J. Cent. South Univ. (2019) 26: 2340-2353

DOI: https://doi.org/10.1007/s11771-019-4178-4

Fractional order PID control for steer-by-wire system of emergency rescue vehicle based on genetic algorithm

XU Fei-xiang(徐飞翔), LIU Xin-hui(刘昕晖), CHEN Wei(陈伟),ZHOU Chen(周晨), CAO Bing-wei(曹丙伟)

School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Aiming at dealing with the difficulty for traditional emergency rescue vehicle (ECV) to enter into limited rescue scenes, the electro-hydraulic steer-by-wire (SBW) system is introduced to achieve the multi-mode steering of the ECV. The overall structure and mathematical model of the SBW system are described at length. The fractional order proportional-integral-derivative (FOPID) controller based on fractional calculus theory is designed to control the steering cylinder’s movement in SBW system. The anti-windup problem is considered in the FOPID controller design to reduce the bad influence of saturation. Five parameters of the FOPID controller are optimized using the genetic algorithm by maximizing the fitness function which involves integral of time by absolute value error (ITAE), peak overshoot, as well as settling time. The time-domain simulations are implemented to identify the performance of the raised FOPID controller. The simulation results indicate the presented FOPID controller possesses more effective control properties than classical proportional-integral-derivative (PID) controller on the part of transient response, tracking capability and robustness.

Key words: steer-by-wire system; emergency rescue vehicle; fractional order proportional-integral-derivative (FOPID) controller; parameter optimization; genetic algorithm

Cite this article as: XU Fei-xiang, LIU Xin-hui, CHEN Wei, ZHOU Chen, CAO Bing-wei. Fractional order PID control for steer-by-wire system of emergency rescue vehicle based on genetic algorithm [J]. Journal of Central South University, 2019, 26(9): 2340-2353. DOI: https://doi.org/10.1007/s11771-019-4178-4.

1 Introduction

The rescue scenes (especially the pedestrian street, the villages inside the city, the narrow roadway, etc.) following a natural disaster are usually complex and narrow. In this context, the limited rescue scenes are too difficult for traditional ECV to enter, which will miss the best rescue time and cause huge losses of property and life. Hence, applying a multi-mode steering system in ECV is of great importance, which can overcome the difficulty of passing through the narrow area and the complex terrain for the ECV.

As the supporting technology of intelligent automobile, steer-by-wire (SBW) system [1] acts a vital function in the developing modern automobile [2, 3]. Compared with the conventional steering system, the SBW system has displaced the mechanical connection between the steering wheel and front wheels with electronic equipment [4]. The SBW system has several advantages: 1) simplifying the design of cab and reducing traffic accidents [5]; 2) improving the vehicle stability, driving force and maneuverability [6]; 3) laying a technical foundation for unmanned driving [7]. As a result, this paper applies the SBW system into the development of ECV. In order to make the SBW system have effective tracking and a quick response, the design of control strategy is vital and urgently needed to be completed.

The electro-hydraulic control method integrates electronic control and hydraulic technology, which has been widely applied in the braking system [8, 9], SBW system [10, 11], steering system [12], hydraulic cylinder [13] construction machinery domain [14] and suspension system [15]. This paper applies the electro- hydraulic control method into the SBW system to control the independent steering of the ECV. In the SBW system, the controller obtains the instruction from the steering wheel firstly, and then the controller produces different corresponding signals to control four proportional electromagnetic valves independently, thus controlling the displacement of the cylinders to achieve the multi-mode steering (front wheel steering, rear wheel steering, four- wheel steering, and in-situ steering) of the ECV. Once the multi-mode steering is achieved by electro-hydraulic control strategy, the ECV can across the narrow and complex rescue scene.

The PID has been applied to electro-hydraulic control for a long time and has achieved good control effect. WANG et al [16] proposed a PID controller to control the electro-hydraulic servo load system. GHAZALI et al [17] designed the PID controller for a nonlinear quarter-car active suspension with electrohydraulic actuator. WANG et al [18] developed the PID controller to control the hydraulic cylinder pressure in electro-hydraulic control system. Nevertheless, the development and the improvement of fractional order theory promote its application in the controller design [19], and many scholars are making efforts to substitute the FOPID controller for the conventional PID controller [20, 21].

