Improved Vibration Suppression Strategy of Fuzzy PI Servo Control for Dual Flexible System with Flexible Joints

Abstract

A high performance manipulator servo drive system is a double flexible system with flexible joints and flexible loads. Flexible joints are composed of elastic connecting elements, and flexible loads are flexible Euler beams with elastic deformation. The dual flexible system has highly nonlinear time-varying characteristics. This kind of characteristic will cause resonance of the double-flexible system and affect the dynamic characteristics of the system. In order to suppress the system resonance, the nonlinear dynamics model of the system with two flexible bodies is established. Then, the servo control method of double flexible body system is designed, and the range of PI controller parameters is determined by the same resistance pole assignment method. Then, a fuzzy control rule is designed to dynamically adjust PI controller parameters based on pole assignment. Finally, the improved fuzzy PI control strategy is simulated numerically. The simulation results show that the vibration of the double-flexible system can be effectively suppressed by establishing the precise dynamic model and designing PI controller parameters.

1. Introduction

The servo system of high-performance flexible manipulator usually includes flexible joints and flexible loads. The servo system with flexible characteristics can be referred to as double flexible system for short [1,2]. The flexible joint is an important part of the high performance manipulator in the double flexible system. The flexible joint is composed of elastic coupling, bearing, ball screw and gear in the transmission mechanism, among which there are many complex nonlinear influencing factors [3,4]. The light and slender flexible load in the double flexible system can be equivalent to the flexible Euler beam. Due to the inertia, the flexible Euler beam will produce elastic deformation during the rotating motion of the double-flexible system, which leads to the fluctuation of the system speed [5,6]. The nonlinear factors brought by flexible joints and flexible Euler beams result in the dynamic time-varying characteristics of double-flexible systems [7]. This dynamic time-varying characteristic is an important factor that leads to resonance of the system. Resonance will not only reduce the dynamic precision of the system, but also damage the mechanical structure of the transmission part of the system. At the same time, it also increases the difficulty for the servo drive control system [8]. Therefore, it is very important to study the vibration suppression of dual flexible system. The establishment of high-precision dynamic model coupled with nonlinear influencing factors and the design of dynamic intelligent control strategy are important means to solve the problem. Many scholars have made contributions to the research of modeling and control.

In terms of dynamic modeling of double flexible body, the literature [9,10] proposed that the flexible joint that transmits power between the motor and the load is equivalent to a small damping spring, and the double inertia system is used to represent the flexibility of the joint. In the literature [11], flexibility and friction factors were considered, and the dynamics of a flexible manipulator with a terminal load was established by using the assumed mode method and Lagrange principle. In the literature [12], the differential orthogonal finite element and the improved Newton–Raphson method were adopted, and based on nonlinear elastic Hertzian contact theory and Timoshenko beam theory, a general five-degree of freedom dynamic model was proposed to study the vibration characteristics of flexible rotor-bearing systems. In the literature [13], a dynamic model of a planar double-link rigid and flexible (TLRF) underactuated manipulator with a passive first joint was established by using the hypothetical mode method and Lagrange modeling method. In the literature [14], a dynamic model of free-floating space robot was established by taking into account uncertainties such as external load variation, external disturbance and joint flexibility in engineering practice. The literature [15] developed an autonomous mobile manipulator driven by a series of designed elastic actuators in combination with the advantages of flexible robots such as high mobility, high load ratio, high torque fidelity and robustness to external interference. Considering the incomplete and complete constraints in mobile operation, the whole body dynamics model is established and simplified.

For the research on control, the literature [16] adopts voltage control strategy to carry out nonlinear tracking control of electrically driven flexible joint manipulator, and the proposed control law is not affected by manipulator dynamics. The literature [17] proposes a new trajectory tracking control method, in which the expected torque is generated by PID control based on manipulator dynamics model. The literature [18] proposes an adaptive fuzzy control strategy for single-link flexible joint robot (SFRM) with specific functions, in which fuzzy logic system is used to identify unknown nonlinear terms. The literature [19] studies a sliding mode neural network fuzzy control method to suppress the vibration of a double flexible beam system. In order to adjust the displacement and clamping force of flexible grippers, a hybrid force/position control strategy based on PID controller and fuzzy sliding mode controller was proposed in the literature [20]. Based on the parameter tuning of online PID controller, a hybrid control structure composed of feedback controller and feedforward controller is proposed in the literature [21], which uses the combination of input shaping and feedforward filter to effectively compensate the heavily damped vibration of hydraulic servo drive system.

