# A Hybrid Control Algorithm Fuzzy-PI with the Second Derivative of the Error Signal for an Active Suspension System

Authors: Duc Ngoc Nguyen; Tuan Anh Nguyen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A vehicle’s oscillation can be improved by using an active suspension system to replace the conventional passive suspension system. The active suspension system operates on the control signal from the controller. This paper introduces the Fuzzy-PI integrated control algorithm to control the system’s efficiency. The Fuzzy algorithm adjusts theKP and KI parameters of the controller. This response depends on the oscillation state of the vehicle. A quarter model with a hydraulic actuator was used in this research. The simulation process takes place in the MATLAB-Simulink environment. In the first case, the maximum and average values of displacement and acceleration of the sprung mass when the vehicle uses the Fuzzy-PI algorithm are only 2.68%, 1.34%, and 1.65%, 0.66%, respectively, compared to the situation of vehicles using the passive suspension system. For the second case, these values are 3.33%, 1.69%, and 2.01%, 0.73%, respectively. As a result, the vehicle’s stability and comfort are greatly enhanced when the integrated controller is used. Besides, the change of KP and KI parameters is also completely suitable for the vehicle’s driving conditions. This research only leads to calculations and simulations; the experimental process is expected to occur soon.


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## Body

## 1. Introduction

The vehicle’s oscillation is regulated by the suspension system. Over the years, the suspension system has been studied and improved a lot. Popular automobiles only use the passive suspension system, also known as the conventional mechanical suspension system. According to [1], the passive suspension system consists of springs and dampers. Besides, the lever arm and stabilizer bar are also considered part of the suspension system [2]. The passive suspension system has a simple structure; however, its oscillation stabilization effect is not high. Therefore, the passive suspension system should be replaced by a more modern system.Recently, many studies on suspension systems have been published. The idea of using air suspension on the vehicle has been proposed by many authors [3–5]. In [6], Gao et al. gave views on the air suspension system. According to them, the stiffness of the pneumatic spring can be changed continuously by varying the pressure inside the air balloon. To improve the efficiency of the air suspension system, Nguyen proposed using hydraulic actuators in combination with air balloons [7]. Therefore, the efficiency of the system is further improved. However, the structure of this suspension system is quite complicated and cumbersome. In [8], Ming et al. introduced using a semi-active suspension system. According to Wei et al., a semi-active suspension system uses electromagnetic dampers instead of conventional passive dampers. When current is supplied to the inside of the damper, a magnetic force will appear. This force changes the arrangement of the iron particles inside the damper. From there, the damping viscosity can be changed [9]. Changing the stiffness of the shock absorbers has not brought high efficiency to the suspension system. Therefore, the active suspension system was used to replace the semi-active suspension system [10]. The active suspension is equipped with a hydraulic actuator. According to Konieczny, this mechanism works based on the opening and closing of the solenoid valves [11]. These valves are controlled based on the current signal supplied by the controller. The efficacy of the active suspension system is very high [12].Anh introduced the PID control algorithm for the active suspension system in [13]. This algorithm has only an input signal and an output signal. Therefore, this algorithm is only suitable for SISO systems. If the integral stage is considered ideal, the PID controller becomes a PD controller. Conversely, if the derivate stage is considered ideal, this controller becomes PI [14]. For systems with many degrees of freedom, the MOPID control algorithm is used to respond to the output signal vectors [15]. The quality of the PID controller depends on the selection of the coefficients KP, KI, and KD. In [16], Al-Mutar and Abdalla used the PSO algorithm to optimize these parameters. Dahunsi et al. introduced Global and Evolutionary Optimization (GEO) techniques in [17]. This algorithm uses eight PID controllers with two in-loops corresponding to four wheels. The techniques used in this paper are quite complex. Besides, the GA algorithm also helps optimize these parameters with high efficiency. This method was presented by Nagarkar et al. [18]. Although the parameters of the PID controller can be optimally calculated through the methods mentioned above, their values are fixed in oscillation situations. Therefore, the