The FOPID controller appends two fractional parameters to acquire a bigger adjustable room, and it makes control system obtain more perfect response properties and robustness [22, 23]. DONG et al [22] designed a FOPID controller for vehicle active suspension system to control the electro-hydraulic actuator of the suspension system accurately. ZHAO et al [24] studied a FOPID controller to save energy of the electro-hydraulic control system. GAO et al [25] combined a FOPID controller with state observer for the actuation of electro-hydraulic servo system. ZHAO et al [26] studied a FOPID controller to manipulate two hydraulic valves to control hydraulic cylinder, thus achieving the force loading system’s control. The electro-hydraulic SBW system proposed in this study encounters some robust stability issues due to the external disturbances and parameter variations. Inspired by previous studies, a FOPID controller is set aside for regulating steering cylinders in the SBW system through making fully use of its great robustness. In order to reduce the bad influence of saturation on the control system, the anti-windup problem is also considered in the FOPID controller design.

The key point of the FOPID controller is to tune the adjustable parameters [27-29]. At present, many optimized methods have been used to tune the FOPID controller’s parameters [30-33], such as particle swarm optimization (PSO), artificial bee colony (ABC). The ABC has the ability to avoid local optimal stagnation (premature convergence) but still suffers from the lacuna of slow convergence over all the problems [34]. The PSO has fast convergence speed relatively, while it is easy to fall into local optimal solution and it is unstable. Compared to these intelligent algorithms mentioned, GA is related to industrial applications closely, which can deal with the optimization of parameters with constraints, targets and dynamic components [35]. Moreover, GA is able to get the global optimal solution based on the natural evolution principle and has quick convergence speed. In the recent years, GA has been used to tune parameters of FOPID controller. ASHU et al [36] developed a FOPID controller based on GA to control motor speed. BUANOVI et al [19] designed a FOPID controller tuned by GA to control the movement of steam turbine in the process of low temperature air separation. In order to achieve the control of seat suspension, GAD et al [37] optimized the FOPID controller based on multi-objective GA. Hence, the FOPID controller’s parameters are optimized based on GA to improve the control effect of the FOPID controller. We propose a time-domain criterion which involves ITAE, overshoot and settling time. This will be done through a fitness function of GA to achieve rise in the performance indices.

Comparing with the literatures that have been studied, the main contributions of this study are highlighted as follows: 1) in order to overcome the difficulty of passing through the narrow area and the complex terrain, the electro-hydraulic SBW system is introduced to control four steering cylinders independently, thus achieving the multi- mode steering of the ECV; 2) based on the fractional theory, the FOPID controller considering the saturation problem is designed to make the control system have better external disturbances rejection, random noise resistance and lower sensitivity to model uncertainty; 3) GA is applied to optimize the FOPID controller’s parameters to achieve the best control performance that minimizes the ITAE, the settling time and the peak overshoot. The rest of this paper is organized as follows: Section 2 briefly introduces the overall structure of the SBW system; Section 3 studies the SBW system modelling; Section 4 focuses on the FOPID controller design; Section 5 describes the simulations; and some concluding remarks are shown in Section 6.

2 Overall framework of SBW system

In the SBW system, the controller obtains the instruction from the steering wheel firstly, and then the controller produces different corresponding signals to four proportional valves independently, thus controlling four steering cylinders’ motion to achieve the steering of the ECV. The displacement sensor installed in the steering cylinder is used to feedback the signal to the controller. As shown in Figure 1, the SBW system consists of three parts: the steering wheel module, the controller module and the hydraulic steering execution module.

2.1 Steering wheel module

The steering wheel module mainly consists of the steering wheel and the potentiometer, which is used to translate the steering signal to the controller. The steering wheel driven by users rotates the potentiometer through small gear to transform the input to the SBW system controller. When the steering wheel moves to the end of the stroke, it cannot be rotated because of the effect of the stop block.

2.2 Controller module

The control scheme of the SBW system is shown in Figure 2. First of all, the values of displacement sensor installed in the steering cylinders and potentiometer of steering wheel are detected by the controller. Secondly, the controller will send corresponding control currents to four proportional electromagnetic valves according to the difference between displacement sensors and potentiometers. Thirdly, the spools of the proportional electromagnetic valves are moved to control the flow rate with the effect of corresponding currents. Finally, the displacements of four steering cylinders are changed to drive four tires to realize the steering of ECV.

