Investigation of the controller parameter optimisation for a servomechanism

Abstract

A novel controller parameter optimisation method to effectively improve the motion accuracy of a servomechanism is proposed in this article. This method overcomes the drawbacks of conventional controller parameter optimisation methods, such as inefficiency and dependence on experience. The servomechanism model was established, and a performance analysis was performed. Moreover, the design requirements were developed. The robust stability was adopted to solve the problem induced by great variations in the modelling elements. Subsequently, a multi-objective nonlinear optimisation problem was developed and solved by an intelligent optimisation algorithm. To achieve the desired performance, the multi-objective optimisation function adopted a comprehensive command response performance function and an external disturbance rejection response function. To validate the effectiveness and feasibility of this method by simulation and experiment, an open servomechanism platform was established. The simulated and experimental results show that the tracking error, protrusion error and performance indicators are all reduced greatly under different working conditions. This method can effectively improve the motion accuracy and has the advantages of high efficiency, high practicality and high flexibility.

Introduction

Servomechanisms are widely used in high-speed and high-precision motion control systems, such as numerical control machine tools, precision positioning stages and robots.^1,2 For a specified servomechanism, matching interactions between the controller subsystem and the mechanical subsystem are essential to achieve the desired performance. ³ In industrial applications, a servomechanism controller is commonly made up of a position loop, a velocity loop and a current loop. However, the mutual influences between these control loops are strongly coupled. ⁴ Direct relationships between the goal of the high-speed, high-precision motion and the controller parameters are difficult to establish for a servomechanism. ⁵

From the perspective of a typical proportional–integral–derivative (PID) controller structure, there are several controller parameter tuning methods, such as Ziegler–Nichol (Z–N), relay feedback, manual tuning and intelligent optimisation algorithms. ⁶ Z–N methods and relay feedback are commonly applied in tuning the controller parameters of single control loop. It is, however, very difficult to tune the controller parameters of a servomechanism. The manual tuning method is widely used to tune the controller parameters of servomechanisms in industrial applications, and the success of manual training depends primarily on the engineers’ experience. ⁷ Some instruments are employed to measure and analyse the tuning results. ⁸ This method has deficiencies such as inefficiency and dependence on experience, which result in large tuning margins. Therefore, this method can rarely obtain the optimal servomechanism dynamics and the desired motion accuracy. Kuo and Yen^9,10 introduced a genetic algorithm to automatically tune the controller parameters of servomechanisms on-line. However, this method is only suitable for a single working condition. Under variable working conditions, the optimised controller parameters for this method cannot always achieve the desired performance. Min-Seok Kim and Sung-Chong Chung^11,12 proposed an integrated servomechanism design methodology. However, external disturbance rejection is ignored during the servomechanism design process. Some experts, such as Yusuf Altinas and K. Erkorkmaz, have attempted to develop complex motion controllers for servomechanisms.^13–15 However, these complex motion controllers are not helpful for most industrial servomechanisms, as they adopt conventional cascade PID controllers. Furthermore, servomechanism stability is affected by the stiffness, damping, inertia and so on. These modelling elements commonly vary with position and other factors. However, the above-mentioned methods neglect these adverse influences. Therefore, optimised controller parameters cannot ensure practical servomechanism stability.

This article proposes a novel controller parameter optimisation method. This method adopts a multi-objective intelligent optimisation algorithm to achieve controller parameter optimisation. This algorithm comprehensively integrates command response and external disturbance rejection. At the same time, the adverse effects caused by stability variations in the modelling elements are considered as well. Therefore, robust stability can be obtained. This method overcomes the drawbacks encountered when using the conventional method, and its advantages include its high efficiency, high practicality, high flexibility and satisfaction of multiple practical design requirements.

The remainder of this article is organised as follows. The servomechanism model is established in section ‘Servomechanism modelling’. The servomechanism performance analysis is described in section ‘Performance analysis and design requirements’. Subsequently, design requirements are developed. The multi-objective intelligent optimisation algorithm is presented in section ‘Multi-objective nonlinear optimisation’. The controller parameter optimisation method proposed in this article is verified in section ‘Simulation and experiment’. Finally, a summary of this article is given in section ‘Conclusion’.

Servomechanism modelling

Most mechanical subsystem models for servomechanisms are primarily based on a rigid body model, that is, the stiffness of the mechanical subsystem is assumed to be infinite, which ignores the effects of stiffness on the dynamic response and stability. To reflect the actual dynamics, an elastic mechanical subsystem model is established, as shown in Figure 1.

Figure 1.

Elastic mechanical subsystem model.

A driving torque T_m generated by the servo motor is transformed into the rotational motion of a ball screw and the linear motion of a worktable. An equivalent deformation θ exists between the motor and the worktable. The torque T_m can be expressed as

$$\{\begin{array}{c}{T}_m=J_m\frac{d{\omega }_m}{\mathrm{dt}}+B_m{\omega }_m+T_l\hfill \\ \multicolumn{1}{c}{T_l=K_e\theta =K_e({\theta }_m-{\theta }_l)}\end{array}$$ (1)

where J_m and B_m are the inertia of the servo motor rotor and the equivalent damping coefficient of the servo motor, respectively. The value of B_m is very small, so it is commonly negligible, that is, B_m = 0; T_l is the load torque; K_e is the equivalent stiffness; ω_m and θ_m are the angular velocity and the rotational angle of servo motor, respectively, and θ_l is the equivalent rotational angle of the load. The equivalent stiffness K_e can be expressed as