In terms of dynamic modeling, the assumed mode method and Lagrange principle are usually used to equalize the flexible joint to the spring damping system, and the light and slender flexible load to the Euler beam or ironwood Cinker beam. However, in the modeling process, the joint flexibility is usually ignored, and the flexible joint is an important factor causing the system output rotation frequency and torque fluctuations, so it is necessary to establish the coupling dynamics model of flexible joint and flexible beam. In the control research, for the high performance manipulator servo system control method, the most widely used is to use PID controller to complete the system adjustment. However, the simple PID control method is unable to track the dynamic change of system output variables. The double flexible system is a highly nonlinear system with time-varying characteristics, so the intelligent control method can improve the working accuracy of the system. Compared with other studies, this paper considers the servo control of a double flexible body system with flexible joints and flexible beams, and uses the AMM method and the Lagrange principle, combined with the vibration mechanics theory, to carry out accurate dynamic modeling of the system. For the control of double flexible servo system, other references often only use pole assignment method or fuzzy control rules to set PI control parameters, but this paper combines the Ziegler–Nichols (ZN) method and pole assignment method with the same damping coefficient to determine the parameter range of PI controller of servo control system. Then, the fuzzy control rules were designed, and the PI parameters of the system were adjusted dynamically and adaptively according to the time-varying output variables of the system, so as to suppress the rotation frequency and torque fluctuation of the vibration system.

In the Section 2 of this paper, the coupling dynamics modeling of the system with two flexible bodies was carried out. The Section 3 aims to design the improved fuzzy PI control strategy for the double-flexible system. The Section 4 presents the numerical simulation analysis of double flexible servo control system. The Section 5 concerns the conclusion obtained.

2. Dynamics Modeling of Two Flexible Body Systems

In this paper, the double flexible body system was composed of flexible Euler beam and flexible joint. The flexible Euler beam in the system is the system load. The flexible Euler beam was connected to the motor end through the flexible joint. The flexible Euler beam rotates under the drive of the servo motor.

2.1. Mathematical Model of Flexible Euler Beam

The mechanical arm with slender structure and light weight will have elastic deformation when moving. This flexible mechanical arm can be equivalent to the Euler–Bernoulli beam, which is hereinafter referred to as flexible beam. The lateral vibration and elastic deformation of the flexible beam are ignored, while the longitudinal vibration and elastic deformation of the flexible beam were not considered. Therefore, based on the theory of vibration mechanics [22], the flexible beam was simplified and can be equivalent to a cantilever beam. As shown in Figure 1, a simplified dynamic model of the flexible Euler beam was established.

According to the analysis of the flexible beam in Figure 1, the vector ${\mathit{r}}_t$ at any point in the static coordinate $XOY$ of the flexible beam can be expressed as Equation (1), as shown below.

$$r_t=\left[\begin{array}{c}x\cos {\theta }_a\left(t\right)-y\left(x,t\right)\sin {\theta }_a\left(t\right)\\ x\sin {\theta }_a\left(t\right)+y\left(x,t\right)\cos {\theta }_a\left(t\right)\end{array}\right],$$

(1)

According to the vibration theory of the beam, the boundary conditions of the flexible beam can be obtained, as shown in Equation (2).

$$\{\begin{array}{c}y\left(0,t\right)=0\\ \frac{\partial y\left(0,t\right)}{\partial x}=0\\ \mathit{E}\mathit{I}\left(x\right)\frac{{\partial }^{2}y\left(l,t\right)}{\partial x^2}=0\\ \mathit{E}\mathit{I}\left(x\right)\frac{{\partial }^{3}y\left(l,t\right)}{\partial x^3}=0\end{array},$$

(2)

Thus, the expression of lateral vibration of the flexible beam is shown in Equation (3).

$$\mathit{\rho }\left(x\right)\mathit{A}\left(x\right)\frac{{\partial }^{2}y\left(x,t\right)}{\partial t^2}\frac{{\partial }^2}{\partial x^2}+\left[\mathit{E}\mathit{I}\left(x\right)\frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}\right]=0,$$

(3)

Among them, $\mathit{\rho }\left(x\right)$ represents the linear density of the flexible beam, $\mathit{A}\left(x\right)$ represents the unit cross-sectional area of the flexible beam, $\mathit{E}\mathit{I}\left(x\right)$ represents the flexural stiffness of the flexible beam. $\mathit{\rho }$, $\mathit{A}$, $\mathit{E}\mathit{I}$ can be expressed in terms of constants.

The elastic deformation of the flexible beam is described by using the hypothetical mode method (AMM) and the method in reference [23,24], as shown in Equation (4).

$$y\left(x,t\right)={\displaystyle \sum _{i=1}^{\infty }{\phi }_i\left(x\right)}{\mathit{\epsilon }}_i\left(x\right)={\phi }^T\mathit{\epsilon },$$

(4)

where ${\phi }_i\left(x\right)$ represents modal function, and ${\mathit{\epsilon }}_i\left(t\right)$ represents modal coordinate.

