Varying Zonotopic tube RMPC with switching logic for lateral path tracking of autonomous vehicle

Highlights

• A complete framework for practical application of AV systems is proposed based on a novel scheme of Varying Zonotopic TRMPC with Switching Logic (SVTMPC).

• The new update mechanism of the present nominal state can determine the unknown internal disturbance set.

• The novel flexible tube parameterization increases QP feasible region and leads to less conservative results.

• The switching logic can reduce the conservatism under conventional conditions.

• Numerical simulation and HIL experiment verify the superiority of the proposed framework.

Abstract

Model mismatch caused by strong nonlinearity and other factors will severely impact the lateral path tracking control in Autonomous Vehicles (AVs) under extreme conditions. Previous studies have focused on guaranteeing robust stability under possible uncertainty realizations through Tube-based Robust Model Predictive Control (TRMPC). However, three deficiencies in TRMPC applications are revealed: unknown disturbance set, simple rigid tube, and excessive conservatism. In this paper, a novel scheme named Varying Zonotopic TRMPC with Switching Logic (SVTMPC) is developed to overcome these limitations. Firstly, a zero steady-state error dynamic model is established, and a new update mechanism of the nominal state is devised to determine the unknown internal disturbance set of the AV system. Secondly, zonotopic representation of all defined sets is used to construct the prediction model, as well as a flexible tube with varying cross-sections is naturally designed to overcome excessive conservation and non-solution of Quadratic Programming (QP). Finally, a switching logic between conservative and radical strategies improves tracking performance under conventional conditions without compromising robust stability. Numerical simulation through three scenarios shows that the SVTMPC controller can comprehensively improve robust stability and adaptability compared with MPC and TRMPC. Hardware-in-the-Loop (HIL) experiment verifies the effectiveness and real-time of the SVTMPC controller.

Introduction

As a key link of Intelligent Transportation System (ITS) [1,2], Autonomous Vehicle (AV) has great potential to improve driving safety, reduce energy consumption, and optimize traffic flow, which has become a research and development hotspot in the current automotive industry. The general hierarchical scheme of an AV consists of three layers: (1) perception layer, (2) planning layer, (3) tracking layer [3]. The tracking layer is the core that controls the AV to follow the reference path pre-planned by the planning layer, which can be decoupled into longitudinal and lateral path tracking [4,5]. In which lateral path tracking makes the vehicle always drive along the reference path through Active Front Steering (AFS) control. As it has more complex dynamics and is the key to ensure vehicle safety and ride comfort, lateral path tracking receives more attention in research activities [5], [6], [7].

State-of-art approaches, including Flatness-based Control (FLAT) [8], Immersion and Invariance Principle (I&I) [9], Sliding Mode Control (SMC) [10,11], feedback control [12,13], robust H∞ [14], and Model Predictive Control (MPC) [15,16], are applied to high speed and large curvature conditions. The main disadvantage of most approaches is that the key safety and actuator constraints of vehicles are not explicitly considered. Moreover, FLAT is sensitive to model parameters, SMC can be aggressive and cause saturated tire forces [17], and robust H∞ requires large computation and behaves too conservative. By contrast, MPC has been widely studied and applied to AV systems due to its ability to systematically utilize predictive information and handle Multiple-Input Multiple-Output (MIMO) under critical constraints [18,19].

Furthermore, to realize the potential of the AV in dealing with complex dynamic scenes that human drivers consider challenging or lack of handling ability, it is necessary to further improve the robust stability and adaptability of the tracking controller under extreme conditions. When the AV drives at handling limits, the nonlinear and multi-dimensional motion coupling characteristics are significantly enhanced, which will result in axle load transfer, tire slip, and time-varying parameters. Meanwhile, rapidly increased computation cost limits the model accuracy. These comprehensive factors will lead to high uncertainty in the dynamic model. Since the traditional MPC cannot handle such high uncertainty, schemes considering the effect of uncertainty in future system evolutions were further studied, mainly including Stochastic MPC (SMPC) and Robust MPC (RMPC) [20]. SMPC schemes [21], including Scenario Generation (SG) [22], Stochastic Tube (ST) [23], Saturation Functions (SF) [24], and Deterministic Equivalents (DE) [25], consider the uncertainty as a noise with expectation and covariance. However, for AV systems, it is difficult to give a full probabilistic description for the uncertainty of model mismatch. By contrast, the uncertainty in RMPC is described as a bounded disturbance set, which is rational and easy to implement in AV systems. Hence we consider RMPC is more promising than SMPC in AV control tasks.

