Moving horizon estimation: Error dynamics and bounding error sets for robust control

doi:10.1016/j.automatica.2013.01.008

Automatica

Volume 49, Issue 4, April 2013, Pages 943-948

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Abstract

In this work we present a method for the derivation of the estimation error dynamics, the bounding set of the estimation error, and the state estimate dynamic equations of the constrained Moving Horizon Estimator (MHE).

Introduction

Moving horizon estimation (MHE) obtains the system state and disturbance estimates for a system by solving a constrained optimization problem (Findeisen, 1997, Rao et al., 2001). In the case that no system constraints are considered in the problem, the feedback controller and the state estimator can be designed separately following the separation principle (Rao, 2000). However, this is not the case if constraints are introduced in the control problem, since the estimation error can significantly degrade the controller performance and result in constraint violations (Mayne et al., 2006, Rawlings and Mayne, 2009). In order to address this problem, the estimation and control of the system have to be considered simultaneously; this is also known as the output-feedback control problem (Mayne et al., 2006).

In order to design a robust controller, which takes the estimation error into account, the bounds on the estimation error $E_{x}$ have to be known, and equivalently the estimation error dynamics, as well as the dynamics of the estimated system. Methods for obtaining these have been developed by Mayne et al. (2006) for the Luenberger observer, by Alessandri, Baglietto, and Battistelli (2003) and Sui, Feng, and Hovd (2008) for the unconstrained MHE with only state estimates and by Voelker, Kouramas, and Pistikopoulos (2010b) for the unconstrained MHE that includes both the estimation of the state and the disturbances.

In this work we present a method for the general case of constrained moving horizon estimation to obtain: (i) the estimation error dynamic equations, (ii) the estimation error bounds, and (iii) the state estimation dynamic equations. This paper presents an overview of the methodology and the interested reader can find more details and the proofs in Voelker (2011).

Section snippets

Problem definition

We consider linear discrete-time dynamic systems with disturbances and measurement noise $x_{k + 1} ≜ A x_{k} + B u_{k} + G w_{k}$ $y_{k} ≜ C x_{k} + v_{k}$ where $x \in R^{n}$ , $y \in R^{s}$ , $u \in R^{m}$ are the states, outputs and inputs of the system, while $w \in R^{p}$ and $v \in R^{q}$ are the process disturbance and measurement noise respectively. The system is subject to the following constraints: $x_{k} \in X ≜ {x_{k} \in R^{n} | D_{x} x_{k} \leq d_{x}},$ $u_{k} \in U ≜ {u_{k} \in R^{m} | D_{u} u_{k} \leq d_{u}},$ $w_{k} \in W ≜ {w_{k} \in R^{p} | D_{w} w_{k} \leq d_{w}},$ $v_{k} \in V ≜ {v_{k} \in R^{q} | D_{v} v_{k} \leq d_{v}} .$

Assumption 1

(i) The pair $(A, B)$ is controllable, (ii) $(A, C)$ is observable, and (iii) the sets $X \subset R^{n}$ , $U \subset R^{m}$ ,

Explicit/multi-parametric MHE

The explicit solution of MHE by using multi-parametric programming methods was first presented in Darby and Nikolaou (2007) for the case of MHE with filtered arrival cost. In this section we revisit their results for the smoothed arrival cost. By considering (i) the estimated values ${[{\hat{x}^{*}}^{T}, \hat{W} {_{T - N}^{T - 1}}^{*}^{T}]}^{T}$ as the optimization variables, (ii) the measurements, the inputs, and the arrival cost as the problem parameters $θ = [{\underline{x}}^{T}, Y^{T}, U^{T}]$ , and (iii) by substituting the state-space formulation of the

Dynamic equations and the bounding set of the estimation error and the state estimate of the constrained MHE

We proceed now to present a method for the determination of (i) the dynamic equations of the estimation error, (ii) the bounding set of the estimating error and (iii) the dynamics of the state estimate for the constrained MHE.

An illustrative example

The calculation of the error dynamics and the bounding error set is demonstrated with the following illustrative example: $x_{k + 1} = [\begin{matrix} 0.9962 & 0.1949 \\ - 0.1949 & 0.3815 \end{matrix}] x_{k} + [\begin{matrix} 1 \\ 1 \end{matrix}] u_{k} + [\begin{matrix} 0.03393 \\ 0.1949 \end{matrix}] w_{k},$ $y_{k} = [1 - 3] x_{k} + v_{k},$ ${\hat{w}}_{k} \in \hat{W} = {\hat{w} \in R | - 0.01 \leq \hat{w}} = W .$ $w_{k} \in W = {w \in R | - 0.01 \leq w \leq 0.4},$ $v_{k} \in V = {v \in R | | v | \leq 0.05} .$ The error bounding set $E_{x}$ is obtained from the estimation error dynamics, as it was shown in Section 4.3. In our simulation the system initial condition is $x 1_{0} = - 0.6741, x 2_{0} = 3.551$ while the estimator initial condition is $\underline{x} 1_{0} = - 0.2741, \underline{x} 2$

Anna Voelker received her Ph.D. in 2011 in Control Engineering from Imperial College London. She obtained an M.Sc. in Automation and Robotics from the University of Dortmund in 2007 and in 2004 a B.Eng. in Information Technology from the Georg-Simon-Ohm University of Applied Sciences Nuremberg.

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Cited by (0)

Konstantinos Kouramas is a Research Associate in the Centre for Process Systems Engineering at the Department of Chemical Engineering, Imperial College London. He obtained a diploma in Electrical Engineering and Computer Technology from Patras University in 1997, an M.Sc. in Control Systems and a Ph.D. in Control Systems from the Department of Electrical and Electronic Engineering, Imperial College London. His research interests include theory and computational algorithms for control and optimization, and their applications in process, energy and automotive/aeronautics systems.

Efstratios N. Pistikopoulos, Director of the Centre for Process Systems Engineering (CPSE) from 2002 to September 2009, is a Professor of Chemical Engineering, at Imperial College London. He obtained a diploma from the Aristotle University of Thessaloniki in 1984, a Ph.D. from Carnegie Mellon University, USA in 1998, and was with Shell Chemicals (Amsterdam, the Netherlands) before joining Imperial in 1991. His research interests focus on the development of theory, algorithms and computational tools for multi-parametric programming and explicit model predictive control, and their applications in biomedical, energy and process systems.

^☆: The financial contribution of EPSRC (EP/E047017/1 and EP/G059071/1) and European Research Council (MOBILE, ERC Advanced Grant No:226462) is gratefully acknowledged. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Lalo Magni under the direction of Editor Frank Allgöwer.

¹: Tel.: +44 2075946620, +44 207594662; fax: +44 2075941129.

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Brief paperMoving horizon estimation: Error dynamics and bounding error sets for robust control☆

Abstract

Introduction

Section snippets

Problem definition

Explicit/multi-parametric MHE

Dynamic equations and the bounding set of the estimation error and the state estimate of the constrained MHE

An illustrative example

Automatica

Automatica

Automatica

Journal of Process Control

Automatica

Receding-horizon estimation for discrete-time linear systems

IEEE Transactions on Automatic Control

Multi-parametric programming, Vol. 1

Brief paper
Moving horizon estimation: Error dynamics and bounding error sets for robust control☆