}\left(u_1\right)$, $g_{l1}\left(u_1\right)$, $g_{r2}\left(u_1\right)$, and $g_{l2}\left(u_1\right)$ are unknown smooth functions, and their derivatives are non-negatively bounded.

Definition 1.

The parameters $b_{r1}$, $b_{r2}$, $b_{l1}$, and $b_{l2}$ are unknown bounded constants.

Definition 2.

The functions $g_{r1}\left(u_1\right)$, $g_{l1}\left(u_1\right)$, $g_{r2}\left(u_1\right)$, and $g_{l2}\left(u_1\right)$ are unknown smooth, and their derivatives are non-negatively bounded. If $u_1\left(t\right)\in \left(b_{r1},+\infty \right)$, $u_2\left(t\right)\in \left(b_{r2},+\infty \right)$, respectively, then ${g^{\prime }}_{r1}\left(u_1\right)$, ${g^{\prime }}_{l1}\left(u_1\right)$ satisfy ${g^{\prime }}_{r1}\left(u_1\right)\in \left({\gamma }_{r10},{\gamma }_{r11}\right)$, ${g^{\prime }}_{r2}\left(u_2\right)\in \left({\gamma }_{r20},{\gamma }_{r21}\right)$. Similarly, if $u_1\left(t\right)\in \left(-\infty ,b_{l1}\right)$, $u_2\left(t\right)\in \left(-\infty ,b_{l2}\right)$, respectively, then ${g^{\prime }}_{r1}\left(u_1\right)$, ${g^{\prime }}_{l1}\left(u_1\right)$ satisfy ${g^{\prime }}_{l1}\left(u_1\right)\in \left({\gamma }_{l10},{\gamma }_{l11}\right)$, ${g^{\prime }}_{l2}\left(u_2\right)\in \left({\gamma }_{l20},{\gamma }_{l21}\right)$.

Secondly, we linearize the dead zone into the following form:

$$\tau \left(t\right)=D_1\left(u_1\right)={K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+d_1\left(u_1\right)$$

(9)

where

$${\mathsf{\Phi }}_1\left(t\right)={\left[{\phi }_{r1}\left(t\right),{\phi }_{l1}\left(t\right)\right]}^T$$

(10)

$${\phi }_{r1}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & u_1\left(t\right)>b_{l1}\\ 0,\hfill & u_1\left(t\right)\le b_{l1}\end{array}\right.$$

(11)

$${\phi }_{l1}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & u_1\left(t\right)<b_{r1}\\ 0,\hfill & u_1\left(t\right)\ge b_{r1}\end{array}\right.$$

(12)

$$K_1\left(t\right)={\left[K_{r1}\left(u_1\left(t\right)\right),K_{l1}\left(u_1\left(t\right)\right)\right]}^T$$

(13)

$$K_{r1}\left(u_1\left(t\right)\right)=\left\{\begin{array}{cc}0& u_1\left(t\right)\le b_{l1}\\ g_{r1}^{\prime }\left({\xi }_{r1}\left(u_1\left(t\right)\right)\right),& b_{l1}<u_1\left(t\right)<+\infty \end{array}\right.$$

(14)

$$K_{l1}\left(u_1\left(t\right)\right)=\left\{\begin{array}{cc}0& u_1\left(t\right)\le b_{r1}\\ g_{l1}^{\prime }\left({\xi }_{l1}\left(u_1\left(t\right)\right)\right),& b_{r1}<u_1\left(t\right)<+\infty \end{array}\right.$$

(15)

$$\begin{array}{c}\hfill \begin{array}{cc}& d_1\left(u_1\right)=\left\{\begin{array}{cc}-g_{r1}^{\prime }\left({\xi }_{r1}\left(u_1\right)\right)b_{r1},& u_1\ge b_{r1}\\ -\left[g_{r1}^{\prime }\left({\xi }_{r1}\left(u_1\right)\right)+g_{l1}^{\prime }\left({\xi }_{l1}\left(u_1\right)\right)\right]u_1,& b_{l1}<u_1<b_{r1}\\ -g_{l1}^{\prime }\left({\xi }_{l1}\left(u_1\right)\right)b_{l1},& u_1\le b_{l1}\end{array}\right.\hfill \end{array}\end{array}$$