stability of the system is still not high.Because the object’s response is a nonlinear function of the excitation; therefore, nonlinear control and intelligent control methods should be used. In [19], Chu et al. proposed using the SMC algorithms for the active suspension system. According to Alves et al., this algorithm must design a suitable sliding surface [20]. From there, the controlled object will move along the sliding surface to return to equilibrium. According to Bai and Guo, the slip control signal consists of two parts: the equivalent control signal and the switching control signal [21]. These signals can be determined based on taking the derivative of the state variables. When using the SMC controller, the “chattering” phenomenon may occur. This phenomenon can cause the control signal to fluctuate continuously [22]. This phenomenon can be reduced by using some new control algorithms presented by Shao et al. [23]. The SMC control algorithm helps the system operate stably in many oscillation situations. This has been demonstrated by Nguyen [24, 25]. Since the system oscillates continuously, the oscillating state of the system will also fluctuate continuously. Therefore, it is necessary to define an intermediate state for the system. In this case, the use of the Fuzzy algorithm is extremely suitable [26]. For this algorithm, the control signal changes continuously based on the input parameters of the controller [27]. The performance of the Fuzzy controller depends on the membership function and the defuzzification rule. This rule largely depends on the knowledge and experience of the designer [28]. In [29], Chao et al. used the Cuckoo optimization algorithm to select the membership function for the Fuzzy controller. The response of the Fuzzy controller is always high, so the system’s stability has been greatly improved [30–32].The use of hybrid controllers or integrated algorithms has recently become a new trend. In [33], Heidari and Homaei combined the PID algorithm and the Neural Network. The artificial neurons were used to predict the parameters KP, KD, and KI accordingly. Similarly, the BPNN algorithm is also used in the paper by Liu et al. [34]. Besides, combining the SMC and Fuzzy algorithms reduces the “chattering” phenomenon [35]. The Fuzzy controller helps continuously adjust the SMC controller’s parameters, corresponding to the vehicle’s moving conditions [36, 37]. Based on these ideas, the method of using the Fuzzy controller to adjust the parameters of the PID controller was born [38]. In [39], Khodadadi and Ghadiri used this hybrid control method. Accordingly, the input values of the controller include displacement and acceleration of the system. These signals are derivatives once before becoming the input signals of the Fuzzy controller. Three membership functions correspond to the three output signals of the PID controller. In addition, the FOPID algorithm can also be combined with the Fuzzy algorithm to control the parameters optimally. In [40], Kumar et al. used the Fuzzy algorithm to adjust the parameters of the FOPID controller. It includes six parameters, which can be adjusted. This association can be further enhanced when the Gravitational Search Algorithm is combined with the Fuzzy-PID algorithm [41]. When using the Fuzzy controller to adjust the parameters of the PID controller, the system margin is stable. This is demonstrated through simulation and experimentation by Liu et al. [42]. Unlike the conventional PID controller, which has only a single input, the Fuzzy-PID controller can have multiple inputs. These inputs are acquired by the Fuzzy algorithm [43]. To improve the efficiency of the hybrid control algorithm in all situations, the ANFIS-PID algorithm has been proposed [44]. In addition, many intelligent algorithms are used to control other mechanical systems. The results of these studies are quite good [45–49].Based on the above-mentioned reviews, this paper focuses on the design of the Fuzzy-PI (FPI) hybrid control method for the active suspension system. The parameters of the PI controller will be continuously adjusted based on the Fuzzy algorithm. This change can meet the vehicle's oscillation conditions. In contrast to previous studies, this paper uses a new model with five state variables. Moreover, the hydraulic actuator of the active suspension system is also included in this model. A new point of the paper is using the second derivative for the input signal of the Fuzzy controller. These contents are considered the main contributions of the paper. The paper structure consists of four sections: in the first section, the control studies of the active suspension system are introduced. In the next section, the dynamic model and control algorithm are designed. In the third section, the simulation is performed. This process occurs in the MATLAB-Simulink environment with specific simulation cases. In the last section, some conclusions and recommendations are made.


