Suboptimal Control Design of Active and Passive Suspensions Based on a Full Car Model

An optimal control design method is introduced and then applied to the optimum design of active and passive suspension systems. A basic three-dimensional 7-DOF car riding model subjected to four correlated random road inputs is considered. The design method is basically developed to allow arbitrary choice of sensors for various car state variables to be used for feedback control of each suspension unit. Previous studies show that full-state control laws and even some limited-state control laws often include feedback gains which are almost zero. Some other gains, although not zero, don't play an important role in improving the system performance measures. With the method proposed in this work, every suspension unit can have its own feedback measurements and the criterion function can be related to all state and control variables. Thus a large number of active and semi-active suspension systems with full- or limited-state control laws based on different measurement combination can be suggested, studied, and compared with each other. Instead of comparing these optimized active and semi-active suspension systems with a basic, passive suspension, the passive system itself is optimized with the same criterion. Simulations in the time domain and frequency analyses are performed, and comparisons are made among the systems in terms of r.m.s. car response measures and ISO riding comfort criterion.     

An insight into linear quarter car model accuracy

The linear quarter car model is the most widely used suspension system model. A number of authors expressed doubts about the accuracy of the linear quarter car model in predicting the movement of a complex nonlinear suspension system. In this investigation, a quarter car rig, designed to mimic the popular MacPherson strut suspension system, is subject to narrowband excitation at a range of frequencies using a motor driven cam. Linear and nonlinear quarter car simulations of the rig are developed. Both isolated and operational testing techniques are used to characterise the individual suspension system components. Simulations carried out using the linear and nonlinear models are compared to measured data from the suspension test rig at selected excitation frequencies. Results show that the linear quarter car model provides a reasonable approximation of unsprung mass acceleration but significantly overpredicts sprung mass acceleration magnitude. The nonlinear simulation, featuring a trilinear shock absorber model and nonlinear tyre, produces results which are significantly more accurate than linear simulation results. The effect of tyre damping on the nonlinear model is also investigated for narrowband excitation. It is found to reduce the magnitude of unsprung mass acceleration peaks and contribute to an overall improvement in simulation accuracy.     

主动悬架系统对汽车加速性能改善分析

针对前轴驱动汽车加速过程中质量转移而使最大驱动力下降的问题，提出了利用主动悬架系统减小汽车车轮动态载荷对其进行改善的方法，并进行了控制系统的设计。模拟分析表明，利用主动悬架系统可以有效地降低前轴驱动汽车由于质量转移而引起的车轮动载荷的改变，是解决加速过程中驱动力下降的一个比较有效的方法。     

Suspension parameter estimation in the frequency domain using a matrix inversion approach

The dynamic lumped parameter models used to optimise the ride and handling of a vehicle require base values of the suspension parameters. These parameters are generally experimentally identified. The accuracy of identified parameters can depend on the measurement noise and the validity of the model used. The existing publications on suspension parameter identification are generally based on the time domain and use a limited degree of freedom. Further, the data used are either from a simulated ‘experiment’ or from a laboratory test on an idealised quarter or a half-car model. In this paper, a method is developed in the frequency domain which effectively accounts for the measurement noise. Additional dynamic constraining equations are incorporated and the proposed formulation results in a matrix inversion approach. The nonlinearities in damping are estimated, however, using a time-domain approach. Full-scale 4-post rig test data of a vehicle are used. The variations in the results are discussed using the modal resonant behaviour. Further, a method is implemented to show how the results can be improved when the matrix inverted is ill-conditioned. The case study shows a good agreement between the estimates based on the proposed frequency-domain approach and measurable physical parameters.     

Robust Stability Analysis of LQG-Controlled Active Suspension with Model Uncertainty Using Structured Singular Value, μ, Method

This paper presents a systematic approach toward robust stability analysis of LQG-con trolled active suspension systems. To perform this task, the paper starts with a brief background information on LQG control, its relation to H₂ method, and showing how H₂ could be formulated to become the frequency domain equivalent of LQG. Then unstructured and structured uncertainties of active suspension are formulated. The paper continues with the definition of maximum singular values and structured singular values of a transfer function matrix. Using these definitions, the robust stability of an active suspension system in the presence of assumed parameter variations are analyzed. These steps are illustrated by means of a numerical example of an active suspension system.     

