Dynamical behavior analysis and control of time-varying lifting system based on fuzzy adaptive back-stepping controller

Abstract

In this article, considering uncertainty and nonlinearity factors, the resonance phenomenon of conveyance is studied and a fuzzy adaptive back-stepping controller is proposed for rope lifting system with time-varying length and distributed parameters. Firstly, the dynamic model is constructed and derived by continuum mechanics, and the dynamical behavior is obtained by time-frequency analysis method. Secondly, the fuzzy adaptive back-stepping control method is proposed to suppress nonstationary vibration. Thirdly, based on Lyapunov function, the asymptotic stability of the system under the proposed control algorithm is verified. Finally, compared with proportion–integration–differentiation (PID) back-stepping controller, the efficiency of the proposed controller is proven by applying excitation with different frequency and uncertain parameters, and the adaptability and robustness are illustrated to validate under mixed frequency disturbance by simulation compared with PID back-stepping controller. Based on the dynamical behavior analysis and control method proposed in this article, the robustness of vibration controller of time-varying lifting system under uncertain nonlinear time-varying disturbance will be greatly improved.

Introduction

Considering the advantages of heavy load and large lifting depth, rope lifting system is widely used in many lifting fields. As shown in Figure 1, the time-varying lifting ropes drive conveyances to move up and down by frictional force. However, caused by the excitation disturbance at drum boundary, there will be large vibration occurred at the conveyance due to the elastic rope spanning a long distance.^1–4 The system will resonate obviously when the natural frequency ⁵ and excitation frequency are superimposed. ⁶ Therefore, it is necessary to propose an effective method to control resonance. ⁷ Considering that the lifting system is a variable length lifting system with distributed mass, the mathematical model is based on the governing equations and boundary conditions, which will contain coupled partial differential equations and ordinary differential equations with geometric boundary conditions. At present, the research on dynamical behavior analysis and control of rope lifting system mainly focuses on active control, passive control, and dynamic solution method. As for the active vibration control, considering the application of control input, boundary control is a good control measure. ⁸ The boundary control of distributed parameter control system is studied by many researchers.^9–26 Considering the uncertainty of disturbance, neural network has been used as a compensator to compensate the disturbance term of the system normally.^13–17 Considering the nonlinear terms in differential equations, back-stepping controller is used to achieve hierarchical static compensation as a common control method.^12,18 As for the transverse vibration control of flexible structure with fixed length and time-varying disturbance, many scholars have proposed a variety of methods. A method combining sliding mode control (SMC) and positive position feedback control is proposed to control of beams subjected to a moving mass. ¹⁹ Considering the speed of the axis of the string, a novel control algorithm that uses the effects of the time-varying axial transport velocity based on the regulation of the axial velocity to suppress the transverse vibration of an axially moving membrane was developed.^20,21 Considering unknown disturbance, robust adaptive boundary control is developed for a class of flexible string-type systems under unknown time-varying disturbance. ²² Considering time-varying disturbances and input constraints in the system, adaptive iterative control learning ²³ and model predictive control ²⁴ are applied to dynamic control and vibration control of flexible bodies, which achieved quite good control effect. As for passive vibration control, some scholars have constructed passive control methods for flexible body vibration control through frequency modulation mass damping systems and verified the effectiveness of vibration control through experiments. ²⁵ Meanwhile, some experiments were conducted to control vibration through flexible bodies. ²⁶ As for the dynamic solution method, there are also many researches about the methods to solve the vibration response of lifting systems.^{1–7,27–30} Assumed mode method, finite element method, mode superposition method, multi-scale analysis method, and absolute node coordinate method are all used to solve the dynamical response of time-varying lifting system.^1–7 Korayem et al.^27–30 have proposed effective solutions and methods for solving the dynamic characteristics, path planning, and motion control of cable suspended parallel and flexible manipulator system.

Figure 1.

System description of traction system with controller.

From above research, researches are focused on the control of time invariant rope lifting system with fixed length and varying velocity, which the time-varying mass and the nonlinear boundary excitation was not considered. The characteristics of time-varying mass and stiffness coefficients of the time-varying lifting system inevitably require the controller to be highly adaptive,^12,18 while the uncertainty of platform parameters and unknown nonlinear disturbances require the controller to be robust and stable. In addition, due to the uncertainty and nonlinear of system parameters and disturbances, the control modeling and numerical solution of the rope lifting system will be more difficult. As for the real applications of the vibration control of flexible body, there are some studies on this. A passive control method to increase the tension wire rope device and an active control method to dissipate the energy of the lifting system are proposed to suppress the vibration of the lifting system. ³¹ An active control strategy based on the fuzzy SMC is developed for controlling the large-amplitude vibrations of an extending cable. ³² Do and Pan ³³ present a design of boundary controllers actuated by hydraulic actuators at the top end for global stabilization of a three-dimensional riser system. Fang et al.^34,35 concentrate on the control issue of a variable length drilling riser under condition of unknown disturbances and output constraint.

In this article, considering the nonlinear excitation and uncertain disturbance, a mathematical control model of time-varying lifting system is established based on the Hamilton's principle. The fuzzy system and nonlinear disturbance observer are introduced to compensate the uncertain and mixed disturbances. Through back-stepping controller, the control law is designed for the lifting system with time-varying length and distributed parameters. The application of the fuzzy adaptive back-stepping controller provides guidance for vibration control and resonance elimination for the lifting system with time-varying length and distributed parameters.

This article is divided into four parts. The first part is about the introduction. The main purpose is to summarize and introduce the research background, significance, and related main content of the article. The second part is the problem formula. The main task is to construct a dynamic control model, propose relevant vibration control methods, and analyze the stability of the designed control rate. The third part is simulation, which mainly verifies the effectiveness and robustness of the fuzzy adaptive back-stepping control method proposed in this article, and compares it with the results of the control without control and proportion–integration–differentiation (PID) control, further demonstrating the progressiveness and effectiveness of the control algorithm proposed in this article. The fourth part is a summary, which summarizes the main research results and innovative points of this article.

Problem formulation

Mathematical model

As Seen in Figure 1(a), the time-varying lifting system is composed of five parts: lifting rope, conveyances, compensating rope, guide, and driving drum. Here, three-axis acceleration sensors and rotation sensors are installed on the lifting conveyance to monitor the vibration acceleration and rotation degree of conveyances and feedback to the controller. The vibration speed and displacement of the system are obtained through integral operation. Based on the control algorithm, the output of the controller is calculated by simulation and comparison.

The actuator of controller is shown in Figure 1(b). The fixed pulleys are fixed on the platform, which plays the role of clamping rope. Then, the movement of the hydraulic cylinder drives the moving pulleys to move horizontally. The vertical displacement of the conveyance is controlled by the pulling of the moving pulleys and the coordination of the fixed pulleys.

