# A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems

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## Abstract

**:**

## 1. Introduction

## 2. Design of Optimal Control Law

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

## 3. Illustrative Examples

#### 3.1. Example 1—Comparative Controller

#### 3.2. Example 2—Proposed Controller with Equation (12)

#### 3.3. Example 3—Proposed Controller with Equation (7)

#### 3.4. Example 4—Proposed Controller for Vibration Control with Random Bump Road Excitation (1)

_{s}and x

_{1}represent the displacement of the sprung mass and driver mass, respectively. The variables k

_{i}and c

_{i}denotes the corresponding coefficients of the spring and damper. Moreover, x

_{0}is the excitation displacement and F

_{MR}represents the magnetic field-dependent damping force of MR damper. The state-space model of the system is derived from the above equations as follows:

#### 3.5. Example 5—Proposed Controller for Vibration Control with Random Step Wave Excitation (2)

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simulation results of the comparative optimal controller in [12]: (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) main control ${u}^{*}\left(t\right)$, (

**c**) state variable (position) ${x}_{e}\left(t\right)$, (

**d**) state variable (velocity) ${\dot{x}}_{e}\left(t\right)$.

**Figure 2.**Simulation results of the proposed control 1 with Equation (12): (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) main control ${u}^{*}\left(t\right)$, (

**c**) state variable ${x}_{e}\left(t\right)$, (

**d**) state variable ${\dot{x}}_{e}\left(t\right)$, (

**e**) large view of state variable ${x}_{e}\left(t\right)$, (

**f**) large view of state variable ${\dot{x}}_{e}\left(t\right)$.

**Figure 3.**Power Spectral Density (PSD) of the proposed control 1 with Equation (12): (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) state variable ${x}_{e}\left(t\right)$.

**Figure 4.**Simulation results of the proposed control 2 with Equation (7): (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) main control ${u}^{*}\left(t\right)$, (

**c**) state variable ${x}_{e}\left(t\right)$, (

**d**) state variable ${\dot{x}}_{e}\left(t\right)$, (

**e**) large view of state variable ${x}_{e}\left(t\right)$, (

**f**) large view of state variable ${\dot{x}}_{e}\left(t\right)$.

**Figure 5.**Power spectral density (PSD) of the proposed control 1 with Equation (12) and the proposed control 2 with Equation (7): (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) state variable ${x}_{e}\left(t\right)$.

**Figure 8.**Control results under random bump road excitation: (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) main control ${u}^{*}\left(t\right)$.

**Figure 9.**Vibration control results at the seat position under random bump road excitation: (

**a**) displacement, (

**b**) velocity.

**Figure 11.**Vibration control results at the driver position under random bump road excitation: (

**a**) displacement, (

**b**) velocity.

**Figure 14.**Control results under random step wave road excitation: (

**a**) gain function ${Q}_{\Delta L}\left(t\right)$, (

**b**) main control ${u}^{*}\left(t\right)$.

**Figure 15.**Vibration control results at the seat position under random step wave road excitation: (

**a**) displacement, (

**b)**velocity.

**Figure 16.**Vibration control results at the driver position under random step wave road excitation: (

**a**) displacement, (

**b**) velocity.

**Figure 17.**Power spectral density of the proposed control system under random step wave road excitation.

Gain Matrix Function | Chosen Matrices | |||
---|---|---|---|---|

Property | Value | Property | Value | |

Example 2 | Positive | Not fixed and change following the system | Not fixed with unit matrix | ${M}_{L\xi}\left(t\right)=1.5,{N}_{L\xi}\left(t\right)=1.2$ |

Example 3 | Positive | Not fixed and change following the system | Not fixed with unit matrix | ${M}_{L\xi}\left(t\right)=1.5,{N}_{L\xi}\left(t\right)=1.2$ |

Example 4 | Positive | Not fixed and change following the system | Not fixed with unit matrix | ${M}_{L\xi}\left(t\right)=20,{N}_{L\xi}\left(t\right)=20$ |

Example 5 | Positive | Not fixed and change following the system | Not fixed with unit matrix | ${M}_{L\xi}\left(t\right)=160000,{N}_{L\xi}\left(t\right)=60000$ |

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**MDPI and ACS Style**

Phu, D.X.; Mien, V.; Choi, S.-B. A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems. *Appl. Sci.* **2019**, *9*, 5050.
https://doi.org/10.3390/app9235050

**AMA Style**

Phu DX, Mien V, Choi S-B. A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems. *Applied Sciences*. 2019; 9(23):5050.
https://doi.org/10.3390/app9235050

**Chicago/Turabian Style**

Phu, Do Xuan, Van Mien, and Seung-Bok Choi. 2019. "A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems" *Applied Sciences* 9, no. 23: 5050.
https://doi.org/10.3390/app9235050