Brief paperNonlinear observers comprising high-gain observers and extended Kalman filters☆
Introduction
Estimation of the state vector of nonlinear systems is a challenging problem and has received considerable attention in the literature; see e.g. Farza, MSaad, Triki, and Maatoug (2011), Sanfelice and Praly (2012), Shen, Shen, Jiang, and Huang (2010) and the references therein. One technique that proved useful in nonlinear feedback control is high-gain observers. A recent survey of its development is presented in Khalil and Praly (2013). However, most of the work on high-gain observers is focused on the problem of partial state estimation. Moreover, previous work on full order high gain observers is limited to minimum phase systems. Memon and Khalil (2009), for instance, proposed a full order observer that employs an open loop observer for the internal dynamics, which limits the validity of the technique to minimum phase systems. Another paper (Ahrens & Khalil, 2007) proposed the use of an EKF-based high gain observer for the estimation of the full state vector of minimum phase systems with linear internal dynamics driven only by the output.
There have been a few techniques that dealt with observer design for non-minimum phase nonlinear systems and achieved non-local convergence results. For example, a method for designing observers for systems affine in the unmeasured states was introduced in Karagiannis, Astolfi, and Ortega (2003) and for a more general class of nonlinear systems in Karagiannis, Carnevale, and Astolfi (2008). The key idea in these results is the construction of an invariant and attractive manifold. This has to be achieved by solving a set of partial differential equations. Shtessel, Baev, Edwards, and Spurgeon (2010) proposed a Higher Order Sliding Mode Observer to estimate the full state vector with a vector of unknown inputs for non-minimum phase nonlinear systems. It considered the case when the internal dynamics are quasilinear and the forcing term can be piece-wise modeled as the output of a dynamical process given by an unknown linear system with a known order.
We consider systems represented by a normal form comprised of the internal dynamics and a chain of integrators that represents the output and its derivatives. We use EHGO to provide estimates of the derivatives of the output in addition to a signal that is used as a virtual output to an auxiliary system based on the internal dynamics. This is indeed possible because of the relative high speed of the EHGO. We choose to use the EKF as an observer for the internal dynamics due to its simplicity and applicability to a wide range of nonlinear systems. In fact, any other suitable observer can be used to estimate the state of the internal dynamics, providing flexibility for the overall observer design.
Extended High Gain Observers have been used in the literature to serve different objectives, such as to estimate model uncertainties and external disturbances (Freidovich & Khalil, 2008) and to develop a Lyapunov-based switching control strategy (Freidovich & Khalil, 2007). More recently, the work in Nazrulla and Khalil (2011) utilizes the EHGO to design a stabilizing controller for a non-minimum phase nonlinear system.
One of the popular approaches for the design of observers for a general class of nonlinear systems is the Extended Kalman Filter. Its popularity is due to the simplicity of the observer design regardless of the system’s complexity. Results on the convergence properties of the EKF appeared in papers such as Baras, Bensoussan, and James (1988), Boutayeb and Aubry (1999), Deza, Busvelle, Gauthier, and Rakotopara (1992), Krener (2003), Reif, Sonnemann, and Unbehauen (1998) and Song and Grizzle (1995). The main drawback of the EKF, however, is the need for linearization. This in turn could limit the region of attraction of the estimator error. Some ideas have been proposed to expand the region of attraction relying mostly on high gain techniques (Deza et al., 1992, Reif et al., 1998). On the other hand, observers, similar to Kalman filters for linear time-varying systems, have been successfully used for systems affine in the unmeasured states; (Besancon et al., 2006, Hammouri and de Leon Morales, 1990, Praly and Kanellakopoulos, 2000).
For systems represented in the normal form, we achieve a convergence result that is local in the estimation error of the internal dynamics but non-local in estimating the chain-of-integrators variables. In the special case when the system is affine in the internal state, we achieve semi-global convergence. This was also a motivation for using the EKF, as it allows us to exploit the linearity in the internal dynamics to achieve a non-local result. In summary, the main contribution of this paper is manifested in two points: First, we propose a full order observer for a class of nonlinear systems that could be non-minimum phase. Second, the observer design procedure is simple and constructive.
The remainder of the paper is organized as follows. Section 2 states the problem formulation and a description of the considered class of systems. Section 3 discusses the problem of designing a full order EHGO observer for linear systems. This serves as a motivation for the main results in Section 4. This section includes two examples, namely, the design of a full order observer for a synchronous generator connected to an infinite bus and for the TORA system. Section 5 includes some conclusions.
