Brief PaperAn observer for systems with nonlinear output map☆
Introduction
In many engineering applications it is desirable to measure quantities without having to install actual measurement equipment for this purpose. In some applications, such as the steel converter process, it is virtually impossible to perform certain measurements online, which necessitates estimation of the sought quantities. To this end, an observer, which combines process knowledge, in the form of a mathematical model, with information, in the form of indirect measurements, is an important tool.
Process models that are valid over a large operating range are, however, often nonlinear. How to construct an observer for such a model is not obvious and therefore this is an active field of research. The following approaches are often applied to observer design for nonlinear systems.
Stochastic approaches: The Kalman filter gives unbiased minimum variance estimates of the state for a linear system utilizing the statistical properties of the disturbances. The extended Kalman filter (EKF) (Gelb, 1974) and the statistically linearized filter (Beaman, 1984) are two methods that seek to somehow extend this property to observers for nonlinear systems.
Transformation methods: It is sometimes possible to find a transformation that brings the nonlinear system into observer canonical form, i.e. linear dynamics with output injection (Krener & Respondek, 1985; Bestle & Zeitz, 1983). The conditions for the existence of this transformation are rather restrictive but several approximate methods are available. Two other transformations, observable form (Gauthier, Hammouri, & Othman, 1992) and triangular form (Gauthier & Kupka, 1994), only require the system to be observable for any inputs.
Extended Linearization: This method, suggested in Baumann and Rugh (1986), gives desired behavior, in terms of eigenvalues of the linearized error system, in the neighborhood of operating points.
Constant feedback: A straightforward approach to nonlinear observer design is to use linear feedback. If the nonlinearities are globally Lipschitz, then it is possible to find a constant feedback gain so that the estimate converges (Raghavan & Hedrick, 1994; Rajamani, 1998). Furthermore, by regarding the nonlinearities as perturbations for a linear system, methods for observer design for uncertain linear systems (Wang, Kuo, & Hsu, 1987) can be used.
Other methods are, e.g. the variable structure (Walcott & Żak, 1987) and sliding mode (Slotine, Hedrick, & Misawa, 1987) approaches. See also Misawa and Hedrick (1989) for a survey of different approaches to observer design for nonlinear systems.
The development of the observer in this paper was motivated by its application to the top blown steel converter. A model of this process can be contained in the process structure:where is the output, is the state vector, and is the input. The sets and are some closed, bounded sets that the state and the input signal are assumed to be confined to.
This process model of the steel converter has four major difficulties that are not addressed by many of the above methods.
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There is a direct coupling from the input to the output via the nonlinear map h(x,u). Transformation to observer canonical or observable form will in this case require derivatives of the input. These derivatives are not available and estimating them, as suggested in Ciccarella, Dalla Mora, and Germani (1993), will increase the dimension of the state vector, which is not desirable. Transformation to triangular form is very complicated since it involves the solution of a fourth-order polynomial.
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The nonlinearity is significant, in the sense that the sensitivity of the output with respect to the state, i.e. varies very much. In the parts of the state space where the sensitivity to a particular variable is low, the feedback gain for this variable must be small to prevent irrelevant information, i.e. disturbances, from affecting its estimate. Constant feedback is therefore not suitable, which will be shown in this paper.
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The process is never in steady state. The extended linearization method is therefore not applicable since it only guarantees properties locally around operating points.
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There are significant uncertainties in the process model, both in the dynamics and in the measurement. Therefore, the trivial observer with no feedbackdoes not provide enough accuracy in the state estimate x̂, even though it is globally convergent in the estimation error . The high-gain technique that is proposed for the observable and triangular forms involves a special choice of constant feedback and is known to be unsuitable for systems with large measurement disturbances (Gauthier & Kupka, 1994).
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For undetectable systems, the state Π of the Riccati equation (4) will diverge.
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The dimension n(n+1)/2 of the state of the Riccati equation makes it computationally heavy.
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The complexity of the EKF makes it difficult to analyze in terms of, e.g. robustness and region of attraction.
The drawbacks with the EKF can, however, by proper selection of time-varying weighting matrices, be relieved for certain classes of processes as will be shown in this paper for the process structure (1). By a special selection of the weighting matrices Ξ and ρ (see Remark 1), Π/ρ will be constant, yielding an observer without the excessive calculations of the Riccati equation but with the desirable property of feedback weighting by the sensitivity of the output with respect to the state. This weighting of the measurements by their relevance for state estimation purposes is the main reason to why the EKF is able to handle disturbances. Furthermore, it will be possible to analyze the resulting observer in terms of region of attraction (Theorem 2, Theorem 3), provided some bounds on the nonlinearity.
Apart from in the EKF, the sensitivity of the output with respect to the state has been used for the purpose of estimation in a number of ways.
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The MIT rule for updating parameters of an adaptive control system (see Remark 2).
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In Tsinias (1990), the sensitivity of the output with respect to the state is suggested as a factor in the feedback weight in the context of deriving conditions for the existence of asymptotically stable, smooth observers for nonlinear systems. The main result, however, applies to a restricted class of systems to which the steel converter process does not belong.
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In many of the transformation based methods, e.g. (Bestle & Zeitz, 1983; Ciccarella et al., 1993), the feedback is weighted by the inverse of the observability matrix. Since the observability matrix contains the sensitivity of the output with respect to the state, this approach is, in some sense, opposite to the ones above.
The top blown converter process (BOS) is the predominant method of steel making. Its purpose is to reduce the contents of impurities, mainly silicon and carbon, in the hot metal coming from the blast furnace.
