Pole location control design of an active suspension system with uncertain parameters

This paper addresses the problem of robust control design for an active suspension quarter-car model by means of state feedback gains. Specifically, the design of controllers that assure robust pole location of the closed-loop system inside a circular region on the left-hand side of complex plane is investigated. Three sufficient conditions for the existence of a robust stabilizing state feedback gain are presented as linear matrix inequalities: (i) the quadratic stability based gain; (ii) a recently published condition that uses an augmented space and has been here modified to cope with the pole location specification; (iii) a condition that uses an extended number of equations and yields a parameter-dependent state feedback gain. Unlike other parameter-dependent strategies, neither extensive gridding nor approximations are needed. In the suspension model, the sprung mass, the damper coefficient and the spring constant are considered as uncertain parameters belonging to a known interval (polytope type uncertainty). It is shown that the parameter-dependent gain proposed allows one to impose the closed-loop system pole locations that in some situations cannot be obtained with constant feedback gains.     

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.     

Pole location control design of an active suspension system with uncertain parameters

This paper addresses the problem of robust control design for an active suspension quarter-car model by means of state feedback gains. Specifically, the design of controllers that assure robust pole location of the closed-loop system inside a circular region on the left-hand side of complex plane is investigated. Three sufficient conditions for the existence of a robust stabilizing state feedback gain are presented as linear matrix inequalities: (i) the quadratic stability based gain; (ii) a recently published condition that uses an augmented space and has been here modified to cope with the pole location specification; (iii) a condition that uses an extended number of equations and yields a parameter-dependent state feedback gain. Unlike other parameter-dependent strategies, neither extensive gridding nor approximations are needed. In the suspension model, the sprung mass, the damper coefficient and the spring constant are considered as uncertain parameters belonging to a known interval (polytope type uncertainty). It is shown that the parameter-dependent gain proposed allows one to impose the closed-loop system pole locations that in some situations cannot be obtained with constant feedback gains.     

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.     

Handling stability improvement through robust active front steering and active differential control

The design of the integrated active front steering and active differential control for handling improvement of road vehicles is undertaken. The controller design algorithm is based on the solution of a set of linear matrix inequalities that guarantee robustness against a number of vehicle parameters such as speed, cornering and braking stiffnesses. Vehicle plane dynamics are first expressed in the generic linear parameter-varying form, where the above-stated parameters are treated as interval uncertainties. Then, static-state feedback controllers ensuring robust performance against changing road conditions are designed. In a first series of simulations, the performance of the integrated controller is evaluated for a fishhook manoeuvre for different values of road adhesion coefficient. Then, the controller is tested for an emergency braking manoeuvre executed on a split-μ road. In all cases, it is shown that static-state feedback controllers designed by the proposed method can achieve remarkable road handling performance compared with uncontrolled vehicles.     

Generalized magnetorheological (MR) damper model and its application in semi-active control of vehicle suspension system

In this paper, analytical characterization of the magneto-rheological (MR) damper is done using a new modified algebraic model. Algebraic model is also more preferable because of its low computational expenses compared to differential Bouc-Wen’s model which is highly computationally demanding. This model along with the obtained model parameters is used as a semi-active suspension device in a quarter car model and the stationary response of the vehicle traversing on a rough road is obtained. The control part consists of two nested controllers. One of them is the system controller which generates the desired damping force and the other is the damper controller which adjusts the voltage level to MR damper so as to track the desired damping force. For the system controller a model reference skyhook Sliding Mode Controller (SMC) is used and for the damper controller a continuous state algorithm is built to determine the input voltage so as to gain the desired damping force. The analytical model is subsequently used in the quarter car vehicle model and the vehicular responses are studied. A simulation study is performed to prove the effectiveness and robustness of the semi-active control approach. Results show that the semi-active controller can achieve compatible performance as that of active suspension controller except for a little deterioration.     

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.     

基于Simulink的车辆主动悬架LQG控制器的设计

建立了二自由度1/4车体的数学模型,并利用线性最优化控制理论进行了汽车主动悬架的LQG控制器设计,并在Matlab/Simulink环境下进行仿真,结果表明具有LQG控制器的主动悬架对车辆行驶平稳性和乘坐舒适性有了很大的改善。     

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.     

Experimental evaluation of a vehicle steering assist controller using a driving simulator

Human-in-the-loop driving simulator experiments are conducted to evaluate a proposed robust steering assist controller that is designed on the basis of driver uncertainty modelling. A nominal controller (NC) that is designed without consideration of driver model uncertainty is also tested for comparison. Two types of experiments are proposed: a long driving task with nominal configurations and a short driving task with initially large lateral position error. The data are analysed using both time domain and frequency domain metrics. In the time domain, the standard deviation of lateral position error and percentage of road departure are used. In the frequency domain, the stability margins and crossover frequency are used. The driving simulator results indicate that statistically, the designed robust controller shows improvements in the short driving experiments. The improvements in the long driving experiments are less evident because of driver adaptation. The non-robust NC suffers from high gain and should be avoided. The benefits of considering driver model uncertainty in the design of vehicle steering assist controllers are, therefore, justified.     

