Design of a fast real-time LPV model predictive control system for semi-active suspension control of a full vehicle

doi:10.1016/j.jfranklin.2018.11.016

Journal of the Franklin Institute

Volume 356, Issue 3, February 2019, Pages 1196-1224

https://doi.org/10.1016/j.jfranklin.2018.11.016 Get rights and content

Highlights

•
This study proposes a fast control scheme aiming the optimal handling and comfort performances of a full vehicle with four semi-active suspensions.
•
The control structure is a Model Predictive Controller with a Linear Parameter Varying model of the (nonlinear) vehicle system.
•
The computation of the control law is sub-optimal, being computed within 5 ms, apt for a real-time implementation.
•
An H 2 extended observer is also designed, in order to estimate the system's states and the road profile the vehicle is subject to, validated in a real experimental test-rig.
•
The studied control method is compared, through realistic nonlinear simulation, to an analytical clipped MPC and to an uncontrolled (passive) damper, in order to illustrate a much improved performance.

Abstract

This article is concerned with the control of a Semi-Active suspension system of a 7DOF Full Vehicle model, equipped with four Electro Rheological (ER) dampers, taking into account their incipient dissipativity constraints. Herein, a real-time, fast, advanced control structure is presented within the Model Predictive Control framework for Linear Parameter Varying (LPV) systems. The control algorithm is developed to provide a suitable trade-off between comfort and handling performances of the vehicle in a very limited sampling period ( $T_{s} = 5 ms$ ), in view of a possible realtime implementation on a real vehicle. The control structure is tested and compared to other standard fast control approaches. Full nonlinear realistic simulation results illustrate the overall good operation and behaviour of the proposed control approach.

Introduction

In order to enhance a vehicle’s driving performance in terms of road handling and ride comfort, one should take a special care to the vehicle suspension system. More and more present in the automotive industry, Semi-Active suspension systems should be highlighted, being efficient and, at the same time, less energy-consuming and less expensive than purely active suspensions.

The use of semi-active suspension systems provides a good trade-off between costs and performance requirements. This type of suspension is present in recent state-of-the-arthigh range-cars and a good deal of academic and industrial research works have been focused on this topic. Further details on semi-active suspension systems are thoroughly discussed in [1], [2].

The main challenge faced by semi-active suspension control problems is how to handle the dissipativity constraints of the semi-active damper. Several control design problems have been worked out with a range of different approaches. In [3], [4], one can find an extensive review of approaches towards semi-active suspension control; the references therein can provide more details and serve for further studies for readers. Indeed, some of the most recent and modern control techniques have been applied for this problem. In [5], an LQ-based clipped optimal control is proposed; an H_∞ control approach is presented in [6]; LPV control approaches, dealing with the dissipativity constraints of these suspension systems, are given in [7], [8]; a robust control approach with input and state constraints is developed in [9]. Nonlinear control methods, such as backstepping and adaptive control, have also been applied [10], [11], [12].

Nevertheless, a more natural approach towards optimal control of processes subject to constraints is the Model Predictive Control (MPC) framework, as thoroughly introduced in [13]. MPC allows to explicitly consider the effect of input (actuator) and state constraints in the control design process [14]. However, it is a known fact that the computation requirements of predictive controllers (MPC) are usually high, due to the complex optimization problem which has to be solved online, at every sampling period.

The control of semi-active suspension systems consists in “manipulating” the damping coefficient of a controlled damper, the actuator from the system’s point-of-view. Semi-active dampers have dissipativity constraints that can be tackled elegantly as an actuator saturation problem within the MPC framework.

Some works have employed an MPC approach for semi-active suspension systems, although most of these studies only consider a simpler quarter-car vehicle model. However, the quarter-car model (and half-car model as well) is not sufficient to describe the dynamics of a full vehicle with four semi-active dampers. The idea of solving the control problem at each corner of the car (four separate controllers) might seem appealing and simple enough, but the effects of coupling and load transfer distribution between corners may not be handled, which should lead to degraded performance, as discussed in [15]. Still on this matter, this solution presents its difficulties for a real-time implementation, given that four control laws have to be computed within the sampling period.

