Elsevier

Neurocomputing

Volume 177, 12 February 2016, Pages 363-372
Neurocomputing

Efficient and accurate computation of model predictive control using pseudospectral discretization

https://doi.org/10.1016/j.neucom.2015.11.020Get rights and content

Abstract

The model predictive control (MPC) is implemented by repeatedly solving an open loop optimal control problem (OCP). For the real-time implementation, the OCP is often discretized with evenly spaced time grids. This evenly spaced discretization, however, is accurate only if sufficiently small sampling time is used, which leads to heavy computational load. This paper presents a method to efficiently and accurately compute the continuous-time MPC problem based on the pseudospectral discretization, which utilizes unevenly spaced collocation points. The predictive horizon is virtually doubled by augmenting a mirrored horizon such that denser collocation points can be used towards the current time step, and sparser points can be used towards the end time of predictive horizon. Then, both state and control variables are approximated by Lagrange polynomials at only a half of LGL (Legendre–Gauss–Lobatto) collocation points. This implies that high accuracy can be achieved with a much less number of collocation points, which results in much reduced computational load. Examples are used to demonstrate its advantages over the evenly spaced discretization.

Introduction

Model predictive control (MPC) is implemented by repeatedly solving an open-loop optimal control problem (OCP) and using the first element of the optimized control sequence as the current control action [1]. This class of controllers has received wide applications in the process industries since 1980s [2]. The main advantage of MPC is its ability to explicitly handle constraints on controls and states and achieve optimized control inside admissible sets. Generally, the plant dynamics are described in the continuous time representation by resorting to first principle equations. For the real-life implementation, however, discrete-time descriptions are required, and the continuous time problem must be converted into a discrete time problem for the computer environment [3].

The typical approach for this conversion is to discretize the plant model, constraints, and cost function at evenly spaced time points. The evenly spaced discretization approach has been widely used in the literature, e.g. Chen and Allgower [4], Muske and Badgwell [5], Qin and Badgwell [6] and Scattolini [7]. In these studies, the system behaviors between two consecutive sampling points are usually approximated by using either zero order hold (ZOH) or first order hold (FOH), which are the two lowest order approximation methods [8], [9]. For reduced computational efforts, some studies used discretization methods with unevenly spaced points, e.g. moving blocking strategy [10], [11]. They, however, share the same approximation methods, ZOH and FOH, which leads to significant state prediction and control errors. Hence, small sampling time is necessary in order to achieve sufficiently high levels of accuracy, which results in high computational load.

In this study, we will discretize the continuous-time MPC with unevenly spaced grid points. The unevenly spaced discretization relies on the known pseudospectral approach, which enables to use high order polynomials to approximate states and controls. The high order approximation provides more flexibility to describe the system behavior between two points, thus effectively avoiding the shortcomings of lower order counterpart. In theory, the discretized NLP converges to the OCP at a spectral rate with the number of collocation points [12].