Firstly, the kinetic energy and potential energy of the time-varying length lifting system are derived, in which the longitudinal vibration of the lifting rope, conveyance, and controller are considered. The specific expression is as follows:

$$\begin{gathered} \begin{array}{rl}{\begin{array}{c}E\end{array}}_k(t)& =\frac{1}{2}{\rho }_1\{{\int }_0^{l_1(t)}\left[v_1(t)+{\displaystyle \frac{D{u}_1(x_1,t)}{Dt}}\right]d{x}_1\\ & {\displaystyle +{\int }_0^{l_4(t)}\left[v_4(t)+{\displaystyle \frac{D{u}_4(x_4,t)}{Dt}}\right]d{x}_4\}} \\ +\sum _{i=1}^2{\displaystyle \frac{1}{2}{m}_{c_i}(t){\dot{\mathit{r}}}_i^{\mathrm{T}}{\dot{\mathit{r}}}_i}.\\ & +{\displaystyle \frac{1}{2}{\mathit{\omega }}_i^{\mathrm{T}}{\mathit{I}}_{p_i}{\mathit{\omega }}_i+\sum _{i=1}^2{m}_{p_i}{\dot{u}}_i^2(0,t)}\end{array} \end{gathered}$$

(1)

Here, D(*)/D(t) stand for the material derivative which can be written as $\mathrm{\partial }(*)/\mathrm{\partial }t+v\mathrm{\partial }(*)/\mathrm{\partial }x$ . $m_{c_i}(t)$ represents the equivalent mass of the conveyance combined with compensating rope, and the specific expression is: $m_{c_i}(t)=m_i+{\rho }_j{l}_j(t),$ $i=1,4,j=2,3$ . Here, $m_i$ , ${\rho }_i$ , and $l_i$ stands for the mass of conveyance, the line density, and length of the compensating, respectively. ${\mathit{I}}_{p_i}$ represents the equivalent moment of inertia of the conveyance combined with compensating. $m_{p_i}$ stands for the mass of controller. $l_i(t),i=1,4$ stands for the length and velocity of ropes. Here, $u_i(0,t),(i=1,2)$ represents the control displacement of vibration controller on ascending and descending side. $u_1(x_1,t)$ and $u_4(x_4,t)$ represent the longitudinal vibration displacement of ropes. ${\mathit{r}}_i$ and ${\mathit{\omega }}_i,(i=1,2)$ indicate the displacement and angular velocity of container along $x,y,z$ directions, and the specific expression is set as follows:

$$\begin{array}{rl}& {\mathit{r}}_i=[u_{c,i,x},u_{c,i,y},u_{c,i,z},],(i=1,2),\\ & {\mathit{\omega }}_i=[{\omega }_{c,i,x},{\omega }_{c,i,y},{\omega }_{c,i,z}{]}^{\mathrm{T}},(i=1,2).\end{array}$$

(2)

The mapping of angular velocity $[{\omega }_{c,i,x},{\omega }_{c,i,y},{\omega }_{c,i,z}],(i=1,2)$ of connected body relative to angular displacement ${\mathit{\phi }}_i=[{\gamma }_i,{\vartheta }_i,{\psi }_i{]}^{\mathrm{T}},(i=1,2)$ of inertial coordinate system is obtained by coordinate transformation, which can be obtained as follows:

$$[{\omega }_{c,i,x},{\omega }_{c,i,y},{\omega }_{c,i,z}{]}^{\mathrm{T}}={\mathit{R}}_i{\dot{\mathit{\phi }}}_i.$$

(3)

In which, ${\mathit{R}}_i$ represents rotation matrix. ${\gamma }_i,{\vartheta }_i,{\psi }_i$ represent the angular displacement around the x-, y-, and z-axes, respectively.

$${\mathit{R}}_i=\left[\begin{array}{ccc}1& 0& -\sin ({\vartheta }_i)\\ 0& \cos ({\gamma }_i)& \sin ({\gamma }_i)\cos ({\vartheta }_i)\\ 0& -\sin ({\gamma }_i)& \cos ({\gamma }_i)\cos ({\vartheta }_i)\end{array}\right].$$

(4)

Taking Equations (2), (3), and (4) into Equation (5), the kinetic energy of the conveyance can be set as follows:

$$E_{k,p}(t)=\sum _{i=1}^2{\displaystyle \frac{1}{2}{m}_{c_i}(t){\dot{\mathit{r}}}_i^{\mathrm{T}}{\dot{\mathit{r}}}_i+{\displaystyle \frac{1}{2}{({\mathit{R}}_i{\dot{\mathit{\phi }}}_i)}^{\mathrm{T}}{\mathit{I}}_{p_i}({\mathit{R}}_i{\dot{\mathit{\phi }}}_i)}}.$$

(5)

The mass, damping, and force terms of the platform can be obtained through the above equation. See Appendix 1 for the specific expression forms. The elastic potential energy of the system is caused by the tension and strain of the lifting rope, and the specific expression is set as follows:

$$K_{u_i}(t)={\int }_0^{l_i(t)}\left[T_i(x_i,t){\epsilon }_i+{\displaystyle \frac{1}{2}EA{\epsilon }_i^2}\right]\mathrm{d}{x}_i,(i=1,2).$$

This, $T_i(x_i,t),i=1,4$ stands for the static tension of lifting ropes.

$$T_i(x_i,t)=[M_{q_i}+{\rho }_i(l_i(t)-x_i(t))]\cdot g,i=1,2.$$

(6)

${\epsilon }_i=u_{i,x}(x_i,t)$ and $i=1,4$ denote the strain of the lifting rope at x, respectively. E represents the elastic modulus of lifting rope, and A represents the cross-sectional area of ropes. $M_{q_i}$ represents the equivalent mass of the conveyance combined with compensating rope, and the specific expression is $M_{q_i}=m_i+{\rho }_i{l}_i(t),i=2,3$ .

The expression of gravitational potential energy of lifting system is

$$E_g=-\sum _{i=1}^2{\int }_0^{l_i(t)}{\rho }_{i}g{u}_i(x_i,t)d{x}_i-\sum _{i=1}^2{M}_{q_i}g\cdot q_i.$$

(7)

The dissipated energy of the lifting rope and the virtual work of the control input of the control actuator can be expressed as follows:

$$\delta W_i=-\sum _{i=1}^2\left(c_i{\int }_0^{l_i(t)}{\displaystyle \frac{D{u}_i(x_i,t)}{Dt}}\delta u_i(x_i,t)dx-{\kappa }_i(t)\delta u_i(0,t)\right).$$

(8)

Here, c_i denotes the system damping coefficient, and ${\kappa }_i(t)$ denotes the controlling force vector. Using Hamilton's principle

$${\int }_{t_1}^{t_2}(\delta E_k-\delta E_e-\delta E_g)dt=0.$$

(9)

Taking Equations (1), (5), and (7) into Equation (9), the governing equations and boundary conditions can be obtained as shown in Equations (10) and (12).

$$\begin{array}{rl}{\rho }_i& \left[{\displaystyle \frac{D{u}_i(x_i,t)}{Dt}+a_i(t)}\right]+{\displaystyle \frac{c_{i}d{u}_i(x_i,t)}{dt}-T_{i,x}(x_i,t)}\\ & -EA{u}_i^{\mathrm{″}}(x_i,t)={\rho }_{i}g.\end{array}$$

(10)

Here, $0\le x_i\le l_i(t)$ , $(i=1,4)$ . The partial derivative of tension for x is obtained as follows:

$$T_{i,x}=-{\rho }_{i}g,(i=1,4).$$

(11)

Taking Equation (11), the upper and lower boundary of the system can be obtained as follows:

$$\begin{array}{rl}& \begin{array}{c}{m}_{p_i}{\ddot{u}}_i(0,t)-EA{u^{\mathrm{\prime }}}_i(0,t)=m_{p_i}g+{\kappa }_i(t)\end{array}(i=1,4)\\ & EA{u^{\mathrm{\prime }}}_i(l_i(t),t)=-{\lambda }_i(t)(i=1,4).\end{array}$$

(12)

${\lambda }_i(t)$ is the connecting tension. The conveyance keeps balance under the tension of the lifting rope and its own gravity. Here, the dynamic equations of the conveyance are obtained as follows:

$$\mathit{M}(\mathit{q})\ddot{\mathit{q}}(t)+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}={\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\lambda }(t)+\mathit{F}(\mathit{q}).$$

(13)

q represents the generalized degree of freedom of conveyance. $\mathit{M}(\mathit{q})$ , $\mathit{C}(\mathit{q},\dot{\mathit{q}})$ , and $\mathit{F}(\mathit{q})$ are expressed as the generalized mass, damping, and force of the conveyance, respectively. $\mathit{G}(\mathit{q})$ represents the derivative of the constraint equations between lifting rope and conveyance. $\mathit{\lambda }(t)$ is the connecting tension. ${\mathit{G}}_q(\mathit{q})$ is the Jacobian matrix of constraint equation for generalized coordinates.

