Feedforward model predictive control

doi:10.1016/j.arcontrol.2011.10.007

Annual Reviews in Control

Volume 35, Issue 2, December 2011, Pages 199-206

https://doi.org/10.1016/j.arcontrol.2011.10.007 Get rights and content

Abstract

This paper examines the role played by feedforward in model predictive control (MPC). We contrast feedforward with preview action. The latter is standard in model predictive control, whereas feedforward has been rarely, if ever, used in contemporary formulations of MPC. We argue that feedforward can significantly improve performance in the presence of measurement noise and certain types of model uncertainty.

Introduction

Model Predictive Control (MPC) has been a major success story in industry. There are two principal reasons for this success. Firstly, the method addresses important practical issues including constraints on inputs and states. Secondly, there exists a rich supporting theory (Goodwin et al., 2005, Maciejowski, 2002, Kouvaritakis and Cannon, 2001, Rawlings and Mayne, 2009).

In classical control theory (Goodwin, Graebe, & Salgado, 2001), a distinction is made between “feedforward”, “preview” and “feedback”. Each of these tools is known to yield performance gains in certain practical problems. By way of contrast, current implementations of MPC are restricted to the use of preview and feedback. This begs the question, “can feedforward be incorporated into MPC and, if so, under what circumstances would it be useful?”. The goal of the current paper is to address this question. The paper builds on earlier work by the same authors presented in Goodwin et al., 2011, Carrasco and Goodwin, 2011. The ideas presented in this paper also have a loose connection to the work of Campi and Garatti (2010). In the latter reference, Campi and Garatti argue that it may be desirable to replace absolute guarantees of performance (including stability) by high probability guarantees, in exchange for a significant improvement in performance. Our work aims at combining an absolute guarantee of robust stability with high performance reference tracking for systems “near” the nominal system. Our examples show that “near” can include large modelling errors. An important departure from the work in Campi and Garatti (2010) is that we augment the control architecture to give a richer set of trade-offs under different probabilistic scenarios rather than optimising the tuning of a given architecture under probabilistic conditions.

To set the framework, we first define the terms preview, feedback and feedforward. We define “Preview” as the inclusion of knowledge regarding future reference or disturbance values into the control law. Preview is helpful in control since it allows the controller to “prepare” for future changes. In particular, it is known (Middleton, Chen, & Freudenberg, 2004) that the availability of preview action weakens the fundamental limitations that apply to set point tracking. We define “Feedback” as the process of observing the plant response and taking corrective action based on the difference between the measured response and the desired response. Finally, we define “Feedforward” as an open loop evaluation of an input without any corrections arising from observations made on the plant. Note that this definition differs from that commonly used in MPC literature (Rossiter & Valencia-Palomo, 2009).

Preview and feedback are standard in MPC. However, feedforward seems to be rarely, if ever, used. We show, for certain types of model uncertainty and measurement noise, that the inclusion of feedforward yields a performance gain. Here we focus on conceptual rather than theoretical issues. Initial theoretical support can be found in Carrasco and Goodwin (2011).

The layout of the remainder of the paper is as follows: In Section 2 we motivate the idea of feedforward using a linear time invariant unconstrained control scenario. In Section 3 we present basic definitions. Section 4 introduces the observer used to estimate the plant state (including unmeasured disturbances). Section 5 outlines the feedforward MPC design. Section 6 describes the design of the feedback component. Section 7 presents numerical examples. Finally, in Section 8 we draw conclusions.

Section snippets

Motivation

By way of introduction to feedforward we consider a simple linear unconstrained control problem. Say the plant transfer function is G. Also say that we have a reference signal r and a measured output disturbance d_n (arising, for example, from an up-stream process or a prediction system Laks, Simley, Wright, Kelley, & Jonkman, 2011). We can then sketch a linear single-input single-output control system incorporating both feedforward and feedback as in Fig. 1. In this figure, y^∗, e, u_ff, u_fb, u, y

Basic definitions

We consider a linear time-invariant single-input single-output system: $y = Gu + d_{n} + d$ $y_{n} = y + n$ where G(q) is a discrete time transfer function (the “true plant” transfer function) and where y, u, d_n, d, n, y_n denote unmeasured plant output, plant input, measured disturbance, unmeasured disturbance, measurement noise and measured plant output respectively. To describe plant uncertainty we write G as $G = G_{o} (1 + G_{Δ})$ where G_o is the nominal model and G_Δ the multiplicative model error.