基于闭环力反馈原理的非线性液压主动阻尼悬架模型的研究及优化设计

本文在理论上建立了一套基于主动力反馈原理的新型主动阻尼悬架的设计和优化方法。首先提出了该主动阻尼悬架的实现模型，该模型是在传统的液力减振器的基础上，应用半主动控制的思想，结合力反馈的原理建立起来的内部液压反馈阻尼网络模型。理论优化分析和计算机仿真表明，通过该液阻网络模型，即可以实现电控的主动阻尼悬架的功能，它可以根据汽车行驶路况的好坏，自适应地调节悬架阻尼大小以实现主动阻尼悬架的最优控制。     

Force Control in Electrohydraulic Active Suspensjons

For a simple vehicle active suspension system complete optimality and zero steady state body displacements may be achieved if the axle and body accelerations, and other easily measured quantities, are included in the performance index. Apart from not requiring an observer, this also allows the optimal feedback gains to be determined for an arbitrary body spring rate. In a theoretical example, model parameters matching those of an experimental test rig are employed. The results of computer simulations, with and without an electrohydraulic servovalve and actuator, are compared to demonstrate the effects of inner loop gain on force control. Aspects of the system behaviour including lockup are commented upon.     

Semiactive Suspension Design with an Optimal Gain Switching Target

The paper presents a two-phase design technique for semiactive suspensions. In the first phase, we use a procedure proposed by Yoshida et al. to compute a target active control law that can be implemented by Optimal Gain Switching. This control law is such that the force generated by the suspension system is bounded within a set U. In the second phase, we approximate this target by controlling the damper coefficient of the semiactive suspension. We also compute the region of the state space in which the force generated by the semiactive suspension is still within the set U. The results of several simulations show that the use of a semiactive suspension leads to minimal loss with respect to optimal performance of an active suspension.     

Virtual durability test rigs for automotive engineering

In recent years so-called ‘virtual test rigs’ have become more and more important in the development process of cars and trucks. Originally, the idea was to substitute expensive durability tests with computer simulation. Meanwhile, the focus has changed towards a more cooperative usage of numerical and laboratory rig simulation. For many safety critical issues laboratory tests remain indispensable. In early development stages, when no physical prototypes are available yet, numerical simulation is used to analyse and optimise the design. In this paper, we show how to build numerical simulation models of complex servo-hydraulic test systems and their test specimen using multi-body simulation for the mechanics in combination with simulation models for the hydraulics and controls. We illustrate this at two industrial application examples: a spindle-coupled passenger car suspension rig and a tyre-coupled full vehicle rig. We show how the simulation models are used to design and optimise better test rigs and to support the test rig operation by preparing the physical tests with new specimen, i.e. by performing numerical simulations including numerical drive file iteration before the physical tests start.     

Semiactive Suspension Design with an Optimal Gain Switching Target

The paper presents a two-phase design technique for semiactive suspensions. In the first phase, we use a procedure proposed by Yoshida et al. to compute a target active control law that can be implemented by Optimal Gain Switching. This control law is such that the force generated by the suspension system is bounded within a set U . In the second phase, we approximate this target by controlling the damper coefficient of the semiactive suspension. We also compute the region of the state space in which the force generated by the semiactive suspension is still within the set U . The results of several simulations show that the use of a semiactive suspension leads to minimal loss with respect to optimal performance of an active suspension.     

Improving crosswind stability of fast rail vehicles using active secondary suspension

Rail vehicles are today increasingly equipped with active suspension systems for ride comfort purposes. In this paper, it is studied whether these often powerful systems also can be used to improve crosswind stability. A fast rail vehicle equipped with active secondary suspension for ride comfort purposes is exposed to crosswind loads during curve negotiation. For high crosswind loads, the active secondary suspension is used to reduce the impact of crosswind on the vehicle. The control input is taken from the primary vertical suspension deflection. Three different control cases are studied and compared with the only comfort-oriented active secondary suspension and a passive secondary suspension. The application of active secondary suspension resulted in significantly improved crosswind stability.     

