Scale Reduction Based Efficient Model Predictive ... - IEEE Xplore

Proceedings of the 16th International IEEE Annual Conference on Intelligent Transportation Systems (ITSC 2013), The Hague, The Netherlands, October 6-9, 2013

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Scale Reduction based Efficient Model Predictive Control and Its Application in Vehicle Following Control* Shengbo Eben Li, Zhenzhong Jia, Keqiang Li, Bo Cheng 

Abstract—The recent progress in advanced vehicle control systems presents a great opportunity for application of model predictive control (MPC) in automotive industry. However, high computational complexity inherently associated with the receding horizon optimization must be addressed to achieve the real-time implementation. This paper presents a general scale reduction framework to reduce the online computational burden of MPC controllers. A lower dimensional MPC algorithm is developed by integrating an existing ‘move blocking’ (MB) strategy with a ‘constraint set compression’ (CSC) strategy, which is first proposed here. Good trade-off between control optimality and computational intensity is achieved by proper design of blocking and compression matrices. Application of the fast algorithm on vehicular following control (e.g. an adaptive cruise control system) was evaluated through real-time simulation. These results indicate that the proposed method significantly improves the computational speed while maintaining satisfactory control optimality without sacrificing the desired performance. I.

INTRODUCTION

Intended to assist drivers during driving process and improve road safety, vehicle control technologies have also progressed from previously basic functional realization to today’s focus on performance maximization and cost minimization. Such a trend presents a high opportunity to use model predictive control (MPC, also called Receding Horizon Control) theory in automotive industry because of its capability in realizing functional optimization while explicitly handling nonlinearities and constraints [1]-[4]. A typical application of MPC in automotive field is the field of vehicular longitudinal automation, e.g. adaptive cruise control (ACC) [3], in particular for some application to balance multiple objectives [5][6]. The high computational burden associated with the receding horizon optimization must be mitigated for real-time implementation in real world [7]. In MPC, an optimization problem is used to optimize the state evolution over a future horizon [7][8]. In fact, it is one of the rarely advanced control techniques that have a significant * Research supported by the National Science Foundation of China under Grant 51205228 and Tsinghua University Initiative Research Program under Grant 2012THZ0. S. Eben Li is currently with the State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 10084, China (e-mail: [email protected]). He was research fellow with the University of Michigan, Ann Arbor, MI 48109, USA. Z. Jia is currently with the Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA. K. Li and B. Cheng are with the State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 10084, China. 978-1-4799-2914-613/$31.00 ©2013 IEEE

impact on industry process control (e.g., chemical, petroleum industry). One critical challenge is the computational burden in numerical optimization. It becomes even worse for applications with fast dynamics and/or limited computing power. The computational issue of MPC motivates researchers to explore more efficient computing techniques [8]. An approach to achieve fast computation is to utilize the structural sparseness of MPC. By properly reordering the manipulated variables related to the sparse matrices, an interior-point method could become more efficient during directional search [9]. Another scheme is called explicit MPC, which is often used for linear or piece-wise affine linear plants with 1-, 2- or ∞- norm based costs and linear constraints [8]. In this method, an explicit solution is generated off-line by using multi-parametric programming [10][11]. One major problem associated with this approach is the large online data storage requirements for high-order controllers due to exponentially increased complexity. In engineering practices, an effective approach to reduce the computational intensity is to reformulate the original MPC problem as a lower order optimization problem by reducing the dimension of manipulated variables. Two examples are the parameterization method [12] [13] and the ‘move blocking (MB)’ method [14] [15] [16]. The former often assumes priori knowledge of the control law, which can be approximated by a parameterized function (with less unknowns to reduce the order of the original problem), e.g. polynomial [12] or exponential [13]. However, this method may change the structure of the original problem due to specific parameterized functions used, thereby resulting in pseudo optimal solutions and losing the degree of control optimality. In the MB method, the number of free variables is reduced by fixing the input or its derivatives to be constant over several time-steps by using a ‘blocking matrix’ [14]. This straight-forward method can be easily integrated with other fast-computing schemes. The purpose of this paper is to extend the MB scale reduction strategy by introducing an additional ‘constraint set compression (CSC)’ strategy, capable of further reducing the computational intensity of MPC for real-time implementation, especially in the fields of engineering practice, e.g. automotive control techniques. The paper is structured as follows. Section II reviews a simplified MPC formulation. Section III presents a generic scale reduction framework for the MPC optimization problem, together with the afore-mentioned MB strategy and the proposed CSC strategy. The MPC-based vehicular adaptive cruise controller is presented in section IV. Application of the proposed fast MPC algorithm to ACC is evaluated through real-time simulation in section V. Section VI comes to a conclusion.

