Multi-parametric Quadratic Programming for Explicit MPC Solutions

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"这篇论文是关于显式模型预测控制（MPC）解决方案的，通过解决多参数二次规划问题（mp-QP）来获得受约束线性模型预测控制问题的显式解。研究了由多参数分段光滑解决方案诱导的状态空间多面体分割，并提出了一种新的mp-QP求解器，其探索策略不同于现有算法，避免了不必要的分区和二次规划问题求解，显著提高了效率。" 在模型预测控制（MPC）中，控制器基于一个动态模型对未来系统行为进行预测，并优化一系列控制动作序列，以达到预期的性能指标。显式MPC解决方案是一种特殊的方法，它允许预先计算控制序列，而无需在线进行复杂的优化计算。这在实时控制中尤其有用，因为可以减少计算负担并提高系统的响应速度。多参数二次规划（mp-QP）是实现显式MPC的关键工具。在这个问题中，二次目标函数和线性约束条件都依赖于状态变量的参数。通过解决mp-QP问题，我们可以得到一个参数化控制策略，该策略在不同状态下的控制输入是预先确定的。论文中，作者对由mp-QP的分段光滑解所诱导的状态空间多面体分割进行了深入研究。这种分割将状态空间划分为多个区域，每个区域对应一个特定的控制输入。这种分割对于理解MPC策略的结构和行为至关重要。作者提出的新算法改进了现有的mp-QP求解策略，采用了一种不同的参数空间探索方法。这种方法避免了在无用的区域内进行分区和二次规划问题的求解，从而节省了计算资源。通过这种方式，新算法能更有效地处理大型的mp-QP问题，特别是在状态空间维度较高或约束条件复杂的情况下。这篇论文的贡献在于提供了一种优化显式MPC解决方案的方法，通过改进mp-QP求解器的设计，减少了计算复杂度，提高了控制系统的实时性能。这对于需要快速响应和高效运算的工业应用，如过程控制、电力系统或自动驾驶汽车等领域具有重要的理论和实践意义。

Available online at www.sciencedirect.com

Automatica 39 (2003) 489 – 497

www.elsevier.com/locate/automatica

Brief Paper

An algorithm for multi-parametric quadratic programming

and explicit MPC solutions



Petter THndel

a;∗

, Tor Arne Johansen

, Alberto Bemporad

Department of Engineering Cybernetics, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Dipartimento di Ingegneria dell’Informazione, University of Siena, 53100 Siena, Italy

Received 8 June 2001; received in revised form 30 May 2002; accepted 22 October 2002

Abstract

Explicit solutions to constrained linear model predictive control problems can be obtained by solving multi-parametric quadratic programs

(mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state

space induced by the multi-parametric piecewise ane solution and propose a new mp-QP solver. Compared to existing algorithms, our

approach adopts a dierent exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem

solving, with a signicant improvement of eciency.

Keywords: Linear quadratic regulators; Piecewise linear controllers; Constraints; Predictive control

1. Introduction

Our motivation for investigating multi-parametric

quadratic programming (mp-QP) comes from linear model

predictive control (MPC). This refers to a class of control

algorithms that compute a manipulated variable trajectory

from a linear process model to minimize a quadratic per-

formance index subject to linear constraints on a prediction

horizon. The rst control input is then applied to the pro-

cess. At the next sample, measurements are used to update

the optimization problem, and the optimization is repeated.

In this way, this becomes a closed-loop approach. There

has been some limitation to which processes MPC could

be used on, due to the computationally expensive on-line

optimization which was required. There has recently been

derived explicit solutions to the constrained MPC prob-

lem, which could increase the area of use for this kind of

controllers. Explicit solutions to MPC problems are not



This paper was not presented at any IFAC meeting. This paper was

recommended for publication in revised form by Associate Editor Kenko

Uchida under the direction of the Editor Tamer Basar.

∗

Corresponding author. Tel.: +47-7359-4348; fax: +47-7359-4399.

E-mail addresses: petter.tondel@itk.ntnu.no (P. THndel),

tor.arne.johansen@itk.ntnu.no (T.A. Johansen), bemporad@dii.unisi.it

(A. Bemporad).

mainly intended to replace traditional implicit MPC, but

rather to extend its area of use. MPC functionality can with

this be applied to applications with sampling rates in the

-sec range, using low cost embedded hardware. Software

complexity and reliability is also improved, allowing the

approach to be used on safety-critical applications. Methods

for ecient online implementation of PWA function eval-

uation in explicit MPC has been developed by exploiting

convexity (Borrelli, Baotic, Bemporad, & Morari, 2001)or

an associated binary search tree data structure (THndel & Jo-

hansen 2002; THndel, Johansen, & Bemporad, 2003). In-

dependent works by Bemporad, Morari, Dua, and Pis-

tikopoulos (2002), Bemporad, Morari, Dua, and Pistikopou-

los (2000b), Johansen, Petersen, & Slupphaug (2002) and

Seron, De Dona, and Goodwin (2000) have reported how a

piecewise ane (PWA) solution can be computed o-line,

while the on-line eort is limited to evaluate this PWA func-

tion. In particular, in Bemporad et al. (2002, 2000b) such a

PWA function is obtained by treating the MPC optimization

problem as a parametric program. Parametric programming

is a term for solving an optimization problem for a range of

parameter values. One can distinguish between parametric

programs, in which only one parameter is considered, and

multi-parametric programs, in which a vector of param-

eters is considered. The algorithm reported in Bemporad

PII: S0005-1098(02)00250-9

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