## 2. Control Model

In this research, a quarter-dynamic model was used (Figure1). This model includes two masses: the sprung mass m1 and the unsprung mass m2. Corresponding to these two masses are two degrees of freedom, z1, and z2. The vehicle's oscillation is caused by the road surface roughness r(t) [50].Figure 1
Model of the vehicle’s dynamics.The system of equations describing the vibration of the vehicle is described as follows:(1)m1z¨1=FK+FC+FA,m2z¨2=FKT−FK−FC−FA.When the vehicle’s body oscillates, the hydraulic actuator will operate. It produces a forceFA acting on both the sprung mass and the unsprung mass. Based on [25], the linearization equation describing the actuator principle is given as follows: (2)F˙A=γ1it−γ2FA+γ3z˙2−z˙1,where,(3)FK=Kz2−z1,FC=Cz˙2−z˙1,FKT=KTr−z2.Let(4)x1=z1,x2=z˙1,x3=z2,x4=z˙2,x5=FA,taking derivatives of state variables(5)x˙1=x2,x˙2=1m1−Kx1−Cx2+Kx3+Cx4+x5,x˙3=x4x˙4=1m2Kx1+Cx2−K+KTx3−Cx4−x5,x˙5=γ1it−γ3x2+γ3x4−γ2x5.The above system of equations can be rewritten in matrix form as follows:(6)x˙=A1x+A2it+A3rt,where, A1 is the matrix of oscillation states, A2 is the matrix of the control signal, and A3 is the matrix of the disturbance signal.(7)A1=01000−Km1−Cm1Km1Cm11m100010Km2Cm2−K+KTm2−Cm2−1m20−γ30γ3−γ2,A2=0000γ1T,A3=000KTm20T.Lety(t) be the output signal of the control model, and ys(t) be the setpoint signal. The difference between these two signals is the error signal e(t).(8)et=yst−yt.If the system is assumed to be SISO, the PID control method is most suitable. Based on the error signale(t), this controller will provide the corresponding output current signal. The I/O model of the PID controller is described as follows:(9)it=KPet+1TI∫0teτdt+TDdetdt.If the derivative stage of the system is ideal, theTD coefficient is assumed to be zero. Therefore, the transfer function of the controller is rewritten more simply.(10)Rs=KP+KIs.The performance of the controller depends on the selection of the coefficientsKP and KI. For traditional controllers, these coefficients will not change. Therefore, the stability of the system in different cases cannot be guaranteed. It is necessary to define an intermediate state. Besides, these coefficients need to change continuously to meet the requirements of the system. Based on the fuzzy theory [51, 52], the Fuzzy-PI integrated control algorithm is proposed. The schematic diagram of this model is shown in Figure 2.Figure 2
Control system.The error signale(t) of the system is derived twice, consecutively. This is the input signal of the Fuzzy controller. Based on the change of this signal, the Fuzzy controller will effectively control the KP and KI coefficients. The algorithm describing the integrated controller is shown through the following equation:(11)it=defuzz1d2etdt2et+defuzz2d2etdt2∫0teτdτ.The membership function describing the Fuzzy control rule is introduced in Figure3. This rule can be determined based on the researcher’s point of view and experience. This is a significant technical challenge when designing a controller. After the controller has been designed, the simulation will take place. This content is covered in the next section of the paper.Figure 3
Fuzzy membership function.