The pseudospectral method was first applied to optimal control problems in the late 1980s. The use of Chebyshev polynomial as interpolation basis is the first method [13]. Recently, the majority have employed Lagrange polynomials as basis functions, which are often categorized into three types: Legendre–Gauss–Lobatto (LGL) method, Legendre–Gauss (LG) method, and Legendre–Gauss–Radau (LGR) method [12]. The first type uses the family of Lobatto quadrature. The collocation points of LGL, defined on the closed interval [1,1], are the roots of the derivative of N th-degree Legendre polynomial, i.e., L̇N(t), together with −1 and 1. Several variants has been presented before, including [14], [15] and [16]. A more general version is the Jacobi pseudospectral method presented by [17]. The second type uses the family of Gauss quadrature [18], [19], [20]. The collocation points of LG, defined on the open interval (1,1), are the zeros of N th-degree Legendre polynomial LN(t), so that the endpoints of integration interval are not included. The third type uses the family of Radau quadrature [21], [22]. The collocation points of LGR, defined on the half-open interval (-1,1 ], are the roots of LN1(t)+LN(t), which only contains one end point. Upon cursory examination it might appear as if LGL, LG and LGR collocation are essentially similar, with only minor difference on whether contains end points. It has been shown by Garg et al. that the differences between three schemes are not merely cosmetic [31]. The LGL collocation leads to a different mathematical form as compared with either LG or LGR. As a result, LGL has different convergence properties from LG and LGR. As pointed out by Garg et al., the main differences are: (1) The differentiation matrices of LG and LGR are rectangular and full-rank, whereas that of LGL is square and singular. Therefore, LG and LGR can be written equivalently in either differential or implicit integral form while LGL does not have an equivalent implicit integral from. (2) The differentiation matrix of LGL is rank deficient. This rank-deficiency leads to a transformed adjoint system that is also rank-deficient, which can yield dual solutions that oscillate about the true solution. Conversely, LG and LGR lead to full-rank transformed adjoint systems which in turn yield approximations that converge to the true solution. (3) There is a fundamental difference on costate estimation between LGL and LGR/LG. The aforementioned oscillatory property also affects the estimation of costate estimation, which naturally attributes to the null space of the LGL transformed adjoint system. This is because that the discrete costate dynamics form a linear system of equations in terms of Lagrange multiplier. In LGL, the matrix in the equations has a null space and therefore there exists a infinite number of solutions to LGL costate dynamics. Despite the null space in LGL, many numerical examples have demonstrated that LGL has convergent approximations to state and control. The convector mapping theorem by Gong et al. [24] has pointed out that any solution of first-order optimality condition for continuous system approximately satisfies the first-order optimality condition for discrete LGL problem, and therefore the error tends to zero as N. Moreover, Gong et al. [24] proposes a closure condition for selecting a good approximation to continuous costate from the infinite number of solutions.

The main study of this paper is to discretize the continuous-time MPC problem by converting the open loop OCP into NLP using the pseudospectral discretization. This unevenly spaced discretization enables more efficient and accurate implementations of MPC over evenly spaced counterpart. This paper mainly relies on the LGL collocation scheme, which makes possible to directly apply initial solution as control input and add constraints for terminal points in the predictive horizon [26]. The remainder of this paper is organized as follows: Section 2 reviews the continuous-time MPC problem. Then, its predictive horizon is designed to be augmented by a mirrored horizon in Section 3. The Legendre pseudospectral method is applied to discretize of the augmented OCP in Section 4. Section 5 renews the known covector mapping principle for the augmented problem; Illustrative examples are given in Section 6. Section 7 concludes this paper.

Section snippets

Continuous-time MPC problem

Let us consider a class of nonlinear continuous-time systems described byẋ(t)=f(x(t),u(t))where xRn is the state vector, uRm is the control, f:(Rn,Rm)Rn is the mapping function. Note that there exists a dual pair (x,u)(X,U)(Rn,Rm) which satisfies f(x,u)=0. The sets XRn and URm are admissible box constraints for the state and input. For narrative convenience, the predictive horizon is assumed to range from t=1 to t=0 for the sake of simplicity, where t=1 is the current time. The

Doubled horizon: augmented by mirrored horizon

In the model predictive control defined on the interval [−1, 1] as shown in Fig. 1, only the first point (t=−1) of optimized sequence in the predictive horizon [−1, 0] is implemented as control input. Thus the accuracy around −1 (beginning of predictive horizon) is more important than that around t=0 (end of predictive horizon). Meanwhile, a point deployment such as “dense beginning and sparse end” is beneficial to enhance the accuracy of MPC. Similar technique has been used in many MPC

Basis of pseudospectral discretization method

The length of the optimal control problem is doubled, and accordingly the orthogonal collocation points should be defined to be in the form of two folds (predictive horizon and mirrored horizon). Since there exists a final constraint at t=0 in the original problem, an odd number of collocation points are necessary for the final constraint to be applied at t=0 for the augmented problem.