$$K_e={(\frac{1}{K_{\theta }}+\frac{1}{K_l}·\frac{4\eta {\pi }^2}{l^2})}^{-1}$$ (2)

where K_θ and K_l are the equivalent torsional stiffness and the equivalent axial stiffness, respectively. l and η are the ball screw lead and the transmission efficiency of the mechanical subsystem, respectively. The load torque T_l overcomes the external disturbance torque T_d and drives the worktable motion. The load torque T_l can be expressed as

$$T_l=J_l\frac{d\omega }{\mathrm{dt}}+B\omega +T_d$$ (3)

where J_l and B are the equivalent inertia and the damping of the load, respectively, and ω is the equivalent angular velocity of the load. The equivalent inertia of the load J_l includes the equivalent inertias of the worktable, the ball screw and the coupling. According to the product specifications of the servomechanism, the above-mentioned parameters can be calculated.

In this article, the servomechanism controller adopts a conventional industrial cascade PID controller and is composed of a position controller G_pc, a velocity controller G_vc and a current controller G_cc to reject both steady-state errors and external disturbances. The controller of the servomechanism can be expressed as

$$\{\begin{array}{c}{G}_{\mathrm{pc}}=K_p\hfill \\ \multicolumn{1}{c}{G_{\mathrm{vc}}=K_{\mathrm{vp}}+\frac{K_{\mathrm{vi}}}{s}}\\ \multicolumn{1}{c}{G_{\mathrm{cc}}=K_t}\end{array}$$ (4)

where K_p is the proportional gain of the position controller. K_vp and K_vi are the proportional gain and the integral gain of the velocity controller, respectively. Because the bandwidth of the current loop is much greater than the bandwidth of the velocity loop or the position loop in a servomechanism, the current controller G_cc is equivalent to a torque constant K_t. The feedback sensor gain R can be expressed as

$$R=\frac{l}{2\pi }$$ (5)

Then, a typical elastic servomechanism model can be constructed, as shown in Figure 2.

Figure 2.

Servomechanism model.

Performance analysis and design requirements

The comprehensive performance analysis of the servomechanism includes an analysis of the dynamic response and a stability analysis. To reveal the nonlinear, coupled relations, the dynamic response and the stability are discussed. Moreover, the design requirements are developed in this section.

Analysis of the dynamic response and stability

The dynamic response analysis of a servomechanism commonly only utilises a command response. However, the external disturbance significantly affects the motion accuracy and should also be suppressed. In this article, the servomechanism dynamic response analysis integrates the command response and the external disturbance rejection response. To evaluate the dynamic response of a servomechanism, the bandwidth B_w is adopted. As shown in Figure 2, the closed-loop transfer function G_c(s) from the command position X_r to the actual worktable position X_l can be expressed as

$$\{\begin{array}{cc}{G}_c(s)\hfill & =\frac{b_{1s}+b_0}{\Delta (s)}\hfill \\ \multicolumn{1}{c}{\Delta (s)}& =a_0+a_{1}s+a_2{s}^2+a_3{s}^3+a_4{s}^4+a_5{s}^5\hfill \end{array}$$ (6)

where

$$\{\begin{array}{cc}{a}_0\hfill & =K_e{K}_p{K}_t{K}_{\mathrm{vi}}R\hfill \\ \multicolumn{1}{c}{a_1}& =K_e{K}_t{K}_{\mathrm{vi}}R+K_e{K}_p{K}_t{K}_{\mathrm{vp}}R\hfill \\ \multicolumn{1}{c}{a_2}& =K_e{K}_t{K}_{\mathrm{vp}}R+{\mathrm{BK}}_t{K}_{\mathrm{vi}}R+{\mathrm{BK}}_e\hfill \\ \multicolumn{1}{c}{a_3}& =K_e(J_l+J_m)+{\mathrm{BK}}_t{K}_{\mathrm{vp}}R+J_l{K}_t{K}_{\mathrm{vi}}R\hfill \\ \multicolumn{1}{c}{a_4}& ={\mathrm{BJ}}_m+J_l{K}_t{K}_{\mathrm{vp}}R\hfill \\ \multicolumn{1}{c}{a_5}& =J_l{J}_m\hfill \\ \multicolumn{1}{c}{b_0}& =K_e{K}_p{K}_t{K}_{\mathrm{vi}}R\hfill \\ \multicolumn{1}{c}{b_1}& =K_e{K}_p{K}_t{K}_{\mathrm{vp}}R\hfill \end{array}$$

According to the closed-loop transfer function G_c(s), the bandwidth of the command response B_w can be calculated. A greater bandwidth B_w indicates that the servomechanism has a faster command response and better dynamics. The disturbance rejection transfer function G_d(s) from the external disturbance T_d to the actual worktable position X_l can be expressed as

$$G_d(s)=\frac{c_3{s}^3+c_2{s}^2+c_{1}s}{\Delta (s)}$$ (7)

where

$$\{\begin{array}{cc}{c}_1\hfill & =R(K_e+K_t{K}_{\mathrm{vi}}R)\hfill \\ \multicolumn{1}{c}{c_2}& =K_t{K}_{\mathrm{vp}}{R}^2\hfill \\ \multicolumn{1}{c}{c_3}& =J_{m}R\hfill \end{array}$$

In this article, the peak error E_p caused by the external disturbance T_d is employed to reflect the dynamics of the external disturbance rejection. The peak error E_p can be obtained by the transfer function G_d(s) and is written as

$$E_p=|\max (E_{\mathrm{rr}})|$$ (8)

where E_rr is the tracking error of the servomechanism that is induced by the external disturbance T_d. A smaller peak error E_p results in a better external disturbance rejection response.

