Elsevier

Automatica

Volume 63, January 2016, Pages 330-337
Automatica

Brief paper
Continuity and monotonicity of the MPC value function with respect to sampling time and prediction horizon

https://doi.org/10.1016/j.automatica.2015.10.042Get rights and content

Abstract

The digital implementation of model predictive control (MPC) is fundamentally governed by two design parameters; sampling time and prediction horizon. Knowledge of the properties of the value function with respect to the parameters can be used for developing optimization tools to find optimal system designs. In particular, these properties are continuity and monotonicity. This paper presents analytical results to reveal the smoothness properties of the MPC value function in open- and closed-loop for constrained linear systems. Continuity of the value function and its differentiability for a given number of prediction steps are proven mathematically and confirmed with numerical results. Non-monotonicity is shown from the ensuing numerical investigation. It is shown that increasing sampling rate and/or prediction horizon does not always lead to an improved closed-loop performance, particularly at faster sampling rates.

Introduction

When designing a model predictive controller (MPC), it is important to consider the influence of various design choices on both the performance and computational complexity of the controller. These design choices could include attributes of the optimal control problem (OCP) associated with the MPC such as sampling time, prediction horizon and model order, as well as attributes of the numerical routine used to solve the OCP. Since sampling and horizon times are quantities that must be chosen in any implementation of MPC, this discussion will focus on the influence of these parameters.

A common practice aimed at improving MPC performance is to decrease sampling and/or increase horizon times, typically whilst meeting computational requirements for real-time controller implementation. There is no guarantee that relying upon this ‘conventional wisdom’ will improve performance. It has been previously established for linear (and nonlinear  Di Palma & Magni, 2007) quadratic regulators (LQR) that decreasing sample time or increasing prediction horizon can degrade closed-loop performance (Di Palma and Magni, 2007, Levis et al., 1971). For MPC, where constraints and finite horizons complicate matters, a preliminary analysis not considering state constraints and a terminal cost (Bachtiar, Kerrigan, Moase, & Manzie, 2015) have echoed the LQR results. A formalized and rigorous analysis of closed-loop MPC performance has yet to be addressed.

This paper considers the continuity, differentiability and monotonicity of control performance in constrained linear systems, measured by an open- or closed-loop value function, with respect to the sample and horizon times. This follows the spirit of related research exploring the open-loop value function sensitivity, and showing that it is piecewise quadratic with respect to the current state of the plant (Baotic et al., 2008, Bemporad et al., 2002). In this paper, an MPC is formulated initially in continuous-time with a given prediction horizon and a zero-order-held input, following  Bini and Buttazzo (2014). The discrete-time equivalence is then formulated, with a residual sampling interval to account for when the prediction horizon is not an integer multiple of the sampling time.

Knowledge of the smoothness properties of the MPC performance with respect to the design parameters is useful to develop systematic methods such as optimization tools, in place of the conventional wisdom, to select sampling and horizon times that achieve the best control performance and alleviate online computational cost. For example, a gradient-based method might be ineffective for a discontinuous optimization surface and a global optimizer is useful if the surface is non-monotonic.

Perturbation analysis is used below to prove continuity of the value function and its differentiability for a given number of prediction steps. Numerical results are presented to confirm the analytical results and demonstrate non-monotonicity, thereby providing examples where conventional wisdom may result in worse MPC designs.

Notational conventions and definitions: For MRn×n, vRn×m and sR with appropriate n and m, vM2vTMv. Superscript + denotes the function value after its argument(s) is perturbed, e.g.  M+M(s+δs). svvs|s is the derivative of a differentiable matrix sv(s) evaluated at s. Big-O notation; f(x)=O(g(x))aR>0,x0R>0:|f(x)|a|g(x)|x[0,x0). Let O(g1(),,gn())=i=1nO(gi()). λi(M) is the ith eigenvalue of the matrix M. () denotes the imaginary part of its argument.

Definition 1 Continuity (Stewart, 2011)

f():RnR is continuous at sRn if limxsf(x)=f(s).

Definition 2 Differentiability (Stewart, 2011)

f():RnR is differentiable at sRn if it can be linearly approximated in the neighbourhood of the point, i.e.f(s+δ)=f(s)+Δ(s)δ+L(s,δ)δ, where the gradient Δ(s)Rn is independent of δ, and limδ0L(s,δ)=0, sRn constitutes higher order terms. Differentiability at point s implies continuity at s.

Section snippets

Definition of OCP

Consider a linear time-invariant dynamic plant model ẋ(t)=Ax(t)+Bu(t) with states x(t)Rnx and inputs u(t)Rnu. Discretization is required for the purpose of digital control; the plant is controlled in a sampled-data fashion at sampling instants ti, iN0, with sampling interval h. A sampling-to-actuation delay of zero is assumed. The control input is restricted to a zero-order-hold (ZOH); u(t)=ui,t[ih,ih+h),iN0.