(16)

In addition, the following equation is available for the end of the manipulator:

$$F\left(t\right)=D_2\left(u_2\right)={K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2+d_2\left(u_2\right)$$

(17)

where

$${\mathsf{\Phi }}_2\left(t\right)={\left[{\phi }_{r2}\left(t\right),{\phi }_{l2}\left(t\right)\right]}^T$$

(18)

$${\phi }_{r2}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & u_2\left(t\right)>b_{l2}\\ 0,\hfill & u_2\left(t\right)\le b_{l2}\end{array}\right.$$

(19)

$${\phi }_{l2}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & u_2\left(t\right)<b_{r2}\\ 0,\hfill & u_2\left(t\right)\ge b_{r2}\end{array}\right.$$

(20)

$$K_2\left(t\right)={\left[K_{r2}\left(u_2\left(t\right)\right),K_{l2}\left(u_2\left(t\right)\right)\right]}^T$$

(21)

$$K_{r2}\left(u_2\left(t\right)\right)=\left\{\begin{array}{cc}0& u_2\left(t\right)\le b_{l2}\\ g_{r2}^{\prime }\left({\xi }_{r2}\left(u_2\left(t\right)\right)\right),& b_{l2}<u_2\left(t\right)<+\infty \end{array}\right.$$

(22)

$$K_{l2}\left(u_2\left(t\right)\right)=\left\{\begin{array}{cc}{g}_{l2}^{\prime }\left({\xi }_{l2}\left(u_2\left(t\right)\right)\right),& -\infty <u_2\left(t\right)<b_{r2}\\ 0,& u_2\left(t\right)\ge b_{r2}\end{array}\right.$$

(23)

$$\begin{array}{c}\hfill \begin{array}{cc}& d_2\left(u_2\right)=\left\{\begin{array}{cc}-g_{r2}^{\prime }\left({\xi }_{r2}\left(u_2\right)\right)b_{r2},& u_2\ge b_{r2}\\ -\left[g_{r2}^{\prime }\left({\xi }_{r2}\left(u_2\right)\right)+g_{l2}^{\prime }\left({\xi }_{l2}\left(u_2\right)\right)\right]u_2,& b_{l2}<u_2<b_{r2}\\ -g_{l2}^{\prime }\left({\xi }_{l2}\left(u_2\right)\right)b_{l2},& u_2\le b_{l2}\end{array}\right.\hfill \end{array}\end{array}$$

(24)

Remark 1.

For ${\xi }_{r1}\left(u_1\right)$, ${\xi }_{l1}\left(u_1\right)$, ${\xi }_{r2}\left(u_2\right)$, and ${\xi }_{l2}\left(u_2\right)$, the following conditions are satisfied:

• If $u_1\left(t\right)$, $u_2\left(t\right)$ satisfy $b_{r1}<u_1\left(t\right)$, $b_{r2}<u_2\left(t\right)$, respectively, then ${\zeta }_{r1}\left(u_1\right)$, ${\zeta }_{r2}\left(u_2\right)$ satisfy ${\zeta }_{r1}\left(u_1\right)\in \left(b_{r1},u_1\right)$, ${\zeta }_{r2}\left(u_2\right)\in \left(b_{r2},u_2\right)$. If $u_1\left(t\right)$, $u_2\left(t\right)$ satisfy $b_{l1}<u_1\left(t\right)<b_{r1}$, $b_{l2}<u_2\left(t\right)<b_{r2}$, respectively, then ${\zeta }_{r1}\left(u_1\right)$, ${\zeta }_{r2}\left(u_2\right)$ satisfy ${\zeta }_{r1}\left(u_1\right)\in \left(u_1,b_{r1}\right)$, ${\zeta }_{r2}\left(u_2\right)\in \left(u_2,b_{r2}\right)$. If $u_1\left(t\right)$, $u_2\left(t\right)$ satisfy $u_1\left(t\right)\le b_{l1}$, $u_2\left(t\right)\le b_{l2}$, respectively, then ${\zeta }_{r1}\left(u_1\right)$, ${\zeta }_{r2}\left(u_2\right)$ satisfy ${\zeta }_{r1}\left(u_1\right)\in \left(u_1,b_{l1}\right)$, ${\zeta }_{r2}\left(u_2\right)\in \left(u_2,b_{l2}\right)$. If $u_1\left(t\right)$, $u_2\left(t\right)$ satisfy $b_{l2}<u_2\left(t\right)<b_{r2}$, respectively, then ${\zeta }_{l1}\left(u_1\right)$, ${\zeta }_{l2}\left(u_2\right)$ satisfy ${\zeta }_{l1}\left(u_1\right)\in \left(b_{l1},u_1\right)$, ${\zeta }_{l2}\left(u_2\right)\in \left(b_{l2},u_2\right)$.