By using the separation of variables method, Equations (3) and (4) are combined, and the expression of Equation (5) can be obtained as

$$\frac{\ddot{\mathit{\epsilon }}\left(t\right)}{\mathit{\epsilon }\left(t\right)}=-\frac{\mathit{E}\mathit{I}}{\mathit{\rho }\mathit{A}}\cdot \frac{{\phi }^{\left(4\right)}\left(x\right)}{\phi \left(x\right)},$$

(5)

It can be concluded that in Equation (5), the left side is independent of x and the right side is independent of t. Then, Equation (5) is equal to a constant, denoted as $-{\Theta }^2$, ${\Theta }^2$ is expressed as Equation (6) below.

$${\Theta }^2=\left[{\tau }_1^2,{\tau }_2^2,\cdots ,{\tau }_n^2\right],$$

(6)

Equation (7) is the modal frequency expression, as shown below.

$$\{\begin{array}{c}{\phi }_i\left(x\right)=\mathrm{ch}{\alpha }_{i}x-\cos {\alpha }_{i}x-{\zeta }_i\left(\mathrm{sh}{\alpha }_{i}x-\sin {\alpha }_{i}x\right)\\ {\zeta }_i=\frac{\mathrm{sh}{\alpha }_{i}l-\sin {\alpha }_{i}l}{\mathrm{ch}{\alpha }_{i}l+\cos {\alpha }_{i}l}\\ {\mathit{\omega }}_i={\alpha }_i^2\sqrt{\frac{\mathit{E}\mathit{I}}{\mathit{\rho }\mathit{A}}}\end{array},$$

(7)

where the length of the flexible beam is expressed as l, ${\mathit{\omega }}_i$ is the modal frequency, and ${\alpha }_i$ is the characteristic root of the modal function equation.

Considering Equations (1)–(7), the expressions of elastic potential energy and kinetic energy of the flexible beam can be obtained, respectively, expressed as Equations (8) and (9), as shown below.

$${\mathit{V}}_1=\frac{1}{2}{\displaystyle {\int }_0^l\mathit{E}\mathit{I}\frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}}\mathrm{d}x,$$

(8)

$${\mathit{T}}_1=\frac{1}{2}{\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}{\dot{\mathit{r}}}_t^T{\dot{\mathit{r}}}_t}\mathrm{d}x=\frac{1}{2}{\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}}\left(\left(x^2+y^2\left(x,t\right)\right){\dot{\theta }}_a^2\left(t\right)+{\left(\frac{\partial y\left(x,t\right)}{\partial t}\right)}^2+2x{\dot{\theta }}_a^2\left(t\right)\frac{\partial y\left(x,t\right)}{\partial t}\right)\mathrm{d}x,$$

(9)

Among them, by combining Equations (8) and (9) and according to the literature [11,13], using the Lagrange principle, dynamic modeling of flexible beam was carried out, and Equation (10) was obtained. As follows.

$$\{\begin{array}{c}{\ddot{\theta }}_a\left(t\right){\displaystyle {\int }_0^l\mathsf{\rho }A{x}^{2}dx}+{\ddot{\theta }}_a\left(t\right){\displaystyle \sum _{i=1}^{\infty }{\mathsf{\epsilon }}_i{\left(t\right)}^2}+2{\dot{\theta }}_a\left(t\right){\displaystyle \sum _{i=1}^{\infty }{\mathsf{\epsilon }}_i\left(t\right){\dot{\mathsf{\epsilon }}}_i\left(t\right)}+{\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathsf{\epsilon }}}_i\left(t\right)}{\displaystyle {\int }_0^l\mathsf{\rho }Ax{\mathsf{\phi }}_{i}dx}=T_a\\ {\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathsf{\epsilon }}}_i\left(t\right)}+{\ddot{\theta }}_a\left(t\right){\displaystyle \sum _{i=1}^{\infty }{\displaystyle {\int }_0^l\mathsf{\rho }Ax{\mathsf{\phi }}_{i}dx}-}{\dot{\theta }}_a{\left(t\right)}^2{\displaystyle \sum _{i=1}^{\infty }{\mathsf{\epsilon }}_i\left(t\right)+\mathrm{\Omega }{\displaystyle \sum _{i=1}^{\infty }{\mathsf{\epsilon }}_i\left(t\right){\mathsf{\epsilon }}_i{\left(t\right)}^2}}=0\end{array},$$

(10)

It can be seen from Equation (10) that there is a nonlinear term in the dynamic model of the flexible beam, which is the modal coordinates and angle coupling of the flexible beam. These nonlinear terms will affect the modeling accuracy of the dynamics model, so they should be removed. Therefore, the simplified dynamic equation is shown in Equation (11).

$$\{\begin{array}{c}{\ddot{\theta }}_a\left(t\right){\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}{x}^2\mathrm{d}x}+{\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right){\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}x{\phi }_i\mathrm{d}x}}={\mathit{T}}_a\\ {\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right)}+{\ddot{\theta }}_a\left(t\right){\displaystyle \sum _{i=1}^{\infty }{\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}x{\phi }_i\mathrm{d}x}}+{\displaystyle \sum _{i=1}^{\infty }{\mathit{\epsilon }}_i\left(t\right){\mathit{\epsilon }}_i{\left(t\right)}^2}=0\end{array},$$

(11)

Define the moment of inertia of the flexible beam and Equations (12)–(14), as shown below.

$${\mathit{I}}_a=\mathit{\rho }\mathit{A}{\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}{x}^2\mathrm{d}x},$$

(12)

$${\mathit{F}}_{ai}={\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}x{\phi }_i\left(x\right)}\mathrm{d}x,$$