The absolute stability can only ensure servomechanism operation under ideal working conditions. However, due to the modelling error and the complex variation in the modelling elements, such as the stiffness, to make the servomechanism work well, both relative stability and robust stability must be simultaneously satisfied. Routh’s stability criterion is applied to analyse absolute stability of a servomechanism. If the roots of the characteristic polynomial Δ(s) are located in the left half of the S plane, the servomechanism will have absolute stability. To ensure servomechanism stability, even if some uncertainties and modelling errors exist during the modelling process, the relative stability and robust stability must also be considered. The relative stability is defined by the magnitude margin A_m and the phase margin P_m. The robust stability is the solution to the problem caused by great variations in the modelling elements, especially the stiffness. The robust stability may vary greatly at different positions. Furthermore, other modelling elements, such as inertia and damping, may produce great fluctuations.

To illustrate the nonlinear and coupled relations that exist in servomechanism, a preliminary simulation study is conducted, and an open servomechanism platform is configured. The main specifications of this open servomechanism platform are shown in Table 1.

Table 1.

Main specifications of the open servomechanism platform.

Parameter	Value
Ball screw lead l (mm)	16
Inertia of servo motor Jm (kg.m2)	0.00178
Equivalent load inertia Jl (kg.m2)	0.0012
Equivalent load damping B (kg.m2.s−1)	0.0288
Equivalent torque constant Kt (N.m.V−1)	2.68
Equivalent torsional stiffness Kθ (N.m.rad−1)	4221
Equivalent axial stiffness Kl (N.μm−1)	102
Equivalent stiffness Ke (N.m.rad−1)	592.6
Feedback sensor gain R (mm.rad−1)	2.5465
Transmission efficiency of mechanical subsystem η	0.98

The complex relationship between the dynamic response and the stability is discussed according to the established servomechanism model. Figure 3(a) shows that the bandwidth B_w improves as the proportional gain K_p of the position controller increases. However, when the proportional gain K_p of the position controller locates the low or high interval, the proportional gain K_vp of the velocity controller’s effect on the bandwidth B_w is obviously different. Figure 3(b) shows the peak error E_p, which is induced by the external disturbance, decreases as the proportional gain K_p of the position controller and the proportional gain K_vp of the velocity controller improve. Meanwhile, the external disturbance rejection increases as well. As shown in Figure 3(a) and (b), the bandwidth B_w and the external disturbance rejection are enhanced as the proportional gain K_p of the position controller increases. However, as shown in Figure 3(c) and (d), with an increase in the proportional gain K_p of the position controller, the magnitude margin A_m and the phase margin P_m decrease simultaneously, and the relative stability worsens. Meanwhile, the magnitude margin A_m decreases as the proportional gain K_vp of the velocity controller increases. Additionally, when the proportional gain K_p of the position controller locates the low or high interval, the proportional gain K_vp of the velocity controller’s effects on the phase margin P_m is obviously different as well. Figure 4 shows that the magnitude margin A_m and the phase margin P_m are reduced as the equivalent stiffness K_e decreases. This result implies that if the equivalent stiffness K_e changes greatly, a stability problem will occur. Therefore, great fluctuations in the equivalent stiffness K_e on the stability should be considered serious. From the discussion above, nonlinear and coupled relationships between the controller parameters, the mechanical subsystem parameters and the performance are inferred to exist. Therefore, tuning the controller parameters for the servomechanism to strike a balance among these parameters is essential.

Figure 3.

Dynamics and stability: (a) relationship between the controller parameters and the bandwidth B_w, (b) relationship between the controller parameters and the peak error E_p, (c) relationship between the controller parameters and the amplitude margin A_m and (d) relationship between the controller parameters and the phase margin P_m.

Figure 4.

Equivalent stiffness K_e and the stability: (a) effects of an equivalent stiffness K_e on the amplitude margin A_m and (b) effects of an equivalent stiffness K_e on the phase margin P_m.

Design requirements

According to practical operation demands, the following servomechanism design requirements should be satisfied simultaneously. To make the servomechanism work well, its stability must be guaranteed. If the roots of the characteristic polynomial Δ(s) are located in the left half of the S plane, according to Routh’s stability criterion, the coefficients of the characteristic polynomial should satisfy the following inequalities

$$\{\begin{array}{c}{a}_0>0\hfill \\ \multicolumn{1}{c}{a_4>0}\\ \multicolumn{1}{c}{a_5>0}\\ \multicolumn{1}{c}{a_4{a}_3-a_5{a}_2>0}\\ \multicolumn{1}{c}{a_5{a}_2^2-a_2{a}_3{a}_4+a_1{a}_4^2-a_0{a}_4{a}_5<0}\\ \multicolumn{1}{c}{a_0^2{a}_5^2-2a_0{a}_1{a}_4{a}_5-a_0{a}_2{a}_3{a}_5}\\ \multicolumn{1}{c}{+a_0{a}_4{a}_3^2+a_1^2{a}_4^2+a_1{a}_5{a}_2^2-a_1{a}_2{a}_3{a}_4<0}\end{array}$$ (9)