Lemma 1.

From Definition 2, we have ${K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)\ge {\beta }_1>0$ and ${K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)\ge {\beta }_2>0$, where ${\beta }_1$ and ${\beta }_2$ are positive constants, and ${\beta }_1\le min\left({\gamma }_{l10},{\gamma }_{r10}\right)$, ${\beta }_2\le min\left({\gamma }_{l20},{\gamma }_{r20}\right)$ are satisfied.

Lemma 2.

From Equations (16) and (24), we know that $d_1\left(u_1\right)$ and $d_2\left(u_2\right)$ are bounded functions.

We define a new function consisting of real perturbations ${\epsilon }_1\left(t\right)$, ${\epsilon }_2\left(t\right)$ and bounded nonlinear error signals $d_1\left(u_1\right)$, $d_2\left(u_2\right)$. We define them as $R_1\left(t\right)$ and $R_2\left(t\right)$:

$$R_1\left(t\right)={\epsilon }_1\left(t\right)+d_1\left(u_1\left(t\right)\right)$$

(25)

$$R_2\left(t\right)={\epsilon }_2\left(t\right)+d_2\left(u_2\left(t\right)\right)$$

(26)

Then, $R_1\left(t\right)$ and $R_2\left(t\right)$ satisfy $\left|R_1\left(t\right)\right|\le R_1$, $\left|R_2\left(t\right)\right|\le R_2$, where $R_1$ and $R_2$ are unknown normal values and are bounded by $R_1\left(t\right)$ and $R_2\left(t\right)$. ${\widehat{R}}_1\left(t\right)$ and ${\widehat{R}}_2\left(t\right)$ are estimates of $R_1$ and $R_2$. We define ${\tilde{R}}_1\left(t\right)=R_1-{\widehat{R}}_1\left(t\right)$, ${\tilde{R}}_2\left(t\right)=R_2-{\widehat{R}}_2\left(t\right)$, where ${\tilde{R}}_1\left(t\right)$ and ${\tilde{R}}_2\left(t\right)$ are the error values.

Then, the system Equations (1)–(4) can be rewritten as

$$\rho \ddot{z}(x,t)=-EI{y}_{xxxx}(x,t)$$

(27)

$${K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+R_1\left(t\right)=I_{\mathrm{h}}\ddot{\theta }\left(t\right)-EI{y}_{xx}(0,t)$$

(28)

$${K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2+R_2\left(t\right)=m\ddot{z}(L,t)-EI{y}_{xxx}(L,t)$$

(29)

$$y(0,t)=y_x(0,t)=y_{xx}(L,t)=0$$

(30)

Definition 3.

Positive constants $k_{b1}$, $k_{b2}$, $k_{b3}$ exist that satisfy

$$\left|e_1\left(0\right)\right|=\left|\theta \left(0\right)-{\theta }_d\right|<k_{b1}$$

(31)

$$\left|{\dot{e}}_1\left(0\right)\right|=\left|\dot{\theta }\left(0\right)-{\dot{\theta }}_d\right|<k_{b2}$$

(32)

$$\left|e_2\left(0\right)\right|=\left|y(L,0)-y_d(L,0)\right|<k_{b3}$$

(33)

$$\left|{\dot{e}}_2\left(0\right)\right|=\left|\dot{y}(L,0)-{\dot{y}}_d(L,0)\right|<k_{b4}$$

(34)

Theorem 1.

