Elsevier

Automatica

Volume 60, October 2015, Pages 79-85
Automatica

Brief paper
Interval observer design for LPV systems with parametric uncertainty

https://doi.org/10.1016/j.automatica.2015.07.001Get rights and content

Abstract

Observer design for dynamical systems with both known and unknown time-varying parameters is of significant interest in a number of real-world applications. This class of systems has been rarely addressed in the existing literature. This paper develops an interval observer design methodology for linear parameter varying (LPV) systems with parametric uncertainty. With information on upper and lower bounds of the uncertain parameters, an interval observer that produces an envelope covering all possible state trajectories is presented. The application of the proposed algorithm in an important vehicle state estimation problem which aims at minimizing the worst-case envelope of the side-slip-angle estimate in the presence of uncertain tire cornering stiffness parameters and varying vehicle speed is presented. The obtained observer is evaluated in simulation using CarSim, a commercial industry-standard vehicle simulation software. The results verify the value of the developed observer design method.

Introduction

Observer design has been a very active research field in the control systems community due to its importance in both feedback controller design and in process monitoring for complex systems. Similar to controller design, the uncertainty coming from either external disturbances or from the mismatch between the model and the real process is a severe handicap in developing a high performance observer. However, the accurate estimation of state variables is a fundamental requirement in many safety critical systems, such as in autonomous driving, automotive active safety systems and fault tolerant flight control systems. All these challenging applications necessitate a robust observer design methodology that explicitly takes model uncertainty into consideration.

The interval observer has become a popular observer design method in the presence of input uncertainty during the last decade (Chebotarev et al., 2013, Cheng et al., 2008, Rapaport et al., 2000). Unlike the Kalman filter, which treats the external disturbance as a stochastic signal with known statistical properties (Simon, 2006), the interval observer ignores any probability distribution of the unmeasured disturbance input signals but assumes that they are always constrained in a known time-invariant or time-varying interval. Instead of a single estimation curve for each state variable, the interval observer computes the lower and upper bounds of all the admissible values of the states in the presence of bounded input uncertainty. For safety critical systems, the interval estimation is an important measure to assess how uncertainty affects the state trajectories (Raka & Combastel, 2013).

A commonly used interval observer design technique is the cooperative observer error approach, which requires that the observer error state matrix is not only a Hurwitz matrix but also a Metzler matrix (i.e. all its off-diagonal elements are nonnegative) for continuous-time case. This property preserves the order of the state variables at any instant of time to be the same with their initial conditions (Bernard et al., 2009, Mazenc and Bernard, 2011, Moisan and Bernard, 2010). However, searching for a qualified observer gain is not a trivial task. The existing LTI methods to obtain an interval observer gain in an analytical way become unsuitable for more complex high order systems and are difficult to be generalized to parameter varying systems (Rapaport & Dochain, 2005). To overcome this difficulty, some methods that release the cooperative constraint for the observer error through a coordinate transformation have been proposed (Chebotarev et al., 2013, Efimov et al., 2012). In spite of the elegant theoretical proof, these coordinate transformation methods often consider the uncertainty in the input channel rather than the model uncertainty, which puts a question mark on the applicability to systems with largely varying parameters.

Interval observer design for systems with model uncertainty is discussed by a few papers (Chebotarev et al., 2013, Cheng et al., 2008, Raissi et al., 2013, Raïssi et al., 2013). However, existing results assume that the uncertain state space matrices are uniformly bounded, such as A¯A(δ)A¯. The structure of the uncertain matrix is not explored, which may result in a conservative interval estimation.

In this paper, a systematic interval observer design method for linear-parameter-varying (LPV) systems with parametric uncertainty will be presented. Unlike most existing methods in the literature that assume the variation of the state matrix is uniformly bounded by constant matrices (Cheng et al., 2008, Raissi et al., 2013), the structure of the parametric uncertainty is explicitly incorporated in the observer design to reduce the conservatism. Furthermore, a gain-scheduled interval observer for LPV systems with both measured and unmeasured time-varying uncertain parameters has never been studied in the past to the best of the authors’ knowledge. This paper constitutes perhaps the first literature discussing interval observers for such systems. It will be shown that the proposed interval observer is indeed a switched LPV system. The well-developed semidefinite programming (SDP) approach (Feron et al., 1994, Scherer and Weiland, 2005) is applied herein to search for the qualified observer gains, which results in a robustly stable and cooperative observer error dynamical system. Then, the proposed design methodology will be applied to the development of an interval observer for a vehicle state estimation problem in automotive active safety systems. Some assumptions used in the previous research in this automotive field are relaxed in this paper.

