Application of Nonlinear Control Theory to Electronically Controlled Suspensions

SUMMARY

This paper illustrates the use of nonlinear control theory for designing electro-hydraulic active suspensions. A nonlinear, “sliding” control law is developed and compared with the linear control of a quarter-car active suspension system acting under the effects of coulomb friction. A comparison will also be made with a passive quarter-car suspension system. Simulation and experimental results show that nonlinear control performs better than PID control and improves the ride quality compared to a passive suspension.     

Performance of Low-bandwidth, Semi-active Damping Concepts for Suspension Control

Active damping has been shown to offer increased suspension performance in terms of vehicle isolation, suspension packaging, and road-tire contact force. It can even approximate the performance of full state feedback control without requiring the difficult measurement of tire deflection. Many semi-active damping strategies have been introduced to approximate the response of active damping with the modulation of passive damping parameters. These strategies have typically required a relatively high bandwidth for actuator response. This paper investigates the simulation performance and “frequency response” of two concepts in low-bandwidth semi-active suspension control, one that sets a damping force directly and another that sets the damping resistance. The electronically controlled bandwidth of these actuators is approximately an order of magnitude less than other semi-active devices; high frequency control is handled mechanically. A quarter-car model is studied with the controlled damping replacing both passive and active damping of typical control schemes. Both low-bandwidth damping strategies perform remarkably well compared to both active and high-bandwidth, semi-active damping. In certain dynamic performances, the new semi-active strategies outperform active damping and what the author calls “nominal” semi-active damping.     

Optimal Linear Active Suspensions with Multivariable Integral Control

In this paper, an optimal suspension system is derived for a quarter-car model using multivariable integral control. The suspension system features two parts. The first part is an integral control acting on suspension deflection to ensure zero steady-sate offset due to body and maneuvering forces as well as road inputs. The second is a proportional control operating on the vehicle system states for vibration control and performance improvement. The optimal ride performance of the active suspensions based on linear full-state feedback control laws with and without integral control together with the performance of passive suspensions are compared.     

DISCUSSION NOTE on Karnopp's article “Active Damping in Road Vehicle Suspension Systems”

SUMMARY

In this paper, an optimal suspension system is derived for a quarter-car model using multivariable integral control. The suspension system features two parts. The first part is an integral control acting on suspension deflection to ensure zero steady-sate offset due to body and maneuvering forces as well as road inputs. The second is a proportional control operating on the vehicle system states for vibration control and performance improvement. The optimal ride performance of the active suspensions based on linear full-state feedback control laws with and without integral control together with the performance of passive suspensions are compared.     

主动汽车悬架的非线性控制

本文采用１／４车模型对天棚阻尼器和主动悬架的动力学性能进行分析，针对执行器的非线性特性，探讨了微分几何法和反馈法线必互法在主动悬架控制中的应用，在系统控制设计中采用了离散滑模法，仿真结果显示非线性控制律能有效地改善主动悬架的隔振特性。     

Performance of Low-bandwidth,Semi-active Damping Concepts for Suspension Control

Abstract

Active damping has been shown to offer increased suspension performance in terms of vehicle isolation, suspension packaging, and road-tire contact force. It can even approximate the performance of full state feedback control without requiring the difficult measurement of tire deflection. Many semi-active damping strategies have been introduced to approximate the response of active damping with the modulation of passive damping parameters. These strategies have typically required a relatively high bandwidth for actuator response. This paper investigates the simulation performance and “frequency response” of two concepts in low-bandwidth semi-active suspension control, one that sets a damping force directly and another that sets the damping resistance. The electronically controlled bandwidth of these actuators is approximately an order of magnitude less than other semi-active devices; high frequency control is handled mechanically. A quarter-car model is studied with the controlled damping replacing both passive and active damping of typical control schemes. Both low-bandwidth damping strategies perform remarkably well compared to both active and high-bandwidth, semi-active damping. In certain dynamic performances, the new semi-active strategies outperform active damping and what the author calls “nominal” semi-active damping.     

Vehicle Dynamic Analysis and Evaluation of Continuously Controlled Semi-Active Suspensions Using Hardware-in-the-loop Simulation

A semi-active suspension system with continuously variable damper is greatly expected to be used mainly in the future as a high-performance suspension system due to its cost-effectiveness, light weight, and low energy consumption. In this paper, to develop a suitable control logic for the semi-active suspension system, the hardware-in-the-loop simulation is performed for the experimental continuously variable damper combined with a quarter-car model, and the simulation results are compared for passive, on/off controlled, and continuously controlled dampers in the aspects of ride comfort and driving safety, assuming each damper to be installed on the vehicle.     

基于微分几何理论的汽车半主动悬架非线性振动控制

针对汽车悬架系统的非线性特性,采用1／4汽车二自由度悬架模型分析半主动悬架控制。应用微分几何理论得到输出-干扰解耦方法,再经适当的坐标变换将该模型由非线性系统简化成一线性系统,并对此系统进行最优控制,然后通过非线性状态反馈实现对原系统的半主动控制。与被动悬架的仿真结果进行了比较,表明这种针对具有非线性特征的半主动悬架的非线性控制方法是可行的。通过功率谱分析,控制后系统的能量比被动悬架更趋于平均,悬架动态性能更稳定。     

DSP-based self-organising fuzzy controller for active suspension systems

This study proposed a self-organising fuzzy controller (SOFC) for controlling an active suspension system to evaluate its control performance. During the control process, the SOFC continually updated the learning strategy in the form of fuzzy rules. The fuzzy rule table of this SOFC could be initially set to zero. This not only overcame the difficulty in finding appropriate membership functions and control rules for designing a fuzzy controller, but also solved the database problem where the fuzzy rules of a fuzzy controller, once determined, remained fixed and could not suitably regulate them in real time to optimise the dynamic response of the system required to gain the desired control performance. To demonstrate the applicability of the proposed SOFC for active suspension systems, a quarter-car hydraulic-servo suspension system was designed and constructed to evaluate the feasibility of active suspension control. Additionally, to conform to real-time application requirements in the vehicular industry, the SOFC was implemented with a digital signal processor to control the hydraulic-servo suspension system so that the control performance could be determined. The SOFC has shown a better control performance in suppressing the vibration amplitude of the vehicle body for enhancing the structural safety of the vehicle and increasing the life of the suspension system. It also effectively suppressed the amplitude of the vehicle body acceleration and reduced the tire deflection for improving the ride and the handling quality of a vehicle better than a passive control, as verified in experimental results.     

