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 H ₂ (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.     

H2 optimal control of disturbance-delayed systems with application to vehicle suspensions

Optimal control of systems with time delays among disturbances, such as vehicle suspensions, is a relatively simple but long-standing problem in time-delayed control. We consider the exact H₂ optimal control of systems with time-delayed disturbances and develop a computationally efficient approach for controller synthesis. We extend the Lyapunov-based H₂ norm computation to systems with time-delayed disturbances and then derive a concise formula to explicitly evaluate the sensitivity of the system H₂ norm with respect to controller gains. Thence, a set of necessary conditions for H₂ optimal control of such systems using static output feedback are obtained in the form of algebraic equations. Gradient-based methods are adapted to optimize the controller gains. The method is also extended to reduced-order and decentralized control. As an application, a passive suspension system for an eight-DOF four-wheel vehicle is designed via structured H₂ optimization. The results are compared with those of a design based on a Pade expansion for the time delays and a design obtained by neglecting the disturbance delays.     

An H2 Formulation for the Design of a Passive Vibration-Isolation System for Cars

Optimal design of an active suspension system for road vehicles can be solved using LQR techniques. Such a problem is equivalent, in the frequency domain, to determine the state feedback gain matrix that minimizes the H₂ norm of a suitable transfer matrix.

A passive suspension system can be seen as the physical realization of a suitable state feedback law whose gains are function of the system parameters. This law, and thus the characteristic elements of the passive suspension, can be determined as an approximation of the H₂ optimal solution. This methodology allows one to choose the best controller from a constrained subset (i.e., all possible passive suspensions of a particular form) of all possible controllers.     

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.     

Design of Nonlinear Controllers for Active Vehicle Suspensions Using Parameter-Varying Control Synthesis

Nonlinear suspension controllers have the potential to achieve superior performance compared to their linear counterparts. A nonlinear controller can focus on maximizing passenger comfort when the suspension deflection is small compared to its structural limit. As the deflection limit is approached, the controller can shift focus to prevent the suspension deflection from exceeding this limit. This results in superior ride quality over the range of road surfaces, as well as reduced wear of suspension components. This paper presents a novel approach to the design of such nonlinear controllers, based on linear parameter-varying control techniques. Parameter-dependent weighting functions are used to design active suspensions that stiffen as the suspension limits are reached. The controllers use only suspension deflection as a feedback signal. The proposed framework easily extends to the more general case where all the three main performance metrics, i.e., passenger comfort, suspension travel and road holding are considered, and to the design of road adaptive suspensions.     

Design of Nonlinear Controllers for Active Vehicle Suspensions Using Parameter-Varying Control Synthesis

Nonlinear suspension controllers have the potential to achieve superior performance compared to their linear counterparts. A nonlinear controller can focus on maximizing passenger comfort when the suspension deflection is small compared to its structural limit. As the deflection limit is approached, the controller can shift focus to prevent the suspension deflection from exceeding this limit. This results in superior ride quality over the range of road surfaces, as well as reduced wear of suspension components. This paper presents a novel approach to the design of such nonlinear controllers, based on linear parameter-varying control techniques. Parameter-dependent weighting functions are used to design active suspensions that stiffen as the suspension limits are reached. The controllers use only suspension deflection as a feedback signal. The proposed framework easily extends to the more general case where all the three main performance metrics, i.e., passenger comfort, suspension travel and road holding are considered, and to the design of road adaptive suspensions.     

基于变传动比的全轮线控转向车辆可拓H∞控制方法研究

为了解决传统固定转向传动比以及鲁棒H_∞控制方法无法很好地改善车辆稳定性的问题,提出全轮线控转向车辆的变传动比和可拓H_∞控制策略。首先,建立八自由度车辆动力学模型和轮胎模型。其次,以车辆方向盘转角和车速为输入信息,基于模糊控制方法设计全轮线控转向车辆的转向传动比,并计算出全轮线控转向车辆的前轮转角。然后,以横摆角速度偏差和偏差微分为特征值,基于可拓控制理论将车辆状态划分为3个区域：经典域、可拓域和非域;在经典域中,采用基于横摆角速度反馈的鲁棒H_∞控制方法,实时获取全轮线控转向车辆的后轮转角;在可拓域和非域中,结合可拓控制和H_∞控制策略,动态调整H_∞控制器的输出信号,在保证控制系统鲁棒性的前提下改善车辆的操纵稳定性。最后,基于MATLAB/Simulink仿真平台和自主研制的全轮线控转向特种消防救援车辆,通过正弦转向、单移线、阶跃转向、双移线等典型工况对所提控制方法进行验证,并以平均绝对误差和均方根误差为评价指标,与无控制和H_∞控制方法进行对比分析。仿真和试验测试结果表明：①变传动比控制方法不仅可以提高车辆低速时的转向灵敏度,也能改善车辆高速时的稳定性;②相比传统鲁棒H_∞控制,可拓H_∞控制策略提高了全轮线控转向车辆的操纵稳定性,改善了车辆全轮线控转向控制系统的鲁棒性。     