Constraint relationship $\mathit{G}(\mathit{q})$ between lifting rope and conveyance is set as

$$u_i(l_i,t)-u_{c,i,x}(t)=0,(i=1,2).$$

(14)

Therefore, the dynamic equations and boundary conditions of the system are shown as Equations (13) and (14).

Preliminaries

To design the control law and prove the feasibility, preliminary theorems should been given firstly. In this section, some related concepts are given to explain the following research.

Law 1: Considering the energy of disturbance $d_z(t)$ is limited, disturb satisfy the following condition: $|d_z(t)|\le {\overline{d}}_z(t)$ , ${\overline{d}}_z(t)\in R^{+}$ .

Law 2: Distribution force $T_i(x_i,t)$ is set as nonnegative. In other words, compared with the acceleration of the lifting rope, $a_i(t)<g$ . Thus, $T_i^{\mathrm{\prime }}(x_i,t)$ and ${\dot{T}}_i(x_i,t)$ are all bounded.

Law 3: The initial state value of the lifting system is bounded in norm $L^2$ .

Law 4: $\dot{\mathit{M}}(\mathit{q})-\mathit{C}(\mathit{q},\dot{q})$ is skew symmetric matrix, and the mass matrix $\mathit{M}(\mathit{q})$ is a positive definite symmetric matrix and bounded.

In Figure 2, the design process of system boundary control is shown.

Figure 2.

Design process of control law.

In Figure 3, the design process of the control law is shown. Boundary control law and stability analysis are designed by Lyapunov function, which is constructed to realize the output constraint.

Figure 3.

The process of boundary control of lifting system.

Based on the dynamic equations established in the previous section, the dynamic equations of the system are organized into the following form:

$$\mathit{A}(t)\ddot{\mathit{x}}=\mathit{B}(t)\mathit{x}+\mathit{C}(t).$$

(15)

In this section, the dynamic equation is a differential-algebraic mixed equation, which contains Lagrange multipliers ${\lambda }_i$ , which is unknown variables. Add two expressions of boundary Equation (12) to obtain the following equation:

$$m_{p_i}{\ddot{u}}_i(0,t)+\epsilon (t)=m_{p_i}g+{\kappa }_i(t)-{\lambda }_i(t).$$

(16)

Here, $\epsilon (t)=EA{u}_i^{\mathrm{\prime }}(l_i(t),t)-EA{u}_i^{\mathrm{\prime }}(0,t)$ . $m_{p_i}$ is the mass of the boundary driven drum. g represents gravitational acceleration. ${\kappa }_i(t)$ is the boundary input. $\epsilon (t)$ is the sum of the elastic force due to gravity and the excitation disturbance of longitudinal vibration. Thus, the Lagrange multiplier ${\lambda }_i(t)$ can be obtained as follows:

$${\lambda }_i(t)={\kappa }_i(t)-{\tilde{F}}_{g,i}(t)+{\tilde{D}}_{d,i}(t).$$

(17)

Here, the specific expression can be obtained as follows:

$$\begin{array}{rl}& {\tilde{F}}_{g,i}(t)=-F_{g,i}(t),\\ & F_{g,i}(t)=M_{c_i}g+\rho l_i(g-a_i)-M_{c_i}{\ddot{u}}_i(0,t),\\ & {\tilde{D}}_{d,i}(t)=-D_{d,i}(t).\end{array}$$

(18)

Introduce Equation (17) into the dynamic equation to replace Lagrange multiplier $\mathit{\lambda }(t)$ :

$$\mathit{M}(\mathit{q})\ddot{\mathit{q}}(t)+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}={\mathit{G}}_q^{\mathrm{T}}(\mathit{q})[\mathit{\kappa }(t\mathbf{)}-{\tilde{\mathit{F}}}_{\mathit{g}}\mathbf{(}t\mathbf{)}+{\tilde{\mathit{D}}}_{\mathit{d}}\mathbf{(}t\mathbf{)}]+\mathit{F}(\mathit{q}).$$

(19)

Here, $\mathit{\kappa }(t\mathbf{)}$ , ${\tilde{\mathit{F}}}_{\mathit{g}}\mathbf{(}t\mathbf{)}$ , and ${\tilde{\mathit{D}}}_{\mathit{d}}\mathbf{(}t\mathbf{)}$ are the collection of input control force vector ${\kappa }_i(t)$ , system force vector ${\tilde{F}}_{g,i}(t)$ , and disturbance vector ${\tilde{D}}_{d,i}(t)$ .

Considering the nonlinear uncertainties factors, it is represented by a nonlinear smooth function $\mathbf{\Delta }N(t)$ . Considering the disturbance term, the control model of the system is as follows:

$$\begin{array}{rl}& {\dot{\mathit{x}}}_{\mathbf{1}}={\mathit{x}}_{\mathbf{2}},\\ & {\dot{\mathit{x}}}_{\mathbf{2}}={\mathit{M}}^{-1}(\mathit{q})[{\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\kappa }(t)\mathbf{+}Q(\mathit{x})+{\mathit{G}}_q^{\mathrm{T}}(\mathit{q}){\tilde{\mathit{D}}}_{\mathit{d}}(t)+\mathbf{\Delta }N(x,t)].\end{array}$$

(20)

Here, $\mathit{Q}(\mathit{x})=-{\mathit{G}}_q^{\mathrm{T}}(\mathit{q}){\tilde{\mathit{F}}}_{\mathit{g}}(t)+\mathit{F}(\mathit{q})-\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}$ .

Approximation principle of fuzzy system

The nonlinear control is a complex problem, in which the nonlinear terms must be approximated by compensation fitting. Fuzzy system can be used as a universal approximator to approximate any nonlinear function with any precision. The selection of product inference engine, single-valued fuzzy, central average defuzzifier, and Gauss membership function as universal approximator is satisfying requirements. In Figure 4, the diagram of fuzzy system composition and deduction procedure is as follows:

Figure 4.

The diagram of fuzzy system composition and deduction procedure.

Here, fuzzy system is the mapping from input vector $x\in U$ to output variable $y\in V$ . By combining a single fuzzy set, the total fuzzy system is as follows:

$$\mathit{y}(x,u)={\displaystyle \frac{\sum _{i=1}^m{\omega }_i\left(\prod _{j=1}^n{\mu }_{F_{ij}}(x_j)\right)}{\sum _{i=1}^m\left(\prod _{j=1}^n{\mu }_{F_{ij}}(x_j)\right)}.}$$

(21)

${\omega }_i$ , $x_j$ , and ${\mu }_{F_{ij}}(x_j)$ represent membership function of fuzzy system, input of membership function, and optimal weight of fuzzy system. Set ${\mathbf{\Phi }}^{\mathrm{T}}=[{\omega }_1,\dots ,{\omega }_m]$ as the weight coefficient of fuzzy system and $\mathbf{P}(x)=[p_1(x),\dots ,p_m(x){]}^{\mathrm{T}}$ as the fuzzy basis function vector. In which, $p_i(x)$ can be expressed as follows:

$$p_i(x)={\displaystyle \frac{\prod _{j=1}^n{\mu }_{F_{ij}}(x_j)}{\sum _{i=1}^m\left(\prod _{j=1}^n{\mu }_{F_{ij}}(x_j)\right)}.}$$