The plant dynamics are

Observer design

For the purpose of observer design, we assume that the unmeasured disturbance d is constant. We then define an observer for the system state and the unmeasured disturbance: ${\hat{x}}^{+} = A \hat{x} + Bu + J_{1} (y - C \hat{x} - d_{n} - \hat{d})$ ${\hat{d}}^{+} = \hat{d} + J_{2} (y - C \hat{x} - d_{n} - \hat{d})$ where $\bar{A} = [\begin{matrix} A - J_{1} C & - J_{1} \\ - J_{2} C & 1 - J_{2} \end{matrix}]$ is stable. The observer (17) will be used as a mechanism to ensure perfect tracking of constant reference signals in the presence of unmeasured disturbances and model errors.

Feedforward design

As discussed previously, we require that this part of the design make no use of actual plant measurements. Hence, we use the nominal plant transfer function G_o. We are interested in designing the feedforward input u_ff so that y_ff = G_o · u_ff tracks the given reference signal with small error (the subscript “ff” refers to the feedforward model).

Of course, we need a mechanism for restraining the bandwidth to a sensible value (otherwise impractical deadbeat type responses will ensue).

Feedback design

We next turn to the design of the feedback component. For this purpose, we define the total plant input as (see Fig. 1) $u (k) = u_{fb} (k) + u_{ff} (k)$ Our goal here is to determine u_fb(k). As before, our main objective is to track y^∗. Here, however, we have two other design goals, namely to achieve robust stability and to reject the (unmeasured) disturbance d. Since we do not measure either the true plant state x or the disturbance d, we must use the observer given earlier in (16), (17).

Examples

In this section we present several examples to illustrate the FFMPC strategy. We utilise the feedback design input weighting λ_fb as a mechanism to ensure robustness by imposing bandwidth restrictions on the feedback component. Since reference tracking and measured disturbance rejection are similar, we restrict attention to reference tracking.

We assume a generic model having transfer function $G = \frac{k}{z^{d} \cdot (z - a)}$ where k/(1 − a) is the d.c. gain, a is the discrete pole and d is a pure delay.

We consider two

Conclusions

In this paper we have discussed the ideas behind the use of feedforward in model predictive control. We have presented the core concepts that underlie the potential advantages of using feedforward. Also, several simulation studies have been provided to illustrate the pros and cons that one might encounter when applying the feedforward scheme.

Acknowledgements

This paper is dedicated to David Mayne on the occasion of his 80th birthday celebration. David has inspired generations of researchers including the second author (who has known David for over 40 years) and the first author (who has read many of his papers).

The topic of feedforward MPC is an appropriate choice for this paper for two reasons: (i) David Mayne’s prominence in the area of MPC and (ii) the genesis of the ideas presented here began during a visit by David Mayne to Australia in

Diego S. Carrasco was born in Valparaı´so, Chile, in 1985. He obtained the Ingeniero Civil Electrónico degree and the M.Sc. degree in Electronics Engineering from the Universidad Técnica Federico Santa Marı´a, Valparaı´so, Chile, in 2009. Since 2010 he has been working towards the Ph.D. degree at The University of Newcastle, Australia. His research interests include model predictive control, robust and optimal control, sampling theory and numerical analysis.

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Graham Goodwin obtained a B.Sc (Physics), B.E (Electrical Engineering), and Ph.D from the University of New South Wales. He is currently Professor Laureate of Electrical Engineering at the University of Newcastle, Australia and is Director of The University of Newcastle Priority Research Centre for Complex Dynamic Systems and Control. He holds Honorary Doctorates from Lund Institute of Technology, Sweden and the Technion Israel. He is the co-author of eight books, four edited books, and many technical papers. Graham is the recipient of Control Systems Society 1999 Hendrik Bode Lecture Prize, a Best Paper award by IEEE Transactions on Automatic Control, a Best Paper award by Asian Journal of Control, and 2 Best Engineering Text Book awards from the International Federation of Automatic Control in 1984 and 2005. In 2008 he received the Quazza Medal from the International Federation of Automatic Control and in 2010 he received the IEEE Control Systems Award. He is a Fellow of IEEE; an Honorary Fellow of Institute of Engineers, Australia; a Fellow of the International Federation of Automatic Control, a Fellow of the Australian Academy of Science; a Fellow of the Australian Academy of Technology, Science and Engineering; a Member of the International Statistical Institute; a Fellow of the Royal Society, London and a Foreign Member of the Royal Swedish Academy of Sciences.

^☆: An earlier version of this paper was presented at the IFAC Workshop on 50 Years of Nonlinear Control and Optimization (London, UK, September 30–October 1, 2010), dedicated to David Q. Mayne on the occasion of his 80th birthday.

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Feedforward model predictive control☆