An H2 Formulation for the Design of a Passive Vibration-Isolation System for Cars

SUMMARY

Optimal design of an active suspension system for road vehicles can be solved using LQR techniques. Such a problem is equivalent, in the frequency domain, to determine the state feedback gain matrix that minimizes the H₂ norm of a suitable transfer matrix.

A passive suspension system can be seen as the physical realization of a suitable state feedback law whose gains are function of the system parameters. This law, and thus the characteristic elements of the passive suspension, can be determined as an approximation of the H₂ optimal solution. This methodology allows one to choose the best controller from a constrained subset (i.e., all possible passive suspensions of a particular form) of all possible controllers.     

Development of Preview Active Suspension Control System and Performance Limit Analysis by Trajectory Optimization

The main role of the suspension system is to achieve ride comfort by reducing vibrations generated by the road roughness. The active damper is getting much attention due to its reduced cost and ability to enhance ride comfort especially when the road ahead is measurable by an environment sensor. In this study a preview active suspension control system was developed in order to improve ride comfort when the vehicle is passing over a speed bump. The control system consists of a feedback controller based on the skyhook logic and a feedforward controller for canceling out the road disturbance. The performance limit for the active suspension control system was computed via trajectory optimization to provide a measure against which to compare and validate the performance of the developed controller. The simulation results indicated that the controller of this study could enhance ride comfort significantly over the active suspension control system employing only the skyhook feedback control logic. Also the developed controller, by displaying similar control pattern as the trajectory optimization during significant time portions, proved that its control policy is legitimate.     

Level and attitude control of the active suspension system with integral and derivative action

A 7-DOF full-car model with optimal active control suspension is utilized to evaluate the vehicle dynamic performances which are achieved through proposed controllers. The optimal controller, which includes the integral action for the suspension deflection, considerably improves the attitude control of a vehicle because the rolling and pitching motion in cornering and braking maneuvers are reduced, respectively. In the viewpoint of level control, the integral control acting on the suspension deflection results in the zero steady-state deflection in response to static body forces and ramp road input. The dynamic characteristics of the suspension control system are evaluated in terms of time domain and frequency domain. The simulations in the time domain demonstrate the advantages of the active suspension system obtained by penalizing the integral and derivative of suspension deflections and the derivative of roll and pitch angles in the performance index. The frequency characteristic curves obtained by simulations regarding integral action or derivative action show the increase of both ride comfort and road-holding performances by maximizing the use of suspension deflections. The potential of derivative control is shown by the performances of the car traveling over a bump and braking.     

Level and attitude control of the active suspension system with integral and derivative action

A 7-DOF full-car model with optimal active control suspension is utilized to evaluate the vehicle dynamic performances which are achieved through proposed controllers. The optimal controller, which includes the integral action for the suspension deflection, considerably improves the attitude control of a vehicle because the rolling and pitching motion in cornering and braking maneuvers are reduced, respectively. In the viewpoint of level control, the integral control acting on the suspension deflection results in the zero steady-state deflection in response to static body forces and ramp road input. The dynamic characteristics of the suspension control system are evaluated in terms of time domain and frequency domain. The simulations in the time domain demonstrate the advantages of the active suspension system obtained by penalizing the integral and derivative of suspension deflections and the derivative of roll and pitch angles in the performance index. The frequency characteristic curves obtained by simulations regarding integral action or derivative action show the increase of both ride comfort and road-holding performances by maximizing the use of suspension deflections. The potential of derivative control is shown by the performances of the car traveling over a bump and braking.     

H8 Control Performance of a Full-Vehicle Suspension Featuring Magnetorheological Dampers

Summary This paper presents an investigation of the feedback control performance of a full-vehicle suspension system featuring magnetorheological (MR) dampers. A cylindrical MR damper is designed and manufactured by incorporating a Bingham model of aMR fluid which is commercially available. After evaluating the field-dependent damping characteristics of the MR damper, a full-vehicle suspension system installed with 4 independent MR dampers is constructed and its governing equations of motion which include vertical, pitch and roll motions are derived. A H 8 controller which has inherent robustness against system uncertainties is formulated by treating the sprung mass of the vehicle as uncertain parameter. This is accomplished by adopting the loop shaping design procedure. For the demonstration of a practical feasibility, control performance characteristics for vibration suppression of the proposed MR suspension system are evaluated under various road conditions through the hardware-in-the-loop simulation (HILS) methodology.     