Indeed, the controller module is used to control the displacement of steering cylinder. This paper mainly focuses on the controller module of the SBW system, where the FOPID control strategy is proposed to manipulate four steering cylinders accurately. Five parameters of the FOPID controller are optimized based on GA.

2.3 Hydraulic steering execution module

The hydraulic steering executive module mainly includes pump, proportional electromagnetic valves, steering cylinders, tank and relief valve. The pump converts the mechanical energy from the engine to the pressure energy of hydraulic oil. The proportional electromagnetic valve receives the signal of the controller, and then it changes the spool displacement to change the direction and value of the flow rate. The steering cylinders are used to realize the movement of steering tire. The tank is used to supply and store hydraulic oil to the hydraulic system. The relief valve ensures the maximum pressure of the hydraulic system.

3 System modelling

In order to simplify the establishment of the mathematical model of the SBW system, this paper only builds the displacement control model for a steering cylinder. The SBW system modelling includes the steering wheel, the proportional electromagnetic valve, the steering cylinder and the feedback.

3.1 Steering wheel model

The steering wheel can realize the transformation from mechanical signal to electrical signal, and its mathematical model is shown as follows:

Figure 1 Overall framework of SBW system

                                (1)

where K_v represents the angle ratio coefficient of the steering wheel; U denotes the output voltage of the steering wheel; ξ represents the angle of the steering wheel.

3.2 Proportional electromagnetic valve model

After the steering wheel’s analog voltage transmits the controller, it is converted into current signal and sent to the electromagnetic proportional valve. This process can be approximated to a proportional cycle. The mathematical model is shown as follows:

                                (2)

where K_a represents the voltage-current gain of the controller; I denotes the control current of the proportional electromagnetic valve.

The proportional electromagnetic valve consists of the proportional electromagnet and the reversal valve. The electromagnetic force of the proportional electromagnet in the effective stroke is shown below.

Figure 2 Control scheme of SBW system

                               (3)

where F_m represents the electromagnetic force; K_m denotes the coefficient of electromagnetic force.

Under the precondition of neglecting the influence of hydrodynamic force, the following equation can be obtained from the force balance of the reversal valve spool.

             (4)

where M_j denotes the spool mass; y represents the spool displacement; B_y is the viscous camping coefficient; K₁ represents the spring stiffness; y₀ denotes the spring’s pre-decrement.

The pressure-flow relationship of the reversing valve is expressed as follows:

,

                  (5)

where Q_L is the outlet flow rate of the reversing valve; n represents the area ratio of the left and right cavity of steering cylinder; ω represents the area gradient; ρ denotes the density of the hydraulic oil; p_s represents the pressure of the hydraulic system; p_L denotes the loading pressure; p_A is the pressure of left cavity of steering cylinder; p_B is the pressure of right cavity of steering cylinder; C_h denotes the reversal valve’s flow coefficient.

Equation (5) can be linearized, which is displayed as below:

,  (6)

where K_qh represents the spool’s zero flow gain; K_ph denotes the spool’s pressure-flow coefficient in the steady state; r_c is the radial clearance of the spool; u denotes the fluid dynamic viscosity.

3.3 Steering cylinder model

The flow equation of the hydraulic cylinder’s control chamber is displayed as follows:

               (7)

where A_c represents the equivalent area of the hydraulic cylinder piston; y_g denotes the piston displacement of the hydraulic cylinder; V_e denotes the equivalent volume of the hydraulic cylinder; C_L represents the leakage ratio; β_e represents the elastic modulus of the liquid equivalent volume.

Under the premise of ignoring the nonlinear loads (such as Coulomb friction), the force balance equation between the piston and the load of the hydraulic cylinder is analyzed as below:

          (8)

where F_L represents the external load force on the piston of the hydraulic cylinder; M_t represents the outright quality of the hydraulic cylinder piston and load; K_L is the spring stiffness of the load; B_t is the viscosity damping coefficient of the load.

