# Computational Validation of the Best Tuning Method for a Vehicle-Integrated PID Controller

Authors: Calequela J. T. Manuel; Max M. D. Santos; Giane G. Lenzi; Angelo M. Tusset
Journal: Modelling and Simulation in Engineering
 (2022)

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## Abstract

This paper validates and analyzes the robustness of the proportional-integral-derivative (PID) action controller from an open transfer function that integrates a proportional-integral (PI) action controller to obtain the response of a robust action control during the automatic parking maneuver of a vehicle where the simulations are based on 3 adjustment methods: Ziegler-Nichols (ZN), Chien-Hrones-Reswick (CHR), and Cohen-Coon (CC), and as a result of the computer simulations, it is determined the best performance index of the PID controller represented by mathematical and graphic equations with the help of MATLAB/Simulink software.


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## Body

## 1. Introduction

The complexity of developing a robust control system [1, 2] that performs adequately when deployed to a vehicle that takes into account noises, disturbances, and adverse conditions [3–5] is a recurrent task in the automotive industry that is increasingly looking more for better solutions and innovations. For the specific problem of aiding the automatic parking maneuver, it is suitable to apply the PID controller that must follow the designated trajectory [6].


### 1.1. Proposed Problem

For the problem proposed in Figure1, the transfer function aggregated by the PI controller and the significant variables of the vehicle system are presented.Figure 1
Vehicle block diagram with a PID controller.Table1 describes the variables that comprise the transfer function of the vehicle system, which allows the extension to proportional-integral-derivative (PID) action controller validation.Table 1
Block diagram variables.
VariablesDescriptionsValuesUnitsmMass of the vehicle2.057kgRWheel radius0.3056mtauplantTime constant0.10sKplantStatic plant gain1—TdistTorque disturbance0.1NmVvehicleVehicle speed—m/savehicleVehicle acceleration—m/s2refReference——s/1+tDDFsFeedforward term——1/sIntegrative constant——etSystem error——arefReference acceleration—m/s2TrefReference torque—NmThe parameters (values) highlighted in Table1 are used in the computational simulations [7, 8].


### 1.2. PID Controller Fundamentals

The industrial proportional-integral-derivative (PID) action controller is a control loop feedback technique widely used in control systems [9]. Feedback control systems [10] are a widely deployed strategy to control the process variable by comparing it to a desired value and applying the resulting difference as an error signal to realize the control output in order to reduce or eliminate the error [11, 12].The proportional-integral-derivative (PID) controller generates its output proportional to the error (P), proportional to the integral of the error (I, integral term), and proportional to the derivative of the error (D, derivative term), considering the position of the parallel PID controller classic [6, 9, 12]. Equation (1) of the algorithm is described below:
(1)ut=KPet+KP1TI∫etdt+KPTDddtet.The block diagram characteristics of the PID controller [12, 13] are represented in Figure 2.Figure 2
PID controller features.The proportional gain also multiplies the integral and derivative terms; the multiplicative factor (TD) is the time derived from the controller, and the derivative action has the function of anticipating the error resulting from the action of the proportional-integral controller in order to eliminate this potential error that is expected and guarantee the stability of the system in less time [6, 11].


## 1.1. Proposed Problem

For the problem proposed in Figure1, the transfer function aggregated by the PI controller and the significant variables of the vehicle system are presented.Figure 1
Vehicle block diagram with a PID controller.Table1 describes the variables that comprise the transfer function of the vehicle system, which allows the extension to proportional-integral-derivative (PID) action controller validation.Table 1
Block diagram variables.
VariablesDescriptionsValuesUnitsmMass of the vehicle2.057kgRWheel radius0.3056mtauplantTime constant0.10sKplantStatic plant gain1—TdistTorque disturbance0.1NmVvehicleVehicle speed—m/savehicleVehicle acceleration—m/s2refReference——s/1+tDDFsFeedforward term——1/sIntegrative constant——etSystem error——arefReference acceleration—m/s2TrefReference torque—NmThe parameters (values) highlighted in Table1 are used in the computational simulations [7, 8].


