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.     

Adaptive Suspension Concepts for Road Vehicles

Most vehicle suspensions are composed of passive spring and damper devices, although improved suspension performance is possible if an active system is used to control forces or relative velocities. The complexity, power requirements, and cost of fully active suspensions have restricted their use. Various partially active suspensions have been proposed and suspensions with slow load levelers and variable dampers are in widespread use. Here we analyze a class of basically passive suspensions the parameters of which can be varied actively in response to various measured signals on the vehicle. These suspensions can come close to optimal performance with simpler means than many of the active or semi-active schemes previously proposed.     

Optimal Performance of Variable Component Suspensions

SUMMARY

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.     

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.     

Adaptive Suspension Concepts for Road Vehicles

SUMMARY

Most vehicle suspensions are composed of passive spring and damper devices, although improved suspension performance is possible if an active system is used to control forces or relative velocities. The complexity, power requirements, and cost of fully active suspensions have restricted their use. Various partially active suspensions have been proposed and suspensions with slow load levelers and variable dampers are in widespread use. Here we analyze a class of basically passive suspensions the parameters of which can be varied actively in response to various measured signals on the vehicle. These suspensions can come close to optimal performance with simpler means than many of the active or semi-active schemes previously proposed.     

Stochastic Optimal Control of Highway Tractors with Active Suspensions

Stochastic optimal control and estimation theories are used to design an active suspension system for a cab ride in a tractor-semitrailer vehicle. A discrete-continuous vehicle model with eleven degrees of freedom is augmented by a stochastic road excitation model and a human perception of vibration shape filter. Both perfect measurement and estimated state cases are considered. The impact of the measurement noise on the design of the optimal controller is demonstrated. The performance of the optimally controlled system is compared with an optimal passive system. It is shown that significant improvements in ride comfort can be achieved through the use of actively controlled cab suspensions.     

An Active Suspension with Optimal Linear State Feedback

In this paper modern optimal control theory is applied to the design of an active suspension system for a motor vehicle. The road profile is assumed to be continuous and random with a power spectral density (P.S.D.) which varies inversely with the square of the frequency. The quadratic integral type performance index employed is a weighted sum of the integral squares of body acceleration, dynamic tyre deflection and relative body-to-axle displacement. A solution is obtained for the infinite time case which is both computationally and physically realizable as an active suspension in which the only continuous measurements required are the body absolute velocity and the body displacement relative to the road. The performance is compared with that of a conventional type passive suspension and found to be significantly better in practically all respects.     

Optimal Preview Control of Rear Suspension Using Nonlinear Neural Networks

The performance of neural networks to be used for identification and optimal control of nonlinear vehicle suspensions is analyzed. It is shown that neuro-vehicle models can be efficiently trained to identify the dynamical characteristics of actual vehicle suspensions. After trained, this neuro-vehicle is used to train both front and rear suspension neuro-controllers under a nonlinear rear preview control scheme. To do that, a neuro-observer is trained to identify the inverse dynamics of the front suspension so that front road disturbances can be identified and used to improve the response of the rear suspension. The performance of the vehicle with neuro-control and with LQ control are compared.     

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.     

An Active Suspension with Optimal Linear State Feedback

SUMMARY

In this paper modern optimal control theory is applied to the design of an active suspension system for a motor vehicle. The road profile is assumed to be continuous and random with a power spectral density (P.S.D.) which varies inversely with the square of the frequency. The quadratic integral type performance index employed is a weighted sum of the integral squares of body acceleration, dynamic tyre deflection and relative body-to-axle displacement. A solution is obtained for the infinite time case which is both computationally and physically realizable as an active suspension in which the only continuous measurements required are the body absolute velocity and the body displacement relative to the road. The performance is compared with that of a conventional type passive suspension and found to be significantly better in practically all respects.     

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.     

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.     

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.     

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。     

Design of passive vehicle suspensions for maximal least damping ratio

This paper studies the use of the least damping ratio among system poles as a performance metric in passive vehicle suspensions. Methods are developed which allow optimal solutions to be computed in terms of non-dimensional quantities in a quarter-car vehicle model. Solutions are provided in graphical form for convenient use across vehicle types. Three suspension arrangements are studied: the standard suspension involving a parallel spring and damper and two further suspension arrangements involving an inerter. The key parameters for the optimal solutions are the ratios of unsprung mass to sprung mass and suspension static stiffness to tyre vertical stiffness. A discussion is provided of performance trends in terms of the key parameters. A comparison is made with the optimisation of ride comfort and tyre grip metrics for various vehicle types.     

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

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.     

Theoretical Limitations in Active Vehicle Suspensions

Vehicle suspensions in which forces are generated in response to feedback signals by active elements obviously offer increased design flexibility compared to conventional suspensions using passive elements such as springs and dampers. It is often assumed that if practical difficulties are neglected, active systems could in principle produce arbitrary ideal, behavior. It is shown, using a simple linear two degree-of-freedom suspension system, model that even using complete state feed back and in the case of in which the system is controllable in the control theory sense, there still are limitations to suspension performance in the fully active case. If the ideal suspension performance is defined based on low-pass filtering of roadway unevenness inputs, an active suspension may not offer much better performance than a partially active or adaptive passive suspension depending upon the values of certain vehicle parameters.     

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.     

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.