Let V be a scalar barrier Lyapunov function. It is defined by the equation $\dot{x}=f\left(x\right)$ on an open set D including the origin. If $V\left(x\right)$ is continuous and positive definite and has continuous first-order partial derivatives on D, then the following properties will hold: (1) $V\left(x\right)\to \infty$, as $x\to \partial D$ (the boundary of D); (2) $V\left(x\left(t\right)\right)\le C$ ( C is a positive constant), $\forall t\ge 0$ along the solutions of $\dot{x}=f\left(x\right)$ for $x\left(0\right)\in D$.

Accordingly, the control laws are designed as

$$\begin{array}{cc}\hfill & u_1\left(t\right)={\displaystyle \frac{1}{{K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)}}\left(-EI{y}_{xx}\left(0,t\right)\right.+\left({\displaystyle \frac{{cos}^2\left(\frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}\right)}{{cos}^2\left(\frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}\right)+1}}\right)\left(-{\displaystyle \frac{e_1}{{cos}^2\left(\frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}\right)}}\right.+EIL{y}_{xxx}(L,t)\hfill \\ \hfill & \left.+EI{y}_{xx}(0,t)-k_1{e}_1-k_3{\dot{e}}_1\right)-\mathrm{sgn}\left({\dot{e}}_1\right){\widehat{R}}_1\left(t\right))\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & u_2\left(t\right)={\displaystyle \frac{1}{{K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)}}\left(-EI{y}_{xxxx}\left(L,t\right)+\alpha L\left({K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+{\widehat{R}}_1\left(t\right)+EI{y}_{xx}\left(0,t\right)\right)\right.\hfill \\ \hfill & +\left({\displaystyle \frac{{cos}^2\left(\frac{\pi {\dot{e}}_1^2}{2k_{b4}^2}\right)}{{cos}^2\left(\frac{\pi {\dot{e}}_1^2}{2k_{b4}^2}\right)+1}}\right)\left(-{\displaystyle \frac{e_2}{{cos}^2\left(\frac{\pi {\dot{e}}_2^2}{2k_{b3}^2}\right)}}+EI{y}_{xxx}\left(L,t\right)-k_2{e}_2-k_4{\dot{e}}_2\right)-\mathrm{sgn}\left({\dot{e}}_2\right){\widehat{R}}_2\left(t\right))\hfill \end{array}$$

(36)

The design-adaptive laws are as follows:

$${\dot{\widehat{R}}}_1=\left|{\dot{e}}_1\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)$$

(37)

$${\dot{\widehat{R}}}_2=\left|{\dot{e}}_2\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)$$

(38)

Notice that ${\tilde{R}}_1\left(t\right)=R_1-{\widehat{R}}_1\left(t\right)$, ${\tilde{R}}_2\left(t\right)=R_2-{\widehat{R}}_2\left(t\right)$, and we have

$${\dot{\tilde{R}}}_1\left(t\right)=-{\dot{\widehat{R}}}_1\left(t\right)=-\left|{\dot{e}}_1\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)$$

(39)

$${\dot{\tilde{R}}}_2\left(t\right)=-{\dot{\widehat{R}}}_2\left(t\right)=-\left|{\dot{e}}_2\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)$$

(40)

Figure 2 shows the control procedure for a flexible manipulator system. The block diagram represents the design of a sophisticated control system for a flexible manipulator.

A flowchart of the control process is shown in Figure 3. It describes the comprehensive control algorithm implementation process that addresses the nonlinearity of the input and state constraint problems, which is crucial for effective control of flexible manipulators.

Theorem 2.

For the system model with dead-zone inputs described by (1)–(4), under Definitions 1–3, Lemmas 1 and 2, Theorem 1, proposed control (35) and (36), and proposed adaptive laws (37) and (38) and given the bounded initial state conditions, we come to the conclusion that

1.