Meanwhile, the relative stability can be satisfied by the following inequalities

$$\{\begin{array}{c}{A}_m\ge A_{\min }\hfill \\ \multicolumn{1}{c}{P_m\ge P_{\min }}\end{array}$$ (10)

where A_min and P_min are the required minimum magnitude margin and minimum phase margin, respectively. The values of these parameters are determined by the practical operation demands. However, the equivalent stiffness K_e and the equivalent load damping B are affected by multiple factors, such as the position and temperature. The values of the equivalent stiffness K_e may vary greatly at different positions and temperatures. Furthermore, modelling errors, for example, in the equivalent stiffness K_e, are inevitable. Therefore, both the modelling errors and the variations in the modelling elements must be given significant consideration. According to the Kharionov rule and considering the variations in the modelling elements, if the following critical characteristic polynomials are both stable, the servomechanism will be robustly stabile

$$\{\begin{array}{c}{\Delta }_1(s)=a_0^{-}+a_1^{-}s+a_2^{+}{s}^2+a_3^{+}{s}^3+a_4^{-}{s}^4+a_5^{-}{s}^5\hfill \\ \multicolumn{1}{c}{{\Delta }_2(s)=a_0^{-}+a_1^{+}s+a_2^{+}{s}^2+a_3^{-}{s}^3+a_4^{-}{s}^4+a_5^{+}{s}^5}\\ \multicolumn{1}{c}{{\Delta }_3(s)=a_0^{+}+a_1^{-}s+a_2^{-}{s}^2+a_3^{+}{s}^3+a_4^{+}{s}^4+a_5^{-}{s}^5}\\ \multicolumn{1}{c}{{\Delta }_4(s)=a_0^{+}+a_1^{+}s+a_2^{-}{s}^2+a_3^{-}{s}^3+a_4^{+}{s}^4+a_5^{+}{s}^5}\end{array}$$ (11)

where $a_0^{-}$ and $a_0^{+}$ are the minimum and maximum of the polynomial coefficient $a_0$ , respectively. $a_1^{-}$ and $a_1^{+}$ are the minimum and maximum of the polynomial coefficient $a_1$ , respectively. $a_2^{-}$ and $a_2^{+}$ are the minimum and maximum of the polynomial coefficient $a_2$ , respectively. $a_3^{-}$ and $a_3^{+}$ are the minimum and maximum of the polynomial coefficient $a_3$ , respectively. $a_4^{-}$ and $a_4^{+}$ are the minimum and maximum of the polynomial coefficient $a_4$ , respectively. $a_5^{-}$ and $a_5^{+}$ are the minimum and maximum of the polynomial coefficient $a_5$ , respectively. The characteristic polynomials Δ₁(s), Δ₂(s), Δ₃(s) and Δ₄(s) are the different configurations of the polynomial coefficients with critical cases. The coefficients $a_0^{-}$ , $a_0^{+}$ , $a_1^{-}$ , $a_1^{+}$ , $a_2^{-}$ , $a_2^{+}$ , $a_3^{-}$ , $a_3^{+}$ , $a_4^{-}$ , $a_4^{+}$ , $a_5^{-}$ and $a_5^{+}$ can be expressed as

$$\{\begin{array}{c}{a}_0^{-}=K_e^{-}{K}_p{K}_t^{-}{K}_{\mathrm{vi}}R\hfill \\ \multicolumn{1}{c}{a_0^{+}=K_e^{+}{K}_p{K}_t^{+}{K}_{\mathrm{vi}}R}\\ \multicolumn{1}{c}{a_1^{-}=K_e^{-}{K}_t^{-}{K}_{\mathrm{vi}}R+K_e^{-}{K}_p{K}_t^{-}{K}_{\mathrm{vp}}R}\\ \multicolumn{1}{c}{a_1^{+}=K_e^{+}{K}_t^{+}{K}_{\mathrm{vi}}R+K_e^{+}{K}_p{K}_t^{+}{K}_{\mathrm{vp}}R}\\ \multicolumn{1}{c}{a_2^{-}=K_e^{-}{K}_t^{-}{K}_{\mathrm{vp}}R+B^{-}{K}_t^{-}{K}_{\mathrm{vi}}R+B^{-}{K}_e^{-}}\\ \multicolumn{1}{c}{a_2^{+}=K_e^{+}{K}_t^{+}{K}_{\mathrm{vp}}R+B^{+}{K}_t^{+}{K}_{\mathrm{vi}}R+B^{+}{K}_e^{+}}\\ \multicolumn{1}{c}{a_3^{-}=K_e^{-}(J_l^{-}+J_m^{-})+B^{-}{K}_t^{-}{K}_{\mathrm{vp}}R+J_l^{-}{K}_t^{-}{K}_{\mathrm{vi}}R}\\ \multicolumn{1}{c}{a_3^{+}=K_e^{+}(J_l^{+}+J_m^{+})+B^{+}{K}_t^{+}{K}_{\mathrm{vp}}R}\\ \multicolumn{1}{c}{+J_l^{+}{K}_t^{+}{K}_{\mathrm{vi}}R}\\ \multicolumn{1}{c}{a_4^{-}=B^{-}{J}_m^{-}+J_l^{-}{K}_t^{-}{K}_{\mathrm{vp}}R}\\ \multicolumn{1}{c}{a_4^{+}=B^{+}{J}_m^{+}+J_l^{+}{K}_t^{+}{K}_{\mathrm{vp}}R}\\ \multicolumn{1}{c}{a_5^{-}=J_l^{-}{J}_m^{-}}\\ \multicolumn{1}{c}{a_5^{+}=J_l^{+}{J}_m^{+}}\end{array}$$ (12)

where $K_e^{-}$ and $K_e^{+}$ are the minimum and maximum of the equivalent stiffness K_e, respectively. $K_t^{-}$ and $K_t^{+}$ are the minimum and maximum of the torque constant K_t, respectively. B⁻ and B⁺ are the minimum and maximum of the equivalent damping B, respectively. $J_m^{-}$ and $J_m^{+}$ are the minimum and maximum inertias of the servo motor J_m, respectively. $J_l^{-}$ and $J_l^{+}$ are the minimum and maximum equivalent inertias of the load J_l, respectively.