3.4 Feedback model

The displacement sensor is used to measure the steering cylinder displacement to give the feedback to the controller module in the form of voltage, and its mathematical model is shown as follows:

                               (9)

where U₀ represents the feedback voltage of the displacement sensor; K₀ denotes the ratio coefficient between the voltage and the displacement and it is set as 1 V/m in this study.

3.5 Analysis of open loop transfer function of SBW system

As shown in Figure 2, the steering wheel model and the feedback model are used to input the signal to the controller module. The open loop transfer function of the SBW system consists of the proportional electromagnetic valve model and the steering cylinder model. Based on the Laplace transform, the transfer functions of the SBW control system are summarized as follows:

    (10)

where K_ce denotes the flow-pressure rate, and

The SBW system is the inertia load system, and the elastic load can be neglected, that is to say K_L=0. In addition, the viscous damping coefficient

B_t is usually small, thus Equation (10)

can be simplified as the following form:

,

  (11)

where ω_h represents the natural frequency of the hydraulic system; ζ_h denotes the damping ratio of the hydraulic system.

Without considering the viscous damping coefficient, the system damping is only generated by the pressure-flow coefficient, and it can be approximately expressed as follows:

                        (12)

The transfer function of the SBW system is shown below:

     (13)

The SBW system’s parameters in the ECV are given in Table 1. Through inverting the actual parameters of Table 1 into Eq. (13), the transfer function can be obtained.

Table 1 SBW system’s parameters in ECV

4 Optimized FOPID control strategy based on GA

4.1 FOPID controller design

The FOPID controller emerges with the increasing application of fractional calculus theory in the control field, and it is the extension of the integer order PID controller in fractional field [38]. Unlike the limit of the PID controller, the FOPID controller can adjust the integral order and differential order arbitrarily within the range of value, which makes the control system more flexible. For the FOPID controller design, the key point is how to calculate the fractional calculus. In general, the fractional calculus can be displayed below:

        (14)

where denotes the differential operator or the integral operator; a and t represent the upper and lower limits of the calculus, respectively; α is the fractional order.

Various definitions of fractional calculus are proposed by many scholars [39, 40]. Riemann Liouville (RL) is widely used in the control field, which is defined as follows [21]:

                   (15)

The fractional integral of RL is defined as follows:

   (16)

The RL definitions mentioned above are turned into a unified form.

                   (17)

In the above Eqs. (15)-(17), Γ₀ is the Gamma function and it is expressed as below:

                (18)

Figure 3 displays the control diagram of the SBW system equipped with the FOPID controller. The FOPID controller’s transfer function is expressed as below:

                  (19)

where K_p, K_i and K_d represent the gains of proportional, integral, and derivative, respectively; U(s) and E(s) represent the control instruction and error signal, respectively; λ and μ denote the orders of the fractional order integrator and differentiator, respectively. Clearly, the classical PID controller is one especial FOPID controller when λ=1, μ=1.

Figure 3 Control diagram of SBW system equipped with FOPID controller

The control output signal u(t) from the FOPID controller is computed as follows:

          (20)

where e(t) is the error signal between the voltage of steering cylinder displacement sensor and the potentiometer voltage.

Because the proportional electromagnetic valve has the maximum working current, the output of the FOPID controller is always subject to the saturation state. In order to reduce the bad influence of saturation on closed-loop response of the control system, the FOPID controller considering anti- windup problem is designed. At present, the design methods of anti-saturation applied into industrial field are divided into two categories: back calculation method and conditional integral method [41]. The disadvantage of the former lies in its insufficient system stability and weak robustness, so the latter is used for developing the FOPID controller. The detailed design process for anti-saturation can be seen in Ref [42].

4.2 Optimal tuning of FOPID by using GA

In recent years, GA has been regarded as an effective method to deal with the optimization issues, which can obtain the global optimization results through making fully use of the natural evolution principle [43]. Hence, we propose using GA for tuning five parameters of the FOPID controller. GA can automatically search the optimal parameters through maximizing the fitness function value, so that the control property verges on the expected value [44]. In order to find suitable fitness function, a novel index J which reflects the performance of the FOPID controller is defined [19], as shown below:

                    (21)

where P₀ represents the peak overshoot; is ITAE; T_s denotes the settling time.