$e_1$, ${\dot{e}}_1$, $e_2$, ${\dot{e}}_2$, ${\zeta }_1\left(t\right)$, ${\zeta }_2\left(t\right)$, $y(x,t)$ are bounded;

2.

The states of $e_1$, ${\dot{e}}_1$, $e_2$, and ${\dot{e}}_2$ satisfy the constraint conditions.

Proof. 

Select the Lyapunov function as

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right)+V_3\left(t\right)+V_4\left(t\right)+V_5\left(t\right)$$

(41)

where each $V_i\left(t\right),(i=1,\dots ,5)$ is defined as

$$V_1\left(t\right)={\displaystyle \frac{k_{b1}^2}{\pi }}tan\left({\displaystyle \frac{\pi e_1^2}{2k_{b1}^2}}\right)+I_h{\displaystyle \frac{k_{b2}^2}{\pi }}tan\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)$$

(42)

$$V_2\left(t\right)={\displaystyle \frac{k_{b3}^2}{\pi }}tan\left({\displaystyle \frac{\pi e_2^2}{2k_{b3}^2}}\right)+m{\displaystyle \frac{k_{b4}^2}{\pi }}tan\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)$$

(43)

$$V_3\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^L\rho {\dot{z}}^2(x,t)dx+{\displaystyle \frac{1}{2}}EI{\int }_0^L{y}_{xx}^2(x,t)dx$$

(44)

$$V_4\left(t\right)={\displaystyle \frac{1}{2}}{k}_1{e}_1^2+{\displaystyle \frac{1}{2}}{k}_2{e}_2^2+{\displaystyle \frac{1}{2}}{I}_h{\dot{e}}_1^2+{\displaystyle \frac{1}{2}}m{\dot{e}}_2^2$$

(45)

$$V_5\left(t\right)={\displaystyle \frac{1}{2}}{\left[{\tilde{R}}_1\left(t\right)\right]}^2+{\displaystyle \frac{1}{2}}{\left[{\tilde{R}}_2\left(t\right)\right]}^2$$

(46)

Differentiating (42)–(46) with respect to t, then

$${\dot{V}}_1\left(t\right)=e_1{\dot{e}}_1{sec}^2\left({\displaystyle \frac{\pi e_1^2}{2k_{b1}^2}}\right)+I_h{\dot{e}}_1{\ddot{e}}_1{sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)$$

(47)

$${\dot{V}}_2\left(t\right)=e_2{\dot{e}}_2{sec}^2\left({\displaystyle \frac{\pi e_2^2}{2k_{b3}^2}}\right)+m{\dot{e}}_2{\ddot{e}}_2{sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)$$

(48)

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)& ={\int }_0^L\rho \dot{z}(x,t)\ddot{z}(x,t)dx+EI{\int }_0^L{y}_{xx}(x,t){\dot{y}}_{xx}(x,t)dx\hfill \end{array}$$

(49)

$${\dot{V}}_4\left(t\right)=k_1{e}_1{\dot{e}}_1+k_2{e}_2{\dot{e}}_2+I_h{\dot{e}}_1{\ddot{e}}_1+m{\dot{e}}_2{\ddot{e}}_2$$

(50)

$${\dot{V}}_5\left(t\right)={\tilde{R}}_1{\dot{\tilde{R}}}_1+{\tilde{R}}_2{\dot{\tilde{R}}}_2$$

(51)

Based on (27)–(30), ${\dot{V}}_3\left(t\right)$ can be rewritten as

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)& ={\int }_0^L\rho \dot{z}(x,t)\ddot{z}(x,t)dx+EI{\int }_0^L{y}_{xx}(x,t){\dot{y}}_{xx}(x,t)dx\hfill \\ \hfill & =-{\int }_0^{L}EI{y}_{xxxx}(x,t)\dot{z}(x,t)dx+EI{\int }_0^L{y}_{xx}(x,t){\dot{y}}_{xx}(x,t)dx\hfill \\ \hfill & ={\dot{e}}_1(-EIL{y}_{xxx}(x,t)-EI{y}_{xx}(0,t))-{\dot{e}}_{2}EI{y}_{xxx}(L,t)\hfill \end{array}$$

