Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4045796, 8 pages

https://doi.org/10.1155/2017/4045796

## Random Modeling and Control of Nonlinear Active Suspension

School of Mathematics and Information Science, Yantai University, Yantai, Shandong Province 264005, China

Correspondence should be addressed to Mingyue Cui

Received 10 March 2017; Accepted 8 May 2017; Published 30 May 2017

Academic Editor: Weihai Zhang

Copyright © 2017 Mingyue Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For a quarter car with nonlinear active suspension in rough road, the problem of random modeling and control is considered. According to the relative motion principle, the influence of rough road can be seen as that force is disturbed by the noise and a random model is constructed. By an appropriate transform, the model is transformed into a lower triangular system, which can be used as backstepping method. Then a controller is designed such that the mean square of the state converges to an arbitrarily small neighborhood of zero by tuning design parameters. The simulation results illustrate the effectiveness of the proposed scheme. Therefore, the active suspension system offers better riding comfort and vehicle handing to the passengers.

#### 1. Introduction

The active suspension is the key technology for vehicles to achieve both ride comfort and control performance. Compared with the passive and semiactive suspension system, the active suspension can supply energy from an external source and generate force to achieve the desired performance. Therefore, the performance of active suspension system is better. In recent decades, the control of active suspension systems has been enthusiastically studied by many researchers. Great efforts have been made in modeling and developing control techniques to obtain the ride comfort. Many control methods, for instance, adaptive control [1], PD control [2], robust control [3], fuzzy logic control [4, 5], intelligent control [6], and sliding mode control [7, 8], have been recently proposed. However, all of these studies are based on deterministic systems.

In the actual life, the car often runs on rough roads and the influence of the rough road is not negligible. Therefore, it is expected that the passenger still feels comfortable in rough road. For increasing the passenger’s comfort, the vertical acceleration of the vehicle caused by road vibrations must be limited which means that the suspension system must absorb the road vibrations and prevent it from transferring to the vehicle body and passengers. It is very necessary to study the random model and control of nonlinear active suspension system in rough road.

In [9–12], the design methods of the controller with different stochastic mechanical systems were studied. However, the stochastic disturbance was described as white noise in these literatures. Because of absorbers, it is more reasonable to describe the final effect of road irregularities as stationary processes. To overcome the conservation, [13] constructed a theoretical framework on stability of random differential equation systems (RDEs) where stochastic disturbance is stationary processes. For a class of Lagrange systems with colored noise, [14] designed a tracking controller such that the mean square of the tracking error converges to an arbitrarily small neighborhood of zero.

Inspired by these, the model and control problem of nonlinear active suspension in rough road are considered in this paper. The main work consists of the following aspects.

Different from the linear deterministic system in [2–5], the active suspension with nonlinear damper in rough road is considered in this paper, which increases the difficulty of modeling and design.

The main difficulty for dynamics modeling is how to transform the effect of the road irregularities to the suspension. In this paper, regardless of the rough road, dynamics model of the system is firstly constructed. According to the dynamic-static method and the relative motion, the road irregularities are transformed to the force disturbed by the stationary processes. Then the random dynamic model is established.

Because the system is an underactuated system which is not the quasi lower triangular structure, one cannot use the design method of vector controller in [9–12]. In this paper, according to special form of the model, the system is transformed into a lower triangular system by a transform. Then using backstepping design method, a state feedback controller is designed such that the state can be made arbitrarily small by tuning design parameters. The simulation results illustrate the effectiveness of the proposed scheme.

This paper is organized as follows. In Section 2, the mathematical preparation is given and the problem is formulated. The random model is constructed in Section 3. In Section 4, the tracking controller design and stability analysis are addressed. A simulation result is given in Section 5. Section 6 concludes the paper.

*Notations*. The following notations are used throughout the paper: For a vector , denotes its transpose; denotes the usual Euclidean norm of “”; denotes the mathematical expectation; denotes the set of all nonnegative real numbers; denotes the real -dimensional space; denotes the real matrix space; denotes the set of all functions with continuous th partial derivative; denotes the set of all functions: , which are continuous, strictly increase, and vanish at zero; denotes the set of all functions which are of class and unbounded. denotes the set of all functions , which is of class- for each fixed and decreases to zero as for each fixed . For simplicity, sometimes the arguments of functions are dropped.

#### 2. Mathematical Preliminaries and Problem Formulation

##### 2.1. Mathematical Preliminaries

Consider the following random nonlinear affine system:where is the state of system, is a stochastic process, and the underlying complete probability space is taken to be the quartet with a filtration satisfying the usual condition (i.e., it is increasing and right continuous while contains all -null sets). Both functions and are locally Lipschitz in piecewise continuous in ; that is, for each , there exists a constant such thatfor any and , . Moreover, and are bounded a.s.

In order to obtain the stability, process satisfies the following assumption.

*Assumption 1. *Process is a -adapted process and piecewise continuous, and there exist positive constants , such that

The following definition, criterion, and inequality are represented now for the stability analysis.

*Definition 2 (see [13]). *System (1) with Assumption 1 is said to be noise-to-state stable in probability (NSS-P) if there exist a function and a function such that, for any , , ,

Lemma 3 (see [13]). *Suppose that for system (1) with conditions (3), there exist parameters and and a function such thatwhere , are functions of class . Then system (1) has a unique solution, and the system is NSS-P.*

*Definition 4 (see [15]). *A stochastic process is said to be bounded in probability if the random variables are bounded in probability uniformly in ; that is,

Lemma 5 (see [13]). *Under Assumption 1, if there exist a positive-definite function and a constant such thatthen system (1) has a unique solution, and the solution is bounded in probability.*

Lemma 6 (see [16]). *Consider the continuous functions , , and they are integrable over every finite interval. If a continuous function satisfies the inequalitythen*

##### 2.2. Problem Formulation

The simplified quarter-car active suspension model is shown in Figure 1 (see [17]). The car runs at a constant speed on a rough road. The road irregularity is described as white noise , which is the acceleration of vertical oscillation. is the mass of car body. The wheel is modeled as an unsprung mass with a linear spring . The actuator is connected in parallel with a linear spring and a nonlinear damper . is the control force from the actuator such as hydraulic actuator. When the components are stationary, the system is in equilibrium. and denote the barycenter displacements of wheel and vehicle body relative to the equilibrium position, respectively.