The following references are emphasized :

Considering quarter-car models:

•
In [16], a fast MPC scheme is designed for a quarter-car vehicle model, but the computation of the control law is sub-optimal, due to conditional constraints;
•
In [17], a methodology is proposed for optimal semi-active suspension control, based on MPC, considering a quarter-car vehicle model and previously-measured road disturbances;
•
Likewise, in [18], the proposition of a clipped-optimal control algorithm is seen, with some experimental results;
•
Finally, in [19], a hybrid MPCcontroller is presented, with some strong discussion on the use of a clipped analytical MPC approach.

half Considering half-car and other models:

•
In [20], a fast MPC scheme is designed for a half-car vehicle model, where the controller is tuned based on a quarter-car suspension model and does not take into account the effect of future disturbances;
•
Finally, in [21], an MPC is formulated aiming safe handling performances and validated by experimental results with a 10 ms sampling period, considering a linear bicycle model and an affine force-input model.

Throughout literature, only few studies have been concerned by multivariable MPC semi-active control techniques considering the full car dynamics. In [22], a nonlinear programming solution approach to this problem is proposed, considering an approximate description of constraints and dependent on the use of a camera to preview future disturbances, which might not be practically implementable due to costly expenses.

Considering the author’s studies in [15], a full vehicle semi-active suspension MPC control is formulated and solved using Mixed Integer constraints and optimization, where simulation results show the interest of this control approach. A more detailed version, presented in the P.h.d dissertation [23], shows that practical implementation on author’s vehicle testbed cannot be achieved, since that the computational time of the MPC (around 50 ms) is much greater than the sampling period.

In this work, a practically implementable semi-active suspension MPC controller is developed for a full vehicle with 4 semi-active dampers, designed in an LPV framework.

Indeed, an LPV system representation is provided here in order to rewrite the nonlinear input constraints of the dampers (dissipativity constraints) into linear constraints dependent on scheduling parameters.

Hence, the contributions are listed below:

(i)
A Fast LPV-MPC suspension control for a Full Car is presented by solving a suboptimal quadratic minimization problem with polyhedral constraints, with explicit mathematical methods. The theoretical innovations of this topic reside in the proposition of a Linear Parameter Varying (fast) predictive controller that can solve the suspension control problem in a sufficiently small computational time (less than 5 ms), allowing possible real-time embedded application, which has not been seen previously in the available literature.
(ii)
An H₂ extended state observer is presented in order to estimate system states and disturbances. It is also used to predict future road profile disturbances. Notice that the performances of MPC controllers are improved with the use of accurate future disturbance information, as clearly seen in a class of applications, e.g. in [24], [25]. This is a also novel practical contribution.
(iii)
The theoretical formulation, simulation results and observer validation are presented with details, showing the interest of the proposed control approach. Comparisons with other simpler control approaches are provided in simulation, considering a full realistic nonlinear vehicle model. The proposed scheme is able to effectively enhanced the driving performances of a vehicle, according to some performance objectives, even when facing bumpy, rough roads.

The structure of this article is given as follows: Section 2 describes the LPV full vertical vehicle model with 4 semi-active suspensions, Section 3 presents the proposed fast LPV MPC controller for the semi-active suspension problem, Section 4 is devoted to the practical implementation of the control scheme, considering an extended H₂ state-observer to estimate future disturbances and system states, validated with the aid of experimental results. In Section 5, results of the global control scheme are presented and analysed in a thorough discussion. Finally, conclusions are drawn in Section 6.