(22)

Then, Equation (21) can be expressed as follows:

$$\mathit{y}(x,u)={\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x).$$

(23)

In order to approximate the disturbance term $\mathbf{\Delta }N(x,t)$ with output of the fuzzy system in above formula, the gain scaling of the disturbance term is carried out to form the state equation of the compensator as $\mathit{N}(x,t)=K_p\mathbf{\Delta }N(x,t)$ , and then the fuzzy system is used to fit the model as follows:

$$\mathbf{\Delta }N(x,t)=K_p^{-1}{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x)+\epsilon .$$

(24)

In which, $\mathbf{\Phi }$ is designated as the optimal approximation weight coefficient vector for fuzzy systems to approximate nonlinear functions $\mathbf{\Delta }N(x,t)$ . By fitting the nonlinear smooth disturbance, the control model of the system in the compensation state is obtained, in which the external disturbance term and the error term of the fuzzy system disturbance are written as a unified form ${\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}={\mathit{G}}_q^{\mathrm{T}}(\mathit{q}){\tilde{\mathit{D}}}_{\mathit{d}}\mathbf{(}t\mathbf{)}+\epsilon$ . Then, the control model becomes the following form:

$$\begin{array}{rl}& {\dot{\mathit{x}}}_{\mathbf{1}}={\mathit{x}}_{\mathbf{2}},\\ & {\dot{\mathit{x}}}_{\mathbf{2}}={\mathit{M}}^{-1}(\mathit{q})[{\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\kappa }(t\mathbf{)}+Q(x)+K_p^{-1}{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x)+{\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}].\end{array}$$

(25)

From above equation, $K_p^{-1}{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x)$ and ${\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}$ are the approximation nonlinear function term of the fuzzy system and the external disturbance term of the system, which contain uncertainties parameters. Therefore, it is necessary to design an auxiliary system as a state observer and adjust the auxiliary parameters in time through the adaptive law.

Let the auxiliary convergence variable $x_2*$ of the output variable of the fuzzy system be $x_2$ , the auxiliary convergence variable of the weight coefficient of the fuzzy system be $\mathbf{\Phi }*$ , and the auxiliary variable ${\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}$ be ${\tilde{\mathit{D}}}_t*\mathbf{(}t\mathbf{)}$ . The output approximation error $e_x$ of the fuzzy system, the approximation error $e_{\varphi }$ of the weight coefficient of the fuzzy system, and the approximation error $e_D$ of the error disturbance term are obtained as follows:

$$\begin{array}{rl}& e_x={\mathit{x}}_2-{\mathit{x}}_2*,\\ & e_{\varphi }=\mathbf{\Phi }-{\mathbf{\Phi }}_{*},\\ & e_D={\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}-{\tilde{\mathit{D}}}_t*\mathbf{(}t\mathbf{)}.\end{array}$$

(26)

In order to make the output approximation error $e_x$ of the fuzzy system, the approximation error $e_{\varphi }$ of the weight coefficient of the fuzzy system and the approximation error $e_D$ of the error disturbance term meet the control accuracy.

It is necessary to design the auxiliary convergence variables $x_2*$ of the approximation error of the output variables of the fuzzy system, the auxiliary convergence variables $\mathbf{\Phi }*$ of the approximation error of the weight coefficients of the fuzzy system, and the auxiliary convergence variables ${\tilde{\mathit{D}}}_t*\mathbf{(}t\mathbf{)}$ of the approximation error of the error disturbance term to obtain the adaptive law. The derivative forms of the auxiliary convergence variables of approximation error are set as follows:

$$\begin{array}{rl}& {\dot{\mathit{x}}}_2*={\mathit{M}}^{-1}[L{\mathit{e}}_x+{\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\kappa }(t\mathbf{)}+Q(x)+K_p^{-1}{\mathbf{\Phi }}^{*\mathrm{T}}\mathbf{P}(x)+{\tilde{\mathit{D}}}_t*\mathbf{(}t\mathbf{)}],\\ & \dot{\mathbf{\Phi }}*=-{\dot{\mathit{e}}}_{\varphi }=-2k\mathbf{\Phi }*+\gamma [({\mathit{e}}_2^{\mathrm{T}}+{\mathit{e}}_x^{\mathrm{T}})K_p^{-1}{\mathbf{P}}^{\mathrm{T}}(x){]}^{\mathrm{T}},\\ & {\dot{\tilde{\mathit{D}}}}_t*\mathbf{(}t\mathbf{)}=K_p{\mathit{e}}_D+{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{P}(x).\end{array}$$

(27)

Take Equation (27) into Equation (26), and simplify as follows:

$${\dot{\mathit{e}}}_x={\dot{x}}_2-{\dot{x}}_2*={\mathit{M}}^{-1}[-L{e}_x+K_p^{-1}{e}_{\varphi }^{\mathrm{T}}\mathbf{P}(x)+e_D(t)],$$

(28)

$${\dot{\mathit{e}}}_{\varphi }=2k\mathbf{\Phi }*-\gamma [(e_2^{\mathrm{T}}+e_x^{\mathrm{T}})K_p^{-1}{\mathbf{P}}^{\mathrm{T}}(x){]}^{\mathrm{T}},$$

(29)

$${\dot{\mathit{e}}}_D={\dot{\tilde{\mathit{D}}}}_t\mathbf{(}t\mathbf{)}-(K_p{e}_D+e_{\varphi }^{\mathrm{T}}\mathbf{P}(x)).$$

(30)

In which, $e_2$ represents the error of the second-order control system in the back-stepping controller design process, and its expression is as follows:

$$e_2=x_2-{\alpha }_1.$$

(31)

In which, ${\alpha }_1$ represents the virtual control law, which is needed to be designed.

Fuzzy adaptive back-stepping controller design

Considering the nonlinear and uncertainty disturbance acting on the system, the control system is designed by the back-stepping control method. Define $x_1=\mathit{q}$ , $x_2=\dot{\mathit{q}}$ , the state space equation of the system can be written as follows:

$$\begin{array}{rl}& {\dot{\mathit{x}}}_1={\mathit{x}}_2,\\ & {\dot{\mathit{x}}}_2={\mathit{M}}^{-1}(\mathit{q})[{\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\kappa }(t\mathbf{)}+Q(x)+K_p^{-1}{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x)+{\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}],\\ & \mathit{y}={\mathit{x}}_1.\end{array}$$

(32)

In order to design the controller using back-stepping method, the coordinate transformation is carried out firstly:

$$e_i=x_i-{\alpha }_{i-1},(i=1,\dots ,n).$$

(33)

In which, ${\alpha }_0=x_d$ , ${\alpha }_{i-1},(i=1,\dots ,n)$ is the virtual control parameter to be designed. It can be seen that the control system is a second-order system, which the control system is separated into two subsystems. The design steps of the controller based on the fuzzy system design in this section are as follows:

1. Design a virtual control law for the first-order subsystem:

$${\dot{e}}_1={\dot{x}}_1-{\dot{x}}_d.$$

(34)

In which, $x_d$ is the desired pose of the conveyance.

2. Define the virtual control law ${\alpha }_1$ as the estimation of $x_2$ , and the error of the second subsystem is

$$e_2=x_2-{\alpha }_1.$$

(35)

Take $x_2={\alpha }_1+e_2$ to equation, obtaining ${\dot{e}}_1={\alpha }_1+e_2-{\dot{x}}_d$ . Designing ${\alpha }_1$ as ${\alpha }_1=-{\lambda }_1{e}_1+{\dot{x}}_d$ , obtain ${\dot{e}}_1=-{\lambda }_1{e}_1+e_2$ .