RMPC schemes consider the worst-case caused by the disturbance set to handle all possible uncertainty realizations [20,26,27]. Min-max RMPC scheme was first proposed by solving a Dynamical Programming (DP) at each time step [28,29]. However, due to the high computational complexity of DP, the Min-max formulation is difficult to apply to the AV system that requires high real-time performance. In recent years, the research focus has shifted to a more efficient Tube-based RMPC (TRMPC) [30], [31], [32], [33]. TRMPC scheme utilizes a separable control policy to separate the real system into a nominal system unaffected by disturbance and a perturbed system constructed by disturbance. The real system is like being tightened in a tube that the nominal system is the center and the perturbed system is the radius, which guarantees the robust attractivity and stability of the constraints satisfaction. Meanwhile, the construction of the tube is computed off-line, and the online computation is only a Quadratic Programming (QP) of MPC for the nominal dynamics. Therefore, compared with traditional MPC, the online computation burden of the TRMPC scheme increases slightly. Due to its high efficiency and satisfactory robust stability, the TRMPC scheme has numerous applications in the AV control field, e.g., lateral control for avoiding obstacles [34], [35], [36], trajectory tracking of autonomous race cars and agricultural tractor-trailers[37,38], adaptive cruise control (ACC) [39], and active safety control of electric vehicles [40].

Nevertheless, it is noticed that there are three deficiencies in the above studies, viz. unknown disturbance set, traditional TRMPC with a simple rigid tube, and excessive conservatism. For the first, in practical applications, the determination of the unknown disturbance set is unclear, and the effects of internal and external disturbance are usually considered together as an additive set. External disturbance, such as sensor noises, can be pre-reduced through Set-Membership Estimation (SME) [41], [42], [43] before being considered in the state-space model, which is not simple additivity. For the second, the rigid tube may cause the whole scheme infeasible when the disturbances are significant at driving limits. The rigid tube is constructed by a pre-computed minimal Robust Positively Invariant set ($mRPI$) [44], in which the shape and size of the tube cross-sections are fixed over the prediction horizon. However, such simplification may lead to excessively conservative behaviour and non-solution of QP. Moreover, commonly used set representations, including vertex polytopes and half-space polytopes [45], will lead to high computation even off-line. The computed polytopes usually cannot be directly used in nominal MPC due to the high complexity accumulated during set operations, and an approximation solution is commonly used to simplify polytopes, bringing more conservatism. For the last, since the tube is pre-computed by considering the worst case, TRMPC will inevitably be conservative under conventional conditions.

Inspired by the existing literature on the set theory and flexible tubes [46], [47], [48], [49], [50], the main contribution of this paper is to propose a complete framework for practical application of AV systems, which is based on a novel scheme of Varying Zonotopic TRMPC with Switching Logic (SVTMPC). In addition, the proposed SVTMPC is a general scheme, so we think that the framework of this paper also has application potential in other automation systems, including but not limited to active suspension and vehicle platoons. The concrete work of this paper is addressed in the following aspects.

• By adding feedforward control to eliminate the road curvature disturbance, a zero steady-state error dynamic model is established. Based on a new update mechanism of the present nominal state, the unknown internal disturbance set of the AV system is determined by co-simulation under various extreme conditions required for autonomous driving.

• A prediction model is constructed efficiently and accurately through the zonotopic representation of all defined sets. A novel flexible tube parameterization is naturally designed, in which each cross-section varies adaptively over the prediction horizon. Compared with the rigid tube, the flexible tube increases the QP feasible region at a negligible increase in the computational cost. Moreover, tube-related constraints are directly substituted into objective function after representation conversion, a reduction in online computation cost is also generated.

• According to vehicle stability flags, a switching logic between conservative and radical strategies is given by switching Linear Quadratic Regulator (LQR) weight matrices of the perturbed system. The switching region is set insider the intersection of feasible regions of two strategies to guarantee robust stability. The switching logic reduces the conservatism under conventional conditions.

• Co-simulation of Carsim and MATLAB/Simulink and Hardware-in-the-Loop (HIL) experiments verify that the proposed SVTMPC controller is feasible, adaptable, robust, and comprehensively improves tracking accuracy and vehicle stability.

Fig. 1 gives a breakdown of the overall block diagram of the proposed SVTMPC scheme for AV systems. The remainder of this research is illustrated as follows. In Section 2, the dynamic model of the AV system is established. TRMPC scheme and the prediction model constructed by zonotopic constraints are given in Section 3. In Section 4, the SVTMPC scheme is proposed, including the novel flexible tube parameterization, the efficient solution to the QP with zonotopic constraints, and the switching logic. Co-simulation and HIL experiment are carried out and analyzed in Section 5. Finally, conclusions are summarized in Section 6.