## 3. Results and Discussions

### 3.1. Simulation Conditions

Vehicle oscillations are generated mainly from pavement stimuli. There are many types of road surface stimuli that a vehicle will encounter in practice. In this research, pulsed stimulation was used. Unlike continuous stimuli such as sine or random stimuli, pulsed stimuli are present only for a short time. This stimulus can be used to assess stability after the excitation signal has ended. There are two types of pulses (single pulse and double pulse) corresponding to the two simulation cases. The amplitude of the pulse in the two cases is different; however, the width of the pulse is the same (Figure4).Figure 4
Roughness on the road.The simulation is performed in the MATLAB-Simulink environment. The results of the simulation include the maximum and average values of displacement and acceleration of the sprung mass. These values are determined according to (12) and (13). Besides, the change of two coefficients, KP and KI, is also shown as the result of the paper. For each case, three situations are presented. For the first situation, the vehicle will use the active suspension with the Fuzzy-PI controller. In the second situation, the passive suspension system is equipped with the vehicle. In contrast, in other situations, the vehicle only uses the active suspension system with the traditional PI controller.(12)ymax=Maxyi,(13)yRMS=1n∑i=1nyi2.Reference parameters of the vehicle are given in Table1.Table 1
Specification of the vehicle.
DescriptionSymbolValueUnitSprung massm1520kgUnsprung massm253kgSpring coefficientK44800N/mDamper coefficientC3950Ns/mTire coefficientKT175000N/mActuator coefficientγ1539561N3/2/kg1/2 m1/2VActuator coefficientγ21s−1Actuator coefficientγ35512500N/m


## 3.1. Simulation Conditions

Vehicle oscillations are generated mainly from pavement stimuli. There are many types of road surface stimuli that a vehicle will encounter in practice. In this research, pulsed stimulation was used. Unlike continuous stimuli such as sine or random stimuli, pulsed stimuli are present only for a short time. This stimulus can be used to assess stability after the excitation signal has ended. There are two types of pulses (single pulse and double pulse) corresponding to the two simulation cases. The amplitude of the pulse in the two cases is different; however, the width of the pulse is the same (Figure4).Figure 4
Roughness on the road.The simulation is performed in the MATLAB-Simulink environment. The results of the simulation include the maximum and average values of displacement and acceleration of the sprung mass. These values are determined according to (12) and (13). Besides, the change of two coefficients, KP and KI, is also shown as the result of the paper. For each case, three situations are presented. For the first situation, the vehicle will use the active suspension with the Fuzzy-PI controller. In the second situation, the passive suspension system is equipped with the vehicle. In contrast, in other situations, the vehicle only uses the active suspension system with the traditional PI controller.(12)ymax=Maxyi,(13)yRMS=1n∑i=1nyi2.Reference parameters of the vehicle are given in Table1.Table 1
Specification of the vehicle.
DescriptionSymbolValueUnitSprung massm1520kgUnsprung massm253kgSpring coefficientK44800N/mDamper coefficientC3950Ns/mTire coefficientKT175000N/mActuator coefficientγ1539561N3/2/kg1/2 m1/2VActuator coefficientγ21s−1Actuator coefficientγ35512500N/m