(52)

Then, we have

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)& ={\dot{V}}_1\left(t\right)+{\dot{V}}_2\left(t\right)+{\dot{V}}_3\left(t\right)+{\dot{V}}_4\left(t\right)+{\dot{V}}_5\left(t\right)\hfill \\ \hfill & ={\dot{e}}_1\left(e_1{sec}^2\left({\displaystyle \frac{\pi e_1^2}{2k_{b1}^2}}\right)+I_h{\ddot{e}}_1{sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)\right.-EI{y}_{xx}(0,t)-EIL{y}_{xxx}(L,t)+k_1{e}_1+I_{\mathrm{h}}{\ddot{e}}_1\hfill \\ \hfill & +{\dot{e}}_2\left(e_2{sec}^2\left({\displaystyle \frac{\pi e_2^2}{2k_{b3}^2}}\right)+m{\ddot{e}}_2{sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)\right.-EI{y}_{xxx}(L,t)+k_2{e}_2+m{\ddot{e}}_2+{\tilde{R}}_1{\dot{\tilde{R}}}_1+{\tilde{R}}_2{\dot{\tilde{R}}}_2\hfill \\ \hfill & ={\dot{e}}_1\left(e_1{sec}^2\left({\displaystyle \frac{\pi e_1^2}{2k_{b1}^2}}\right)-EIL{y}_{xxx}(L,t)\right.-EI{y}_{xx}(0,t)+k_1{e}_1\left.+\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)I_{\mathrm{h}}{\ddot{e}}_1\right)\hfill \\ \hfill & +{\dot{e}}_2\left(e_2{sec}^2\left({\displaystyle \frac{\pi e_2^2}{2k_{b3}^2}}\right)-EI{y}_{xxx}(L,t)\right.\left.+k_2{e}_2+\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)m{\ddot{e}}_2\right)+{\tilde{R}}_1{\dot{\tilde{R}}}_1+{\tilde{R}}_2{\dot{\tilde{R}}}_2\hfill \end{array}$$

(53)

From the system model of Equations (27)–(30), we acquire

$$I_{\mathrm{h}}{\ddot{e}}_1\left(t\right)=I_{\mathrm{h}}\ddot{\theta }\left(t\right)={K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+R_1\left(t\right)+EI{y}_{xx}(0,t)$$

(54)

Then,

$$\begin{array}{cc}\hfill m{\ddot{e}}_2& =m\ddot{y}\left(L,t\right)\hfill \\ \hfill & =m\ddot{z}\left(L,t\right)-mL\ddot{\theta }\hfill \\ \hfill & ={K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2+R_2\left(t\right)+EI{y}_{xxx}(L,t)-mL\ddot{\theta }\hfill \\ \hfill & ={K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2+R_2\left(t\right)+EI{y}_{xxxx}\left(L,t\right)-m\left({\displaystyle \frac{L}{I_h}}\tau \left(t\right)+{\displaystyle \frac{L}{I_h}}EI{y}_{xx}\left(0,t\right)\right)\hfill \\ \hfill & ={K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2+R_2\left(t\right)+EI{y}_{xxxx}\left(L,t\right)-\alpha L\left({K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+R_1\left(t\right)+EI{y}_{xx}(0,t)\right)\hfill \end{array}$$