The fitness function is defined based on the J, which is shown below:

            (22)

It is obvious that when the fitness function reaches the maximum, the FOPID controller has the best control performance, and then five parameters (K_p, K_i, K_d, λ, μ) of the FOPID controller are optimal. To reduce tuning time, the scopes of FOPID parameters are defined as follows:

K_p[0, 20], K_i[0, 20], K_d[0, 20], λ[0, 2],

μ[0, 2]                             (23)

Figure 4 shows the working mechanism flowchart of GA.

Figure 4 Working mechanism flowchart of GA

Step 1: The population size is determined. A small population size causes poor performance of GA, and a big population size increases calculation time.

Step 2: The initial population code is generated to obtain a number of genetic strings based on the binary coding rules.

Step 3: The population is decoded, and the maximum value of fitness function is selected. The individual with a maximum value of fitness function is copied to next generation of new species, and then population selection, crossover and mutation to the parent individual are operated to reproduce the next generation of new species.

Step 4: If reaching the set value of genetic generations, the best gene string will be achieved, and then the optimal parameters of the FOPID controller are obtained. If not, the third step will be carried out.

In the process of the GA-based optimization, the related parameters are defined in Table 2.

Table 2 Parameters related to GA-based optimization algorithm

5 Simulations

For the sake of verifying the control property of the optimized FOPID controller based on GA, the time-domain simulations using MATLAB software are implemented in this paper. Moreover, the GA based PID controller is also developed to contrast the proposed FOPID controller.

Figure 5 shows the relationship between genetic generation and fitness function values in the process of GA optimization. In the GA optimization process, the maximum fitness function value of FOPID controller is higher than that of PID controller, which shows that the proposed FOPID controller possesses more perfect control property than classical PID controller. Both tuning parameters of the FOPID controller as well as classical PID controller based on GA are presented in Table 3.

The optimal parameters of Table 3 are put into the PID and FOPID controllers, and then the time-domain simulation tests are carried out in the three aspects: transient response, tracking performance and robustness.

5.1 Transient response test

Based on MATLAB, the unit step response curves of the optimized PID controller and FOPID controllers are displayed in Figure 6. On one hand,we obtain with optimized FOPID controller that overshoot is 0.032%, settling time is 0.102 s and rise time is 0.093 s; on the other hand, with optimized PID controller, we have 5.912% overshoot, 0.203 s settling time and 0.17 s rise time. The FOPID controller almost eliminates overshoot by comparison with common PID controller. Moreover, the rise time and the settling time of the FOPID controller are smaller than classical PID controller largely, where the rise time reduces by 45.294% and the settling time decreases by 49.753%. We can come to the conclusion that the proposed FOPID controller owns more excellent transient response than classical PID controller.

Figure 5 Relationship between genetic generation and fitness function value:

Table 3 Tuning parameters of FOPID controller as well as common PID controller based on GA

Figure 6 Step responses of SBW system equipped with optimal FOPID and classical PID controllers

5.2 Tracking performance test

Figure 7 displays the steering cylinder’s following curves of the SBW system equipped with optimal FOPID controller and common PID controller for the standard sinusoidal signal.Figure 8 shows the steering cylinder’s displacement tracking error of the SBW system. The common judging indexes about error are integral of squared error (ISE) criterion, ITAE, and integral of time by squared error (ITSE) [19], which are set as follows:

           (24)

Based on Eq. (24), the comparisons of PID and FOPID controller about displacement tracking error are shown in Table 4. It can be seen obviously that the SBW system with the FOPID controller holds more outstanding tracking property than that with classical PID controller. The delay of tracking curve is normal, which is caused by the large inertia characteristic of the SBW system.

Figure 7 Sinusoidal following curve of steering cylinder using optimized FOPID and PID controllers

Figure 8 Steering cylinder’s displacement tracking error of SBW system

Table 4 Comparisons of PID and FOPID controllers about displacement tracking error

5.3 Robustness test

An excellent controller must keep the control system stable in the presence of interference signal. Hence, the robustness of the controller should be verified. In this section, simulations are used to implement model uncertainties test, external disturbance test and random noise test to demonstrate the robustness of the proposed FOPID controller.