To achieve robust stability, the characteristic polynomials Δ₁(s), Δ₂(s), Δ₃(s) and Δ₄(s) should be stable. The coefficients of the characteristic polynomials introduced above should satisfy the following inequality group

$$\begin{array}{cc}\multicolumn{1}{c}{}& \{\begin{array}{c}{a}_0^{-}>0\hfill \\ \multicolumn{1}{c}{a_4^{-}>0}\\ \multicolumn{1}{c}{a_5^{-}>0}\\ \multicolumn{1}{c}{a_4^{-}{a}_3^{+}-a_5^{-}{a}_2^{+}>0}\\ \multicolumn{1}{c}{a_5^{-}{(a_2^{+})}^2-a_2^{+}{a}_3^{+}{a}_4^{-}+a_1^{-}{(a_4^{-})}^2-a_0^{-}{a}_4^{-}{a}_5^{-}<0}\\ \multicolumn{1}{c}{{(a_0^{-})}^2{(a_5^{-})}^2-2a_0^{-}{a}_1^{-}{a}_4^{-}{a}_5^{-}-a_0^{-}{a}_2^{+}{a}_3^{+}{a}_5^{-}}\\ \multicolumn{1}{c}{+a_0^{-}{a}_4^{-}{(a_3^{+})}^2+{(a_1^{-})}^2{(a_4^{-})}^2+a_1^{-}{a}_5^{-}{(a_2^{+})}^2}\\ \multicolumn{1}{c}{-a_1^{-}{a}_2^{+}{a}_3^{+}{a}_4^{-}<0}\end{array}\hfill \\ \multicolumn{1}{c}{}& \{\begin{array}{c}{a}_0^{-}>0\hfill \\ \multicolumn{1}{c}{a_4^{-}>0}\\ \multicolumn{1}{c}{a_5^{-}>0}\\ \multicolumn{1}{c}{a_4^{-}{a}_3^{-}-a_5^{-}{a}_2^{+}>0}\\ \multicolumn{1}{c}{a_5^{-}{(a_2^{+})}^2-a_2^{+}{a}_3^{-}{a}_4^{-}+a_1^{+}{(a_4^{-})}^2-a_0^{-}{a}_4^{-}{a}_5^{-}<0}\\ \multicolumn{1}{c}{{(a_0^{-})}^2{(a_5^{-})}^2-2a_0^{-}{a}_1^{+}{a}_4^{-}{a}_5^{-}-a_0^{-}{a}_2^{+}{a}_3^{-}{a}_5^{-}}\\ \multicolumn{1}{c}{+a_0^{-}{a}_4^{-}{(a_3^{-})}^2+{(a_1^{+})}^2{(a_4^{-})}^2+a_1^{+}{a}_5^{-}{(a_2^{+})}^2}\\ \multicolumn{1}{c}{-a_1^{+}{a}_2^{+}{a}_3^{-}{a}_4^{-}<0}\end{array}\hfill \\ \multicolumn{1}{c}{}& \{\begin{array}{c}{a}_0^{+}>0\hfill \\ \multicolumn{1}{c}{a_4^{-}>0}\\ \multicolumn{1}{c}{a_5^{-}>0}\\ \multicolumn{1}{c}{a_4^{+}{a}_3^{+}-a_5^{-}{a}_2^{-}>0}\\ \multicolumn{1}{c}{a_5^{-}{(a_2^{-})}^2-a_2^{-}{a}_3^{+}{a}_4^{+}+a_1^{-}{(a_4^{+})}^2-a_0^{+}{a}_4^{+}{a}_5^{-}<0}\\ \multicolumn{1}{c}{{(a_0^{+})}^2{(a_5^{-})}^2-2a_0^{+}{a}_1^{-}{a}_4^{+}{a}_5^{-}-a_0^{+}{a}_2^{-}{a}_3^{+}{a}_5^{-}}\\ \multicolumn{1}{c}{+a_0^{+}{a}_4^{+}{(a_3^{+})}^2+{(a_1^{-})}^2{(a_4^{+})}^2+a_1^{-}{a}_5^{-}{(a_2^{-})}^2}\\ \multicolumn{1}{c}{-a_1^{-}{a}_2^{-}{a}_3^{+}{a}_4^{+}<0}\end{array}\hfill \\ \multicolumn{1}{c}{}& \{\begin{array}{c}{a}_0^{+}>0\hfill \\ \multicolumn{1}{c}{a_4^{+}>0}\\ \multicolumn{1}{c}{a_5^{+}>0}\\ \multicolumn{1}{c}{a_4^{+}{a}_3^{-}-a_5^{+}{a}_2^{-}>0}\\ \multicolumn{1}{c}{a_5^{+}{(a_2^{-})}^2-a_2^{-}{a}_3^{-}{a}_4^{+}+a_1^{+}{(a_4^{+})}^2-a_0^{+}{a}_4^{+}{a}_5^{+}<0}\\ \multicolumn{1}{c}{{(a_0^{+})}^2{(a_5^{+})}^2-2a_0^{+}{a}_1^{+}{a}_4^{+}{a}_5^{+}-a_0^{+}{a}_2^{-}{a}_3^{-}{a}_5^{+}}\\ \multicolumn{1}{c}{+a_0^{+}{a}_4^{+}{(a_3^{-})}^2+{(a_1^{+})}^2{(a_4^{+})}^2+a_1^{+}{a}_5^{+}{(a_2^{-})}^2}\\ \multicolumn{1}{c}{-a_1^{+}{a}_2^{-}{a}_3^{-}{a}_4^{+}<0}\end{array}\hfill \end{array}$$ (13)