For the first-order subsystem, in order to reach the system converge, the Lyapunov function $F_1(t)$ is set as follows:

$$F_1(t)={\displaystyle \frac{1}{2}{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_1.}$$

(36)

The derivative of ${\mathit{F}}_1(t)$ can be obtained as ${\dot{F}}_1(t)={\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_2-{\mathit{e}}_1^{\mathrm{T}}{\mathit{\lambda }}_1{\mathit{e}}_1$ , where ${\mathit{\lambda }}_1>0$ . Here, $-{\mathit{\lambda }}_1{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_1$ is negative definitely. When ${\mathit{e}}_2=0$ , the first-order subsystem must be stable. In order to make the error ${\mathit{e}}_2$ of the second-order subsystem tend to 0, we need to design the total Lyapunov function $F_2(t)$ to make it tend to be stable. Positive definite function of output error $e_2$ , output approximation error $e_x$ of fuzzy system, weight approximation error $e_{\varphi }$ of fuzzy system, and approximation error $e_D$ of error disturbance term are all contained in the total Lyapunov function $F_2(t)$ , which all need to obtain the corresponding positive definite function to verify and construct the corresponding Lyapunov function. So taking Equations (33), (34), (35), and (36) into Equation (37), $F_2(t)$ is set as follows:

$$\begin{array}{r}{F}_2(t)={\displaystyle \frac{1}{2}{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_1+{\displaystyle \frac{1}{2}{\mathit{e}}_2^{\mathrm{T}}M{\mathit{e}}_2+{\displaystyle \frac{1}{2}{\mathit{e}}_x^{\mathrm{T}}M{\mathit{e}}_x+{\displaystyle \frac{1}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }+{\displaystyle \frac{1}{2}{\mathit{e}}_D^{\mathrm{T}}{\mathit{e}}_D.}}}}}\end{array}$$

(37)

Differential Equation (37) $F_2(t)$ with t, obtaining

$$\begin{array}{rl}& {\dot{F}}_2(t)=\left({\mathit{e}}_1^{\mathrm{T}}{\dot{\mathit{e}}}_1+{\displaystyle \frac{1}{2}{\mathit{e}}_2^{\mathrm{T}}\dot{M}{\mathit{e}}_2+{\mathit{e}}_2^{\mathrm{T}}M{\dot{\mathit{e}}}_2+{\displaystyle \frac{1}{2}{\mathit{e}}_x^{\mathrm{T}}\dot{M}{\mathit{e}}_x+{\mathit{e}}_x^{\mathrm{T}}M{\dot{\mathit{e}}}_x+{\displaystyle \frac{1}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\dot{\mathit{e}}}_{\varphi }+{\mathit{e}}_D^{\mathrm{T}}{\dot{\mathit{e}}}_D}}}\right).\end{array}$$

(38)

Put ${\dot{e}}_2={\dot{x}}_2-{\dot{\alpha }}_1$ into equation, and put ${\dot{\mathit{x}}}_{\mathbf{2}}$ in Equation (32), obtaining:

$$\begin{array}{rl}& {\dot{F}}_2(t)=\left(\begin{array}{c}(e_2^{\mathrm{T}}[\tau \mathbf{+}Q(x)+K_p^{-1}{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{P}(x)+{\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}]-e_2^{\mathrm{T}}M{\dot{\alpha }}_1)+{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_2-{\mathit{e}}_1^{\mathrm{T}}\mathit{\lambda }{\mathit{e}}_1\\ +{\displaystyle \frac{1}{2}{e}_x^{\mathrm{T}}\dot{M}{e}_x+e_x^{\mathrm{T}}M{\dot{e}}_x+{\displaystyle \frac{1}{2}{e}_2^{\mathrm{T}}\dot{M}{e}_2+{\displaystyle \frac{1}{\gamma }{e}_{\varphi }^{\mathrm{T}}{\dot{e}}_{\varphi }+e_D^{\mathrm{T}}{\dot{e}}_D}}}\end{array}\right).\end{array}$$

(39)

By constructing the intermediate virtual control variables of each order system, the total control output can be obtained to achieve the stabilization of the vibration system and suppress the amplitude of the longitudinal vibration.

Here, $\tau ={\mathit{G}}_q^{\mathrm{T}}(\mathit{q})\mathit{\kappa }(t\mathbf{)}$ . Design the output variables of the control τ as the following form:

$$\tau =C_1+C_2+C_3+C_4.$$

(40)

In which, the specific expressions is as follows:

$$\begin{array}{rl}& C_1=-{\mathit{\lambda }}_2{\mathit{e}}_2-{\mathit{e}}_1-C{\mathit{e}}_2,C_2=-\mathit{Q}(x),C_3=\mathit{M}{\dot{\alpha }}_1,\\ & C_4=-K_p^{-1}{\mathbf{\Phi }}^{*\mathrm{T}}\mathbf{P}(x)-{\tilde{\mathit{D}}}_t*\mathbf{(}t\mathbf{)}\end{array}.$$

Here, C₁ is used to compensate the partial error-related terms produced by the state variables. C₂ is used to compensate the dynamic characteristic term caused by the factors of the lifting system. C₃ is used to compensate partial error correlation term and trajectory acceleration disturbance term produced by virtual intermediate control variables. C₄ represents the compensation term of the lifting system due to the time-varying uncertainty of platform parameters and external nonlinear disturbance. λ₂ is a positive definite diagonal matrix.

Stability analysis

Based on previous analysis, the Lyapunov function of the lifting system is considered as Equation (37). Take Equation (40) into Equation (39), and based on Law 4, ${\dot{F}}_2(t)$ can be simplify as follows:

$${\dot{F}}_2(t)=\left(\begin{array}{c}-{\mathit{e}}_2^{\mathrm{T}}{\mathit{\lambda }}_2{\mathit{e}}_2-{\mathit{e}}_1^{\mathrm{T}}{\mathit{\lambda }}_1{\mathit{e}}_1+{\mathit{e}}_2^{\mathrm{T}}{K}_p^{-1}{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{P}(x)+{\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_D\\ +{\displaystyle \frac{1}{2}{\mathit{e}}_x^{\mathrm{T}}\dot{M}{\mathit{e}}_x+{\mathit{e}}_x^{\mathrm{T}}M{\dot{\mathit{e}}}_x+{\displaystyle \frac{1}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\dot{\mathit{e}}}_{\varphi }+{\mathit{e}}_D^{\mathrm{T}}{\dot{\mathit{e}}}_D}}\end{array}\right).$$

(41)

Based on Equation (28), obtaining:

$${\mathit{e}}_x^{\mathrm{T}}M{\dot{\mathit{e}}}_x=[-{\mathit{e}}_x^{\mathrm{T}}L{\mathit{e}}_x+{\mathit{e}}_x^{\mathrm{T}}{K}_p^{-1}{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{P}(x)+{\mathit{e}}_x^{\mathrm{T}}{\mathit{e}}_D(t)].$$

(42)

Based on the Young inequalities, the parameters in the above formula are obtained by inequality transformation as follows:

$${\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_D\le \left({\displaystyle \frac{1}{2}{\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_2+{\displaystyle \frac{1}{2}{\mathit{e}}_D^{\mathrm{T}}{\mathit{e}}_D}}\right),$$

(43)

$${\mathit{e}}_x^{\mathrm{T}}M{\dot{\mathit{e}}}_x\le -{\mathit{e}}_x^{\mathrm{T}}\left(L-{\displaystyle \frac{I}{2}}\right){\mathit{e}}_x+{\mathit{e}}_x^{\mathrm{T}}{K}_p^{-1}{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{P}(x)+{\displaystyle \frac{1}{2}{\mathit{e}}_D^{\mathrm{T}}{\mathit{e}}_D.}$$