(13)

$${\mathit{F}}_a={\left[\begin{array}{cccc}{\mathit{F}}_{a1}& {\mathit{F}}_{a2}& \cdots & {\mathit{F}}_{an}\end{array}\right]}^T,$$

(14)

Equations (12)–(14) are used to express the content in Equation (11), and then the final expression of the dynamic equation of the flexible beam is shown in Equation (15).

$$\{\begin{array}{c}{\mathit{I}}_a{\ddot{\theta }}_a\left(t\right)+{\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right)}{\mathit{F}}_{ai}={\mathit{T}}_a\\ {\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right)}+{\ddot{\theta }}_a\left(t\right){\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right)}{\mathit{F}}_{ai}+{\displaystyle \sum _{i=1}^{\infty }{\mathit{\epsilon }}_i\left(t\right){\mathit{\epsilon }}_i{\left(t\right)}^2=0}\end{array},$$

(15)

2.2. Mathematical Model of Flexible Joint

In the field of servo system transmission, there is widely a system of two inertias connected by an elastic coupling. According to the literature [1], such joint containing nonlinear factors can be called flexible joint. The flexible joint can be equivalent to a spring structure. Therefore, the potential energy and kinetic energy equations of the flexible joint are established, as shown in Equations (16) and (17) below.

$${\mathit{V}}_2=\frac{1}{2}{\displaystyle {\int }_0^l{\mathit{J}}_m{\dot{\theta }}_m^{2}dx},$$

(16)

$${\mathit{T}}_2=\frac{1}{2}{{\displaystyle {\int }_0^l{k}_s\left({\theta }_l-{\theta }_m\right)}}^{2}dx,$$

(17)

As shown in Figure 1, the dynamic modeling of the flexible joint is shown in Equation (18).

$$\{\begin{array}{c}{\mathit{J}}_m{\dot{\theta }}_m-k_s\left({\theta }_l-{\theta }_m\right)-C_{\mathsf{\omega }}\left({\dot{\theta }}_l-{\dot{\theta }}_m\right)={\mathit{T}}_m\\ {\mathit{J}}_l{\ddot{\theta }}_l+k_s\left({\theta }_l-{\theta }_m\right)+C_{\mathsf{\omega }}\left({\dot{\theta }}_l-{\dot{\theta }}_m\right)+{\mathit{T}}_a=0\end{array},$$

(18)

Since the damping coefficient has little influence, the damping effect of the rotating system can be ignored. Then, the dynamic equation of the flexible joint is shown in Equation (19) below.

$$\{\begin{array}{c}{\mathit{J}}_m{\dot{\mathit{\omega }}}_m={\mathit{T}}_m-k_s\left({\theta }_m-{\theta }_l\right)={\mathit{T}}_m-{\mathit{T}}_s\\ {\mathit{J}}_l{\dot{\mathit{\omega }}}_l=k_s\left({\theta }_m-{\theta }_l\right)-{\mathit{T}}_a={\mathit{T}}_s-{\mathit{T}}_a\\ {\mathit{T}}_s=k_s\left({\theta }_m-{\theta }_l\right)\end{array},$$

(19)

where the torque of the connecting shaft is expressed as ${\mathit{T}}_s$, the speed of the motor is expressed as ${\mathit{\omega }}_m$, and the speed of the load is expressed as ${\mathit{\omega }}_l$.

2.3. Lagrangian Dynamics Modeling of a System with Two Flexible Bodies

In servo transmission systems, there is usually a flexible Euler beam and a flexible joint. As shown in Figure 1, the so-called double-flexible system takes into account the dual characteristics of the flexible Euler beam and flexible joint. The flexible beam will produce elastic deformation in the process of rotation and the flexible joint will cause the rotation angle error when the servo motor drives the load.

From Equations (8) and (16), the expression of potential energy and kinetic energy of a double flexible rotating Euler beam can be obtained, as shown in Equations (20) and (21) below.

$$\mathit{V}=\frac{1}{2}{\displaystyle {\int }_0^l\mathit{E}\mathit{I}\frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}\mathsf{d}x+\frac{1}{2}}{\displaystyle {\int }_0^l{J}_m{\dot{\theta }}_m^2}\mathsf{d}x,$$

(20)

$$\mathit{T}=\frac{1}{2}{\displaystyle {\int }_0^l\mathit{\rho }\mathit{A}\left(\left(x^2+y^2\left(x,t\right)\right){\dot{\theta }}_a^2\left(t\right)+{\left(\frac{\partial y\left(x,t\right)}{\partial t}\right)}^2+2x{\dot{\theta }}_a^2\left(t\right)\frac{\partial y\left(x,t\right)}{\partial t}\right)}\mathsf{d}x+\frac{1}{2}{\displaystyle {\int }_0^l{k}_s{\left({\theta }_l-{\theta }_m\right)}^2\mathsf{d}x},$$

(21)

Based on Lagrange’s principle, the dynamics equation of a system with two flexible rotating Euler beams can be established. The coupled nonlinear term in the equation is removed, and the system dynamics equation is finally obtained, as shown in Equation (22).