## 1.2. PID Controller Fundamentals

The industrial proportional-integral-derivative (PID) action controller is a control loop feedback technique widely used in control systems [9]. Feedback control systems [10] are a widely deployed strategy to control the process variable by comparing it to a desired value and applying the resulting difference as an error signal to realize the control output in order to reduce or eliminate the error [11, 12].The proportional-integral-derivative (PID) controller generates its output proportional to the error (P), proportional to the integral of the error (I, integral term), and proportional to the derivative of the error (D, derivative term), considering the position of the parallel PID controller classic [6, 9, 12]. Equation (1) of the algorithm is described below:
(1)ut=KPet+KP1TI∫etdt+KPTDddtet.The block diagram characteristics of the PID controller [12, 13] are represented in Figure 2.Figure 2
PID controller features.The proportional gain also multiplies the integral and derivative terms; the multiplicative factor (TD) is the time derived from the controller, and the derivative action has the function of anticipating the error resulting from the action of the proportional-integral controller in order to eliminate this potential error that is expected and guarantee the stability of the system in less time [6, 11].


## 2. Mathematical Modeling

The 3 tuning methods determined to calibrate the PID controller are aimed at evaluating the best system performance index based on the response from the identification of the process frequency (reaction curve).


### 2.1. Ziegler-Nichols Method

The adjustment method for the PID controller proposed by Ziegler and Nichols [14] is mainly used in industries. This is a manual mode process based on an open-loop step response from a system, and it is indicated for a first-order plant model plus the dead time represented by
(2)Gs=k1+sTe−sL,where L is the delay time, T is the time constant, and K stands for the static gain of the controller. If the output of the step response is measured through an experiment, then the parameters K, L, and T (or a, where a=KL/T) can be extracted from the reaction curve graph. The calculation of the PID controller is based on the results of the reaction curve proposed by Ziegler-Nichols [14, 15], as shown in Table 2.Table 2
Controller parameters for the Ziegler-Nichols method.
Controller typeKPTITDPTKL∞0PI0.9TKLL0.30PID1.2TKL2L0.5LThe values ofKP, TI, and TD are 1.2/a, 2L, and L/2, respectively.


### 2.2. Chien-Hrones-Reswick Method

Derived from the original Ziegler-Nichols open-loop method, the Chien-Hrones-Reswick (CHR) method gives a faster response without overshoots and also with 20% [16, 17]; the adjustment parameters are based on Table 3.Table 3
Controller parameters for the Chien-Hrones-Reswick method.
Controller typeWith 0% overshootWith 20% overshootKPTITDKPTITDPID0.6aT0.5L0.95a1.4T0.47LThis method has the ability to adjust the set point and the disturbance. To tune the controller according to the CHR method,L and T parameters must be set [16].


### 2.3. Cohen-Coon Method

The Cohen-Coon (CC) tuning technique [18] of the PID controller is based on the Ziegler-Nichols method. The Ziegler-Nichols method presents a slow response at a steady state, whereas the Cohen-Coon tuning method can overcome this limitation in the parameter values for adjusting the PID controller [18, 19] based on Table 4.Table 4
Controller parameters for the Cohen-Coon method.
Controller typeKPTITDPID1.35TKL1+0.18τ1−τ2.5−2τ1−0.39τL0.37−0.37τ1−0.81τLThis method takes the parameters of the PID controller and tries the open-loop transfer function. If there is a long delay in the process in relation to the open-loop time constant, it presents a better answer about the Ziegler-Nichols method, and the three initial adjustment parameters are as follows: steady output state (K), effective time constant of the first-order response (s), and dead time (t) [20, 21].


## 2.1. Ziegler-Nichols Method

The adjustment method for the PID controller proposed by Ziegler and Nichols [14] is mainly used in industries. This is a manual mode process based on an open-loop step response from a system, and it is indicated for a first-order plant model plus the dead time represented by
(2)Gs=k1+sTe−sL,where L is the delay time, T is the time constant, and K stands for the static gain of the controller. If the output of the step response is measured through an experiment, then the parameters K, L, and T (or a, where a=KL/T) can be extracted from the reaction curve graph. The calculation of the PID controller is based on the results of the reaction curve proposed by Ziegler-Nichols [14, 15], as shown in Table 2.Table 2
Controller parameters for the Ziegler-Nichols method.
Controller typeKPTITDPTKL∞0PI0.9TKLL0.30PID1.2TKL2L0.5LThe values ofKP, TI, and TD are 1.2/a, 2L, and L/2, respectively.