Section snippets

An LPV control-oriented vehicle model

Firstly, this Section presents the dynamical model of a vehicle’s vertical behaviour. This is a classic 7 degrees of freedom (DOF) suspension model, as seen in Fig. 1 and adapted from [26]; it will be used for control design purposes. This model involves the chassis dynamics (vertical displacement (z_s), roll angle (θ) and pitch angle (ϕ)) and the vertical displacements of the wheels ( $z_{u s_{i j}}$ ) at the front/rear - left/right corners ( $i = (f, r)$ and $j = (l, r)$ ). This $7 - D O F$ model is governed by the

A fast LPV model predictive control solution for semi-active suspension control

The main objective of the semi-active automotive suspensions is to isolate the body from the road disturbances and improve road handling [2]. These two objectives can be referred to as comfort performance and handling performance, respectively, and can be described through the vehicle’s COG acceleration (given by ${\ddot{z}}_{s}$ ) and roll angle (given by θ), as seen in [27].

For control design purposes, two perfomance indexes, with respect to each control objective, are considered: $\begin{matrix} J_{c o m f o r t} & = & \int_{0}^{τ} {\ddot{z}}_{s}^{2} (t) d t \end{matrix}$ $\begin{matrix} and \\ J_{r o} \end{matrix}$

Practical implementation

Assumptions 3.1 and 3.2 consider that the proposed LPV-FMPC controller has to be fed with x[k] and $\hat{w} [k]$ by some other scheme. Indeed, to ensure the feasibility and accuracy of the control objectives, this work considers an extended observer approach to estimate the system states and road disturbances. Hereafter, the experimental testbed and the observer design methodology are presented.

Results, analysis and discussion

Simulation results are presented and discussed, considering the problem of controlling a Full Vehicle’s Semi-Active Suspension, within the Model Predictive Control framework.

In order to improve to truthfulness of the simulations, bringing them closer to realistic conditions, the following simulation results consider a full nonlinear vehicle model, presented in [38], with the parameters given in Table 1. This means that simulation runs mimic the real life conditions of the INOVE 1/5-sized car.

Conclusion and future works

This article presented a complete control system for a semi-active suspension system. As shown by realistic simulation results and experimental tests, the proposed scheme is able to effectively enhance the driving performances of a vehicle, according to performance objectives, despite the presence of bumpy roads. Some key points of this work should be stressed:

•
The proposed controller is a fastLinear Parameter Varying MPC scheme developed for a real-timeapplication with a sampling frequency of

Acknowledgements

This work has been partially supported by the LabEx PERSYVAL-Lab ( $A N R - 11 - L A B X - 0025 - 01$ ), funded by the French program Investissements d’avenir. The authors also thank CAPES for financing project BRAFITEC ECoSud.

References (40)

D. Fischer et al.
Mechatronic semi-active and active vehicle suspensions
Control Eng. Pract.
(2004)
C. Poussot-Vassal et al.
Survey and performance evaluation on some automotive semi-active suspension control methods: a comparative study on a single-corner model
Ann. Rev. Control
(2012)
A. Unger et al.
Application of LQ-based semi-active suspension control in a vehicle
Control Eng. Pract.
(2013)
H. Du et al.
Semi-active h_∞ control of vehicle suspension with magneto-rheological dampers
J. Sound Vibr.
(2005)
C. Poussot-Vassal et al.
A new semi-active suspension control strategy through LPV technique
Control Eng. Pract.
(2008)
M.Q. Nguyen et al.
A state feedback input constrained control design for a 4-semi-active damper suspension system: a quasi-LPV approach
IFAC-PapersOnLine
(2015)
W. Sun et al.
Multi-objective control for uncertain nonlinear active suspension systems
Mechatronics
(2014)
P. Brezas et al.
A clipped-optimal control algorithm for semi-active vehicle suspensions: theory and experimental evaluation
Automatica
(2015)
A. Pawlowski et al.
Predictive control with disturbance forecasting for greenhouse diurnal temperature control
IFAC Proc. Vol.
(2011)
C. Poussot-Vassal et al.
Attitude and handling improvements through gain-scheduled suspensions and brakes control
Control Eng. Pract.
(2011)

Y. Wang et al.

Fast model predictive control using online optimization

IFAC Proc. Vol.

