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.     

Output feedback H ∞/GH 2 preview control of active vehicle suspensions: a comparison study of LQG preview

This study concerns with multi-objective H _∞/GH ₂ preview control of active vehicle suspensions. This control scheme has two main aspects: first, it allows constrained outputs of the system to vary freely as long as they remain within their given bounds, in order that the best possible performance could be delivered. Secondly, the optimisation as well as constraint fulfilment is done for the worst-case road disturbances to cover all road types. To design a system to perform satisfactorily for a wide range of road irregularities, H _∞-norm is used wherever minimisation is required, and generalised H ₂-norm is used to care for the constraints on suspension working space. Moreover, to ensure desired stability margins for the system, pole location constraints are considered in the design. The proposed approach is evaluated on a quarter-car model and compared with the state-of-the-art preview control algorithm in the literature, namely, Linear quadratic Gaussian preview. Simulation results demonstrate the effectiveness of the proposed approach.     

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.     

Linear matrix inequality based control of vehicle active suspension system

Linear matrix inequality (LMI) methods, novel techniques in solving optimisation problems, were introduced as a unified approach for vehicle's active suspension system controller design. LMI methods were used to provide improved and computationally efficient controller design techniques. The active suspension problem was formulated as a standard convex optimisation problem involving LMI constraints that can be solved efficiently using recently developed interior point optimisation methods. An LMI based controller for a vehicle system was developed. The controller design process involved setting up an optimisation problem with matrix inequality constraints. These LMI constraints were derived for a vehicle suspension system. The resulting LMI controller was then tested on a quarter-car model using computer simulations. The LMI controller results were compared with an optimal PID controller design solution. The LMI controller was further tested by incorporating a nonlinear term in the vehicle's suspension model; the LMI's controller degraded response was enhanced by using gain-scheduling techniques. The LMI controller with gain-scheduling gave good results in spite of the unmodelled dynamics in the suspension system, which was triggered by large deflections due to off-road driving.     

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.     

Robust control and actuator dynamics compensation for railway vehicles

A robust controller is designed for active steering of a high speed train bogie with solid axle wheel sets to reduce track irregularity effects on the vehicle’s dynamics and improve stability and curving performance. A half-car railway vehicle model with seven degrees of freedom equipped with practical accelerometers and angular velocity sensors is considered for the H_∞ control design. The controller is robust against the wheel/rail contact parameter variations. Field measurement data are used as the track irregularities in simulations. The control force is applied to the vehicle model via ball-screw electromechanical actuators. To compensate the actuator dynamics, the time delay is identified online and is used in a second-order polynomial extrapolation carried out to predict and modify the control command to the actuator. The performance of the proposed controller and actuator dynamics compensation technique are examined on a one-car railway vehicle model with realistic structural parameters and nonlinear wheel and rail profiles. The results showed that for the case of nonlinear wheel and rail profiles significant improvements in the active control performance can be achieved using the proposed compensation technique.     

Nonlinear H∞ control for semi-active suspension

This paper describes the development of a damping control system for semi-active suspension which is based on nonlinear H_∞ control theory instead of conventional linear control theory. A two degrees of freedom system is used as the structure for the vehicle suspension model. Since the structure is bilinear, it's not easy to design the controller. We designed the controller based on the Hamilton-Jacobi inequality by solving a linear Riccati equation. We were able to verify by simulation that nonlinear H_∞ control theory made it possible to control vehicle vibration optimally and smoothly.     

Nonlinear robust control of integrated vehicle dynamics

A new methodology to design the vehicle GCC (global chassis control) nonlinear controller is developed in this paper. Firstly, to handle the nonlinear coupling between sprung and unsprung masses, the vehicle is treated as a mechanical system of two-rigid-bodies which has 6 DOF (degree of freedom), including longitudinal, lateral, yaw, vertical, roll and pitch dynamics. The system equation is built in the yaw frame based on Lagrange's method, and it has been proved that the derived system remains the important physical properties of the general mechanical system. Then the GCC design problem is formulated as the trajectory tracking problem for a cascade system, with a Lagrange's system interconnecting with a linear system. The nonlinear robust control design problem of this cascade interconnected system is divided into two H _∞ control problems with respect to the two sub-systems. The parameter uncertainties in the system are tackled by adaptive theory, while the external uncertainties and disturbances are dealt with the H _∞ control theory. And the passivity of the mechanical system is applied to construct the solution of nonlinear H _∞ control problem. Finally, the effectiveness of the proposed controller is validated by simulation results even during the emergency manoeuvre.     