## 2.2. Chien-Hrones-Reswick Method

Derived from the original Ziegler-Nichols open-loop method, the Chien-Hrones-Reswick (CHR) method gives a faster response without overshoots and also with 20% [16, 17]; the adjustment parameters are based on Table 3.Table 3
Controller parameters for the Chien-Hrones-Reswick method.
Controller typeWith 0% overshootWith 20% overshootKPTITDKPTITDPID0.6aT0.5L0.95a1.4T0.47LThis method has the ability to adjust the set point and the disturbance. To tune the controller according to the CHR method,L and T parameters must be set [16].


## 2.3. Cohen-Coon Method

The Cohen-Coon (CC) tuning technique [18] of the PID controller is based on the Ziegler-Nichols method. The Ziegler-Nichols method presents a slow response at a steady state, whereas the Cohen-Coon tuning method can overcome this limitation in the parameter values for adjusting the PID controller [18, 19] based on Table 4.Table 4
Controller parameters for the Cohen-Coon method.
Controller typeKPTITDPID1.35TKL1+0.18τ1−τ2.5−2τ1−0.39τL0.37−0.37τ1−0.81τLThis method takes the parameters of the PID controller and tries the open-loop transfer function. If there is a long delay in the process in relation to the open-loop time constant, it presents a better answer about the Ziegler-Nichols method, and the three initial adjustment parameters are as follows: steady output state (K), effective time constant of the first-order response (s), and dead time (t) [20, 21].


## 3. Experimental Data Used in PI Controller Simulation

From the block diagram proposed in Figure1, the exit speed and acceleration of the vehicle are estimated, and an approach and simplification for the study of the transfer function associated with the PI controller [22] responsible for controlling the plant of a given vehicle are determined. The plant’s open-loop [23] acceleration transfer function is referred to as Gs, according to
(3)Gs=mR+TdistRmKplant1+stauplant.Using the MATLAB software [24] and replacing the values in Table 1 in the plant transfer function, the resolution of the transfer function description in Equation (4) is presented:
(4)Gs=1.0030.1s+1.The current simulator is controlled by a proportional-integral (PI) action designed with minimum proportional gain (KPmini=2) and minimum integrative gain (TImini=1) and maximum proportional gain KPmax=2 and maximum integrative gain (TImax=3). The plant operates with minimum gain (Kplant=1) and maximum gain (Kplant=3). The open-loop acceleration transfer function [25] after the step signal is applied has the following behavior, as shown in Figure 3.Figure 3
Reaction curve oraref tangent line in open-loop PI (1).The first step was to consider all the minimum values of the plant in the open loop and the PI controller gain, as shown in Figure3.The resolution was presented by MATLAB [26], where the PI controller obtained the following transfer function, according to
(5)GPs=2s+2s.The output variables do not track input variables; however, the system is not controlled by the proposed proportional-integral (PI) controller output that does not have the desired gain. When considering a closed loop, the acceleration and velocity [27] responses are, respectively, shown in Figure 4.Figure 4
Vehicle acceleration (a) and velocity (b) responses, respectively, in PI (1).
(a)(b)The acceleration (Figure4(a)) shows a fast output signal in relation to time, but at no time is it stable. This directly influences the behavior of the velocity signal (Figure 4(b)); therefore, it does not guarantee any stability in the system.The second step was to consider all the maximum values of the gain of the plant and the PI controller: in this case, the open-loop transfer function [28] is represented by
(6)Gs=3.0030.1s+1.The open-loop acceleration transfer function after the step signal [29] is applied has the following behavior, as shown in Figure 5.Figure 5
Reaction curve oraref tangent line in open-loop PI (3).The resolution was presented by MATLAB, where the PI controller [30] obtained the following transfer function, according to
(7)GPs=6s+23s.When considering a closed loop, the acceleration and velocity responses are, respectively, shown in Figure6.Figure 6
Vehicle acceleration (a) and speed (b) responses, respectively, in PI (3).
(a)(b)The output variables do not track input variables; however, the system is not controlled by the proposed proportional-integral (PI) controller output that does not have the desired gain.


### 3.1. Considerations

In this case, the output of the closed-loop PI controller is similar to that of the open-loop system response for the minimum and maximum gains of the controller, as shown in Figures3–6; the acceleration (Figure 6(a)) does not present an adequate output, with sudden changes in relation to time. The speed (Figure 6(b)) is not constant in relation to the estimated time. Both variables do not perform satisfactorily; this reflects the lack of control of the system.


