Two controllers for an automotive suspensions with Magneto-Rheological (MR) dampers are proposed. One is a model-based using the Linear Parameter Varying (LPV) approach, and the other is a model-free controller with a Frequency Estimation Based (FEB) principle. The LPV controller includes an experimental nonlinear model of an MR damper using only one scheduling parameter. A comparison with a several semiactive controllers for comfort and road holding is discussed. The FEB controller is the best option based on frequency and time response analysis for comfort (10–20%), suspension deflection (30–50%), and road holding (1–5%).
An MR damper is a nonlinear component used in semi-active suspensions. The accurate modelling of the The controller output is not the end damper manipulation, and it is needed because the computation of the desired force of the MR damper or the damping coefficient. This demands a mapping algorithm from control output to manipulation units. Free-model strategies are more efficient and feasible for implementation than complex controllers. The antiwindup mechanism is assumed by applying inverse MR damper models, but controllers do not have a feedback of a windup effect.
This paper deals with the (a) design of an LPV controller with one scheduling parameter based on a simple nonlinear MR damper model, (b) design of a free-model controller based on the deflection frequency estimation, and (c) a comparison with several well known comfort, and road-holding controllers. The comparison uses a
This paper is organized as follows. Section
The lumped parameter QoV model describes the sprung mass (
Passive and controlled suspension schemes.
Passive suspension (Baseline)
Controlled suspension
The suspension tuning in automotive applications is used to achieve good comfort, road holding, and safety suspension deflections. These goals demand several industrial specifications in the span of [0–20] Hz, Poussot-Vassal et al. [
The controlled suspension uses a semi-active damper (
The ideal damping coefficient (
Four state-of-the-art controllers were chosen for comparison: Sky Hook (SH) and Mix-1-Sensor (M1S) controllers for comfort, Groundhook (GH) controller for road-holding, and the
The aim of the SH control, Karnopp et al. [
The key principle is the selection of high/low damping at each sampling time to achieve a comfortable ride condition according to the resonance frequency of the QoV, Savaresi and Spelta [
It uses a fictitious damping element between the wheel and the ground parallel with the tire, Valasek et al. [
The combination of SH and GH is called hybrid control technique. The corresponding semi-active damping can be expressed as
The passive damper model is the baseline. It is represented by the
Nonlinear passive damping force used as baseline.
The nonlinear QoV model with a semi-active suspension is
The model describes a passive linear damping and stiffness coefficients and it includes a lineal varying damping of the preyield and postyield modes of operation of the MR fluid using a receding horizon computed from the velocity of piston
The representation of a QoV in the LPV framework (
The input of the damper model will be positive
The control input provided by the semi-active damper has a finite interval
The damping coefficient always must be positive. The term of (
The generalized system for the
Model with a semi-active bounded input saturation.
To meet the control specifications, two
Parameters for
|
| ||||
---|---|---|---|---|---|
|
|
|
|
|
|
33.29 | 28.54 | 52.32 | 5.26 | 1.95 | 51.1 |
| |||||
|
| ||||
|
|
|
|
|
Constant |
| |||||
1.29 | 0.072 | 4.09 | 0.136 | 0.152 |
|
LPV-based controller.
By assuming a harmonic motion (
Look-up table for selection of electrical current based on frequency estimation.
|
|
|
|
|
---|---|---|---|---|
|
|
|
|
|
The experimental training path for identification is a sinusoidal with variable amplitude
Parameters model.
Symmetric |
|||||
---|---|---|---|---|---|
|
|
|
| ||
1061 | −1307 | 401 | 128 samples | ||
| |||||
Asymmetric full modified SP model | |||||
| |||||
Parameters for |
|||||
|
|
|
|
|
|
| |||||
607 | 5370 | 4 | 127 | 398 | 90 |
|
|
| |||
441 | 7.8 | −20 | |||
| |||||
Parameters for |
|||||
|
|
|
|
|
|
| |||||
604 | −2836 | 4.2 | 128 | 530 | 265 |
|
|
| |||
421 | 7.4 | 1.2 |
ESR index for MR damper models.
Step | Learning | Testing | |||
---|---|---|---|---|---|
Experiment | 1 | 1 | 2 | 3 | 4 |
Full model SP | 0.031 | 0.016 | 0.013 | 0.025 | 0.073 |
|
0.089 | 0.211 | 0.247 | 0.123 | 0.194 |
Top plots correspond to the full modified SP model, and bottom to
The lumped parameters of the QoV have been taken from a commercial vehicle model, Table
QoV parameters.
Component of the QoV | Value | Units |
---|---|---|
Sprung mass ( |
522 | Kg |
Unsprung mass ( |
50 | Kg |
Spring stiffness ( |
83 | kN/m |
Tire stiffness ( |
230 | kN/m |
Suspension stroke | [0.05, −0.07] | m |
Suspension maximum velocity | [1.2, −1.5] | m/s |
Motion ratio |
0.614 | — |
|
1.2 | Hz |
|
11.5 | Hz |
The values of
Damping coefficients.