$$\{\begin{array}{c}{\mathit{J}}_m{\ddot{\theta }}_m={\mathit{T}}_m-k_s\left({\theta }_l-{\theta }_m\right)={\mathit{T}}_m-{\mathit{T}}_s\\ {\mathit{J}}_l{\ddot{\theta }}_l+{\displaystyle \sum _{i=1}^{\infty }{\ddot{\mathit{\epsilon }}}_i\left(t\right){\mathit{F}}_{ai}}=k_s\left({\theta }_l-{\theta }_m\right)-{\mathit{T}}_a={\mathit{T}}_s-{\mathit{T}}_a\\ {\ddot{\mathit{\epsilon }}}_i\left(t\right)+{\ddot{\theta }}_a\left(t\right){\mathit{F}}_{ai}+{\mathit{\epsilon }}_i\left(t\right){\mathit{\epsilon }}_i{\left(t\right)}^2=0\end{array},$$

(22)

3. Improved Fuzzy PI Control Strategy for Dual Flexible System

3.1. Transfer Function of Two Flexible Body System

In the double flexible servo system, two structural components of the flexible beam and flexible joint are considered. The flexible joint will cause the fluctuation and error of load output angle, and the elastic deformation of the flexible beam will affect the working accuracy of the end of the system. Therefore, the key to improve the flutter problem is to establish the precise coupling dynamics model of the dual-flexible system. To obtain the transfer function of the double flexible system is also the premise of the control stability of the servo system. According to the literature [25], it can be concluded that the influence of second-order modes on system output is only 1% of that of first-order modes. Therefore, only the first-order modal function of the system can be taken, and the transfer function of the system can be obtained according to Equation (22), as shown in Equation (23).

$$\frac{{\omega }_s\left(s\right)}{{\omega }_R\left(s\right)}=\frac{E{s}^5+F{s}^3+Gs}{A{s}^6+{\rm B}{s}^4+C{s}^2+D},$$

(23)

where ${\omega }_R$ is the system input frequency, and ${\omega }_s$ is the system frequency after PI control strategy setting. The coefficient of Equation (23) is expressed in the following Equations (24)–(30).

$$A={\mathit{J}}_m{\mathit{J}}_l-{\mathit{J}}_m{\mathit{F}}_{a1}^2,$$

(24)

$$B={\mathit{J}}_m{\mathit{J}}_l{\mathit{\epsilon }}_1^2+\left({\mathit{J}}_m+{\mathit{J}}_l\right)k_s-k_s{\mathit{F}}_{a1}^2,$$

(25)

$$C=\left({\mathit{J}}_m+{\mathit{J}}_l\right)k_s{\mathit{\epsilon }}_1^2,$$

(26)

$$D=2k_s{\mathit{\epsilon }}_1^2,$$

(27)

$$E={\mathit{J}}_l-{\mathit{F}}_{a1}^2,$$

(28)

$$F={\mathit{J}}_l{\mathit{\epsilon }}_1^2+k_s,$$

(29)

$$G={\mathit{\epsilon }}_1^2,$$

(30)

3.2. The Same Damping Pole Assignment Method Improves PI Parameters

For the double flexible rotating Euler beam system, the drive is completed by the servo motor. Because the control bandwidth of the current loop in the system is much larger than the modal frequency of the double-flexible system, the influence of the current loop can be ignored and regarded as constant 1. In reference [26], the system transfer function was reduced by Pade method, and Equation (31) can be obtained as follows.

$$\frac{{\omega }_s\left(s\right)}{{\omega }_R\left(s\right)}=\frac{\alpha s}{\beta s^2+\gamma },$$

(31)

Among them, α, β and γ are the system transfer functions after reduced order. PI controller is selected to adjust the system speed ring, and the control block diagram is shown in Figure 2.

If the influence of friction on the double-flexible system is not considered, the closed-loop transfer function of the system is shown in Equation (32).

$$G_s\left(s\right)=\frac{\alpha k_{p}s+\alpha k_i}{\beta s^2+\alpha k_{p}s+\alpha k_i+\gamma },$$

(32)

$k_p$ and $k_i$ are the parameters of the PI controller.

When PI parameters are fixed by pole assignment method with the same damping coefficient, the closed-loop transfer function of the system has the same damping coefficient. According to the literature [27], the Ziegler–Nichols(ZN for short) method was used to determine the initial traditional PI parameters. Then, the traditional PI parameters of the system were improved by pole assignment method with the same damping coefficient, and the PI parameters were determined in a range. The improved PI parameter range is used as the basis for fuzzy controller to set PI parameters of the system, which is referred to as improved PI parameter for short. According to the literature [23], the pole assignment method with the same damping coefficient was used to set PI parameters of the system, and Equation (33) below was established.

$$\{\begin{array}{c}\beta s^3+\alpha k_p{s}^2+\left(\alpha k_i+\gamma \right)s=\beta \psi \\ \psi =s^3+2{\xi }_1^{\ast }{\omega }_a^{\ast }{s}^2+{\omega }_a^{\ast 2}s\end{array},$$

(33)

In the equation, ${\xi }_1^{*}$ denotes pole damping coefficient of the system, ${\omega }_a^{*}$ denotes natural frequency of the system.