Motorcycle suspension design using matrix inequalities and passivity constraints

This paper presents a design methodology for the suspension system of a novel aerodynamically efficient motorcycle. Since the machine’s layout and the rider’s seating position are unconventional, several aspects of the machine design, including the suspension, must be reviewed afresh. The design process is based on matrix inequalities that are used to optimise a road-grip objective function – others could be used equally well. The design problem is cast as the minimisation of an H ₂ cost with passivity constraints imposed on the suspension transference. The resulting bilinear matrix inequality problem is solved using a locally optimal iterative algorithm. The matrix inequality-type characterisation of positive real functions permits the optimisation of the suspension system over an entire class of passive admittances. Torsional springs, dampers and inerters are then used to construct networks corresponding to the optimal (positive real) admittances. Networks of first, second, third and fourth orders are considered, and an argument based on the compromise between complexity and improved grip is made for the most suitable suspension configuration. Finally, the effects of improved road grip on the stability of the vehicle’s lateral dynamics are analysed.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> preview control of an active suspension system with norm-bounded uncertainties

A robust H _∞ preview control is investigated for an active suspension system with look-ahead sensors. The uncertain system is described by a state-space model with linear nominal parts and additional nonlinear time-varying norm-bounded uncertainties. Proof of robust stability and a feedback-type robust H _∞ preview controller are derived by augmenting the dynamics of the original system and previewed road input. As, however, the augmented previewed road input gives the system a much larger dimension than the original system, much more computation time is required for solving of Riccati equations. To resolve this problem, a decomposed robust H _∞ preview controller is proposed. Robust stability and performance variations for system uncertainties are shown using a numerical example of a quarter-car model.     

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.     

Dynamic model for automotive side impact crashes

A rigid body model to represent a side impact crash is constructed using five degrees-of-freedom (dof) for the vehicle and three dof for each occupant in the vehicle. Nonlinear stiffness and damping elements and the presence of physical gaps between several components make the model highly nonlinear. The model is validated using experimental crash test data from a National Highway Traffic Safety Administration (NHTSA) database. To simplify the parameter identification process and reduce the number of parameters to be identified at each stage, a two-step process is adopted in which the vehicle is first assumed to be unaffected by the presence of the occupants, and its model parameters are identified. Subsequently, the parameters in the occupant models are identified.

The active set method with a performance index that includes both the L₂ and L_∞ norms is used for parameter identification. A challenge is posed by the fact that the optimisation problem involved is non-convex. To overcome this challenge, a large set of random initial values of parameter estimates is generated and the optimisation method is applied with all these initial conditions. The values of parameters that provide the minimal performance index from the entire set of initial conditions are then chosen as the best parameter values. The optimal parameters values thus identified are shown to significantly improve the match between the model responses and the experimentally measured sensor signals from the NHTSA crash test.     

H 2 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 H ₂ optimal control of systems with time-delayed disturbances and develop a computationally efficient approach for controller synthesis. We extend the Lyapunov-based H ₂ norm computation to systems with time-delayed disturbances and then derive a concise formula to explicitly evaluate the sensitivity of the system H ₂ norm with respect to controller gains. Thence, a set of necessary conditions for H ₂ 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 H ₂ 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.     

Two degrees of freedom PID multi-controllers to design a mathematical driver model: experimental validation and robustness tests

This paper proposes a mathematical driver model based on PID multi-controllers having two degrees of freedom. Each PID controller making up this model is synthesised by the Ziegler–Nichols oscillation method, using the linear time invariant models which are obtained around their nominal operating points. Different PID controllers are combined using nonlinear optimisation and the H _∞ constraint. To demonstrate its robustness, it was tested on two models: a linear parameter variant model and a nonlinear four-wheel model. It was also tested in situations of high dynamic demand. The driver model showed good performance, stability and trajectory tracking. The performance tests were carried out using experimental data acquired by a Laboratory Peugeot 307 developed by INRETS-MA. This driver model was developed for an application known as ‘Itinerary Rupture DIagnosis’ (DIARI), which aims to evaluate the physical limits of a vehicle negotiating a bend. DIARI requires a tool to determine the steering commands to be applied to a vehicle model, making extrapolations with respect to speed.     