Damping |
|
|
|
|
---|---|---|---|---|
|
0.6 |
|
|
4,826 |
|
0.6 |
|
|
7,225 |
|
0.15 |
|
|
1,207 |
|
0.05 |
|
|
1,084 |
|
1 |
|
|
4,826 |
|
0.15 |
|
|
1,206 |
Since the benchmark controllers output has
Control schemes.
Control strategy for changing force to electric current. The strategy acts on a nonlinear MR damper model
Output of the proposed controller is the input of the nonlinear MR damper model
The evaluation of the QoV model was done in open- and close-control systems based on industrial specifications, Sammier et al. [
Six open-control system simulations with
Open-control system. Pseudo-Bode of:
Six control system simulations, each one with the control strategies SH, GH, Hybrid, M1S, FEB, and LPV were compared, Figure
Control system. Pseudo-Bodes for transfer functions (a) sprung-mass acceleration, (b) suspension deflection, and (c) road holding versus road profile.
Transfer function of
Transfer function of
Transfer function of
Figure
Improved percentages of PSD for (a) sprung-mass acceleration transfer function in comfort bandwidths, (b) suspension deflection in all bandwidths, and (c) tire deflection in road holding bandwidths.
Ride comfort and road holding normally occurs in 1-2 Hz and 10–15 Hz. Figure
Transient response for (a) sprung-mass acceleration at
Electric current of transient response for a sinusoidal road under resonance frequencies oscillation.
Improved RMS in transient response for (a) sprung-mass acceleration at
Thecomfort condition depends on
The suspension deflection transfer function improves by holding higher current below 2 Hz and between 5–20 Hz. In the span from 2–5 Hz and 16–20 Hz, the electric current has not influence on this performance, Figure
Transfer functions
Look-up table for the best performance in comfort and road holding.
|
0–2 | 2–12 | 12–14 | 14–20 |
---|---|---|---|---|
|
2.5 | 0 | 2.5 | 0 |
The baseline suspension is not optimized for comfort (i.e., it is a hard suspension) see Figure
The best controllers are FEB and M1S, Figure
The best performance corresponds to the baseline suspension. The best controller is the FEB. The M1S, GH, and LPV have similar performances. The SH and hybrid are not well suited, Figure
The hybrid does not achieve a good performance for comfort/road holding. The baseline suspension is optimized for suspension deflection and road holding. The controller with the best tradeoff is the FEB.
The proposed controllers have an improved performance of 10% and 17% in the primary ride frequencies (BW1), while the SH and M1S are the best (20% and 22%), Figure
The FEB controller shows optimum performances for sprung-mass acceleration, Figure
The FEB controller uses two damping coefficients, related to
Results show how the FEB controller obtains the comfort and road holding, while the hybrid controller does not achieve the minimum compromise. Also, the FEB controller follows the performance of SH and M1S in comfort (Figure
Two controllers for an automotive suspension with Magneto-Rheological (MR) dampers are proposed: one is based on the model using Linear Parameter-Varying (LPV) approach, and the other is a free of model using a frequency estimation of the road profile. A comparison with several semi-active control strategies for comfort and road holding was presented. Both controllers exhibit important features for practical applications: (1) the controller output is the electric current through MR damper coil, (2) the controllers achieve the objectives with a bounded output, (3) the scheduling parameter is based on one measurement, (4) there is no real-time computation of derivatives of matrix, hence the controllers allow good sampling time, (5) the LPV controller is linear combination of matrices, and the FEB computes the 1-norm of two signals over n samples, and (6) the controllers can modify their goal performances with a set of matrices for LPV, and a look-up table of electric current for (FEB).
MR damping coefficient (
Applied MR damping coefficient (
Max/Min damping coefficient (
Passive damping coefficient (
Skyhook/groundhook coefficient (
Gain of
Gain of
Preyield damping gain (s/m)
Postyield damping (N/A)
Preyield gain due to stiffness (1/m)
Virtual mass of the MR damper (kg)
Estimated frequency by FEB controller (Hz)
Internal stiffness coefficient (N/m)
Receding horizon in order to compute
Determines the damping coefficient magnitude when the device is in tension or compression.
Perturbation shaped as chirp sinusoidal
Piston deflection (m)
Piston deflection velocity (m/s)
Minimum/maximum measured
Piston deflection acceleration (
Sprung mass velocity, acceleration (
Unsprung mass displacement (m)
Unsprung mass velocity, acc (
Upper/lower limits in suspension (
Matrices of state space representation according to Do et al. [
Electric current, maximum
Vertical damper, steering force (N)
Vertical spring force (N)
MR damping force (N)
MR damping force (N) due to
Skyhook, groundhook force (N)
Mixed-1-sensor, hybrid Force (N)
Sinusoidal amplitude (m)
Weighting function road profile, sprung-mass acceleration, unsprung-mass displacement
Absolute maximum deflection velocity,
Motion ratio in wheel axis
Tradeoff parameter for comfort/road holding
Effect of the mechanical and hydraulic properties of the damper on
Semiactiveness of the LPV output controller
Dynamic saturation of the electric current limited to a maximum value
Scheduling parameter for LPV controller
Frequency (rad/s).
Authors thank the PCP 06/2007 program (CONACyT) and Autotronics Research Chair (Tecnologico de Monterrey).