## 4. Results

The results of the simulation process are shown in two cases below.Case 1.
In the first case, the excitation from the pavement takes the form of a single pulse. The displacement of the sprung mass concerning time is shown in Figure5. According to this graph, the maximum displacement of the vehicle body will be achieved when the vehicle uses only a passive suspension system. Its maximum value can be up to 151.42 (mm). When the stimulus ended abruptly, the sprung mass continued to decrease to a value of –51.03 (mm). This oscillation process takes place for more than 2 (s) and then ends. The average value of displacement, in this case, calculated in terms of RMS, is 76.87 (mm). The excitation from the pavement only lasts for 2 (s), while the average value is calculated for 10 (s). Therefore, the average displacement value in the situation of vehicles using passive suspension is quite large. This problem can be improved when the active suspension system is used to replace the conventional mechanical suspension. The maximum and average values of the displacement of the sprung mass are significantly reduced when the active suspension system is controlled by a two-component linear controller PI. These measurements are 71.73 mm and 46.74 mm, respectively. Compared with the passive suspension system, it is only about 47.37% and 60.80%. The effectiveness of the active suspension system comes into play when the Fuzzy-PI integrated controller is used. According to this result, the maximum and average displacement of the vehicle body is 2.68 (mm) and 1.65 (mm), respectively. This is an ideal result. This value is only approximately 1.77% and 2.15% compared to the situation of vehicles using the passive suspension system.
The smoothness of the vehicle is expressed through the value of the acceleration of the sprung mass. If this value is too large, a loss of smoothness will occur. According to the results of Figure6, the maximum value and the average value of the acceleration for the two situations are equivalent. The vehicle using the passive suspension system and the active suspension system controlled by the PI controller is equivalent. These values are 21.39 (m/s2), 21.48 (m/s2), 4.19 (m/s2), and 4.11 (m/s2), in that order, even in some situations, the vehicle’s acceleration when using the active suspension system is greater than in other situations. Therefore, smoothness and comfort cannot be guaranteed. In contrast, a Fuzzy-PI integrated controller can greatly reduce these values. Based on the simulation performed, the result of this situation is 1.34 (m/s2) and 0.66 (m/s2), respectively, with its maximum value and average value. Compared with the situation where the vehicle only uses a passive suspension system, these two values are only 6.26% and 15.75%. Therefore, stability and comfort have been greatly enhanced.
When the pavement excitation ends, the value of the acceleration of the sprung mass continues to vary for a short time. This happens because the coefficientsKp and Ki values will slowly decrease to zero without an immediate decrease. This process may take about 1.5(s). However, its value is extremely small. This does not affect the oscillation of the whole vehicle.
Compared with recent studies, the response of the Fuzzy-PI control algorithm is better. Its response time is faster. Besides, phase differences will not occur like in [13, 53].
For the traditional PI controller, the values of the coefficients are constant (KP = const, KI = const). Because their values cannot be changed, the response of the system is therefore poor. In many cases, vehicle instability persists. In contrast, the Fuzzy-PI integrated controller can change these values continuously. The change depends on the oscillation condition of the vehicle. The process of variation of these two values is shown in Figures 7 and 8. Accordingly , when the stimulus has not appeared, the values of KP and KI are both zero. It is only when the stimulus is present that these values begin to change. When the stimulus ended, the values decreased again and fluctuated around the zero thresholds for a short time, before officially returning to zero (its change principle was pre-established by the Fuzzy algorithm, which the author proposed. The input of the Fuzzy algorithm is the oscillator signal. Once the oscillation does not exist, the controller will no longer function). The maximum value and the average value of the KP coefficient reached 60.42 and 29.74, respectively. For the KI coefficient, these are 120.85 and 59.45. Based on this change, the vehicle's stability when traveling on the road has been greatly improved.Figure 5
Displacement of the sprung mass (Case1).Figure 6
Acceleration of the sprung mass (Case1).Figure 7
Value of theKp (Case 1).Figure 8
Value of theKi (Case 1).Case 2.
In the second case, pulsed excitation continues to be used. However, this is a double pulse with a larger amplitude. Like the above case, the maximum and average values of the displacement of the sprung mass when the vehicle is using the passive suspension system are the largest (Figure9). These two measurements are 181.70 (mm) and 92.35 (mm), respectively. Meanwhile, for the active suspension system, these two values have been significantly improved. If the conventional PI controller is used, the displacement values reach 86.49 (mm) and 56.69 (mm), respectively. If the Fuzzy-PI integrated controller is used, the values are only 3.33 (mm) and 2.01 (mm). They are only 1.83% and 2.18% of the first situation.
Similarly, the value of the acceleration of the sprung mass has been effectively ensured when the Fuzzy-PI controller is used to control the active suspension system. According to the results shown in Figure10, the values of the maximum and average accelerations are 1.69 (m/s2) and 0.73 (m/s2), respectively. Meanwhile, the values corresponding to the situation of vehicles using mechanical suspension systems are quite large, reaching the thresholds of 25.67 (m/s2) and 4.90 (m/s2). It is up to 15.19 and 6.71 times higher than the results mentioned above. For the vehicle situation where the Fuzzy-PI controller is used, the oscillation of the vehicle body's acceleration after the excitation is terminated is not large. At the same time, it is extinguished after only a small amount of time. The smoothness and stability of the vehicle are still highly effective.
To improve the oscillation efficiency of the vehicle, the controller parameters need to be continuously variable. This change is the result of the combination of the Fuzzy control algorithm and the PI control algorithm. The process of changing coefficients depends on the oscillation state of the vehicle. These values are shown in Figure11 and 12. According to these results, the change is continuous. The change of parameter values can well meet the vehicle's oscillation conditions. The delay of the signal does exist. However, its effect is not significant.
The results of the simulation are summarized in Table2–5. These results are graphed in Figure 13–16.Figure 9
Displacement of the sprung mass (Case2).Figure 10
Acceleration of the sprung mass (Case2).Figure 11
Value of theKp (Case 2).Figure 12
Value of theKi (Case 2).Table 2
The maximum value (Case1).
DisplacementAccelerationKPKIPassive151.4221.39PI71.7321.48FPI2.681.3460.42120.85Table 3
The average value (Case1).
DisplacementAccelerationKPKIPassive76.874.19PI46.744.11FPI1.650.6629.7459.45Table 4
The maximum value (Case2).
DisplacementAccelerationKPKIPassive181.7025.67PI86.4925.77FPI3.331.6976.14152.26Table 5
The average value (Case2).
DisplacementAccelerationKPKIPassive92.354.90PI56.694.81FPI2.010.7332.8765.71Figure 13
Percent of the displacement of the sprung mass (Case1).Figure 14
Percent of the acceleration of the sprung mass (Case1).Figure 15
Percent of the displacement of the sprung mass (Case2).Figure 16
Percent of the acceleration of the sprung mass (Case2).


