Robust Control for an Electric Power Steering with unconsidered Modes and Parameter Uncertainties

chassis.tech 2019, Munich, 25-26 June 2019

M. Irmer, H. Henrichfreise,
Technische Hochschule Köln
Labor für Mechatronik (Cologne Laboratory of Mechatronics, CLM),
Betzdorfer Str. 2
50679 Köln
info@clm-online.de
www.clm-online.de


H. Briese, M. Haßenberg,
DMecS Development of Mechatronic Systems GmbH und Co. KG,
Gottfried-Hagen-Straße 20
51105 Köln
info@dmecs.de
www.dmecs.de

Abstract

For the control of electric steering systems, there is still a need for a robust implementation due to unconsidered degrees of freedom, nonlinear spring stiffnesses and gear ratios. Therefore, this article describes the modeling, control design and system analysis of an electric power steering system. It is focused on the degree of robustness of the possible controls. Two LQG-controllers are designed which are able to perform active vibration damping and disturbance compensation. This allows a high control bandwidth. Both resulting control systems are robust against unconsidered degrees of freedom, nonlinear system behavior and variations of plant parameters. Thus, the presented controllers fulfill the requirements of a modern steering system and allow the adaptation of the steering feel to the current driving situation.

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DMecS_RobustControlForEPS.pdf (571 KB)

1 Introduction

An LQG/LTR-design to control the driver’s steering torque for electromechanical power steering (EPS) systems was presented in [1,2]. There, a simplified plant model for the steering system with two degrees of freedom, which only considers the elasticity of the torsion bar, is used for the controller resp. observer design. However, several other elastic components exist in a real steering mechanism. These elasticities were considered in [3,4] in a new detailed model of the steering system.

Disregarding degrees of freedom (eigenmodes) of the plant model for the controller resp. observer design can lead to stability problems (spillover instability) in the closed-loop system [5]. The effects of unconsidered (residual) eigenmodes on the behavior of the feedback control of a steering system were investigated in [3,4]. In this article, further reduced (simplified) design models are developed using a model-based approach, and a systematic analysis in the time and frequency domain is performed. For this, in section 2 the detailed model developed in [3,4] is briefly described. It contains all relevant elasticities occurring in a steering mechanism. Based on this detailed model, in section 3 reduced design models for the controller resp. observer design are derived. In section 4 the closed-loop systems resulting from these different design models are analyzed and compared with respect to their dynamic behavior as well as their robustness against unconsidered eigenmodes and parameter uncertainties in the plant model. Finally, further measures for improvement are described and a summary is given in section 5.

All investigations are done based on an axially parallel EPS. The methods can also be applied to other configurations and systems.

2 Detailed Model of the Steering Mechanism

Fig. 1 depicts the physical model of the detailed steering mechanism with eight degrees of freedom (8DOF). The individual rigid bodies are labeled with the indices S (steering), P (pinion), R (rack), N (nut), M (motor), C (casing), V (vehicle), WL (left wheel) and WR (right wheel). The elasticities in the model are considered by the torsion bar stiffness ctb (torque sensor), stiffness cNR of the ball screw drive, belt drive stiffness cMN, stiffness cNC of the axial nut bearing, stiffness cCV of the casing attachment and the stiffnesses cRWL and cRWR of the attachments of the left and right wheel.

Figure 1 Physical Model of the Steering Mechanism with eight Degrees of Freedoms

The equations of motion of this model have been derived, linearized and converted into state space representation. Based on this model with eight degrees of freedom, further simplified models are generated by model order reduction. Each of these different models for the steering mechanism is extended by a simplified, linear model of a current-controlled EPS motor in form of a first order lag system. Combined, they form the plant model. The plant model consisting of the current-controlled EPS motor and the detailed model of the steering mechanism with eight degrees of freedom will be called “8DOF” in the subsequent sections.

The frequency response of the control transfer path of the detailed plant model “8DOF” from the reference current iref to the torsion bar torque Ttb is shown in Fig. 2. There, the frequency response of the undamped model is illustrated to highlight the locations of the eigenfrequencies (black circled digits) and the notch frequencies (grey circled digits). The stiffness ctb of the elastic torsion bar dominantly affects the oscillation with the eigenfrequency ① of about 51 rad/s (8 Hz) [1,2]. The eigenfrequencies ② and ③ as well as the notch frequencies ① and ② of the control transfer path of the plant model are characterized by the in-phase and counter-phase displacement of the wheels [6]. Thus, the locations of these eigenfrequencies and notch frequencies depend essentially on the parameters of the wheels and the wheel attachments. A stiffer connection cRW of the wheels to the rack increases the eigenfrequencies and notch frequencies, whereas a higher moment of inertia JW of the wheels reduces them. A change in the gear ratio iRW between rack and wheel also mainly affects these eigenfrequencies and notch frequencies.


Furthermore, the stiffness cNC of the axial nut bearing influences the eigenfrequencies ①, ②, ④ and ⑤ as well as the notch frequency ④, whereas the stiffness cCV of the casing attachment affects the eigenfrequencies ②, ④ and ⑤ as well as the notch frequencies ① and ③. The belt drive stiffness cMN influences the eigenfrequency ⑥ and the stiffness cNR of the ball screw drive the eigenfrequency ⑦. As a stiffness increases, the respective eigenfrequencies and notch frequencies increase and vice versa.

Figure 2 Frequency Response of the Control Transfer Path of the detailed Plant Model

3 Control Design

Fig. 3 shows the block diagram of the closed-loop system consisting of the detailed plant model “8DOF” and the dynamic LQG-compensator. The compensator contains the linear optimal static state space controller (LQR) and the linear optimal state space observer (LQE). The controller and the observer are designed with reduced plant models. These are based on the detailed plant model “8DOF” described in section 2.