(55)

where $\alpha ={\displaystyle \frac{m}{I_h}}$. Combining (54) and (55), we have

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)& ={\dot{e}}_1\left(e_1{sec}^2\left({\displaystyle \frac{\pi e_1^2}{2k_{b1}^2}}\right)-EI{y}_{xx}(0,t)\right.-EIL{y}_{xxx}(L,t)+k_1{e}_1+\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)\hfill \\ \hfill & \left.\ast \left({K_1}^T\left(t\right){\mathsf{\Phi }}_1\left(t\right)u_1+R_1\left(t\right)+EI{y}_{xx}(0,t)\right)\right)+{\dot{e}}_2\left(e_2{sec}^2\left({\displaystyle \frac{\pi e_2^2}{2k_{b3}^2}}\right)-EI{y}_{xxx}(L,t)+k_2{e}_2\right.\hfill \\ \hfill & +\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)\left({K_2}^T\left(t\right){\mathsf{\Phi }}_2\left(t\right)u_2\right.+R_2\left(t\right)+EI{y}_{xxxx}\left(L,t\right)-\alpha L\left({K_1}^T\left(t\right)\right.\hfill \\ \hfill & \left.\left.\left.\ast {\mathsf{\Phi }}_1\left(t\right)u_1+R_1\left(t\right)+EI{y}_{xx}(0,t)\right)\right)\right)+{\tilde{R}}_1{\dot{\tilde{R}}}_1+{\tilde{R}}_2{\dot{\tilde{R}}}_2\hfill \end{array}$$

(56)

From (9), (17), and (35), (36), we obtain

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)={\dot{e}}_1\left(-k_3{\dot{e}}_1\right)+{\dot{e}}_1\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)R_1\left(t\right)-\left|{\dot{e}}_1\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right){\widehat{R}}_1\left(t\right)\hfill \\ \hfill & +{\dot{e}}_2\left(-k_4{\dot{e}}_2\right)+{\dot{e}}_2\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)R_2\left(t\right)-\left|{\dot{e}}_2\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right){\widehat{R}}_2\left(t\right)\hfill \\ \hfill & +{\tilde{R}}_1{\dot{\tilde{R}}}_1+{\tilde{R}}_2{\dot{\tilde{R}}}_2\hfill \end{array}$$

(57)

Combining (37)–(40), we conclude that

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)& \le -k_3{{\dot{e}}^2}_1-k_4{{\dot{e}}^2}_2-\left|{\dot{e}}_1\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_1^2}{2k_{b2}^2}}\right)+1\right)\left(R_1-\left|R_1\left(t\right)\right|\right)\hfill \\ \hfill & -\left|{\dot{e}}_2\right|\left({sec}^2\left({\displaystyle \frac{\pi {\dot{e}}_2^2}{2k_{b4}^2}}\right)+1\right)\left(R_2-\left|R_2\left(t\right)\right|\right)\hfill \\ \hfill & \le 0\hfill \end{array}$$

(58)

Integrating both ends of (58), we obtain

$${\int }_0^t\dot{V}\left(x\right)dx\le {\int }_0^{t}0dx$$

$$V\left(t\right)\le V\left(0\right)$$

(59)

Thus, we deduce that $V\left(t\right)$ is bounded. It further follows that $V_1\left(t\right)$ to $V_5\left(t\right)$ are also bounded. And thus $e_1$, ${\dot{e}}_1$, $e_2$, ${\dot{e}}_2$, ${\zeta }_1\left(t\right)$, ${\zeta }_2\left(t\right)$, and $y(x,t)$ are bounded. That is, all of the signals are bounded. Now that $V_1\left(t\right)$ and $V_2\left(t\right)$ are known to be bounded, if $e_1\to k_{b1}$, ${\dot{e}}_1\to k_{b2}$, $e_2\to k_{b3}$, or ${\dot{e}}_2\to k_{b4}$ holds, this contradicts the boundedness. Then, we can deduce conversely that $e_1\ne k_{b1}$, ${\dot{e}}_1\ne k_{b2}$, $e_2\ne k_{b3}$, and ${\dot{e}}_2\ne k_{b4}$. And the initial conditions of the systems $e_1$, ${\dot{e}}_1$, $e_2$, and ${\dot{e}}_2$ correspondingly are in $\left(-k_{b1},k_{b1}\right)$, $\left(-k_{b2},k_{b2}\right)$, $\left(-k_{b3},k_{b3}\right)$, and $\left(-k_{b4},k_{b4}\right)$, respectively. This completes the proof of the theorem. □

3. Simulation Verification

This section will further verify the effect of the controller and the adaptive rate on system (1)–(4) through numerical simulation. Here are the parameters.