## 5. Conclusions

The stimuli from the road surface cause the vehicle to oscillate. These oscillations affect passengers and cargo on the vehicle. The suspension system is used to quell these oscillations. An active suspension system is used to replace the conventional passive suspension system to improve stability and ride comfort. This research focuses on establishing a quarter-dynamic model with five state variables, including hydraulic actuators. This model has been linearized with high accuracy. To improve the efficiency of the system, the Fuzzy-PI integrated control algorithm has been proposed. The simulation is performed in the MATLAB-Simulink environment. This process is based on two proposed oscillation cases. For each case, three situations are investigated, including the vehicle's using a passive suspension system, an active suspension system controlled by a PI controller, and an active suspension system controlled by a Fuzzy-PI controller. The research results include the maximum and average values of displacement and acceleration of the sprung mass. Besides, the changes in the coefficientsKP and KI are also shown in this paper.According to the results of the simulation process, the vehicle's oscillation has been significantly reduced when the Fuzzy-PI algorithm is used to control the active suspension system. Both the maximum and average values of the displacement and acceleration of the sprung mass are greatly reduced. In the first case, these values are only 1.77% and 2.15% of displacement values and 6.26% and 15.75% of acceleration values, respectively. In the other case, these values are 1.83% and 2.18%, and 6.58 and 14.90%, respectively. Besides, since the change of the two coefficients,KP and KI, is continuous, it responds well to the change of the stimulus from the road surface. Therefore, the vehicle's oscillation has been improved.This paper has come up with a completely novel and original algorithm. Besides, its efficiency is very high. However, this research has only been carried out based on computational and simulation methods. The actual effects have not been fully considered in the simulation. In the future, an experimental process is needed to evaluate the algorithm's performance more accurately.



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