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II. PROBLEM DEFINITION

III. FRAMEWORK OF PROBLEM SCALE REDUCTION

Consider a nonlinear plant model in the discrete-time domain:

The scale reduction framework consists of two parts: ‘move blocking (MB)’ and ‘constraint-set compression (CSC)’. The former aims to reduce the number of manipulated input variables in the predictive horizon while the latter aims to reduce the number of inequality constraints.

x k 1  f

 xk   g  xk   u k ,

(1) where u∈R is the control input, x∈Rl is the system state, and f(∙) and g(∙) are nonlinear functions of x. The receding horizon optimization problem in MPC has the following form: m

P 1

m in J  u

 lx

, u k  i |k      k , 2

k  i  1| k

(2)

i0

subject to Eq. (1) and u k  i | k  u m ax   k  v m ax u

 i  0, ..., P  1  ,

(3) where P is the length of the prediction horizon, xk+i|k is the predicted state using the measured (or estimated) state xk, ε∈R+ is the slack variable, ρ∈R+ is the weighting coefficient, umax∈Rm is the upper bound of u, vumax∈Rm is the relaxing coefficient [7]. Other constraints on x and u are omitted for simplicity. Pm

We introduce three column vectors, Uk∈R , Umax∈R and Vk∈RPm, defined as: T  T T  U k   u k | k , ..., u k  P  1| k   T T T .  U k   u m ax , ..., u m ax   T T u V   v u T , ..., v m ax     k m ax      

Pm

(4)

Plugging Eq. (1), (4) into Eq. (2) and (3), the MPC problem is transformed into a nonlinear programming (NP) problem [7]: m in J  L  U k    k , 2

A. Move Blocking (MB) Strategy A common strategy to reduce the computational complexity of optimal control is to reduce degrees of freedom by fixing the control input (or the control increment) to be constant over several steps. This policy is referred to as ‘move blocking’ [14]. Instead of solving for the optimal Uk∈RPm, problem (5) is restated in terms of solving for a lower-order vector: T Z k   z k | k ,

 U k   u V m ax     U m ax  ,   k   

U k  M T  Zk ,

978-1-4799-2914-613/$31.00 ©2013 IEEE

(7)

M T  T  Im

Recall that only the first entry uk+0|k in Uk will be applied to the plant while all other entries (i.e., uk+i|k, i=1,…,P-1) are not used for direct control [7]. Thus, it is not necessary to calculate the exact value of every entry in Uk except for the first element uk+0|k. Hence, a blocking matrix (Fig. 1) can be formulated as: 1  0  T   0  0 

(6)

For a NP problem, its computational intensity depends on three aspects: (a) problem style; (b) problem scale; and (c) optimization algorithm. The problem style relies on the form of the cost function and the constraints; generally, when given a MPC problem, it is not easy to be changed, except some linearization methods [7]. It, together with optimization algorithm, is not considered in this paper. The effective strategies for scale reduction are preferred in engineering practice due to its easy-to-use and in-field flexibility. The scale of a commonly-constructed optimization problem, e.g. NP, relies on two critical factors, namely, the number of manipulated variables NA and the number of inequality constraints NB [17]. Taking one typical numerical optimization algorithm, Dantzig-Wolfe algorithm, as an example, the required iteration is often no less than Niter, which equals the sum of NA and NB. Hence, the scale reduction is expected to effectively reduce the computational complexity, thereby enabling on-line implementation of MPC.

 P,

. (8) In Eq. (8), ⊗ denotes the Kronecker product and T∈RQ×P is a so-called ‘blocking matrix’ with Q