Simplified fatigue durability assessment for rear suspension structure

The eight-channel test rig is widely used in durability assessment of vehicle components while for some cases of rear suspension, this costly instrument is unnecessary. Based on the analysis of structure and forces, a simpler one-channel testing approach is presented for the durability calculation of a dependent rear suspension. Taking a punched rear shock tower as the study object, a FEA strain-stress analysis was first performed to determine the risk area. Then, the entire vehicle test system was created, and the proving ground tests were carried out so that the real strain on the part could be measured. Based on the road test data and the P-S-N curve of the component, the cumulative fatigue damage of a 15,000-kilometer proving ground test road was calculated, and the computational result indicates that the modified structure was safe for durability analysis. Moreover, a standard 50% S-N survival fraction curve was plotted using Corten and Dolan’s method, which can be utilized in the durability analysis for other similar components. Finally, the road test for this modified suspension structure was carried out, and the test result certified that the punched shock tower can be subjected to a 15,000-kilometer proving ground test road without the appearance of fatigue failure.     

Designing H∞/GH 2 static-output feedback controller for vehicle suspensions using linear matrix inequalities and genetic algorithms

This paper presents an approach to design the H_∞/GH ₂ static-output feedback controller for vehicle suspensions by using linear matrix inequalities (LMIs) and genetic algorithms (GAs). Three main performance requirements for an advanced vehicle suspension are considered in this paper. Among these requirements, the ride-comfort performance is optimized by minimizing the H_∞ norm of the transfer function from the road disturbance to the sprung mass acceleration, while the road-holding performance and the suspension deflection limitation are guaranteed by constraining the generalized H₂ (GH ₂) norms of the transfer functions from the road disturbance to the dynamic tyre load and the suspension deflection to be less than their hard limits, respectively. At the same time, the controller saturation problem is considered by constraining its peak response output to be less than a given limit using the GH ₂ norm as well. A four-degree-of-freedom half-car model with active suspension system is applied in this paper. Several kinds of H_∞/GH ₂ static-output feedback controllers, which use the available sprung mass velocities or the suspension deflections as feedback signals, are obtained by using the GAs to search for the possible control gain matrices and then resolving the LMIs together with the minimization optimization problem. These designed H_∞/GH ₂ static-output feedback controllers are validated by numerical simulations on both the bump and the random road responses which show that the designed H_∞/GH ₂ static-output feedback controllers can achieve similar or even better active suspension performances compared with the state-feedback control case in spite of their simplicities.     

Observer Design for Semi-Active Suspension Control

This paper presents an observer for automotive semi-active suspension control. Automotive suspensions are disturbance affected dynamic systems and semi-active suspensions can be represented as a bilinear model. An observer for semi-active suspensions is formulated such that the estimation error is independent of unknown external disturbance. The proposed observer uses easily accessible measurements such as accelerations and guarantees exponentially convergent state estimation for suspension deflections and velocities. Absolute sprung mass and unsprung mass velocities can be estimated using the proposed observer. Simulations and experimental rig tests show that all states of a semi-active suspension can be estimated only with acceleration measurements. The estimated states are used to improve ride quality in a semi-active suspension.     

A method for testing railway wheel sets on a full-scale roller rig

Full-scale roller rigs for tests on a single axle enable the investigation of several dynamics and durability problems related with the design and operation of the railway rolling stock. In order to exploit the best potential of this test equipment, appropriate test procedures need to be defined, particularly in terms of actuators’ references, to make sure that meaningful wheel –rail contact conditions can be reproduced. The aim of this paper is to propose a new methodology to define the forces to be generated by the actuators in the rig in order to best reproduce the behaviour of a wheel set and especially the wheel –rail contact forces in a running condition of interest as obtained either from multi-body system (MBS) simulation or from on-track measurements. The method is supported by the use of a mathematical model of the roller rig and uses an iterative correction scheme, comparing the time histories of the contact force components from the roller rig test as predicted by the mathematical model to a set of target contact force time histories. Two methods are introduced, the first one considering a standard arrangement of the roller rig, the second one assuming that a differential gear is introduced in the rig, allowing different rolling speeds of the two rollers. Results are presented showing that the deviation of the roller rig test results from the considered targets can be kept within low tolerances (1% approximately) as far as the vertical and lateral contact forces on both wheels are concerned. For the longitudinal forces, larger deviations are obtained except in the case where a differential gear is introduced.     