According to Equation (33), the following Equation (34) is valid.

$$\{\begin{array}{c}2{\xi }_1^{\ast }{\omega }_a^{\ast }=\frac{\alpha }{\beta }{k}_p\\ {\omega }_a^{\ast 2}=\frac{\alpha }{\beta }{k}_i+\frac{\gamma }{\beta }\end{array},$$

(34)

Therefore, PI parameters of the system controller can be obtained, as shown in Equation (35) below.

$$\{\begin{array}{c}{k}_p=\frac{2{\xi }_1^{\ast }{\omega }_a^{\ast }\beta }{\alpha }\\ k_i=\frac{{\omega }_a^{\ast 2}\beta -\gamma }{\alpha }\end{array},$$

(35)

When using the same damping pole assignment method to design PI controller parameters, it is necessary to consider the value of pole damping coefficient ${\xi }_1^{*}$ and natural frequency ${\omega }_a^{*}$ on the influence of system adjustment time and system overshoot index, as shown in Figure 3.

It can be seen from Figure 3a that with the value of pole damping coefficient ${\xi }_1^{*}$ increasing gradually, the system adjustment time is shortened. When the value range of ${\xi }_1^{*}$ is between 0.4 and 0.7, the system adjustment time is short, and the system stability is high. When the value range of ${\xi }_1^{*}$ is between 0.3 and 0.4, the system needs a long adjustment time to reach a stable state, and the stability of the system is poor. The natural frequency ${\omega }_a^{*}$ has little influence on the adjustment time of the system.

It can be seen from Figure 3b that with the value of pole damping coefficient ${\xi }_1^{*}$ increasing gradually, the overshoot of the system decreases gradually. When the value range of ${\xi }_1^{*}$ is between 0.4 and 0.7, the overshoot of the system is small and the dynamic performance of the system tends to be stable. When the value range of ${\xi }_1^{*}$ is between 0.3 and 0.4, the system overshoot is large and the system instability is high. The value of natural frequency ${\omega }_a^{*}$ has little influence on the overshoot of the system, but when the value of ${\omega }_a^{*}$ is about 5 and the value of ${\xi }_1^{*}$ is about 0.4, the overshoot of the system reaches the minimum and the system is stable.

3.3. Fuzzy Rule Setting Improves PI Control Strategy

The elastic deformation of the flexible Euler beam is caused by its vibration during rotation. When the flexible joint connects the motor end to the flexible beam, the torsion angle error will be generated in the transmission process. To deal with the elastic deformation and angle error, this paper adopted a method of adjusting PI controller parameters with the same damping pole assignment method, and used fuzzy control strategy to control the rotating flexible Euler beam system with flexible joints in real time. The fuzzy control strategy can dynamically adjust the PI parameters of the system, improve the stability of the system, and effectively restrain the vibration and deformation of the double flexible system during rotation.

When designing the fuzzy PI controller, double input and double output mode is adopted. The double input variables of the fuzzy rule are the error e and the error change rate ec of the double flexible system. The Gaussian function is selected as the membership function of the double input variables. The double output variables of the fuzzy rule are $\Delta k_p$ and $\Delta k_i$, and the triangular function is selected as the membership degree of the double output variables. The input variable domain of fuzzy PI controller is set to [−3,3], and the output variable domain is also set to [−3,3]. Set the input error domain to [−1,1], error change rate domain to [−2,2], and actual output controller domain to [−0.5, 0.5]. However, in the design of fuzzy controller, it is impossible to distinguish these four parameters directly. Therefore, these four parameters should be fuzzy, as shown in Equation (36).

$$e,ec,\Delta k_p,\Delta k_i=\left[\begin{array}{ccccccc}\mathrm{NB}& \mathrm{NM}& \mathrm{NS}& \mathrm{ZO}& \mathrm{PS}& \mathrm{PM}& \mathrm{PB}\end{array}\right],$$

(36)

In subsets, N represents negative definite, P represents positive definite, ZO represents zero, and B, M, and S represent large, medium, and small, respectively. The fuzzy rules corresponding to dual output variables $\Delta k_p$ and $\Delta k_i$ are 49, respectively. It can be seen that

Table 1. Eigenroot values of the first three orders of flexible loads.

ec/e	NB	NM	NS	ZO	PS	PM	PB
NB	PB/NB	PB/NB	PB/NM	PM/NM	PS/NS	PS/NS	ZO/ZO
NM	PB/NB	PB/NB	PM/NM	PS/NS	PS/NS	ZO/ZO	ZO/PS
NS	PB/NM	PM/NM	PM/NS	PS/NS	ZO/ZO	ZO/PS	NS/PS
ZO	PM/NM	PM/NS	PS/ZO	ZO/ZO	NS/PS	NS/PS	NM/PM
PS	PM/NS	PS/NS	ZO/ZO	NS/PS	NM/PM	NM/PM	NB/PM
PM	PS/NS	ZO/ZO	NS/PS	NM/PS	NM/PM	NB/PB	NB/PB
PB	ZO/ZO	ZO/PS	NM/PS	NM/PM	NM/PM	NB/PB	NB/PB

is the design parameter rule table of fuzzy PI controller and Figure 4 is the reasoning diagram of fuzzy rule design.