Optimal Performance of Variable Component Suspensions

Most vehicle suspension systems use fixed passive components that offer a compromise in performance between sprung mass isolation, suspension travel, and tireroad contact force. Recently, systems with discretely adjustable dampers and air springs been added to production vehicles. Active and semi-active damping concepts for vehicle suspensions have also been studied theoretically and with physical prototypes. This paper examines the optimal performance comparisons of variable component suspensions, including active damping and full-state feedback, for “quartercar” heave models. Two and three dimensional optimizations are computed using performance indicators to find the component parameters (control gains) that provide “optimal” performance for statistically described roadway inputs. The effects of performance weighting and feedback configuration are examined. Active damping is shown to be mainly important for vehicle isolation. A passive vehicle suspension can control suspension travel and tire contact force nearly as well as a full state feedback control strategy.     

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.     

汽车主动悬架系统的自适应模糊控制方法

本文提出了一种汽车主动悬架系统的自适应模糊控制方法，该模糊控制方法可以有线自适应调整模糊控制的有关参数。1/4车辆模型作为系统仿真对象，模糊逻辑控制器可以显著发减小车辆的振动及干扰，提高车辆受路面激励时车辆的舒适性。仿真结果清楚地表明该模糊控制方法的有效性。另外，当主动悬架系统模型参数发生变化时该模糊控制器表现出良好的鲁棒性。     

油气主动悬架非线性模型的建立、仿真与试验验证

本文通过分析油气悬架中存在的气体弹簧刚度非线性特性和摩擦力，建立了主动悬架的非线性模型。分析了悬架的非线性特性对悬架运动的影响。仿真和试验结果表明，非线性模型比线性模型更接近油气悬架的实际情况，根据非线性模型设计的车身高度控制策略比根据线性模型设计的控制策略具有更好的控制效果。     

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.     

Spectral Decomposition Methods for the Computation of RMS Values in an Active Suspension

The theory of the spectral decomposition of a matrix with distinct eigenvalues is reviewed briefly and then applied to the closed-loop system matrix of a quarter-car active suspension with preview. RMS values of the state variables and the control force are derived for an integrated white noise type random road input with the assumption of an optimised control system. The explicit formulæ obtained are easily computed numerically using MATLAB with accurate results at all speeds.     

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.     

Active Control of Non-stationary Response of Vehicles with Nonlinear Suspensions

SUMMARY

Active control of non-stationary response of a single degree of freedom vehicle model with nonlinear passive suspension elements is considered in this paper. The method of equivalent linearization is used to derive the equivalent linear model and the optimal control laws are obtained by using stochastic optimal control theory based on full state information. Velocity squared quadratic damping and hysteresis type of stiffness nonlinearities are considered. The effect of the nonlinearities on the active system performance is studied. The performance of active suspensions with nonlinear passive elements is found to be superior to the corresponding passive suspension systems.     

Linear Quadratic Gaussian Control of a Quarter-Car Suspension

This paper presents a method for designing linear multivariable controllers in the frequency-domain for an intelligent controlled suspension system for a quarter-car model. The design methodology uses singular value inequalities and optimal control theory. The vehicle system is augmented with additional dynamics in the form of an integrator to affect the loop shapes of the system. The measurements are assumed to be obtained in a noisy state, and the optimal control gain and the Kalman filter gain are derived using system dynamics and noise statistics. A combination of singular value analysis, eigenvalue analysis, time response, and power spectral densities of random response is used to describe the performance of the active suspension systems.     

Dual objective active suspension system based on a novel nonlinear disturbance compensator

This paper proposes an active suspension system to fulfil the dual objective of improving ride comfort while trying to keep the suspension deflection within the limits of the rattle space. The scheme is based on a novel nonlinear disturbance compensator which employs a nonlinear function of the suspension deflection. The scheme is analysed and validated by simulation and experimentation on a laboratory setup. The performance is compared with a passive suspension system for a variety of road profiles.     

Adaptive sliding control of active suspension systems with uncertain hydraulic actuator dynamics

An adaptive sliding controller is proposed in this article to control the active suspension systems of a quarter-car model with hydraulic actuator. The highly nonlinear actuator dynamics is assumed to have some time-varying uncertainties with unknown bounds. Owing to its time-variant nature, traditional adaptive designs are not feasible. As the variation bounds are not given, the conventional robust controllers cannot be applied either. In this article, we use the function approximation technique to represent the uncertainties with finite combinations of some basis functions, and the Lyapunov method is employed to find update laws for the coefficients of the approximating series. The actuator force can track the desired force generated from the skyhook dynamics with ultimately bounded performance. If a sufficient number of basis functions are used and the approximation error can be ignored, asymptotic convergence performance can be proved. If the bound of the approximation error is available, asymptotic convergence of the output error still can be obtained with some modifications of the proposed control law. Simulation results show that the controller proposed can give significant improvement of ride comfort when compared with the performance of its passive counterpart.