(44)

${\varpi }_1$ is introduced to give the range of Euclidean norm of fuzzy basis function vector P(x), which can be expressed as $\Vert \mathbf{P}(x)\Vert \le {\varpi }_1$ . Using Young's inequalities, obtaining:

$$\begin{array}{r}{\mathit{e}}_D^{\mathrm{T}}{\dot{\mathit{e}}}_D\le {\displaystyle \frac{1}{2}{\mathit{e}}_D^{\mathrm{T}}(I+\gamma {\varpi }_1^{2}I-2K_p){\mathit{e}}_D+{\displaystyle \frac{1}{2}{\dot{\tilde{\mathit{D}}}}_t^{\mathrm{T}}\mathbf{(}t\mathbf{)}{\dot{\tilde{D}}}_t\mathbf{(}t\mathbf{)}+{\displaystyle \frac{1}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }^{.}}}}\end{array}$$

(45)

Based on Law 1, the energy of disturbance $d_z(t)$ is limited and the disturbance is satisfied $|d_z(t)|\le {\overline{d}}_z(t)$ , ${\overline{d}}_z(t)\in R^{+}$ . So ${\tilde{\mathit{D}}}_t\mathbf{(}t\mathbf{)}$ is also limited. Set the magnitude of its Euclidean norm as ${\xi }_1$ . Thus, Equation (45) can be simplified as follows:

$${\mathit{e}}_D^{\mathrm{T}}{\dot{\mathit{e}}}_D\le {\displaystyle \frac{1}{2}{\mathit{e}}_D^{\mathrm{T}}(I+\gamma {\varpi }_1^{2}I-2K_p){\mathit{e}}_D+{\displaystyle \frac{1}{2}{\xi }_1^2+{\displaystyle \frac{1}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }^{.}}}}$$

(46)

Substituting Equation (29) into Equation (41), obtaining:

$${\displaystyle \frac{1}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\dot{\mathit{e}}}_{\varphi }=\left({\displaystyle \frac{2k}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{\Phi }*-{\mathit{e}}_{\varphi }^{\mathrm{T}}{[({\mathit{e}}_2^{\mathrm{T}}+{\mathit{e}}_x^{\mathrm{T}})K_p^{-1}{\mathbf{P}}^{\mathrm{T}}(x)]}^{\mathrm{T}}}\right).}$$

(47)

Take ${\mathit{e}}_{\varphi }^{=}\mathbf{\Phi }-\mathbf{\Phi }*$ into $\frac{2k}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{\Phi }*$ and simplify equations as follows:

$${\displaystyle \frac{2k}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{\Phi }*\le {\displaystyle \frac{k}{\gamma }(-{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }-{\mathbf{\Phi }}^{*\mathrm{T}}\mathbf{\Phi }*+2{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }).}}$$

(48)

Based on Young's inequalities, obtaining:

$${\displaystyle \frac{2k}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}\mathbf{\Phi }*\le \left(-{\displaystyle \frac{k}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }+{\displaystyle \frac{2k}{\gamma }{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }}}\right).}$$

(49)

Substituting Equation (49) into Equation (47), and simplified as follows:

$${\displaystyle \frac{1}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\dot{\mathit{e}}}_{\varphi }\le \left(-{\displaystyle \frac{k}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }+{\displaystyle \frac{2k}{\gamma }{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }-{\mathit{e}}_{\varphi }^{\mathrm{T}}{[({\mathit{e}}_2^{\mathrm{T}}+{\mathit{e}}_x^{\mathrm{T}})K_p^{-1}{\mathbf{P}}^{\mathrm{T}}(x)]}^{\mathrm{T}}}}\right).}$$

(50)

Substituting Equations (43), (44), (46), and (50) into Equation (41), obtained:

$$\begin{array}{rl}& {\dot{\mathit{V}}}_2(t)\le \left(\begin{array}{c}-{\mathit{e}}_1^{\mathrm{T}}{\mathit{\lambda }}_1{\mathit{e}}_1+{\mathit{e}}_2^{\mathrm{T}}\left({\displaystyle \frac{I}{2}-{\mathit{\lambda }}_2}\right){\mathit{e}}_2+{\mathit{e}}_D^{\mathrm{T}}\left({\displaystyle \frac{3}{2}I+{\displaystyle \frac{1}{2}\gamma {\varpi }_1^{2}I-K_p}}\right){\mathit{e}}_D\\ +{\mathit{e}}_x^{\mathrm{T}}\left({\displaystyle \frac{I}{2}+{\displaystyle \frac{1}{2}\dot{M}-L}}\right){\mathit{e}}_x+{\displaystyle \frac{(1-k)}{2\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }+{\displaystyle \frac{2k}{\gamma }{\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }+{\displaystyle \frac{1}{2}{\xi }_1^2}}}\end{array}\right).\end{array}$$

(51)

${\mathit{\lambda }}_1,{\mathit{\lambda }}_2,K_p$ is the control gain matrix. I is the identity matrix. L is the positive definite symmetric gain matrix. M is the generalized mass matrix. $\gamma$ is positive real number. k is a positive number. ${\xi }_1$ is a unknown bounded positive number. Based on $0<\mathit{M}(q)\le {\sigma }_0\mathit{I}$ , it can be obtained as $-\frac{1}{\mathit{M}}\le -\frac{\mathit{I}}{{\sigma }_0}$ . Here, in order to prove the astringency of ${\mathit{V}}_2(t)$ , ${\dot{\mathit{V}}}_2(t)$ is arranged in the form shown below:

$$\begin{array}{rl}& {\dot{\mathit{V}}}_2(t)\le \left({\mathit{H}}_1{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_1+{\displaystyle \frac{{\mathit{H}}_2}{{\sigma }_0}{\mathit{e}}_2^{\mathrm{T}}\mathit{M}{\mathit{e}}_2+{\mathit{H}}_3{\mathit{e}}_D^{\mathrm{T}}{\mathit{e}}_D+{\displaystyle \frac{{\mathit{H}}_4}{{\sigma }_0}{\mathit{e}}_x^{\mathrm{T}}\mathit{M}{\mathit{e}}_x+{\displaystyle \frac{{\mathit{H}}_5}{\gamma }{\mathit{e}}_{\varphi }^{\mathrm{T}}{\mathit{e}}_{\varphi }+{\mathit{H}}_6}}}\right).\end{array}$$

(52)

Here, $H=min\left[\lambda (2{\mathit{H}}_1),\lambda (\frac{2{\mathit{H}}_2}{{\sigma }_0}),\lambda (2{\mathit{H}}_3),\lambda (\frac{2{\mathit{H}}_4}{{\sigma }_0}),\lambda (2{\mathit{H}}_5)\right]$ , ${\mathit{b}}_0={\mathit{H}}_6$ . ${\mathit{H}}_i,i=1,2,3,4,5,6$ represents the expression definition in the proof of control law convergence. The specific expression can be found in Appendix 1. So

$${\dot{\mathit{V}}}_2(t)\le -H{\mathit{V}}_2(t)+{\mathit{H}}_6.$$

(53)

Therefore, the designed control rate can make the system converge exponentially, and the vibration suppression and stabilization of the system can be realized by selecting appropriate gain variables and parameters.

Simulations

Based on the control model established in the previous section, the dynamic response results of the lifting system can be obtained through numerical solution algorithm. The operating parameters are shown in Table 1.

Table 1.

Parameters of the lifting system.