Let L2N(t) denote the Legendre polynomial of order 2N. The orthogonal collocation points are selected to be the

Costate estimation for augmented problem

The study of the relationship between the costate variables and the Lagrange multipliers is critical to optimality examination. As a byproduct, this section renews the known covector mapping principle (CMP) for the augmented problem. The CMP actually provides conditions under which dualization can be commuted with pseudospectral discretization [24]. Alternatively, with CMP, the costates can be determined accurately at the orthogonal collocation points by simply dividing the Karush–Kuhn–Tucker

Illustrative examples

In this section, we use two MPC examples to illustrate the superior accuracy of pseudospectral (PS) discretization over the evenly spaced (ES) discretization. The first example is a simple linear quadratic optimal control problem with control constraint, and the second example is an adaptive cruise control problem of ground vehicle. Using both PS and ES discretization, the augmented OCP in the predictive horizon is converted into NLP at each control step. The converted NLP is then numerically

Conclusions

This paper presents a method to efficiently and accurately compute the continuous-time MPC problem by using the pseudospectral discretization, which utilizes unevenly spaced collocation points. The conventional discretization used evenly spaced method to implement MPC. This method is easy to use, but often causes significant error because of the lower order approximation of system behavior between two sampling points. In the pseudospectral discretization, the predictive horizon of open loop OCP

Acknowledgment

This research work is supported by the National Natural Science Foundation of China under Grant 51205228 and 51575293. Special thanks should be given to Prof. Bo Cheng in Tsinghua University for his suggestions on the controller design.

Shengbo Eben Li received the M.S. and Ph.D. degrees from Tsinghua University in 2006 and 2009. He worked at Stanford University in 2007, University of Michigan from 2009 to 2011, and University of California, Berkeley, in 2015. He is currently an associate professor in Department of Automotive Engineering at Tsinghua University. His active research interests include autonomous vehicle control, driver modeling and driver assistance, control topics of battery, optimal control and multi-agent

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    Shengbo Eben Li received the M.S. and Ph.D. degrees from Tsinghua University in 2006 and 2009. He worked at Stanford University in 2007, University of Michigan from 2009 to 2011, and University of California, Berkeley, in 2015. He is currently an associate professor in Department of Automotive Engineering at Tsinghua University. His active research interests include autonomous vehicle control, driver modeling and driver assistance, control topics of battery, optimal control and multi-agent control, etc. He is the author of more than 80 journal/conference papers, and the co-inventor of more than 20 patents. Dr. Li was the recipient of Award for Science and Technology of China ITS Association (2012), Award for Technological Invention in Ministry of Education (2012), National Award for Technological Invention in China (2013), Honored Funding for Beijing Excellent Youth Researcher (2013), NSK Sino-Japan Outstanding Paper Prize in Mechanical Engineering (2013/2015), Best Student Paper Award in 2014 IEEE Intelligent Transportation System Symposium, Top 10 Distinguished Project Award of NSF China (2014), Best Paper Award in 14th ITS Asia Pacific Forum, 2015. He also served as the Associate editor of IEEE Intelligent Vehicle Symposium (2012/2013), Chairman of organization committee of China ADAS forum (2013), Guest Editor of Mathematical Problem in Engineering (2014), etc.

    Shaobing Xu received the B.S. degree in automotive engineering from China Agricultural University, Beijing, China, in 2011. He is currently working on the Ph.D. degree in automotive engineering at Tsinghua University, Beijing, China. His research interest is the optimal control theory and vehicle dynamics control. His awards and honors include National Scholarship, President Scholarship, first prize of Chinese 4th Mechanical-Design Contest & first prize of 19th Advanced Mathematical Contest.

    Dongsuk Kum received his Ph.D. degree in mechanical engineering from the University of Michigan, Ann Arbor, in 2010. He is currently an Assistant Professor at the Graduate School for Green Transportation in Korea Advanced Institute of Science & Technology (KAIST), and the Director of the Vehicle Dynamics and Controls (VDC) Laboratory. His research centers on the modeling, control, and design of advanced vehicular systems with particular interests in hybrid electric vehicles and autonomous vehicles. Prior to joining KAIST, Professor Kum had worked for the General Motors R&D Propulsion Systems Research Laboratory in Warren, MI as a visiting research scientist. His works at General Motors focused on advanced propulsion system technologies including hybrid electric vehicles, flywheel hybrid, and waste heat recovery systems.

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