Finally, using $\Delta k_p$ and $\Delta k_i$, the parameters of the improved PI controller are adjusted adaptively, and the precise results of the output variables are obtained by defuzzification. To sum up, the improved fuzzy PI control block diagram of the double flexible body system is shown in Figure 5 below.

4. Numerical Simulation Analysis

Combined with the coupling dynamics model of the servo-driven dual flexible system, the improved fuzzy PI control strategy was used to track the rotation frequency and torque of the system output in real time, and to reduce the input error of the system. In order to verify the accuracy of the coupling dynamics of the established dual-flexible system and the effectiveness and stability of the proposed improved fuzzy PI control strategy, Simulink was used for numerical simulation analysis of the system. In this paper, two flexible systems with three different parameters were selected, as shown in

Table 2. System parameters of two flexible bodies under different conditions.

Determinants	Condition 1	Condition 2	Condition 3
$\mathrm{Flexible}\mathrm{Euler}\mathrm{beam}\mathrm{length}\mathit{l}∕\mathrm{m}$	1.1	1.6	2.1
$\mathrm{Flexible}\mathrm{Euler}\mathrm{beam}\mathrm{quality}\mathit{m}∕\mathrm{kg}$	2.5	2.5	2.5
$\mathrm{Flexural}\mathrm{stiffness}\mathrm{of}\mathrm{flexible}\mathrm{Euler}\mathrm{beam}\mathit{E}\mathit{I}∕{\mathrm{Nm}}^2$	103	103	103
$\mathrm{Flexible}\mathrm{Euler}\mathrm{beam}\mathrm{density}\mathit{\rho }\mathit{A}∕\mathrm{kg}∕{\mathrm{m}}^2$	2.2727	1.5406	1.1954
$\mathrm{Moment}\mathrm{of}\mathrm{inertia}\mathrm{of}\mathrm{a}\mathrm{flexible}\mathrm{Euler}\mathrm{beam}{\mathit{J}}_l∕{\mathrm{kgm}}^2$	1.0083	2.1523	3.5416
$\mathrm{Modal}\mathrm{coupling}\mathrm{coefficient}{\mathit{F}}_{a1}∕{\mathrm{kg}}^{1/2}\mathrm{m}$	32.8567	55.1766	81.5530
$\mathrm{Modal}\mathrm{frequency}{\mathit{\omega }}_1∕\mathrm{rad}∕\mathrm{s}$	23.8961	28.7083	33.2122
$\mathrm{Eigenroots}\mathrm{of}\mathrm{modal}\mathrm{functions}{\mathit{\alpha }}_1$	1.875	1.875	1.875
$\mathrm{Torsional}\mathrm{stiffness}\mathrm{of}\mathrm{flexible}\mathrm{joints}{\mathit{k}}_s∕\mathrm{Nm}∕\mathrm{rad}$	400	400	400
$\mathrm{Moment}\mathrm{of}\mathrm{inertia}\mathrm{of}\mathrm{the}\mathrm{motor}{\mathit{J}}_m∕{\mathrm{kgm}}^2$	3.025	6.275	10.875

below. The influence of flexible beams of different lengths on the output variables of the system was especially emphasized. The ZN method, the pole assignment method with the same damping coefficient and the improved fuzzy adaptive method were respectively used to adjust the PI controller parameters of the double-flexible system with each parameter, and the output variables of the system under the three control methods were compared. In Figure 6, Figure 7 and Figure 8, the ZN PI represents the PI control strategy set by Ziegler–Nichols method, the PO-PLACE PI represents the PI control strategy set by pole assignment method with the same damping coefficient, and the FC-POLE PI is represented as a fuzzy setting PI control strategy combined with pole assignment method with the same damping coefficient.

According to the parameters of three kinds of dual-flexible systems in Table 2, the influence of the control strategies of ZN method, pole assignment PI method with the same damping coefficient and improved fuzzy PI method on the output rotation frequency and torque changes of the dual-flexible system is obtained in the case of flexible Euler beams with different lengths of flexible joints. The numerical simulation results obtained are shown in Figure 6, Figure 7 and Figure 8 below. In Figure 6a, Figure 7a, Figure 8a, Figure 6c, Figure 7c, Figure 8c, Figure 6e, Figure 7e and Figure 8e, respectively, represent the fluctuation trend of system rotation frequency, torque and angle. Figure 6b, Figure 7b and Figure 8b are the local enlarged image of the marked part in Figure 6a, Figure 7a and Figure 8a. Figure 6d, Figure 7d and Figure 8d are the local enlarged image of the marked part in Figure 6c, Figure 7c and Figure 8c. Figure 6f, Figure 7f and Figure 8f are the local enlarged image of the marked part in Figure 6e, Figure 7e and Figure 8e.