Furthermore, to ensure that the servomechanism is effectively stabile, the absolute stability, relative stability and robust stability should be satisfied simultaneously. Additionally, the overshoot O_p is a key indicator for the servomechanism and should satisfy the following inequality

$$O_p\le O_{\mathrm{pmax}}$$ (14)

where O_pmax is the maximum overshoot of the servomechanism, and its value is determined by the practical operation demands. Additionally, the mechanical resonance is one of the most pervasive problems in motion control. ¹⁶ Most often resonance is caused by the compliance between the servo motor and the load. ¹⁷ Therefore, the bandwidth B_w should be much less than the mechanical resonance frequency of the servomechanism, M_r. The mechanical resonance and the bandwidth limitation can be described as

$$\{\begin{array}{c}{M}_r=\sqrt{K_e(\frac{1}{J_m}+\frac{1}{J_l})}\hfill \\ \multicolumn{1}{c}{B_w<{\eta }_b{M}_r}\end{array}$$ (15)

where η_b is the ratio between the bandwidth B_w and the mechanical resonance frequency M_r. Generally, η_b = 1/5–1/3.

Multi-objective nonlinear optimisation

Complex nonlinear relationships exist between the controller parameters and the performance and design requirements of the servomechanism. The controller parameter optimisation is performed to obtain the desired performance by satisfying the above-mentioned design requirements. The optimisation is essentially a multi-objective nonlinear optimisation problem, and a single objective optimisation function cannot achieve the desired optimisation performance for the servomechanism. The particle swarm optimisation (PSO) algorithm is used to solve the nonlinear optimisation problems. This algorithm has the advantages of good accuracy, fast convergence and easy implementation and is adopted in this article. ¹⁸ Each particle represents a set of controller parameters. The algorithm first randomly generates a particle swarm, and the optimisation performance of each particle is evaluated using a multi-objective function. In this article, a multi-objective optimisation function Q_e is proposed and comprehensively adopts a command response performance function Q ₁ and an external disturbance rejection response performance function Q ₂. These functions can be formulated as

$$\{\begin{array}{cc}{Q}_e\hfill & =\frac{w_1{Q}_1}{Q_c}+\frac{w_2{Q}_2}{Q_d}+F_p\hfill \\ \multicolumn{1}{c}{Q_1}& =\underset{0}{\overset{t_a}{\int }}t|E_{\mathrm{rr}}(t)|\mathrm{dt}\hfill \\ \multicolumn{1}{c}{Q_2}& =E_p\hfill \\ \multicolumn{1}{c}{F_p}& =\{\begin{array}{cc}0\hfill & \mathrm{meet}\mathrm{design}\mathrm{requirements}\hfill \\ \multicolumn{1}{c}{{10}^5}& \mathrm{otherwise}\hfill \end{array}\hfill \end{array}$$ (16)

where t_a is the transition time. E_rr is the tracking error with the unit step command during the transition time t_a. E_p is the peak error caused by the external unit step disturbance. Generally, when using function Q ₁, both the transition time t_a and tracking error E_rr will be reduced. The function Q ₂ primarily focuses on the peak error E_p caused by the external unit step disturbance. A penalty function F_p is employed to achieve the design requirements. Q_c and Q_d are the values of first randomly generated particle in the first iteration for the command response performance function Q ₁ and the external disturbance response performance function Q ₂, respectively. w ₁ and w ₂ are the weight factors of the command response performance function Q ₁ and the external disturbance response rejection performance function Q ₂, respectively. The values of w ₁ and w ₂ are determined by the practical design requirements, where w ₁, w ₂∈ [0, 1] and w ₁ + w ₂ = 1. Generally, w ₁ =w ₂ = 0.5. A smaller value of the multi-objective optimisation function Q_e indicates a servomechanism that has better integrated dynamics. Therefore, the nature of the controller parameter optimisation is to search for the smallest value of the multi-objective optimisation function Q_e under conditions that satisfy the design requirements.