Torque coordinating robust control of shifting process for dry dual clutch transmission equipped in a hybrid car

For a hybrid car equipped with dual clutch transmission (DCT), the coordination control problems of clutches and power sources are investigated while taking full advantage of the integrated starter generator motor's fast response speed and high accuracy (speed and torque). First, a dynamic model of the shifting process is established, the vehicle acceleration is quantified according to the intentions of the driver, and the torque transmitted by clutches is calculated based on the designed disengaging principle during the torque phase. Next, a robust H_∞ controller is designed to ensure speed synchronisation despite the existence of model uncertainties, measurement noise, and engine torque lag. The engine torque lag and measurement noise are used as external disturbances to initially modify the output torque of the power source. Additionally, during the torque switch phase, the torque of the power sources is smoothly transitioned to the driver's demanded torque. Finally, the torque of the power sources is further distributed based on the optimisation of system efficiency, and the throttle opening of the engine is constrained to avoid sharp torque variations. The simulation results verify that the proposed control strategies effectively address the problem of coordinating control of clutches and power sources, establishing a foundation for the application of DCT in hybrid cars.     

Vehicle dynamic stability improvements through gain-scheduled steering and braking control

This paper is concerned with the synthesis of a robust gain-scheduled ?_∞ MIMO vehicle dynamic stability controller (VDSC) involving both steering and rear braking actuators. This VDSC aims at improving automotive vehicle yaw stability and lateral performances. The aim of this work is to provide a methodology to synthesise such a controller while taking into account the braking actuator limitations and use the steering actuator only if it is necessary. These objectives are treated in an original way by the synthesis of a parameter-dependent controller built in the LPV framework and by the solution of an LMI problem. The proposed solution is coupled with a local ABS strategy to guarantee slip stability and make the solution complete. Nonlinear time and frequency domain simulations on a complex full vehicle model (which has been validated on a real car), subject to critical driving situations, show the efficiency and robustness of the proposed solution.     

System identification and attitude control of motorcycle by computer-aided dynamics analysis

System identification of the motorcycle model constructed by computer-aided dynamics analysis is introduced to design a control system for attitude stabilization of the motorcycle. The identified model can be reduced to the coupled mode system between the roll and the front steering. The front-steering control system using the roll angle is designed by H_∞ control theory, based on the reduced-order model and the full-order model, respectively. It is verified from simulation results that the motorcycle attitude against disturbance is stabilized by the H_∞ controller, and that the reduced-order controller exhibits efficient stabilization performance in comparison with the full-order controller.     

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.     

New energy recovery H∞ robust controller for electric bicycles

The electric controller is one of the most crucial components in an electric bicycle. The overall performance of the whole system heavily depends on the properties of the controller. The authors use the robust control theory to design a new H_∞ robust controller for the closed speed-current dual-loop driving and braking system. The designed controller also incorporates the driving and energy recovery braking circuit. Therefore, it has energy recovery ability, which coverts the kinetic energy wasted in braking into electric energy to recharge the battery. This prolongs the driving distance per battery charge. The simulations and experiments show that the new H_∞ robust controller out-performs the traditional PID controller in many respects including stability, error, responding speed and driving distance per battery charge.     

电磁主动悬架的设计及仿真研究

基于电磁学原理,利用电磁铁作为主动悬架的作动器,构造出电磁作动器的一般结构。在1/4汽车悬架的基础上,建立了电磁主动悬架的非线性模型,并应用现代控制理论设计了该模型的次优控制器,对该模型进行分析、仿真。模拟结果表明,电磁悬架能够实现一般主动悬架的功能,满足车辆平顺性的要求,可以适用于汽车的悬架系统。