LMI-based robust model predictive control for a quarter car with series active variable geometry suspension

TL;DR

This work addresses robust suspension control for a quarter-car equipped with Series Active Variable Geometry Suspension (SAVGS). It develops an uncertain linear equivalent model to bridge the nonlinear SAVGS dynamics and uses an LMI-based robust model predictive controller (RMPC) that computes state-feedback gains online while explicitly enforcing actuator constraints. A parallel PI controller is incorporated to ensure zero steady-state error at low frequencies, and a systematic constraint mapping links nonlinear high-fidelity dynamics to the linear RMPC framework. Numerical simulations under ISO road profiles show that the proposed RMPC outperforms passive suspension and a prior $H_{\infty}$ controller in ride comfort (reduced $\ddot{z}_{s}$) and road holding (reduced $\Delta l_{t}$), achieving better actuator utilization and respecting all constraints. Overall, the approach demonstrates increased robustness and performance for SAVGS in real-world driving scenarios with bounded disturbances and model uncertainties.

Abstract

This paper proposes a robust model predictive control-based solution for the recently introduced series active variable geometry suspension (SAVGS) to improve the ride comfort and road holding of a quarter car. In order to close the gap between the nonlinear multi-body SAVGS model and its linear equivalent, a new uncertain system characterization is proposed that captures unmodeled dynamics, parameter variation, and external disturbances. Based on the newly proposed linear uncertain model for the quarter car SAVGS system, a constrained optimal control problem (OCP) is presented in the form of a linear matrix inequality (LMI) optimization. More specifically, utilizing semidefinite relaxation techniques a state-feedback robust model predictive control (RMPC) scheme is presented and integrated with the nonlinear multi-body SAVGS model, where state-feedback gain and control perturbation are computed online to optimise performance, while physical and design constraints are preserved. Numerical simulation results with different ISO-defined road events demonstrate the robustness and significant performance improvement in terms of ride comfort and road holding of the proposed approach, as compared to the conventional passive suspension, as well as, to actively controlled SAVGS by a previously developed conventional H-infinity control scheme.

Figures (11)

• Figure 1: SAVGS application to a quarter car double-wishbone suspension yu2017quarter. $\theta_{SL}$ denotes the single-link angle with respect to the horizontal plane (as shown in the diagram it has a negative value). The superscript $(se)$ refers to the static equilibrium state with zero torque $T_{SL}=0$ applied on the single link. The superscript $(ne)$ denotes the nominal equilibrium state, where $\theta_{SL}^{(ne)}-\theta_{SL}^{(se)}=\Delta\theta_{SL}^{(ne)}=90\degree$ (Note that $\theta_{SL}-\theta_{SL}^{(se)}=\Delta\theta_{SL}$).
• Figure 2: (a) Schematic of SAVGS linear equivalent quarter car model ($z_s$ and $z_u$ are the linear equivalent displacements of the sprung and unsprung mass in the vertical direction, respectively, $z_r$ is the vertical displacement of road surface, and ${l_{{SD}}^{(eq)}}$ is the equivalent spring-damper length); (b) transformation between linear equivalent and multi-body models of the SAVGS quarter caryu2017quarter. The function $a$ converts the control input $u=\dot{z}_{lin}$ to the rotary single-link velocity $(\omega_{SL}=\dot{\theta}_{SL})$, and the function $\beta$ converts the SL angle $\Delta \theta_{SL}$ to the linear actuator displacement $\Delta z_{lin}$. $d$ is the derivative of the road displacement profile and is considered an exogenous unbounded disturbance, and $y$ represents the measurable system's state $x(t)=[\dot{z}_{s},\dot{z}_{u},\Delta l_{s},\Delta l_{t}, \Delta z_{lin}]^T$ .
• Figure 3: RMPC scheme with PI incorporated plant model of linear equivalent quarter car SAVGS, where $\bar{\mathbf{d}}$ corresponds to the stacked vector for disturbance bound, ${P}'$ to the new uncertain system with parallel PI incorporated, and $\Delta z_{lin}^{(ref)}$ to the exogenous reference signal of linear actuator displacement.
• Figure 4: Plots of functions $\alpha$ (top) and $\beta$ (bottom) for conversion between the multibody and linear equivalent models.
• Figure 5: Overall simulation block diagram, which contains the high fidelity nonlinear model of quarter car and actuator ('QC SAVGS' and 'Single-link actuator', respectively, where $\omega_{SL}^*$ is the reference of $\omega_{{SL}}$ tracked by the actuator) and the proposed coupled closed-loop control scheme. The 'Online RMPC Scheme' uses the linear equivalent uncertain quarter car model and initial feasible sets $(Y^*,\tilde{Y}^*)$ based on offline created lookup tables. The first four elements of the state $x$ as defined in the subsection \ref{['sec:2-2']} are measurable at the output of the high-fidelity model and the last state component ($\Delta z_{lin}$), which is also used as input to the PI controller, is computed through the function $\beta$ and the measurable output $\Delta \theta_{SL}$ of the high fidelity model, as explained in subsection \ref{['subsec:ConstrainsAssociation']}.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5