The flexible manipulator’s parameters are set as $EI=2.0$, $\rho =7.4$, $m=5.5$, $I_h=10.5$, and $L=1.0$.

The initial parameters are as follows: ${\epsilon }_1\left(t\right)={\epsilon }_2\left(t\right)=cos\left(2\pi t\right)$, ${\theta }_d=0.3$, ${\dot{\theta }}_d=0$, and $y_d\left(L,t\right)=0$.

The controller’s parameters are given as $b_{r1}=20$, $b_{l1}=-1$, $b_{r2}=20$, $b_{l2}=-1$, $k_1=60$, $k_2=100$, $k_3=100$, $k_4=60$, $k_{b1}=0.5$, and $k_{b2}=k_{b3}=k_{b4}=0.2$.

Figure 4 describes the deflection of vibrations under the boundary controller. It is shown that the vibrational deflection converges completely within a small range from zero. Figure 5 and Figure 6 illustrate that the angle $\theta \left(t\right)$ and the angular speed $\dot{\theta }\left(t\right)$, the boundary deflection $y\left(L,t\right)$ constraint, and the boundary deflection speed $\dot{y}\left(L,t\right)$ constraint are kept strictly within the constraints. Both the angle $\theta \left(t\right)$ and the angular speed $\dot{\theta }\left(t\right)$ have reached the desired values, as shown in Figure 7. And the vibrational deflection converges completely to within a small range from 0. The control inputs shown in Figure 8 indicate that the inputs remain bounded and adjust dynamically to counteract dead-zone effects. Figure 9 shows the dead-zone inputs of the system, which illustrate the piecewise nature of the dead zone’s nonlinearity. If $D_1\left(u_1\right)$ satisfies $b_{l1}<D_1\left(u_1\left(t\right)\right)<b_{r1}$, $D_2\left(u_2\right)$ satisfies $b_{l2}<D_2\left(u_2\right)<b_{r2}$, and then we set $D_1\left(u_1\right)$ and $D_2\left(u_2\right)$ to 0, then the simultaneous control effect is acceptable. Figure 10 gives the adaptive laws of the system. This rapid convergence ensures that uncertainties are continuously compensated for, maintaining system stability. To illustrate detailed information about the selection of the design parameters, we provide some examples in Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15. Specifically, one can see that when the parameter ${\theta }_d$ is chosen to fall within the open interval $(-0.4,0.34)$, it can be ensured that the constraint bound is not violated. When the parameter is selected as ${\theta }_d=0.2$, Figure 11 shows that the constraint boundary is not violated, while as ${\theta }_d=0.8,$ as shown in Figure 12, it becomes unbounded, violating the constraints. This highlights the importance of the initial constraint bounds in ensuring stability. When the parameter is set to ${\theta }_d=0.34$ and ${\theta }_d=-0.4$, respectively, Figure 13 and Figure 14 show the violation of the constraint bounds. When the parameter ${\theta }_d$ is set to 0.64, then this will lead to system instability, as shown in Figure 15. In addition, under the parameters in this paper, Figure 16 and Figure 17 were obtained by using the logarithmic-type BLF method in reference [34]. It can be seen that the system is unstable under the given parameters, and the logarithmic BLF fails to stabilize the system, with the deflections oscillating around non-zero values. Compared with references [34,35], the simulation results show that the system for which the boundary inputs and state constraints are considered simultaneously in this paper can quickly achieve stability in a short time, and this validates the effectiveness of the method.

4. Conclusions

This paper considers control design accounting for the impact of nonlinear factors in mechanical structural components on the performance of a flexible manipulator system. Taking both the constraint effects and external disturbances into account, the tangent-type BLF controller is designed to prove the system’s stability and ensure all states remain within the specified constraint boundary. It is worth noting that the boundary control method studied in this paper still does not have universal applicability. In practical applications, when the system is affected by nonlinearity and parameter uncertainty, this will bring substantial difficulties. In future work, vibration suppression problems for flexible manipulators can also be studied while applying position or speed constraints.