(2008)

A. Núñez-Reyes et al.

A smith predictive based MPC in a solar air conditioning plant

J. Process control

(2005)

A. Pawlowski et al.

Improving feedforward disturbance compensation capabilities in generalized predictive control

J. Process Control

(2012)

M.J. Khosrowjerdi et al.

A mixed h₂/h_∞ approach to simultaneous fault detection and control

Automatica

(2004)

S.M. Savaresi et al.

Semi-active Suspension Control Design for Vehicles

(2010)

H.E. Tseng et al.

State of the art survey: active and semi-active suspension control

Veh. Syst. Dyn.

(2015)

A.-L. Do et al.

An LPV control approach for semi-active suspension control with actuator constraints

American Control Conference (ACC), 2010

(2010)

W. Sun et al.

Saturated adaptive robust control for active suspension systems

IEEE Trans. Ind. Electron.

(2013)

W. Sun et al.

Vibration isolation for active suspensions with performance constraints and actuator saturation

IEEE/ASME Trans. Mechatron.

(2015)

E.F. Camacho et al.

Model Predictive Control

(2013)

Cited by (60)

Model predictive control of a semi-active suspension with a shift delay compensation using preview road information
2023, Control Engineering Practice
This paper presents a model predictive control of a semi-active suspension with a shift delay compensation using preview road information. A Model Predictive Control (MPC) methodology has been developed to optimize both ride comfort and road handling performance, where suspension states have been estimated through a model-based Kalman filter. The use of MPC in semi-active suspension control has proven to be suitable due to its ability to consider constraints when optimizing the objective function. A feasible region for the control inputs into the semi-active suspension has been determined from its mechanical limitations. However, the proposed Model Predictive Controller carries a sizeable computational load. Occurrences of “shift time delays” have been observed from the vehicle’s Electronic Control Unit processing the algorithm. To combat this, a model prediction system, in which a full-car dynamic model and road preview information are utilized to predict vehicle suspension states, has been further proposed to compensate for the shift time delay. Preview road information has been obtain through the computation of relative vehicle motion and a temporal–spatial conversion. The overall algorithm has been evaluated via computer simulation studies. Simulation results have shown that the shift compensation algorithm provides considerable improvements with regards to ride comfort. A noteworthy section of this study is the successful implementation of the proposed algorithms into an actual vehicle, where the real world performance of the proposed algorithms have been assessed. It has been shown via vehicle tests that significant improvement in ride comfort can be obtained by the proposed MPC damping control.
Gain-adaptive Skyhook-LQR: a coordinated controller for improving truck cabin dynamics
2023, Control Engineering Practice
Truck manufacturers place more emphasis on drivers’ health, safety, and comfort in recent decades. Some progresses have been made in reducing drivers’ vibration exposure levels by adding semi-active cabin suspension systems. Considering various driving conditions of a commercial vehicle, an advanced cabin suspension control needs to be capable of solving multi-objective optimization problems in real-time and adapting to different road conditions. To minimize the undesired vibrations in vertical, pitch, and roll directions, a new coordinated control system is introduced in this paper, namely integrated Skyhook-LQR. Specifically, the controller integrates Skyhook and Linear Quadratic Regulator (LQR) in order to attenuate the vibrations transmitted to the cabin, by which the computational cost is affordable for real-time implementation on microcontrollers. Moreover, a gain-adaptive algorithm is designed to intelligently adjust the controller gains based on the higher modes of the cabin vibrations. The performance of the proposed control system is evaluated through the co-simulation between ADAMS/Car and MATLAB/Simulink. The results demonstrate that the gain-adaptive Skyhook-LQR takes the advantage of the Skyhook control when the vertical disturbances are dominate and utilizes the benefits of the LQR algorithm when all modes of vibrations (pitch, roll, and vertical) are present. As a result, the proposed coordinated control system achieves optimum ride comfort under various driving and road conditions with minimum computational costs.