1) Model uncertainty test

For the model uncertainty test, we assume that the parameter K_m increases by 20% and K₁ reduces by 20%. Figure 9 presents the comparison of step responses under perturbed system parameters K_m and K₁, respectively. Clearly, the designed FOPID controller owns more perfect robustness than traditional PID controller on the part of model uncertainty. In addition, the hydraulic oil temperature rises because of long time work of the SBW system, which will increase the leakage coefficient C_L and decrease the elastic modulus of the liquid equivalent volume β_e. Assuming that the rising oil temperature causes that β_e decreases by 20% and C_L increases by 20%. Figure 10 shows the comparison of the robustness under the rising oil temperature. As can be seen from Figure 10, although the SBW system equipped with both controllers shows larger overshoot because of the rising oil temperature, the FOPID controller can respond more quickly to maintain system stability than the PID controller.

Figure 9 Comparison of robustness under perturbed system parameters:

Figure 10 Comparison of robustness under rising hydraulic oil temperature

2) External disturbance test

In this test, we assume that the SBW system receives a sudden external disturbance which is shown in Figure 11. Through simulation of the step response, Figure 12 shows the comparison of the robustness of designed FOPID controller and classical PID controller under the case of external disturbance. As can be observed from Figure 12, the designed FOPID controller shows the capability to reject the disturbance and maintain the stability of the SBW system, and it has better robustness than the traditional PID controller in terms of external disturbance.

Figure 11 External disturbances signal

Figure 12 Step responses of steering cylinder displacement of SBW system equipped with optimal FOPID controller and common PID controller under case of external disturbance

3) Random noise test

A white Gaussian random noise (shown in Figure 13) is generated and injected in the feedback loops to simulate the steering cylinder displacement sensor noise. Simulation result in Figure 14 shows that the proposed FOPID and PID controllers have great capability of suppressing noise.

As can be seen from the enlarged drawing in Figure 14, the FOPID controller can keep the steady-state error to zero ignoring the effect of sensor noise. Hence, the FOPID controller possesses more excellent robustness than common PID controller on the part of random sensor noise.

Figure 13 Noise signal of steering cylinder displacement sensor

Figure 14 Step responses of steering cylinder displacement using optimized FOPID and PID controllers under case of sensor noise

6 Conclusions

This paper designs a FOPID controller which is used to control the SBW system. The anti-windup problem is considered in the FOPID controller design to reduce the bad influence of saturation. Furthermore, GA is applied to optimize the FOPID controller parameters (proportional constant, integral constant, derivative constant, integral order and derivative order) to achieve the best control performance that minimizes the ITAE, the settling time and the peak overshoot while maintaining the steering cylinder in the normal operation limits. The PID controller is also tuned by GA to contrast the performance of the optimized FOPID controller. The simulations based on MATLAB are performed, and the simulation results indicate that the optimized FOPID controller has better transient response, tracking performance and robustness than the traditional PID controller.

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(Edited by YANG Hua)

中文导读

基于遗传算法的应急救援车辆线控转向系统的分数阶PID控制方法

摘要：针对传统应急救援车辆很难顺利通过狭小地域的救援现场，本文设计电液线控转向系统，实现应急救援车辆的多模式转向。首先，详细描述了电液线控转向系统的总体架构和数学模型。其次，设计基于分数阶微积分理论的分数阶PID控制器，控制线控转向系统中转向油缸的运动。进一步，在分数阶PID控制器设计中考虑抗饱和问题。然后，通过遗传算法最大化适应度函数，优化分数阶PID控制器的5个参数。该适应度函数包括时间乘绝对值误差积分，最大超调和调节时间。最后，通过时域仿真来判定分数阶PID控制器的性能。仿真结果表明，本文所提出的分数阶PID控制器在瞬态响应，跟踪能力和鲁棒性方面比传统PID控制器具有更好的控制特性。

关键词：线控转向系统；应急救援车辆；分数阶PID控制器；参数优化；遗传算法

Foundation item: Project(2016YFC0802904) supported by the National Key Research and Development Program of China

Received date: 2018-06-25; Accepted date: 2019-05-08

Corresponding author: CHEN Wei, PhD, Associate Professor; Tel: +86-18643077790; E-mail: chenwei_1979@jlu.edu.cn; ORCID: 0000-0001-9953-2629