To improve convergence speed and optimisation performance, a linear decreasing weight (LDW) strategy is adopted. The position and the speed of a particle can be expressed as

$$\{\begin{array}{c}{v}^{j+1}(i,k)=w(k)v^j(i,k)+c_{1}r[p_{\mathrm{best}}^j(i,k)-x^j(i,k)]\hfill \\ \multicolumn{1}{c}{+c_{2}r[g_{\mathrm{best}}^j(k)-x^j(i,k)]}\\ \multicolumn{1}{c}{x^{j+1}(i,k)=x^j(i,k)+v^j(i,k)}\\ \multicolumn{1}{c}{w(k)=w_{\min }+\frac{w_{\max }-w_{\min }}{N}\times (N-j)}\end{array}$$ (17)

where i is the ith particle. k is the kth coordinate component of a particle. j is the jth iteration. w is the weight factor. c ₁ and c ₂ are the acceleration factors. r is a random number such that r∈ [0, 1]. x^j and v^j are the position and the speed of a particle in the jth iteration, respectively. $p_{\mathrm{best}}^j$ is the best value of a particle in the jth iteration. $g_{\mathrm{best}}^j$ is the best value of the particle swarm. w_max and w_min are the maximum and minimum of the weight factor w, respectively. N is the maximum number of iterations.

Simulation and experiments

The validity of the controller parameter optimisation method proposed in this article is verified on an open servomechanism platform, as shown in Figure 5. This open servomechanism platform comprises a servo motor, an amplifier, a mechanical subsystem and an open computer numerical control (CNC) system. The worktable is driven by a ball screw coupled to a servo motor and its amplifier. The open CNC system is composed of a host computer and a slave computer. The host computer is responsible for the operation, data sampling, motion monitoring and so on. The motion controller that generates the control signals sent to the amplifier was developed on the slave computer. Some logical control operations can be implemented by the logical signals, such as the emergency stop operation. The position feedback signal is generated by the linear scale installed on the worktable. The velocity feedback signal is obtained by the encoder directly coupled to the servo motor. These feedback signals are sampled by the motion controller. Additionally, the related signals, such as the position, velocity and torque command, are transmitted to the host computer by data bus in real time.

Figure 5.

Open servomechanism platform.

This open servomechanism platform is primarily used for precision positioning. To achieve the desired positioning accuracy, overshoot and oscillations are not allowed during the positioning process. External disturbance should be avoided as well. To achieve the above-mentioned practical demands, the initial controller parameters used with the manual tuning method are as follows: K_p = 83.1792 s⁻¹, K_vp = 0.0951 V.s.mm⁻¹ and K_vi = 13.428 V.s.mm⁻¹. According to the practical demands, the related design requirements are as follows: A_min = 10 dB, P_min = 30°, O_pmax = 0 and η_b = 0.2. The main parameters of the PSO algorithm are as follows: w_min = 0.4, w_max = 0.9, N = 200, c ₁ = 2 and c ₂ = 0.2.

The controller parameter optimisation method proposed by this article is adopted. The equivalent stiffness K_e is an influential variation factor on this platform and can be identified as follows: K_e ⁻ = 533.4 N.rad⁻¹ and K_e ⁺ = 654.2 N.rad⁻¹. Other modelling elements have little variation on this open platform. The controller parameter optimisation results are as follows: K_p = 137.889 s⁻¹, K_vp = 0.2109 V.s.mm⁻¹ and K_vi = 21.1 V.s.mm⁻¹. Figure 6 shows the optimal value of the multi-objective optimisation function Q_e, which decreases in each iteration and gradually converges as the number of iterations increases. By simply considering the command response item of multi-objective optimisation function Q_e, that is, w ₁ = 1, w ₂ = 0, the controller parameter optimisation results are as follows: K_p = 140.1792 s⁻¹, K_vp = 0.198 V.s.mm⁻¹ and K_vi = 8.89 V.s.mm⁻¹. The controller parameter optimisation simulation results, considering and not considering external disturbance rejection, are shown in Figure 7. Figure 7(a) shows the peak error E_p induced by the external unit step disturbance when considering and not considering the external disturbance response. By comprehensively integrating the command and disturbance rejection response, both the peak error E_p and transition time t_a decrease. Furthermore, as shown in Figure 7(b), the bandwidth B_w increases simultaneously. Therefore, the multi-objective optimisation function Q_e should include an external disturbance rejection parameter. The controller parameter optimisation results are shown in Figure 8. Figure 8(a) shows the closed-loop frequency response before and after controller parameter optimisation. The bandwidth B_w is improved greatly. After the controller parameter optimisation, the bandwidth B_w increases from 204 to 286 rad s⁻¹. The external disturbance rejection response is shown in Figure 8(b) before and after controller parameter optimisation. After controller parameter optimisation, the external disturbance rejection response is greatly improved for most frequencies, and the maximum amplitude of the external disturbance rejection response is reduced from −30 to −39 dB. Figure 8(c) shows the actual worktable position X_l with the unit step command before and after controller parameter optimisation. After controller parameter optimisation, the rise time t_r decreases from 0.021 to 0.016 s. The overshoot O_p does not appear. The tracking error E_rr caused by the external unit step disturbance is shown in Figure 8(d) before and after controller parameter optimisation. After controller parameter optimisation, the peak error E_p is reduced from 0.0202 to 0.096 mm. Furthermore, at the same time, the transition time t_a decreases from 0.06 to 0.042 s. Combining the simulation in the time and frequency domains, the desired performance of this open servomechanism platform can be obtained using the method described in this work.

Figure 6.

Variation curve of the multi-objective optimisation function Q_e.

Figure 7.

Simulated results from the controller parameter optimisation with and without considering external disturbance rejection: (a) comparison of the peak error E_p and (b) comparison of the bandwidth B_w.

Figure 8.

Controller parameter optimisation results: (a) closed-loop frequency response, (b) external disturbance rejection frequency response, (c) actual worktable position X_l with the unit step command and (d) tracking error E_rr caused by the external unit step disturbance.