H8 Control Performance of a Full-Vehicle Suspension Featuring Magnetorheological Dampers

Summary This paper presents an investigation of the feedback control performance of a full-vehicle suspension system featuring magnetorheological (MR) dampers. A cylindrical MR damper is designed and manufactured by incorporating a Bingham model of aMR fluid which is commercially available. After evaluating the field-dependent damping characteristics of the MR damper, a full-vehicle suspension system installed with 4 independent MR dampers is constructed and its governing equations of motion which include vertical, pitch and roll motions are derived. A H 8 controller which has inherent robustness against system uncertainties is formulated by treating the sprung mass of the vehicle as uncertain parameter. This is accomplished by adopting the loop shaping design procedure. For the demonstration of a practical feasibility, control performance characteristics for vibration suppression of the proposed MR suspension system are evaluated under various road conditions through the hardware-in-the-loop simulation (HILS) methodology.     

Using the lead vehicle as preview sensor in convoy vehicle active suspension control

Both ride quality and roadholding of actively suspended vehicles can be improved by sensing the road ahead of the vehicle and using this information in a preview controller. Previous applications have used look-ahead sensors mounted on the front bumper to measure terrain beneath. Such sensors are vulnerable, potentially confused by water, snow, or other soft obstacles and offer a fixed preview time. For convoy vehicle applications, this paper proposes using the overall response of the preceding vehicle(s) to generate preview controller information for follower vehicles. A robust observer is used to estimate the states of a quarter-car vehicle model, from which road profile is estimated and passed on to the follower vehicle(s) to generate a preview function. The preview-active suspension, implemented in discrete time using a shift register approach to improve simulation time, reduces sprung mass acceleration and dynamic tyre deflection peaks by more than 50% and 40%, respectively. Terrain can change from one vehicle to the next if a loose obstacle is dislodged, or if the vehicle paths are sufficiently different so that one vehicle misses a discrete road event. The resulting spurious preview information can give suspension performance worse than that of a passive or conventional active system. In this paper, each vehicle can effectively estimate the road profile based on its own state trajectory. By comparing its own road estimate with the preview information, preview errors can be detected and suspension control quickly switched from preview to conventional active control to preserve performance improvements compared to passive suspensions.     

浅埋大断面公路隧道渐进破坏规律与安全控制

围绕浅埋大断面公路隧道渐近破坏过程与机制，以大永高速公路甸头隧道下穿大西二级公路工程为背景，开展了室内相似模型试验与现场监测分析。针对不同围岩级别条件（Ⅳ₃，Ⅴ₁和Ⅴ₂），分析了隧道毛洞开挖过程中围岩应力和位移变化规律，提出了基于变形差率的隧道施工安全控制指标，隧道施工现场监测验证了该指标的科学性，保障了隧道施工安全。研究结果表明：①隧道的破坏首先发生在拱顶位置，随着隧道的不同分部的开挖，破坏区向拱肩和地表扩展；围岩级别为Ⅳ₃时，隧道开挖后围岩形成一定厚度的塌落拱，塌落拱高度约为0.67M（M为隧道最大跨度）；围岩级别为Ⅴ₁和Ⅴ₂时，隧道开挖将引起围岩坍塌，形成塌方等较严重事故。②隧道开挖过程中，在开挖卸荷作用下，隧道产生不同程度的收敛与沉降，Ⅳ₃比Ⅴ₁，Ⅴ₁比Ⅴ₂，Ⅳ₃比Ⅴ₂最大地表变形分别减少了63.0%、20.0%和70.4%；隧道开挖应力变化方面，Ⅴ₁比Ⅳ₃，Ⅴ₂比Ⅴ₁，Ⅴ₂比Ⅳ₃最大应力变化量分别减少了43.5%、23.0%和56.5%，且Ⅳ₃，Ⅴ₁和Ⅴ₂围岩级别下隧道开挖过程应力和变形影响范围逐个增大。③采用模型试验手段，通过计算典型测点沉降差与测点距离的比值——变形差率，分别探讨Ⅴ₂，Ⅴ₁和Ⅳ₃围岩级别的浅埋隧道施工安全控制标准。隧道施工现场监测结果表明，对于Ⅴ₁级围岩浅埋隧道，当隧道地表横向和纵向变形差率均小于10 mm·m^-1，可防止公路地表裂缝的产生。研究成果对于浅埋大断面公路隧道施工与安全控制具有重要意义。     