Design of shifting state-feedback controllers for LPV systems subject to time-varying saturations via parameter-dependent Lyapunov functions
2022, ISA Transactions
This paper considers the problem of designing a shifting state-feedback controller via quadratic parameter-dependent Lyapunov functions (QPDLFs) for systems subject to symmetric time-varying saturations. By means of the linear parameter varying (LPV) framework and the use of the shifting paradigm and the ellipsoidal invariant theory, it is shown that the solution to this problem can be expressed with linear matrix inequalities (LMIs) which can efficiently be solved via available solvers. Specifically, three hyper-ellipsoidal regions are defined in the state-space domain for ensuring that the control action remains in the linearity region of the actuators where saturation does not occur. Furthermore, the closed-loop convergence speed is regulated online according to the instantaneous saturation limit values through the shifting paradigm concept. The main characteristics of the proposed approach are validated by means of two illustrative examples.
Road classification using built-in self-scaling method of Bayesian regression
2022, Journal of Sound and Vibration
This paper proposes the use of the built-in self-scaling (BS) method for ISO classification of road roughness. The technique employs the transfer function between the vehicle body acceleration as input and the suspension travel as output. This transfer function has a nonzero dc gain, which is important for application of the BS method. Frequency response magnitude patterns corresponding to this transfer function are estimated via Bayesian regression, capitalizing on the inherent properties of the BS method where the prior dc gain is incorporated into the formulation. This strategy leads to high classification accuracy. The proposed approach requires only low-cost sensors. It possesses a short detection time of 0.5s and a short training time of 5s for each road class. The method is model-free and does not require recalibration when the load carried by the vehicle changes. Additionally, it is capable of handling varying vehicle velocity and is effective for both passive and active suspensions. A laboratory-scale experiment shows that the proposed technique increases the percentage of correct classification by an average of 34% in the case of constant road profiles, compared with a state-of-the-art method using augmented Kalman filtering. A corresponding value of 24% is achieved for a varying road profile. The significant improvement in the accuracy of road classification is impactful as it will enable controller design for suspension systems to be enhanced resulting in more comfortable ride and higher vehicle stability.
Practical Solution for Attenuating Industrial Heavy Vehicle Vibration: A New Gain-Adaptive Coordinated Suspension Control System
2024, SSRN
Event-triggered based finite-frequency control for vehicle electrohydraulic suspension with force tracking
2024, International Journal of Robust and Nonlinear Control

View all citing articles on Scopus

View full text

Education ArticleDesign of a fast real-time LPV model predictive control system for semi-active suspension control of a full vehicle

Highlights

Abstract

Introduction

Section snippets

An LPV control-oriented vehicle model

A fast LPV model predictive control solution for semi-active suspension control

Practical implementation

Results, analysis and discussion

Conclusion and future works

Acknowledgements

Control Eng. Pract.

Ann. Rev. Control

Control Eng. Pract.

J. Sound Vibr.

Control Eng. Pract.

IFAC-PapersOnLine

Mechatronics

Automatica

IFAC Proc. Vol.

Control Eng. Pract.

IFAC Proc. Vol.

J. Process control

J. Process Control

Automatica

Semi-active Suspension Control Design for Vehicles

State of the art survey: active and semi-active suspension control

Veh. Syst. Dyn.

An LPV control approach for semi-active suspension control with actuator constraints

American Control Conference (ACC), 2010

Saturated adaptive robust control for active suspension systems

IEEE Trans. Ind. Electron.

Vibration isolation for active suspensions with performance constraints and actuator saturation

IEEE/ASME Trans. Mechatron.

Model Predictive Control

Education Article
Design of a fast real-time LPV model predictive control system for semi-active suspension control of a full vehicle