Parameter	Description	Vales
a 1	Acceleration	0.72 m/s2
tup（tdown）	Acceleration time	19.2 s
vmax	Velocity	13.66 m/s
tv	Uniform time	79.4 s
eA	Perturbation amplitude	0.005 mm

Physical parameters of the lifting system are shown below. Here, the line density of ropes is 9.6 kg/m. The mass of ascending and descending conveyance is 61t and 36t, respectively. Axial modulus of elasticity of ropes is 1.2 × 1011 pa. The diameter of the rope is 48 mm. The initial lengths of ropes at ascending and descending side are 853 and 30 m, respectively. The fundamental frequency of the disturbance is v/R, in which v represents the lifting speed and R represents drum radius. The finite difference method is used to solve the equation of controlling model. The defined control parameters include ${\mathit{\lambda }}_1,{\mathit{\lambda }}_2,\gamma ,k$ , and $K_p$ . In order to meet the requirements of exponential convergence of the designed control law's stability, based on inequalities (52) and (53), the range of control parameter values are obtained. After the parameters of the selected controller are brought into the inequalities (52) and (53), they must be established to meet the control requirements. Furthermore, considering excessive control gain can easily lead to system divergence, the control parameters are limited to a certain range. Moreover, there are many control parameters satisfying the inequality (52) and (53), and it is necessary to find the optimal control parameters by trying to calculate. By introducing different control parameters into the control model for solution, different vibration control effects can be obtained. By continuously introducing parameters into the calculation, the optimal control parameters can be selected as follows. Therefore, the parameters of the system controller are set as follows:

$$\begin{array}{rl}& {\mathit{\lambda }}_1=\mathrm{diag}(300,0,0,1000,1000,0);\\ & {\mathit{\lambda }}_2=\mathrm{diag}(500,0,0,1000,1000,0);\\ & \gamma =\mathrm{diag}(200,20,20,400,400,20);\\ & k=100;K_p=\mathrm{diag}(1000,1000,1000,1000,1000,1000).\end{array}$$

(54)

According to the variation range of the input quantity of the fuzzy system, the fuzzy membership function is in the following form:

$$\begin{array}{rl}& {\mu }_{F_i^1}=\exp [-0.5{((x_i+2.5)/0.6)}^2],\\ & {\mu }_{F_i^2}=\exp [-0.5{((x_i+1.25)/0.6)}^2],\\ & {\mu }_{F_i^3}=\exp [-0.5{(x_i/0.6)}^2],\\ & {\mu }_{F_i^4}=\exp [-0.5{((x_i-1.25)/0.6)}^2].\\ & {\mu }_{F_i^5}=\exp [-0.5{((x_i-2.5)/0.6)}^2]\end{array}$$

(55)

Here, （i = 1,2,3,…,n）represents the number of fuzzy sets of a fuzzy system. Based on adaptive fuzzy approximation and disturbance observer, the longitudinal vibration of conveyance under excitation disturbance and active control intervention of the top boundary controller is given in the following figures. Here, the fundamental and 0.8 times fundamental frequency excitation disturbance is selected as the frequency of the excitation disturbance. Here, another control algorithm and corresponding controller: PID back-stepping controller are designed to verify the progressiveness and effectiveness of the fuzzy adaptive back-stepping controller proposed in this article. In Figure 5, the longitudinal displacement of conveyance under different excitations is shown as follows.

Figure 5.

The longitudinal displacement of conveyance under fundamental frequency and 0.8 times fundamental frequency excitation: (a, b) descending side and (c, d) ascending side.

The figures above show the longitudinal displacement of the conveyance under external excitation disturbance with (red and blue line) or without (black line) the participation of controller, in which (a), (b), (c), and (d) represent the response of ascending and descending conveyance under the fundamental and 0.8 fundamental frequency excitation. By comparing the displacement tracking results of the two controllers after application and without application, it can be seen that there is a significant difference in displacement tracking between the PID back-stepping control method and the fuzzy adaptive control method proposed in this article, especially in the middle position during operation, the displacement tracking effect of the PID back-stepping controller is very poor. However, the fuzzy adaptive back-stepping controller proposed in this article can effectively track the set ideal curve with minimal error. By comparison, the vibration of the conveyances is obvious due to nonlinear disturbance without controller acting. With proposed controller and PID back-stepping controller acting on the system, the results show that the vibration amplitudes are obviously attenuated, which proves the effectiveness of the control algorithm. Compare the control curve obtained by applying two control algorithms with the ideal set curve to obtain the displacement control error, as shown in Figure 6.

Figure 6.

The displacement error of conveyance by PID back-stepping control and proposed control: (a, c) descending side and (b, d) ascending side.

From the displacement error results, it can be seen that the fuzzy adaptive back-stepping controller proposed in this article has significant advantages in displacement tracking compared to the PID back-stepping controller. The controller proposed in this article can effectively control displacement error within a very small range throughout the entire operation process, while the PID back-stepping controller cannot, especially in the middle stage of operation.

After applying different control algorithms, the vibration velocity and acceleration of conveyance under different excitation can be obtained as the following figures. The results are shown in Figures 7 and 8.

Figure 7.

The vibration velocity of conveyance under fundamental and 0.8 times fundamental frequency excitation: (a, b) descending side and (c, d) ascending side.

Figure 8.

The vibration acceleration of conveyance under fundamental and 0.8 times fundamental frequency excitation: (a, b) descending side and (c, d) ascending side.

Considering two kinds of excitation disturbance characteristics and by contrast, both the proposed controller and PID back-stepping controller can effectively suppress the vibration of velocity and acceleration caused by the disturbance at both fundamental and 0.8 times frequencies. Especially in the resonance interval, where the excitation frequency crosses the natural frequency, the resonance can be completely eliminated by the implementation of the adaptive fuzzy back-stepping controller and PID controller. The two controllers can effectively suppress system vibration at both fundamental and 0.8 times frequencies. However, as can be seen from the above figure, the vibration control effect of the PID back-stepping controller is not as good as that of the fuzzy adaptive back-stepping controller proposed in this article.

To further analyze the differences between the two controllers, fast Fourier transformation (FFT) is applied to analyze the amplitude and frequency response characteristics of the vibration acceleration under control and uncontrolled conditions. Furthermore, the signal autocorrelation function transformation is carried out to obtain the power spectral density transformation of the system. Please see Figure 9. Here, the red and black lines represent the power spectral density of the corresponding frequency of the vibration acceleration with and without control.

Figure 9.

The power spectral density diagram of the acceleration of conveyance under fundamental and 0.8 times fundamental frequency excitation: (a, b) descending side and (c, d) ascending side.

In Figure 9, x-axial corresponds to the frequency point within the frequency range, and y-axial represents the power spectral density under the frequency value. The amplitude of the power spectral density represents vibration energy from the time domain corresponding to the frequency domain.

Compared with the power spectral density diagram, it can be seen that the application of fuzzy adaptive back-stepping and PID control algorithm can reduce the power spectral density of the vibration acceleration at the conveyance largely, which is greatly reduced and basically tends to 0, which can achieve the stabilization of the system. However, after amplifying the FFT results of the proposed controller and PID controller, it can be seen that the vibration control effect of the controller proposed in this article is better than that of PID control under fundamental frequency disturbances and 0.8 times fundamental frequency disturbances.

After normalizing the vibration displacement and velocity to −1 to 1, the phase diagram of the vibration response of the system with and without control can be obtained. The phase diagrams and sum of squared acceleration amplitudes of conveyances under 0.8 times and fundamental frequency excitation are shown in Figures 10 and 11.

Figure 10.

The phase diagram and sum of squared acceleration amplitudes of conveyance under 0.8 times fundamental frequency excitation.

Figure 11.

The phase diagram and sum of squared acceleration amplitudes of conveyance under fundamental frequency excitation.