Observer Design for Semi-Active Suspension Control

This paper presents an observer for automotive semi-active suspension control. Automotive suspensions are disturbance affected dynamic systems and semi-active suspensions can be represented as a bilinear model. An observer for semi-active suspensions is formulated such that the estimation error is independent of unknown external disturbance. The proposed observer uses easily accessible measurements such as accelerations and guarantees exponentially convergent state estimation for suspension deflections and velocities. Absolute sprung mass and unsprung mass velocities can be estimated using the proposed observer. Simulations and experimental rig tests show that all states of a semi-active suspension can be estimated only with acceleration measurements. The estimated states are used to improve ride quality in a semi-active suspension.     

Robust Stability Analysis of LQG-Controlled Active Suspension with Model Uncertainty Using Structured Singular Value, μ, Method

SUMMARY

This paper presents a systematic approach toward robust stability analysis of LQG-con trolled active suspension systems. To perform this task, the paper starts with a brief background information on LQG control, its relation to H ₂ method, and showing how H ₂ could be formulated to become the frequency domain equivalent of LQG. Then unstructured and structured uncertainties of active suspension are formulated. The paper continues with the definition of maximum singular values and structured singular values of a transfer function matrix. Using these definitions, the robust stability of an active suspension system in the presence of assumed parameter variations are analyzed. These steps are illustrated by means of a numerical example of an active suspension system.     

A new active variable stiffness suspension system using a nonlinear energy sink-based controller

This paper presents the active case of a variable stiffness suspension system. The central concept is based on a recently designed variable stiffness mechanism which consists of a horizontal control strut and a vertical strut. The horizontal strut is used to vary the load transfer ratio by actively controlling the location of the point of attachment of the vertical strut to the car body. The control algorithm, effected by a hydraulic actuator, uses the concept of nonlinear energy sink (NES) to effectively transfer the vibrational energy in the sprung mass to a control mass, thereby reducing the transfer of energy from road disturbance to the car body at a relatively lower cost compared to the traditional active suspension using the skyhook concept. The analyses and simulation results show that a better performance can be achieved by subjecting the point of attachment of a suspension system, to the chassis, to the influence of a horizontal NES system.     

Achievable Dynamic Response for Automotive Active Suspensions *

A complete set of constraints is derived for the road disturbance transfer functions in a quarter car model of an automotive active suspension, for typical choices of measured outputs. It is shown that any road disturbance responses which are achievable using “full state feedback” can be achieved, to within an arbitrary small tolerance, using a dynamic compensator measuring suspension deflection only. Also considered are the disturbance responses to loads acting on the sprung mass, and a complete set of constraints is derived for these. It is shown that road disturbance and load disturbance responses can be determined independently if suspension deflection and sprung mass velocity are measured. Indeed, any responses achievable separately with “full measurements” can be approximated together to an arbitrary small tolerance. Certain integral relationships are shown to follow from the derived transfer function constraints. These relationships imply fundamental limitations for certain responses (e.g. tyre deflection) no matter what measurements are available for feedback.     

Nonlinear Backstepping Active Suspension Design Applied to a Half-Car Model

A fresh nonlinear backstepping design scheme, which is developed for the control of half-car active suspension systems to improve the inherent tradeoff between ride quality and suspension travel, is proposed in this paper. Since ride quality is dependent on a combination of vertical and angular displacements of a vehicle body, the design of active suspensions must have the potential to minimize heave and pitch movements in order to guarantee the ride comfort of passengers. The other important factor to be emphasized in the design of active suspensions is the suspension travel which means the space variation between the car body and the tires. In order to avoid damaging vehicle components and generating more passenger discomfort, the active suspension controllers must be capable of preventing the suspension from hitting its travel limits. Our design strategy, with two intentionally additional nonlinear filters, shows the potential to achieve these conflicting control objectives. The novelty of our active suspension design is in the use of two particular nonlinear filters at both the front and rear wheels. The effective bandwidths of these two nonlinear filters depend on the magnitudes of the front and rear suspension travels, individually. When suspension travel is small, the proposed controllers soften the suspension for enhancing passenger comfort. However, our control design shifts its attention to rattlespace utilization by stiffening the suspension when suspension travel approaches its limits. As a result, the improvement of tradeoff between ride quality and suspension travel can be guaranteed and is then demonstrated through comparative simulations.