Multi-objective control of vehicle active suspension systems via load-dependent controllers

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Abstract

This paper presents a load-dependent controller design approach to solve the problem of multi-objective control for vehicle active suspension systems by using linear matrix inequalities. A quarter-car model with active suspension system is considered. It is assumed that the vehicle body mass resides in an interval and can be measured online. This approach of designing controllers, whose gain matrix depends on the online available information of the body mass, is based on a parameter-dependent Lyapunov function. Since the parameter-dependent idea is fully exploited, the proposed controller design approach can yield much less conservative results compared with previous approaches that design robust constant controllers in the quadratic framework. The usefulness and the advantages of the proposed controller design methodology are demonstrated via numerical simulations.

Introduction

Vehicle suspensions have been a hot research topic for many years due to its important role in ride comfort, vehicle safety, road damage minimization and the overall vehicle performance. To meet these requirements, many types of suspension systems, ranging from passive [1], [2], semi-active [3], [4], to active suspensions [5], [6], are currently being employed and studied. It has been well recognized that active suspensions have a great potential to meet the tight performance requirements demanded by users. Therefore, in recent years more and more attention has been devoted to the development of active suspensions and various approaches have been proposed to solve the crucial problem of designing a suitable control law for these active suspension systems (see, for instance Refs. [7], [8], [9], [10], [11], [12] and the references therein).

The linear quadratic regulator (LQR) has been used as one of the main control techniques for dealing with active suspension design [13]. In this framework, an optimal state feedback gain minimizing a quadratic cost function is obtained. It is noted that the model parameters are assumed to be precisely known and the optimal control strategies may collapse in the face of some uncertain parameters. However, a suspension system often contains parameters that are intrinsically uncertain, such as the sprung mass, whose value is dependent on the total load of the vehicle. Therefore, the use of robust control techniques has become a major requirement in the further development of active suspension systems, and many useful results on robust control for active suspension systems have been reported [14], [15], [16], [17].

To achieve a compromise between several performance requirements for uncertain active suspension systems, very recently a robust multi-objective controller was designed for a quarter-car model whose system matrices are subject to parameter uncertainties characterized by a given polytope [18]. The main objective is to use a robust state-feedback controller to achieve multiple performance objectives for different controlled output signals. It is worth mentioning that the solutions are given in the quadratic framework. Although being adequate to ensure stability for systems with arbitrarily fast time-varying parameters, methods based on quadratic stability can produce conservative results since the same parameter-independent Lyapunov function must be used for the entire uncertainty domain. In addition, it is noted that the gain matrix for the designed controllers keeps constant for all the uncertain parameters. However, for active suspension systems with parameters, such as the sprung mass, that can be measured online without difficulty, the online available information of these parameters can be utilized in the realization of control strategy. This would generally allow less conservative designs to be achieved.

Motivated by the above discussion, in this paper we present a load-dependent controller design approach to solve the problem of multi-objective control for vehicle active suspension systems. A quarter-car model with active suspension system is considered and the linear matrix inequality (LMI) technique is employed to cast the controller designs into convex optimizations. It is assumed that the vehicle body mass (whose value changes with the vehicle load) resides in an interval and can be measured online. This approach of designing controllers, whose gain matrix depends on the online available information of the body mass, is based on a parameter-dependent Lyapunov function. Since the parameter-dependent idea is fully exploited, the proposed controller design approach can yield much less conservative results compared with previous approaches that design robust constant controllers in the quadratic framework. The usefulness and advantage of the proposed controller design methodology are demonstrated via numerical simulations.

The remainder of this paper is organized as follows. The problem of multi-objective load-dependent controller design for uncertain active suspension systems is formulated in Section 2. Sections 3 presents controller synthesis results by using LMI techniques. A design example illustrating the usefulness and advantage of the proposed methodology is given in Section 4 and we conclude the paper in Section 5.

Notations: The superscript T stands for matrix transposition; Rn denotes the n-dimensional Euclidean space, Rm×n is the set of all m×n real matrices and the notation P>0 means that P is symmetric and positive definite; I and 0 represent identity matrix and zero matrix; the notation ||·|| refers to the Euclidean vector norm. In addition, in symmetric block matrices or long matrix expressions, we use * to represent a block in a matrix that is induced by symmetry and {} stands for a block-diagonal matrix. For simplicity, we use sym(M) to represent M+MT. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Section snippets

Problem formulation

Consider the quarter-car model shown in Fig. 1, which has been used extensively in the literature due to its simplicity while capturing many essential characteristics of a real suspension system. In this figure, ms is the sprung mass, which represents the car chassis whose value changes with the vehicle load; mu is the unsprung mass, which represents mass of the wheel assembly; ks and cs are stiffness and damping of the uncontrolled suspension, respectively; kt serves to model the

Load-dependent controller design

In this section, we will investigate the problem of multi-objective control through load-dependent controllers formulated in the above section. First, according to Ref. [24], the closed-loop system in Eq. (19) is asymptotically stable with T1(s)2γ1 and Tj(s)Gγl(l=2,3,4) if and only if there exist matrix functions P(ms)>0 and S(ms)>0 satisfyingTr(S(ms))<γ12,[A¯T(ms)P(ms)+P(ms)A¯(ms)P(ms)Bw(ms)*-I]<0,[-S(ms)C¯1(ms)*-P(ms)]<0,[-γl2IC¯l(ms)*-P(ms)]<0,l=2,3,4.

In addition, in order to obtain

A design example

In this section, we use an example to illustrate the usefulness and advantage of the load-dependent controller design method proposed in the above sections. Model parameters are borrowed from Ref. [20] and listed in Table 1. The values listed in Table 1 are for the nominal system. We assume that the sprung mass ms changes with the vehicle load, which is expressed asms=(320+λ)kg,where λ is a parameter satisfying |λ|λ¯. In this case, the state-space model (13) can be represented by a two-vertex

Concluding remarks

A load-dependent controller design approach has been proposed to solve the problem of multi-objective control of active suspension systems with uncertain parameters. This approach designs controllers whose gain matrix depends on the online available information of the body mass based on parameter-dependent Lyapunov functions. Compared with previous approaches that design robust constant controllers, the proposed load-dependent approach can yield much less conservative results. The usefulness

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that helped them improve the presentation of the paper. This work was partly supported by RGC HKU 7103/01P.

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