In the context of optimal regulation, the control command can be obtained by

Differentiability of transition and cost matrices

The matrices A¯, B¯,Q¯,R¯ and S¯ are differentiable w.r.t.  h and independent of T. The Taylor series expansion for the perturbation h+h+δh as δh0 is A¯+A¯(h+)=A¯(h)+hA¯(h)δh+O(δh2) and similarly for B¯,Q¯,R¯  and  S¯. The matrices A¯r, B¯r,Q¯r,R¯r and S¯r are differentiable w.r.t. hr, thus differentiable w.r.t. (h,T), for all hr(0,h) since hrTNh.

Optimality of the OCP solution

Optimality is identified by the Karush–Kuhn–Tucker (KKT) conditions (Wright & Nocedal, 1999). z is a solution to (10) iff μ:{Hz=GTμ(a)μ0(b)

Open-loop

The following analysis considers the optimal control at a given sampling instant w.r.t. sampling and prediction horizon times. This extends investigations such as  Bini and Buttazzo (2014) that solely analyse sampling time.

In subsequent analyses, non-degeneracy of the KKT conditions is assumed.

Assumption 12

Non-degeneracy of the OCP (2)

If the equivalent OCP (10) is non-degenerate per Definition 5, then the OCP (2) is non-degenerate.

To show smoothness properties, a perturbation h+h+δh and/or T+T+δT as δh0 and/or δT0 is performed.

Conclusions and future work

We have investigated the smoothness properties of the MPC value function in open- and closed-loop as performance measures of constrained linear systems. The main results presented are: (1) The open- and closed-loop value functions are continuous w.r.t. sampling and horizon times, as well as differentiable for a given number of prediction steps, if the OCP is non-degenerate. If the OCP is degenerate, the guarantee of continuity is lost, as demonstrated numerically. (2) The open- and closed-loop

Vincent Bachtiar obtained a degree in Engineering Science with First Class Honours at the University of Auckland in 2012 and was included in the Dean’s Honours List in 2011 and 2012. He has been a Ph.D. candidate at the University of Melbourne since 2013. His doctoral research investigates the optimization, calibration and multi-objective design of model-based controllers with applications in defence-related aerospace control and guidance. His research interests include model-based controller

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Vincent Bachtiar obtained a degree in Engineering Science with First Class Honours at the University of Auckland in 2012 and was included in the Dean’s Honours List in 2011 and 2012. He has been a Ph.D. candidate at the University of Melbourne since 2013. His doctoral research investigates the optimization, calibration and multi-objective design of model-based controllers with applications in defence-related aerospace control and guidance. His research interests include model-based controller design, as well as multi-objective optimization theory and algorithms.

Eric C. Kerrigan  has been with Imperial College London since 2006. His research is on efficient numerical methods and computer architectures for solving optimization, control and estimation problems in real-time, with applications in aerospace, renewable energy and computing systems. He is Chair of the United Kingdom Automatic Control Council and on the editorial boards of the IEEE Transactions on Control Systems Technology, Control Engineering Practice and IEEE Control Systems Society Conferences.

William H. Moase received the B.E. (with honours) and B.C.S. degrees and Ph.D. in Mechanical Engineering from the University of Melbourne, Melbourne, Australia in 2004 and 2009, respectively. He is currently an Honorary Fellow at the University of Melbourne and a Guidance, Navigation and Control Engineer at BAE Systems Australia. His research interests include extremum-seeking, systems optimization and developing practical calibration procedures for modern controllers. Applications of interest include automotive, aerospace and traffic networks.

Chris Manzie is a Professor in the Department of Mechanical Engineering at the University of Melbourne. He is also an Assistant Dean (Research Training) and the Mechatronics Discipline coordinator in the Melbourne School of Engineering. His research interests are in model-based and model-free control and optimization, with applications in a range of areas including systems related to energy, transportation and mechatronics. His work has been recognized with awards including an Australian Research Council Future Fellowship 2011–2014, and has been supported by strong industry collaborations with a number of companies including Toyota Motor Corporation, Ford Australia, BAE Systems, ANCA Motion and DSTO. He was a Visiting Scholar with the University of California, San Diego in 2007 and a Visiteur Scientifique at IFP Energies Nouvelles, Rueil Malmaison in 2012. Professor Manzie is an Associate Editor for Elsevier Control Engineering Practice; IEEE/ASME Transactions on Mechatronics and the Australian Journal of Electrical and Electronic Engineering. He is also a Member of the IEEE and IFAC Technical Committees on Automotive Control.

The material in this paper was presented at the 10th IEEE Asian Control Conference, May 31–June 3, 2015, Kota Kinabalu, Malaysia. This paper was recommended for publication in revised form by Associate Editor Riccardo Scattolini under the direction of Editor Ian R. Petersen.

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