## 3.1. Considerations

In this case, the output of the closed-loop PI controller is similar to that of the open-loop system response for the minimum and maximum gains of the controller, as shown in Figures3–6; the acceleration (Figure 6(a)) does not present an adequate output, with sudden changes in relation to time. The speed (Figure 6(b)) is not constant in relation to the estimated time. Both variables do not perform satisfactorily; this reflects the lack of control of the system.


## 4. Results: Experimental Data Used in PID Controller Simulation

For proper control of the vehicle plant, the system must be considered a closed loop; the PID controller [31] calculation is based on Table 2 proposed by Ziegler-Nichols (ZN), where the PID controller gains are found, respectively.


### 4.1. Minimum Plant Gain (ZN)

From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 2 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(8)GPs=0.001157s2+0.2512s+13.640.01842s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure7.Figure 7
Reaction curve oraref tangent line in closed-loop PID-ZN (1).The output of the PID controller (reference acceleration) starts to be identified from the amplitude (gainK=0,5). At gain K=1, the controller output has an overshoot and a low stationary error, and at time 0.1 s, the system starts to stabilize. The acceleration response and vehicle speed, respectively, are shown in Figure 8.Figure 8
Acceleration (a) and vehicle velocity (b), respectively, in closed-loop PID-ZN (1).
(a)(b)Compared to the response of the PI controller, the results of the PID controller are better and guarantee stability in the system.


### 4.2. Maximum Plant Gain (ZN)

From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 2 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(9)GPs=0.0003855s2+0.08374s+4.5470.01842s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure9.Figure 9
Reaction curve oraref tangent line in closed-loop PID-ZN (3).The PID controller output signal (reference acceleration) does not reach the desired gain (K=3) to stabilize the system; in this case, it produces a large stationary error [32, 33] and consequently instability in the system. The behavior of the acceleration response and vehicle speed, respectively, is shown in Figure 10.Figure 10
Acceleration (a) and vehicle speed (b), respectively, in closed-loop PID-ZN (3).
(a)(b)The acceleration output signal initially presents an overshoot during the time of 0.1 s and an acceleration of 1.1 m/s2 and a low stationary error; then, the system is controlled [34]. The speed output signal relates to time, and each second tends to increase.


### 4.3. Minimum Plant Gain (CHR)

The PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively, to be no response overshoot (0%); from Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(10)GPs=0.003287s2+0.7139s+6.8210.1047s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure11.Figure 11
Reaction curve oraref tangent line in closed-loop PID-CHR (1).The PID controller does not track input variables, which makes the system slow and uncontrolled; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure12.Figure 12
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (1).
(a)(b)The acceleration and velocity apparently have an adequate output, but in this case, the controller takes a long time to respond and this can cause instability in the system.With 20% of the overshoot in response, the PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively. From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(11)GPs=0.006849s2+1.583s+10.80.1465s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure13.Figure 13
Reaction curve oraref tangent line in closed-loop PID-CHR (20%) (1).The output signal from the PID controller (reference acceleration) reaches the recommended gain; in this case, it does not produce a large stationary error and consequently stabilizes the system from the time of 0.28 s; the behavior of the acceleration response and vehicle velocity, respectively, is shown in Figure14.Figure 14
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (20%) (1).
(a)(b)Acceleration response and velocity perform well, and this is due to the stable control of the system.


### 4.4. Maximum Plant Gain (CHR)

The PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively, to be no response overshoot (0%); from Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(12)GPs=0.001096s2+0.238s+2.2740.1047s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure15.Figure 15
Reaction curve oraref tangent line in closed-loop PID-CHR (3).The system was not controlled due to the fact that the PID controller output does not track its input in this sampling time; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure16.Figure 16
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (3).
(a)(b)The acceleration and velocity apparently have an adequate output, but in this case, the controller takes a long time to respond and this can cause instability in the system.With 20% of the overshoot in response, the PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively. From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(13)GPs=0.002283s2+0.5275s+3.60.1465s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure17.Figure 17
Reaction curve oraref tangent line in closed-loop PID-CHR (20%) (3).The PID controller (reference acceleration) does not reach the recommended gain, and it creates an interval between the input and output variables, which produces a large stationary error at all times; the system is not stable. The behavior of the acceleration response and vehicle velocity, respectively, is shown in Figure18.Figure 18
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (20%) (3).
(a)(b)The system is not controlled; the controller takes a long time to respond. The acceleration and speed apparently have an adequate output.


