# A New Adaptive Fuzzy PID Controller Based on Riccati-Like Equation with Application to Vibration Control of Vehicle Seat Suspension

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. New Adaptive Fuzzy PID Controller

#### 2.1. Interval Type 2 Fuzzy Neural Network Model

#### 2.2. Adaptive Fuzzy PID Control

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof**

**1.**

## 3. Application to Vibration Control

#### 3.1. Seat Suspension Model

#### 3.2. Computer Simulations

^{2}for the initial acceleration. In this simulation, the fourth-order Runge–Kutta method was applied for solving the differential equation. Simulation results are shown in Figure 2, Figure 3, Figure 4 and Figure 5. In Figure 2, the displacement of the seat before and after controlling is presented. The controlled displacement of the seat is less than the initial vibration. The initial amplitude of vibration before and after using the control is in range of −0.06 to 0.08 m, as shown in Figure 2a, and −0.002 to 0.004 m, as shown in Figure 2c, respectively. Subsequently, the velocity shown in Figure 2b,d of the seat is also decreased with the stability. The power spectral density of these vibrations and the driver position is shown in Figure 3. It is clearly observed that the proposed control is very efficient for controlling the random bump road excitation. This result directly indicates that the vibration with small difference of amplitude and trivial disturbance can be controlled well by using the proposed controller. The simulated results under the random step wave road are shown in Figure 4 and Figure 5. In Figure 4, the displacement of the seat is much lower than the initial vibration. The initial amplitude of vibration before and after using the control is in range of −1.5 to 1.5 m, as shown in Figure 4a, and −0.004 to 0.004 m, as shown in Figure 4c, respectively. The velocity shown in Figure 4b,c of the proposed control is also stable without large vibration. These results can be seen in Figure 5 with the power spectral density. The simulation results show very effective vibration control of the seat suspension system subjected to severe vibration with mixed disturbances. This is from the new modification of the Riccati-like equation with embedded parameters of the PID controller, and the combination of the sliding surface of sliding mode control with the fuzzy neural networks model to take account for the uncertainties and disturbances.

## 4. Experimental Results and Discussions

^{2}. It is seen in Figure 9a that three controllers are effective in control vibration, but the proposed controller clearly provides lower values than the other controllers [9,23]. Based on the decrease of the acceleration of the seat, the acceleration at the driver position is also decreased, as shown in Figure 9b. The acceleration of the proposed control and the comparative controller 2 [9] is nearly similar. This is originated from the damping ratio of the seat frame which is nonlinear. To compare three controllers, quantification was undertaken in this work: the performance of three controllers was continuously evaluated using the ToD (Transmissibility of Displacement) and ToA (Transmissibility of Acceleration) values of the seat [9]. The ToD values of the random step wave excitation were 0.213565, 0.358171, and 0.246641 for the proposed controller, Comparative Controller 1 [23], and Comparative Controller 2 [9], respectively. Similarly, from the acceleration responses, the ToA values of random step wave excitation were identified as 0.170182, 0.396618, and 0.257991 for the proposed controller, Comparative Controller 1 [23], and Comparative Controller 2 [9], respectively. From the above results, the proposed controller is the best, showing outstanding vibration control with the stability of the system under severe road disturbance.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Simulation results of displacement of seat under random bump road excitation: (

**a**,

**b**) general view; and (

**c**,

**d**) large view.

**Figure 3.**Simulation result of power spectral density of displacements under random bump road excitation.

**Figure 4.**Simulation results of displacement of seat under random step wave road excitation: (

**a**,

**b**) general view; and (

**c**,

**d**) large view.

**Figure 5.**Simulation result of power spectral density of displacements under random step wave road excitation.

**Figure 7.**Experiment results under random-step-wave excitation: (

**a1**) displacement; (

**a2**) large view of the displacement; (

**b1**) velocity; (

**b2**) large view of the velocity; (

**c1**) acceleration at the seat; (

**c2**) large view of the acceleration at the seat; (

**d1**) acceleration of the driver; and (

**d2**) large view of the acceleration of the driver.

**Figure 9.**Experiment results of power spectral density: (

**a**) acceleration of seat; and (

**b**) acceleration of driver.

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**MDPI and ACS Style**

Phu, D.X.; Choi, S.-B. A New Adaptive Fuzzy PID Controller Based on Riccati-Like Equation with Application to Vibration Control of Vehicle Seat Suspension. *Appl. Sci.* **2019**, *9*, 4540.
https://doi.org/10.3390/app9214540

**AMA Style**

Phu DX, Choi S-B. A New Adaptive Fuzzy PID Controller Based on Riccati-Like Equation with Application to Vibration Control of Vehicle Seat Suspension. *Applied Sciences*. 2019; 9(21):4540.
https://doi.org/10.3390/app9214540

**Chicago/Turabian Style**

Phu, Do Xuan, and Seung-Bok Choi. 2019. "A New Adaptive Fuzzy PID Controller Based on Riccati-Like Equation with Application to Vibration Control of Vehicle Seat Suspension" *Applied Sciences* 9, no. 21: 4540.
https://doi.org/10.3390/app9214540