# SCIENCE CHINA Information Sciences, Volume 60 , Issue 9 : 092201(2017) https://doi.org/10.1007/s11432-016-0269-y

## A fast algorithm for nonlinear model predictive control applied to HEV energy management systems • AcceptedOct 31, 2016
• PublishedMar 13, 2017
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### Abstract

This paper presents a fast algorithm for nonlinear model predictive control. In real-time implementation, a nonlinear optimal problem is often rewritten as a nonlinear programming (NLP) problem using the Euler method, which is based on dividing the prediction horizon into $N$ steps in a given time interval. However, real-time optimization is usually limited to slow processes, since the sampling time must be sufficient to support the task's computational needs. In this study, by combining the Gauss pseudospectral method and model predictive control, a fast algorithm is proposed using fewer discrete points to transcribe an optimal control problem into an NLP problem while ensuring the same computational accuracy as traditional discretization methods. The approach is applied to the torque split control for hybrid electric vehicles (HEV) with a predefined torque demand, and its computational time is at least half that of the Euler method with the same accuracy.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61520106008, 61522307, 61374046) and Graduate Innovation Fund of Jilin University (Grant No. 2016188).

• Figure 1

(Color online) Discrete points and their corresponding Euler method and Gauss pseudospectral method weights with different numbers of discrete points. The constant weight of the Euler method is defined as $A_{\rm Euler}=2/N$, and the Gauss weights are Legendre-Gauss (LG) points. Here, $N$ represents the number of discrete points.

• Figure 2

Control inputs of MPC using GPM in real time.

• Figure 3

(Color online) State, control input, and error trajectories during the time horizon using the GPM. From top to bottom, the figures show the results using different LG points, as (a) $N$=10, and (b) $N$=20, respectively.

• Figure 4

(Color online) The maximum error of the control variables and states using the GPM and Euler method with different numbers of discrete points.

• Figure 5

(Color online) The redefined maximum errors of the control variables and states using the GPM and Euler method with different numbers of discrete points.

• Figure 6

Final cost using different control intervals and sampling time.

• Figure 7

Topology of the parallel hybrid electric vehicle powertrain .

• Figure 8

A driving scenario for evaluating the computational efficiency of the two methods.

• Figure 9

(Color online) Errors of the control variables and states using the GPM and Euler method in a driving scenario.

• Figure 10

(Color online) State-of-charge and torque trajectories using the GPM-MPC ($N_{\rm g}=16$).

• Figure 11

(Color online) Trajectories of simulation results using the GPM-MPC and Euler-MPC in NEDC. From top to bottom, the figures show the trajectories of (a) vehicle speed, (b) torque demand, (c) torque-split ratio, (d) state-of-charge, (e) motor torque, and (f) engine torque, respectively.

• Figure 12

(Color online) Trajectories of simulation results using the GPM-MPC and Euler-MPC in UDDS. From top to bottom, the figures show the trajectories of (a) vehicle speed, (b) torque demand, (c) torque split ratio, (d) state-of-charge, (e) motor torque, and (f) engine torque, respectively.

• Figure 13

(Color online) Computational efficiency of the proposed method (NEDC). The second figure shows the results in the time horizon $t \in [800, 1180]$ s.

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