The remaining of this paper is organized as follows. First, some background knowledge is reviewed in Section  2. Next, the design of the gain-scheduled interval observer for LPV systems with parametric uncertainty is presented in Section  3. Finally, we apply this design methodology to a vehicle state estimation problem in Section  4. Section  5 contains the final conclusions.

Section snippets

Notation and background

This section presents some theoretical background related to interval observer design from the literature. The symbol R represents the set of real numbers. Rm×n denotes the set of m×n matrices whose elements all belong to R. For matrices X,YRm×n,XY or YX indicates that each element in X is no smaller than its counterpart in Y. For symmetric matrices A,BRn×n,AB or BA means that AB is a positive definite matrix.

Interval observer design for LPV systems

In many real-world applications, the dynamical model of the process contains some time-varying parameters that can be measured online, such as the vehicle speed in automotive active safety systems. Hence, the controller and observer gains can be updated accordingly to achieve robust stability and performance. This is the so-called gain-scheduling design technique. The linear-parameter-varying (LPV) modeling and design methodology are widely accepted as a robust way to develop a gain-scheduled

Robust slip angle estimation

In this section, a slip angle estimation problem in automotive active safety systems will be used to illustrate the application of the gain-scheduled interval observer design methodology.

Conclusion

In this paper, we present an interval observer design methodology for LPV systems with parametric uncertainty. It has been shown that uncertain parameters can be treated as a disturbance input for the observer error. Then, a simple analytical method to estimate the interval bounding this input uncertainty is developed. The search for the qualified observer gains, such that the observer error is robustly stable and cooperative, is cast as a convex semidefinite programming problem. A simulation

Acknowledgments

The authors sincerely acknowledge the anonymous reviewers for critical and valuable comments that helped them improve the quality of the paper.

Yan Wang earned his M.S. from Delft University of Technology (The Netherlands) in 2006, and Ph.D. from Auburn University (USA) in 2014 in mechanical engineering. He worked as a control engineer in TRW Automotive from 2006 to 2009. In Oct. 2014, he joined the Department of Mechanical Engineering at University of Minnesota as a Postdoc research associate. Yan’s research interests include robust control analysis, convex optimization, sensor fusion, and nonlinear observer design.

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    It was firstly used in the biotechnological domain [1]. In recent years, for safety-critical systems (such as autonomous driving vehicle), interval observer has been introduced to evaluate the uncertainty effects on state trajectory in advance [5]. The effective estimated boundary by the interval observer can also be used for controller design [8].

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Yan Wang earned his M.S. from Delft University of Technology (The Netherlands) in 2006, and Ph.D. from Auburn University (USA) in 2014 in mechanical engineering. He worked as a control engineer in TRW Automotive from 2006 to 2009. In Oct. 2014, he joined the Department of Mechanical Engineering at University of Minnesota as a Postdoc research associate. Yan’s research interests include robust control analysis, convex optimization, sensor fusion, and nonlinear observer design.

David M. Bevly is the McNair Endowed Professor in the Department of Mechanical Engineering and director of the GPS and Vehicle Dynamics Laboratory at Auburn University. David received his B.S. from Texas A&M University in 1995, M.S. from Massachusetts Institute of Technology in 1997, and Ph.D. from Stanford University in 2001 in mechanical engineering. He joined the faculty of the Department of Mechanical Engineering at Auburn University in 2001 as an assistant professor. Dr. Bevly’s research interests include control systems, sensor fusion, GPS, state estimation, and parameter identification. His research focuses on vehicle dynamics as well as modeling and control of vehicle systems. Specifically, Dr. Bevly has developed algorithms for control of off-road vehicles and methods for identifying critical vehicle parameters using GPS and inertial sensors.

Rajesh Rajamani obtained his M.S. and Ph.D. degrees from the University of California at Berkeley in 1991 and 1993 respectively and his B.Tech. degree from the Indian Institute of Technology at Madras in 1989. Dr. Rajamani is currently Professor of Mechanical Engineering at the University of Minnesota. His active research interests include sensors and estimation systems for automotive and biomedical applications.

Dr. Rajamani has co-authored over 100 journal papers and is a co-inventor on 9 patent applications. He is the author of the popular book “Vehicle Dynamics and Control” published by Springer Verlag. Dr. Rajamani has served as Chair of the IEEE Technical Committee on Automotive Control and on the editorial boards of the IEEE Transactions on Control Systems Technology and the IEEE/ASME Transactions on Mechatronics. Dr. Rajamani is a Fellow of ASME and has been a recipient of the CAREER award from the National Science Foundation, the 2001 Outstanding Paper award from the journal IEEE Transactions on Control Systems Technology, the Ralph Teetor Award from SAE, and the 2007 O. Hugo Schuck Award from the American Automatic Control Council.

The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen.

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