Section snippets
Problem formulation
We consider a single-input, single-output nonlinear system with a well defined relative degree represented in the form Isidori (1997): where , is the measured output and is the control input. Eqs. (1), (2), (3), (4) can be written in a compact form where the matrix , the matrix and the matrix represent a chain of integrators.
Assumption 1 The functions , and are known and
Linear systems
We briefly visit the problem of designing a full-order extended high-gain observer for linear systems as a motivation for our main result. We consider a single-input–single-output linear system where the dimension of the system is and its relative degree is . It is always possible to represent this system in the normal form Isidori (1999) We assume that the pair is observable. This is equivalent to observability of
General case
We consider the system (1), (2), (3), (4). We begin by first considering an observer, we call it the internal observer, for the auxiliary system in which is considered as a known input. We shall utilize an EHGO to estimate the state vector and the signal . Since we anticipate that the EHGO will provide these signals in a relatively fast time, we can assume that they are available for the internal observer. We choose the EKF as an observer for this system. Thus, the
Conclusions
We have proposed a full order observer for a class of nonlinear systems that could be non-minimum phase. The observer is based on the use of the EHGO to estimate, in a relatively fast time, the derivatives of the output and a signal that is used as a virtual output to an auxiliary system comprised of the internal dynamics. Accordingly, we chose to design an EKF observer for this system. The reason for this choice is primarily because of the EKF’s simplicity and wide use in practice. If the
AlMuatazbellah M. Boker received his B.Eng degree in Mechatronics Engineering from the University of Leeds, UK, in 2002 and M.S. degree in Control Systems Engineering from the University of Sheffield, UK, in 2003. He worked as a Teaching Assistant at the University of Benghazi, Libya, 2004–2008. Since 2008 he has been a Ph.D. student at Michigan State University, USA. His research interests include estimation and control of nonlinear systems and hybrid control of networked systems. Mr. Boker
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AlMuatazbellah M. Boker received his B.Eng degree in Mechatronics Engineering from the University of Leeds, UK, in 2002 and M.S. degree in Control Systems Engineering from the University of Sheffield, UK, in 2003. He worked as a Teaching Assistant at the University of Benghazi, Libya, 2004–2008. Since 2008 he has been a Ph.D. student at Michigan State University, USA. His research interests include estimation and control of nonlinear systems and hybrid control of networked systems. Mr. Boker received the 2002 Smallpeice Trust Prize, the University of Leeds, for “an outstanding work on mechatronics”, and best poster award, 2012, Graduate Research Symposium, Michigan State University.
Hassan K. Khalil received the B.S. and M.S. degrees in Electrical Engineering from Cairo University, Egypt, in 1973 and 1975, respectively, and the Ph.D. degree from the University of Illinois, Urbana–Champaign, in 1978, all in Electrical Engineering.
Since 1978, he has been with Michigan State University (MSU), where he is currently University Distinguished Professor of Electrical and Computer Engineering. He has consulted for General Motors and Delco Products, and published several papers on singular perturbation methods and nonlinear control. He is the author of Nonlinear Systems (Macmillan 1992; Prentice Hall 1996 & 2002) and coauthor of Singular Perturbation Methods in Control: Analysis and Design (Academic Press 1986; SIAM 1999).
Dr. Khalil was named an IEEE Fellow in 1989 and an IFAC Fellow in 2007. He received the 1989 IEEE-CSS George S. Axelby Outstanding Paper Award, the 2000 AACC Ragazzini Education Award, the 2002 IFAC Control Engineering Textbook Prize, the 2004 AACC O. Hugo Schuck Best Paper Award, and the 2009 AGEP Faculty Mentor of the Year Award. At MSU he received the 2003 Teacher Scholar Award, the 1994 Withrow Distinguished Scholar Award, and the 1995 Distinguished Faculty Award. He served as Associate Editor of the IEEE Transactions on Automatic Control, Automatica, and Neural Networks, and as Editor of Automatica for nonlinear systems and control. He was Registration Chair of the 1984 CDC, Finance Chair of the 1987 ACC, Program Chair of the 1988 ACC, and General Chair of the 1994 ACC.
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This work was supported in part by NSF under grant number ECCS-1128467. The material in this paper was partially presented at the 51st IEEE Conference on Decision and Control (CDC), December 10–13, 2012, Maui, Hawaii, USA. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor Andrew R. Teel.
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