Hitting the correct carbon content in the final product is, however, a difficult task, since no continuous measurement of this quantity is available. Being able to accurately estimate the metal analysis, especially the content of carbon is important to ensure the quality of the steel, minimize the consumption of resources and avoid re-blowing. The use of an analytical process model is a promising way of achieving this goal although existing process models are not good enough to predict the carbon content with sufficient accuracy and therefore require feedback of the output, which will be shown in the sequel.
The following notation is used in the paper. Let λi(M) be an eigenvalue of . For symmetric matrices, the eigenvalues are ordered so thatFurthermore, |·| is the euclidean vector norm, ||·|| is the corresponding induced matrix 2-norm, while and are defined as the open and closed ball, respectively, with radius r and center in x, i.e. and . Finally, the nullspace of M is denoted .
Section snippets
Problem formulation
Consider the process structure (1) where h(x,u) is assumed to be piecewise continuous in u and twice differentiable in x. For the state matrix A, two cases are considered Case 1 A is Hurwitz, i.e. Re{λi(A)}<0 for i=1,2,…,n. Case 2 A is allowed to have one eigenvalue with Re{λj(A)}=0 and all other eigenvalues Re{λi(A)}<0.
Linear analysis
The following result states that the observer error process (7) is asymptotically stable if the linear part of (1) is asymptotically stable. Theorem 1 Given system(1)and observer(5), the erroris locally asymptotically stable in the neighborhood ofif the state matrix A is Hurwitz, i.e.Re{λi(A)}<0, fori=1,2,…,n. Proof Linearizing the error dynamics (7) around yields where F(x,u)=A−Kh′T(x,u)h′(x,u). Assume the Lyapunov function , which is positive definite since K−1 is the
The converter process
The operation of the converter process (Fig. 1) is started by charging hot metal, scrap, and slag-forming agents into the converter. Other additives can be added throughout the blow.
Oxygen (O2) is blown at a supersonic rate onto the metal surface and oxidizes the metal components, mainly iron (Fe), silicon (Si), and carbon (C). Carbon monoxide (CO) and carbon dioxide (CO2) are thus produced and form the off-gases of the process.
The process takes approximately 15– and is stopped by the
State estimation in the converter process
A comparison between the simulated carbon content and the measured final value indicates that the open-loop model, i.e. the trivial observer (2), does not produce sufficiently accurate estimates of the carbon content (Johansson, Medvedev, & Widlund, 2001). Feedback of the measured signals should therefore be applied to improve the results.
Conclusions
An observer for linear systems with nonlinear output map and injection is proposed and analyzed. The central feature of this observer is that the feedback is weighted by the sensitivity of the output with respect to the state. It has been shown that when using this feedback, the error dynamics are exponentially stable and a region of attraction can be calculated.
The observer has been applied to the top blown steel converter process and shown to produce accurate estimates of the carbon content.
Acknowledgements
The authors want to thank the personnel of SSAB Oxelösund for supplying the data and Prof. Torkel Glad of Linköping University for valuable suggestions regarding the proofs of Theorem 2, Theorem 3. The deep and insightful comments and suggestions by the anonymous reviewers are very much appreciated. Financial support provided by the Swedish National Board for Industrial and Technical Development, NUTEK, is also gratefully acknowledged.
Andreas Johansson was born in Luleå, Sweden, in 1972. He received his M.S. degree in Computer Science and Ph.D degree in Automatic Control from Luleå University of Technology in 1997 and 2002, respectively and is, since then, associate professor at the Department of Computer Science and Electrical Engineering at the same university. His current research interest is estimation and fault detection in nonlinear and uncertain systems.
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Andreas Johansson was born in Luleå, Sweden, in 1972. He received his M.S. degree in Computer Science and Ph.D degree in Automatic Control from Luleå University of Technology in 1997 and 2002, respectively and is, since then, associate professor at the Department of Computer Science and Electrical Engineering at the same university. His current research interest is estimation and fault detection in nonlinear and uncertain systems.
Alexander Medvedev was born in Leningrad (St. Petersburg), USSR (Russia), in 1958. He received his M.Sc. (Honors) and Ph.D. degrees in control engineering from Leningrad Electrical Engineering Institute (LEEI) in 1981 and 1987, respectively. From 1981 to 1991, he subsequently kept positions as a system programmer, Assistant Professor, and Associate Professor (docent) in the Department of Automation and Control Science at the LEEI. He was with the Process Control Laboratory, Åbo Akademi, Finland during a long research visit in 1990–1991. In 1991 he has received Docent degree from the LEEI. In the same year, he joined the Computer Science and Electrical Engineering Department at Luleå University of Technology, Sweden as an Associate Professor in the Control Engineering Group (CEG). In February 1996 he has been promoted to Docent.
From October 1996 to December 1997, he served as Acting Professor of Automatic Control in the CEG. Since January 1998, he has been Full Professor of Automatic Control at the same institution. Starting October 2001, A. Medvedev is also Professor of Automatic Control at Uppsala University, Sweden. He is presently involved in research on fault detection, time-delay systems, time-varying and alternative parameterization methods in analysis and design of dynamic systems. In 1997, in co-operation with a number of leading Swedish process industry companies, he established the Center for Process and System Automation (ProSA) at LTU and until 2002 supervised its activities.
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An abridged version of this paper was presented at the fifth IFAC Symposium Nonlinear Control Systems NOLCOS’01 in Saint-Petersburg, Russia, July 2001. This paper was recommended for publication in revised form by Associate Editor Carlos Canudas de Wit under the direction of Editor Hassan Khalil.