### 4.5. Minimum Plant Gain (CC)

The PID controller calculation is based on Table4 proposed by Cohen-Coon (CC), where the PID controller gains are found, respectively. From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 4 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(14)GPs=0.001161s2+0.3466s+15.590.02223s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure19.Figure 19
Reaction curve oraref tangent line in closed-loop PID-CC (1).The system was not controlled because the PID controller does not track its input in this sampling time; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure20.Figure 20
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CC (1).
(a)(b)For this sampling time, the PID controller response is slow, and this causes instability in the system.


### 4.6. Maximum Plant Gain (CC)

From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 4 with the help of MATLAB software; the values of L, tau, and T are obtained; the PID controller is represented by
(15)GPs=0.0003871s2+0.1155s+5.1960.02223s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure21.Figure 21
Reaction curve oraref tangent line in closed-loop PID-CC (3).The system was not controlled because the PID controller output does not track its input in this sampling time. The behavior of the acceleration response and vehicle speed, respectively, is shown in Figure22.Figure 22
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CC (3).
(a)(b)Apparently, the acceleration response and vehicle speed are great, and it is notable that the controller output takes a long time to respond (Figure21), which makes the system uncontrollable.


## 4.1. Minimum Plant Gain (ZN)

From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 2 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(8)GPs=0.001157s2+0.2512s+13.640.01842s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure7.Figure 7
Reaction curve oraref tangent line in closed-loop PID-ZN (1).The output of the PID controller (reference acceleration) starts to be identified from the amplitude (gainK=0,5). At gain K=1, the controller output has an overshoot and a low stationary error, and at time 0.1 s, the system starts to stabilize. The acceleration response and vehicle speed, respectively, are shown in Figure 8.Figure 8
Acceleration (a) and vehicle velocity (b), respectively, in closed-loop PID-ZN (1).
(a)(b)Compared to the response of the PI controller, the results of the PID controller are better and guarantee stability in the system.


## 4.2. Maximum Plant Gain (ZN)

From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 2 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(9)GPs=0.0003855s2+0.08374s+4.5470.01842s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure9.Figure 9
Reaction curve oraref tangent line in closed-loop PID-ZN (3).The PID controller output signal (reference acceleration) does not reach the desired gain (K=3) to stabilize the system; in this case, it produces a large stationary error [32, 33] and consequently instability in the system. The behavior of the acceleration response and vehicle speed, respectively, is shown in Figure 10.Figure 10
Acceleration (a) and vehicle speed (b), respectively, in closed-loop PID-ZN (3).
(a)(b)The acceleration output signal initially presents an overshoot during the time of 0.1 s and an acceleration of 1.1 m/s2 and a low stationary error; then, the system is controlled [34]. The speed output signal relates to time, and each second tends to increase.


## 4.3. Minimum Plant Gain (CHR)

The PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively, to be no response overshoot (0%); from Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(10)GPs=0.003287s2+0.7139s+6.8210.1047s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure11.Figure 11
Reaction curve oraref tangent line in closed-loop PID-CHR (1).The PID controller does not track input variables, which makes the system slow and uncontrolled; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure12.Figure 12
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (1).
(a)(b)The acceleration and velocity apparently have an adequate output, but in this case, the controller takes a long time to respond and this can cause instability in the system.With 20% of the overshoot in response, the PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively. From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(11)GPs=0.006849s2+1.583s+10.80.1465s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure13.Figure 13
Reaction curve oraref tangent line in closed-loop PID-CHR (20%) (1).The output signal from the PID controller (reference acceleration) reaches the recommended gain; in this case, it does not produce a large stationary error and consequently stabilizes the system from the time of 0.28 s; the behavior of the acceleration response and vehicle velocity, respectively, is shown in Figure14.Figure 14
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (20%) (1).
(a)(b)Acceleration response and velocity perform well, and this is due to the stable control of the system.