On the achievable performance using variable geometry active secondary suspension systems in commercial vehicles

There is a need to further improve driver comfort in commercial vehicles. The variable geometry active suspension offers an interesting option to achieve this in an energy efficient way. However, the optimal control strategy and the overal performance potential remains unclear. The aim of this paper is to quantify the level of performance improvement that can theoretically be obtained by replacing a conventional air sprung cabin suspension design with a variable geometry active suspension. Furthermore, the difference between the use of a linear quadratic (LQ) optimal controller and a classic skyhook controller is investigated. Hereto, an elementary variable geometry actuator model and experimentally validated four degrees of freedom quarter truck model are adopted. The results show that the classic skyhook controller gives a relatively poor performance while a comfort increase of 17–28% can be obtained with the LQ optimal controller, depending on the chosen energy weighting. Furthermore, an additional 75% comfort increase and 77% energy cost reduction can be obtained, with respect to the fixed gain energy optimal controller, using condition-dependent control gains. So, it is concluded that the performance potential using condition-dependent controllers is huge, and that the use of the classic skyhook control strategy should, in general, be avoided when designing active secondary suspensions for commercial vehicles.     

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.     

Design of robust-stable and quadratic finite-horizon optimal controllers with low trajectory sensitivity for uncertain active suspension systems

This paper presents a design method for designing the robust-stable and quadratic-finite-horizon-optimal controllers of uncertain active suspension systems. The method integrates a robust stabilisability condition, the orthogonal functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA). Using the integrative computational method, a robust-stable and quadratic-finite-horizon-optimal controller with low-trajectory sensitivity can be obtained such that (i) the active suspension system with elemental parametric uncertainties is stabilised and (ii) a quadratic-finite-horizon-integral performance index including a quadratic trajectory sensitivity term for the nominal active suspension system is minimised. The robust stabilisability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the OFA, an algebraic algorithm only involving the algebraic computation is derived for solving the nominal active suspension feedback dynamic equations. By using the OFA and the LMI-based robust stabilisability condition, the dynamic optimisation problem for the robust-stable and quadratic-finite-horizon-optimal controller design of the linear uncertain active suspension system is transformed into a static-constrained-optimisation problem represented by the algebraic equations with constraint of LMI-based robust stabilisability condition; thus greatly simplifies the design problem. Then, for the static-constrained-optimisation problem, the HTGA is employed to find the robust-stable and quadratic-finite-horizon-optimal controllers of the linear uncertain active suspension systems. A design example is given to demonstrate the applicability of the proposed integrative computational approach.     

Structured H2 Optimization of Vehicle Suspensions Based on Multi-Wheel Models

Summary Various control techniques, especially LQG optimal control, have been applied to the design of active and semi-active vehicle suspensions over the past several decades. However passive suspensions remain dominant in the automotive marketplace because they are simple, reliable, and inexpensive. The force generated by a passive suspension at a given wheel can depend only on the relative displacement and velocity at that wheel, and the suspension parameters for the left and right wheels are usually required to be equal. Therefore, a passive vehicle suspension can be viewed as a decentralized feedback controller with constraints to guarantee suspension symmetry. In this paper, we cast the optimization of passive vehicle suspensions as structure-constrained LQG/H2 optimal control problems. Correlated road random excitations are taken as the disturbance inputs; ride comfort, road handling, suspension travel, and vehicle-body attitude are included in the cost outputs. We derive a set of necessary conditions for optimality and then develop a gradient-based method to efficiently solve the structure-constrained H2 optimization problem. An eight-DOF four-wheel-vehicle model is studied as an example to illustrate application of the procedure, which is useful for design of both passive suspensions and active suspensions with controller-structure constraints.     

Multi-objective control of a full-car model using linear-matrix-inequalities and fixed-order optimisation

This paper studies multi-objective control of a full-vehicle suspension excited by random road disturbances. The control problem is first formulated as a mixed ?₂/?_∞ synthesis problem and an output-feedback solution is obtained by using linear-matrix-inequalities. Next, the multi-objective control problem is re-formulated as a non-convex and non-smooth optimisation problem with controller order restricted to be less than the vehicle model order. For a range of orders, controllers are synthesised by using the HIFOO toolbox. The efficacy of the presented procedures are demonstrated by several design examples.