From Figure 6a, Figure 7a and Figure 8a, it can be seen that with the increase of the length of the flexible beam, the vibration amplitude and frequency of the output rotation frequency of the system continued to increase, and the system instability gradually increased. The same PI controller parameters could no longer satisfy the stable output of the flexible beam system with different parameters. The improved fuzzy PI control strategy proposed in this paper is a variable parameter control strategy. From Figure 6a, Figure 7a and Figure 8a, it can be concluded that the improved fuzzy PI control could adapt to the stability of the output of the double flexible system under different parameters. Using the ZN method to set PI parameters of controller will cause large amplitude and high frequency fluctuation of system output. On this basis, the PI controller parameters of the system were improved by the pole assignment method with the same damping coefficient. Although the fluctuation of system output rotation frequency can be improved by adjusting controller PI parameters by the pole assignment method, it was found that this control method has no obvious improvement effect on system output by observing local enlarging Figure 6b, Figure 7b and Figure 8b. Therefore, the fuzzy control method was used to adjust the PI controller parameters based on the pole assignment method. In this way, the time-varying characteristics of the system can be tracked. It can be seen from Figure 6b, Figure 7b and Figure 8b that the fluctuation amplitude and rotation error of the system output rotation frequency were significantly reduced.

Similarly, according to Figure 6c, Figure 7c, Figure 8c, Figure 6e, Figure 7e and Figure 8e, it can be seen that with the increase of the length of the flexible beam, the system output torque and angle error also increased. Based on the partial amplification of Figure 6d, Figure 7d, Figure 8d, Figure 6f, Figure 7f and Figure 8f, the improved fuzzy PI control strategy could reduce the errors of the output torque and angle of the system and improve the stability of the system. As can be seen from the overall simulation diagram, with the improved fuzzy PI control strategy, the output torque and angle fluctuation of the system tend to be stable. In conclusion, the improved fuzzy PI control strategy proposed in this paper significantly improved the control stability of the double flexible system. By dynamically adjusting the PI control parameters of the double flexible system, real-time tracking and feedback of the output variables of the system were realized. The elastic deformation of the flexible Euler beam and the angle error of the flexible joint were reduced during the rotation of the double-flexible system. The output fluctuation amplitude of the system was reduced, and the working precision of the double flexible system was improved. The effectiveness of the improved fuzzy PI control strategy was verified.

According to Figure 6b, Figure 7b and Figure 8b, Figure 6d, Figure 7d, Figure 8d, Figure 6f, Figure 7f and Figure 8f, representative absolute values of errors can be observed. As the length of the flexible beam increases, the fluctuation of rotation frequency, output torque and output angle of the double-flexible system will increase. After using the pole assignment method to set PI control parameters of the system, the fluctuations of rotation frequency, output torque and output angle of the system will decrease by about 10%. After using the improved fuzzy PI control strategy proposed in this paper, the system rotation frequency, output torque and output angle fluctuations will be reduced by about 20%. By analyzing the absolute value of the error, the effectiveness of the proposed improved fuzzy PI control strategy on vibration suppression of the dual-flexible system can be further verified.

5. Conclusions

In this paper, the coupling dynamics modeling of a flexible Euler beam with flexible joints was considered, and an improved fuzzy PI control strategy was used to track the dual-flexible system in real time. By using the numerical simulation method, the rotational frequency, torque and angle of the output of the double flexible system were compared and simulated. The simulation results showed that compared with the ZN method and the same damping pole assignment method, the improved fuzzy PI control strategy proposed in this paper has obvious effect on improving the motion accuracy and stability of the system, and reduces the errors of rotation frequency, torque and Angle of the system output. Specific conclusions are as follows:

(1)

In this paper, a dynamic model of a double-flexible system including flexible joints and flexible Euler beams was established. Considering the rotational frequency error of the system caused by flexible joints, flexible Euler beams vibrate during rotation, resulting in elastic deformation. It is an important prerequisite to improve the accuracy of the system to establish the coupling dynamics model of the two-flexible system by combining two factors.

(2)

In this paper, an improved fuzzy PI control strategy is proposed for the control of two flexible systems. To improve the fuzzy PI control strategy, the initial value of PI controller parameters was obtained by using the ZN method, and then the same damping pole assignment method was used to optimize the range of PI parameters. Finally, the fuzzy control rules were designed, and the fuzzy control was applied to the setting of the improved PI parameters, so that the PI controller was transformed into a dynamic adjustment process. The real-time monitoring and real-time adjustment of the output of the double flexible system were realized.

(3)

In this paper, the improved fuzzy PI control strategy of the two-flexible system was numerically simulated in Simulink simulation environment. The simulation results showed that the proposed control strategy can effectively reduce the rotation frequency, torque and angle error of the system output, restrain the elastic deformation of the system in the process of movement, and improve the terminal working accuracy of the double-flexible system.