From the phase diagram, it can be seen that with the introduction of fuzzy adaptive controllers, the final response phase diagram is gradually concentrated at the origin position, and the vibration energy is significantly reduced, gradually tending to an absolute stable state. This can also be well verified through the graph of the sum of root mean square amplitudes. By zooming in on the phase diagram of the PID and the controller proposed in this article, it can be seen that the phase diagram of the vibration response controlled by the controller proposed in this article is more concentrated at the origin than the PID controller, indicating that the controller proposed in this article has higher stability and better control effect in vibration control.

In order to verify the adaptability and robustness of the proposed control algorithm, the superposition of fundamental and multiple frequency excitation are considered to act on the lifting system. The expression of the external excitation at the boundary is

$$E_t=A_1\sin (vt/R)+A_2\sin (2vt/R).$$

(56)

The disturbance of excitation $E_t$ is introduced into the control model. The model with adaptive fuzzy back-stepping controller, PID controller, and without controller is calculated and simulated as follows. Please see Figure 14.

Figure 14.

Comparison of acceleration response with PID back-stepping and proposal control or without control under mixed frequency excitation.

As shown in Figures 12–14, under the mixed excitation of the superposition of fundamental and multiple frequency excitation, the proposed controller also has a good control effect in tracking displacement trajectory and vibration control, which has excellent performance in tracking accuracy and control stability. However, in comparison, the displacement trajectory tracking of the PID controller is still very poor, and its vibration control effect is far better than the previous control effect where the disturbance is a single frequency component. It is proven that the proposed fuzzy adaptive back-stepping controller has good adaptability and high robustness than PID controller.

Figure 12.

Comparison of displacement response with PID back-stepping and proposal control or without control under mixed frequency excitation.

Figure 13.

Comparison of vibration velocity response with PID back-stepping and proposal control or without control under mixed frequency excitation.

After normalizing, the phase diagrams and sum of squared acceleration amplitudes of conveyances under mixed frequency excitation are shown in Figure 15.

Figure 15.

The phase diagram and sum of squared acceleration amplitudes of conveyance under mixed frequency excitation.

From the results of the phase diagram and the sum of squared amplitudes, it can be seen that the robustness and adaptability of the fuzzy adaptive control method proposed in this article are much higher than those of the PID control method.

Comparing the control effectiveness of the fuzzy adaptive control method proposed in this article with the PID back-stepping control method, using attenuation rate of the sum of the vibration response amplitude's squares under fundamental and mixed frequency conditions as evaluation indicators, the amplitude attenuations of different control methods are examined. The results obtained are shown in Table 2.

Table 2.

Attenuation rate of the sum of the vibration response amplitude's squares by PID back-stepping control and fuzzy adaptive back-stepping control method.

Controller	PID back-stepping control	Fuzzy adaptive back-stepping control method
Indicator (descending/ascending)
Disturbance frequency
0.8 times fundamental frequency	7.2%/3.9%	5.1%/1.6%
Fundamental frequency	4.8%/2.2%	3.7%/1.2%
Mixed frequency	16.1%/3.1%	4.2%/1.2%

From the table above, it can be seen that the vibration control attenuation rate of the PID back-stepping controller is even as high as 16.1% during mixed frequency disturbance control. However, the vibration control attenuation rate of the fuzzy back-stepping adaptive controller proposed in this article remains basically stable, basically maintained at 5% or even 1% or below. Under the same parameters, the vibration control attenuation rate achieved by the controller proposed in this article is not only much smaller than that of the PID back-stepping controller, Moreover, the controller maintains good robustness characteristics in response to interference from different disturbance sources, and its robustness is much higher than PID back-stepping controller.

To further analyze the result, FFT is applied to analysis the amplitude and frequency response characteristics of the vibration acceleration under two controllers, as seen in Figure 16.

Figure 16.

The power spectral density diagram of the acceleration of conveyance under mixed frequency excitation: (a, b) descending side and (c, d) ascending side.

From the suppression of the corresponding frequency and amplitude in the FFT results in the above figure, it can be seen that under the influence of mixing disturbance, the main advantage of the controller proposed in this article is not only the vibration control of the fundamental frequency disturbance in mixing, but more importantly, compared to the PID controller, the control effect of the controller proposed in this article is more prominent in the vibration control of the second harmonic disturbance, which cannot be achieved by the PID back-stepping controller. This is also the advantage of the fuzzy back-stepping controller proposed in this article.

Figure 17 shows the phase diagram of the lifting system, in which figure (a) represents the ascending side, figure (b) represents the descending side. X-axis represents time, Y-axis represents the modal displacement amplitude, and Z-axis represents the modal velocity amplitude. Here, red line represents the phase diagram under the controller applied, and blue line represents the phase diagram before the controller applied (without control).

Figure 17.

Phase of the acceleration of conveyance under fundamental and 0.8 times fundamental frequency excitation: (a, b) descending side and (c, d) ascending side.

By comparison, it can be seen that the red line is completely enveloped by the blue curve, and compared with the blue line, the fluctuation of the red line decreases significantly and close to the three-dimensional center line, indicating that the convergence speed of the phase diagram is significantly accelerated and the stability of the lifting system is significantly enhanced by applying the proposed control algorithm.

By extracting the control force and the control drive displacement at the top boundary, the control input is obtained. Please see Figure 18. The blue and red line represents the control force output on the ascending and descending side.

Figure 18.

Control output force curve: (a) fundamental frequency excitation disturbance and (b) 0.8 times fundamental frequency excitation disturbance.

Compared with the above curves in Figure 18, the control force will occur a step mutation phenomenon due to the influence of acceleration and deceleration. The adjustment amplitude of the output driving displacement of the ascending side is larger than that of descending side hydraulic cylinder, and the adjustment range of the control driving displacement is proportional to the length of rope.

Conclusion

In this article, considering the nonlinear boundary disturbance and the uncertainty factors, a fuzzy adaptive back-stepping controller is designed to control the longitudinal vibration of the conveyance in lifting system with distributed mass time-varying length using fuzzy approximation and disturbance observer. The simulation results show that under different disturbance, the proposed controller can eliminate the disturbance nonstationary vibration caused by nonlinear disturbance and system parameter uncertain and effectively eliminate the system vibration. By comparing the fuzzy adaptive back-stepping controller proposed in this article with the PID back-stepping controller, it is evident that the fuzzy adaptive back-stepping controller proposed in this article has obvious advantages in trajectory tracking and vibration control. From results, it can be seen that the vibration control attenuation rate of the fuzzy back-stepping adaptive controller proposed in this article remains basically stable, basically maintained at 5% or even 1% or below. Compared with previous works, the main contributions of the article can be stated as follows:

A vibration control model for a distributed mass flexible lifting system under multiple complex excitations was established, simplifying the nonmatching control model of a nonlinear multi-input multi-output system. The longitudinal vibration of the lifting rope was transformed into an internal state, achieving the effect of controlling the pose of the end effector and the tension of the lifting rope through boundary control. A back-stepping adaptive control algorithm was constructed for its control indicators, improving the effectiveness of vibration control.

Based on the boundary control method, through boundary energy input, the characteristics of fuzzy systems that can approximate any nonlinear continuous function are utilized, combined with nonlinear disturbance observers, the nonlinear terms of the system can be compensated, which is well adapted to the disturbance observation requirements under different excitation effects, providing support and basis for vibration control of systems under multi-dimensional disturbances.

Based on the research of this article, the vibration of the time-varying lifting system can be fast suppressed and eliminated by the control method proposed in this article, which increases system stability and reliability and promotes the development of flexible body boundary vibration control theory. In this article, the control parameters are still selected by trial, and the search for optimal control parameters by multi-objective optimization method will be considered in the future.