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The road disturbance rejection problem for vehicle active suspension involving the nonlinear characteristics is researched in this paper. A continuous-time state space of nonlinear vehicle active suspension is established first, in which the road disturbance is generated from the output of an introduced exosystem based on the ground displacement power spectral density. After that, based on the dynamics of road roughness and the internal model principle, a disturbance compensator with zero steady-state error is designed, which is related to the dynamic characteristics of road disturbance and independent of the control system model. By combining the vehicle active suspension system and the designed road disturbance compensator, an augmented system is obtained without explicit indication of road disturbance. Then by solving a series of decoupled nonlinear two-point-boundary-value problem and employing an iterative computing algorithm, an approximation optimal road disturbance rejection controller is obtained. Finally, the simulation results illustrate that the proposed approximation optimal road disturbance rejection controller can reduce the values of sprung mass acceleration, tire deflection, suspension deflection, and energy consumption and compensate the nonlinear behaviors of vehicle active suspension effectively.

With the development of advanced actuator technologies, vehicle active suspension provides the basic support for active safety technology of ground vehicle [

Vehicle active suspension is usually a typical nonlinear system. While designing the control strategies, the tire lift-off phenomenon, the spring nonlinearity, and the piece-wise linear behavior of the damper must be taken into consideration [

From the perspective of isolating the road-induced vibration, the precise estimated information of road disturbance plays an important role in designing the feedforward component of control strategies [

Based on the above analysis, this paper studies the optimal road disturbance rejection problem for nonlinear vehicle active suspension. The contribution of this paper includes twofold. On the one hand, the road disturbance rejection problem is formulated as a nonlinear two-point-boundary-value problem for an augmented system, which is constructed by combining a designed road disturbance compensator and a nonlinear vehicle active suspension. On the other hand, by solving a series of decoupled two-point-boundary-value problem, an approximation optimal disturbance rejection control scheme is proposed for the augmented system, which includes feedback component and nonlinear compensation component. Finally, simulation results are given to illustrate the effectiveness of improving the control performance and compensating the nonlinear behaviors of vehicle active suspension.

The rest of the article is organized as follows. The nonlinear vehicle active suspension model is established under persistent road disturbance in Section

In this paper, a simplified structure of quarter vehicle active suspension with an ideal active actuator is considered, which is presented in Figure

The simplified structure of quarter vehicle active suspension.

Defining

It is assumed that the road disturbance

By designing the road disturbance state as the vector

While designing the optimal disturbance rejection controller, the performance requirements of vehicle active suspension must be taken into account, including the sprung mass acceleration

To obtain the main results, the following assumptions are given first.

The pair

Each eigenvalues

In order to compensate the road disturbance, the following disturbance compensator is designed by using the internal model principle, which is described as

By integrating the dynamic model of road disturbance into the augmented system (

Then, the road disturbance rejection problem for vehicle active suspension is formulated to find an optimal road disturbance rejection controller

Based on the necessary conditions of optimal control theory, the following nonlinear two-point-boundary-value problem for the above road disturbance rejection problem can be formulated as

However, it is difficult to solve the analytical solution

In order to design the approximation optimal road disturbance rejection controller, the following lemma is introduced first.

The nonlinear system is described as

Then the approximation optimal road disturbance rejection controller is given in the following theorem.

Consider the road disturbance rejection problem for a nonlinear vehicle active suspension (

In order to solve the nonlinear two-point-boundary-value problem (

It should be pointed that, at the

Noting the values of

Therefore, in the

Based on Lemma

Due to the infinite item

By employing the approximation optimal road disturbance rejection controller (

Parameters of active vehicle suspension.

Parameters | Variable Symbol | Value | Unit |
---|---|---|---|

Mass of Sprung | | 972.2 | |

Mass of Unsprung | | 113.6 | |

Damping of Passive Suspension | | 1095 | |

Stiffness of Passive Suspension | | 42719.6 | |

Compressibility of Pneumatic Tire | | 101115 | |

Damping of Pneumatic Tire | | 14.6 | |

Stiffness coefficient of Spring | | 1 | - |

Spring Nonlinearity | | 1 | - |

Then the matrices of nonlinear vehicle active suspension in (

Meanwhile, the parameters of road disturbance

Parameters of random road roughness.

Variable | | | | | | | |

| |||||||

Value | 2 | 1.4 | 0.45 | 5 | 2.12 | 20 | 400 |

According to the road displacement

The curve of the road displacement

The curve of the road disturbance

Applying the proposed approximation optimal road disturbance rejection controller (

The curve of the sprung mass acceleration

The curve of the suspension deflection

The curve of the tire deflection

By analyzing Figures

The curve of the control energy consumption

An approximation optimal disturbance road rejection controller was proposed for a nonlinear vehicle active suspension under persistent road disturbance, which constitutes of the feedback terms and the compensation terms for nonlinear behaviors. First, a disturbance compensator with zero steady-state error was introduced based on the ground displacement power spectral density. After that, an augmented system was designed without explicit indication of road disturbance by combining the vehicle active suspension and the designed disturbance compensator. By solving a decoupled nonlinear two-point-boundary-value problem, an approximation optimal road disturbance rejection controller was obtained from a Riccati equation and a vector sequence of nonlinear compensation terms. Applying the proposed approximation optimal road disturbance rejection controller to a nonlinear vehicle active suspension, the performance requirements were satisfied significantly with small energy consumption.

The main contribution of this paper is under the assumption that the actuator of vehicle active suspension is idealized and the dynamic road disturbance can be obtained from a road roughness with known values. One aspect of our future work will focus on the vibration controller for vehicle active suspension considering the dynamic behaviors of actuator. On the other hand, the intelligent sensors for road disturbance will be designed based on the road recognition methods by using intelligent pattern recognition theory.

The simulation results are from MATLAB. Readers can request the results of this article by emailing the corresponding author.

The authors declare that they have no conflicts of interest.

This work is supported by the Natural Science Foundation of Shandong Province (ZR2017MF044), the Shandong Province Key Research and Development Program (2018GGX101016, 2018GGX101048, and 2017GGX10144), the Shandong Province Higher Educational Science and Technology Program (J17KA047, J16LN07, J16LB06, and J15LN13), and the Natural Science Foundation of China (61671220 and 61702217).