Adaptive impedance control of a hydraulic suspension system using particle swarm optimisation

This paper presents a novel active control approach for a hydraulic suspension system subject to road disturbances. A novel impedance model is used as a model reference in a particular robust adaptive control which is applied for the first time to the hydraulic suspension system. A scheme is introduced for selecting the impedance parameters. The impedance model prescribes a desired behaviour of the active suspension system in a wide range of different road conditions. Moreover, performance of the control system is improved by applying a particle swarm optimisation algorithm for optimising control design parameters. Design of the control system consists of two interior loops. The inner loop is a force control of the hydraulic actuator, while the outer loop is a robust model reference adaptive control (MRAC). This type of MRAC has been applied for uncertain linear systems. As another novelty, despite nonlinearity of the hydraulic actuator, the suspension system and the force loop together are presented as an uncertain linear system to the MRAC. The proposed control method is simulated on a quarter-car model. Simulation results show effectiveness of the method.     

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.     

Design of rollover prevention controller with linear matrix inequality-based trajectory sensitivity minimisation

This paper presents a method to design a rollover prevention controller for vehicle systems. The vehicle rollover can be prevented by a controller that minimises the lateral acceleration and the roll angle. Rollover prevention capability can be enhanced if the controlled vehicle system is robust to the variation of the height of the centre of gravity and the speed of the vehicle. For this purpose, a robust controller is designed with linear matrix inequality-based trajectory sensitivity minimisation. Differential braking and active suspension are adopted as actuators that generate yaw and roll moments, respectively. The newly proposed method is shown to be effective in preventing rollover by the simulation on a non-linear multibody dynamic simulation software, CarSim^®.     

Robust optimization design method for powertrain mounting systems based on six sigma quality control criteria

This paper presents a robust optimization design method based on Six Sigma quality control criteria to improve the design of a powertrain mounting system (PMS). The powertrain is modeled as a rigid body having six degrees of freedom (DOF) connected to a rigid base by four rubber mounts, and each mount is simplified as a three-dimensional spring-damper element in its local coordinate system (LCS). The calculation method based on energy decoupling is used to estimate the decoupling ratios of a PMS. The location and static stiffness of each mount and the orientations of the two anti-torsion mounts are selected as uncertain design variables, and the nominal values of these design variables are optimized to obtain a robust Six Sigma design for a PMS. The uncertain design variables are characterized by a perturbation or percent variation around their nominal values. The generalized reduced gradient (LSGRG2) optimization method is employed to solve the robust optimization problem, and a second-order Taylor series expansion is used to estimate the statistical properties of the performance constraints and objectives. The optimization results show that the robust design ensures good robustness or high reliability for the natural frequencies, decoupling ratios, and frequency separation constraints of a PMS.     

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.     

Design of Robust Controllers for Active Vehicle Suspension Using the Mixed µ Synthesis

Summary The mixed µ synthesis for active suspension problems is proposed. Applying this method the real parametric uncertainties can be taken into consideration, which is more realistic than the traditional approaches, and the design process yields a less conservative compensator than other robust control design methods. The concept of the active suspension design using full-car models to handle the uncertain components is presented. The result of the mixed µ method is compared with the complex µ synthesis, and the passive system.     

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.     

Robust state feedback stabilization of articulated steer vehicles

In this work, a full-state feedback controller is designed to prevent the oscillatory instability or snaking behaviour of an articulated steer vehicle. To design the controller, first, a linearized model of the vehicle is developed and analyzed to identify the most important uncertain tire parameters with regard to the snaking mode. By using this linearized model, the equations of motion are represented in the form of a polytopic system, which depends affinely on the most important uncertain tire parameters. Then, by solving some linear matrix inequalities, both the Lyapunov and state feedback matrices for the robust stabilization of the vehicle are found. The performance of the resulting controller is evaluated by conducting several simulations based on the linearized model. To verify the results from the linearized model analysis, some simulations are also done by a virtual prototype of the vehicle in ADAMS. The results based on the linearized model are reasonably consistent with those from the simulations in ADAMS. They show that the controller can effectively stabilize the vehicle during the snaking mode in different driving conditions.     

Design of Robust Controllers for Active Vehicle Suspension Using the Mixed µ Synthesis

Summary The mixed µ synthesis for active suspension problems is proposed. Applying this method the real parametric uncertainties can be taken into consideration, which is more realistic than the traditional approaches, and the design process yields a less conservative compensator than other robust control design methods. The concept of the active suspension design using full-car models to handle the uncertain components is presented. The result of the mixed µ method is compared with the complex µ synthesis, and the passive system.