## 4.4. Maximum Plant Gain (CHR)

The PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively, to be no response overshoot (0%); from Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(12)GPs=0.001096s2+0.238s+2.2740.1047s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure15.Figure 15
Reaction curve oraref tangent line in closed-loop PID-CHR (3).The system was not controlled due to the fact that the PID controller output does not track its input in this sampling time; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure16.Figure 16
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (3).
(a)(b)The acceleration and velocity apparently have an adequate output, but in this case, the controller takes a long time to respond and this can cause instability in the system.With 20% of the overshoot in response, the PID controller calculation is based on Table3 proposed by Chien-Hrones-Reswick (CHR), where the PID controller gains are found, respectively. From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 3 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(13)GPs=0.002283s2+0.5275s+3.60.1465s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure17.Figure 17
Reaction curve oraref tangent line in closed-loop PID-CHR (20%) (3).The PID controller (reference acceleration) does not reach the recommended gain, and it creates an interval between the input and output variables, which produces a large stationary error at all times; the system is not stable. The behavior of the acceleration response and vehicle velocity, respectively, is shown in Figure18.Figure 18
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CHR (20%) (3).
(a)(b)The system is not controlled; the controller takes a long time to respond. The acceleration and speed apparently have an adequate output.


## 4.5. Minimum Plant Gain (CC)

The PID controller calculation is based on Table4 proposed by Cohen-Coon (CC), where the PID controller gains are found, respectively. From Equation (3), the minimum plant gain Kplant=1 was replaced together with the parameters of Tables 1 and 4 with the help of MATLAB software; the values of L, tau, and T are obtained; however, the PID controller is represented by
(14)GPs=0.001161s2+0.3466s+15.590.02223s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure19.Figure 19
Reaction curve oraref tangent line in closed-loop PID-CC (1).The system was not controlled because the PID controller does not track its input in this sampling time; the behavior of the acceleration response and vehicle speed, respectively, is shown in Figure20.Figure 20
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CC (1).
(a)(b)For this sampling time, the PID controller response is slow, and this causes instability in the system.


## 4.6. Maximum Plant Gain (CC)

From Equation (3), the maximum plant gain Kplant=3 was replaced together with the parameters of Tables 1 and 4 with the help of MATLAB software; the values of L, tau, and T are obtained; the PID controller is represented by
(15)GPs=0.0003871s2+0.1155s+5.1960.02223s.The PID controller output can be evaluated by the reaction curve method; the system is excited by a step input signal, as shown in Figure21.Figure 21
Reaction curve oraref tangent line in closed-loop PID-CC (3).The system was not controlled because the PID controller output does not track its input in this sampling time. The behavior of the acceleration response and vehicle speed, respectively, is shown in Figure22.Figure 22
Vehicle acceleration (a) and speed (b) responses, respectively, in PID-CC (3).
(a)(b)Apparently, the acceleration response and vehicle speed are great, and it is notable that the controller output takes a long time to respond (Figure21), which makes the system uncontrollable.


## 5. Performance Index of the PID Controller

The Ziegler-Nichols and Chien-Hrones-Reswick (20%) tuning methods are the most indicated to be used in helping maneuver the vehicle system.Some performance indexes, such as overshoot (MO), settling time (ts), and rise time (tr), can evaluate the performance of the controller in the system. Table 5 shows the performance of the tuning method of the PID controller based on the results of the simulations already highlighted.Table 5
Transient response to a single-step entry.
PID controllerOvershootMO (%)Settling timets (s)Rise timetr (s)Peak amplitudeZiegler-Nichols11.1000.0730.0151.110Chien-Hrones-Reswick (20%)00.1280.0350.999These values were extracted from the PID controller response of Figure7 (Ziegler-Nichols) and Figure 13 (Chien-Hrones-Reswick (20%)), both responses with minimum plant gain, that is, Kplant=1. The performance index of the Ziegler-Nichols tuning method has an overshoot, a shorter settling time and a shorter rise time, and a greater amplitude than the desired amplitude it is considered. The performance index of the Chien-Hrones-Reswick (20%) tuning method has the opposite response, that is, desired amplitude, no overshoot, and larger settling and larger rise time. Both answers are acceptable because the PID controller showed stability even with time variation.


## 6. Discussion and Final Considerations

PI controller does not recognize its output variables to estimate its input values in order to have proper control of the vehicle system during the parking maneuver; the proposed PID controller guarantees stability in the system and classifies the operation action as robust according to the simulations presented, highlighting the 2 best tuning methods (ZN and CHR (20%)), both with minimal system gains (ZN method: aggressive tuning method but with tune adjustments, better control of the system achieved, and small values of settling time; CHR method (20% over value): faster system response with 20% over value and higher value requirement of settling time). However, the PID controller is recommended to control the parking maneuver assistance system integrated into the vehicle.



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*Source: 3873639-2022-06-25.xml*