To further verify the effectiveness of the controller parameter optimisation method proposed in this article, a controller parameter optimisation experiment is performed on the open servomechanism platform. Moreover, to validate the conclusion that this method can be applied to different working conditions, different S-shaped trajectories S₁ and S₂, which are based on a trapezoidal velocity profile, are adopted. The trajectory parameters of S₁ and S₂ are set as follows: the accelerations in the acceleration–deceleration sections are 10 and 20 mm s⁻², respectively; the velocities in the constant-velocity sections are 10 and 20 mm s⁻¹, respectively; the motion distances are both 50 mm. Figure 9(a) and (b) shows the motion trajectories S₁ and S₂, respectively.

Figure 9.

Motion trajectory: (a) motion trajectory S₁ and (b) motion trajectory S₂.

Figure 10 shows the experimental and simulation results with different working conditions. Figure 10(a) and (c) shows the simulated and experimental tracking error E_rr before and after controller parameter optimisation with the motion trajectory S₁, respectively. Figure 10(b) and (d) shows the simulated and experimental tracking error E_rr before and after controller parameter optimisation with the motion trajectory S₂, respectively. As shown in Figure 10, the simulated and experimental tracking error E_rr is effectively reduced, and the motion accuracy improves greatly after controller parameter optimisation. Protrusion error appears during the velocity reversal and significantly affects the motion accuracy.^19,20 The protrusion error is induced by the nonlinear friction force, which can be considered an external disturbance. Figure 10(a)–(d) indicates that the simulated and experimental protrusion error is effectively suppressed, which is a result of the external disturbance rejection enhancement after controller parameter optimisation.

Figure 10.

Simulated and experimental results for the different working conditions: (a) simulated tracking error E_rr with motion trajectory S₁, (b) simulated tracking error E_rr with motion trajectory S₂, (c) experimental tracking error E_rr with motion trajectory S₁ and (d) experimental tracking error E_rr with motion trajectory S₂.

In this article, the controller parameter optimisation performance in the case of the simulations and the experiments is comprehensively evaluated by a set of performance indicators:

• A_e: absolute maximum of the tracking error;

• E_r: root mean square of the tracking error;

• M_e: absolute mean of the tracking error;

• M_p: maximal percentage of the tracking error.

Table 2 shows that the performance indicators are both significantly reduced after controller parameter optimisation under different working conditions. Moreover, after controller parameter optimisation in the case of the simulations, the absolute maximum of the tracking error A_e, the root mean square of the tracking error E_r and the absolute mean of the tracking error M_e with the motion trajectories S₁ and S₂ decrease by 69%, 72% and 73%, respectively. After controller parameter optimisation in the case of the experiments, the absolute maximum of the tracking error A_e, the root mean square of the tracking error E_r and the absolute mean of the tracking error M_e with the motion trajectories S₁ and S₂ decrease by 64%, 65% and 62%, respectively. Meanwhile, the motion accuracy improves greatly. The effectiveness and feasibility of the method proposed by this article are verified by the simulations and the experiments. Moreover, these optimised controller parameters can be applied to different working conditions.

Table 2.

Comparison of the performance indicators before and after controller parameter optimisation.

Trajectory	BCOS				ACOS				BCOE				ACOE
	Ae (μm)	Er (μm)	Me (μm)	Mp (%)	Ae (μm)	Er (μm)	Me (μm)	Mp (%)	Ae (μm)	Er (μm)	Me (μm)	Mp (%)	Ae (μm)	Er (μm)	Me (μm)	Mp (%)
S1	11.2	0.58	0.069	0.022	3.4	0.16	0.025	0.006	14.10	0.92	0.48	0.028	5.0	0.32	0.18	0.01
S2	12.2	0.6	0.03	0.024	3.9	0.18	0.028	0.007	15.80	0.96	0.49	0.031	6.10	0.36	0.19	0.012

BCOS: before controller parameter optimisation with simulation; ACOS: after controller parameter optimisation with simulation; BCOE: before controller parameter optimisation with experiment; ACOE: after controller parameter optimisation with experiment.

Conclusion

Servomechanism modelling and performance analysis were conducted in this article. Moreover, according to the practical operation demands, design requirements were developed. Using the multi-objective optimisation method proposed in this article, the optimised servomechanism controller parameters were obtained. The main conclusions can be summarised as follows:

1. Nonlinear coupled relations exist between the controller parameters and the performance and design requirements. The controller parameter optimisation was performed to obtain the desired performance by satisfying the design requirements. This optimisation is essentially a multi-objective nonlinear optimisation problem, and manual tuning can rarely obtain the desired performance. The stability problem induced by large variations in the modelling elements is also considered. In this article, robust stability is proposed to ensure the stability of the servomechanism.

2. Combined with the servomechanism model, performance analysis and design requirements, the controller parameter optimisation is converted into a multi-objective nonlinear optimisation problem. The multi-objective optimisation function comprehensively adopts a command response performance function and an external disturbance response rejection performance function. The PSO algorithm is used to solve this problem.

3. The effectiveness and feasibility of the controller parameter optimisation method proposed in this article are verified through simulations and experiments. The experimental and simulated results show that the tracking error, protrusion error and performance indicators are significantly reduced, and the motion accuracy is greatly improved. This method has several advantages, such as high efficiency, high practicality and high flexibility. Moreover, the optimised controller parameters generated by this method can be applied to varied working conditions.