Unifying approach to observer-filter design. (English) Zbl 1165.93318
Summary: The paper examines similarities between observer design as introduced in automatic control theory and filter design as established in signal processing. It is shown in the paper that there are obvious connections between them in spite of different aims for their design. Therefore, it is prospective to make them be compatible from the structural point of view. Introduced error invariance and error convergence properties of both of them are unifying tools for their design. Lyapunov’s stability theory, signal power, system energy and a power balance relation are other basic terms used in the paper.
MSC:
| 93C10 | Nonlinear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 93B07 | Observability |
References:
| [1] | A. N. Atassi and H. K. Khalil: A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Automat. Control 44 (1999), 1672-1687. · Zbl 0958.93079 · doi:10.1109/9.788534 |
| [2] | A. N. Atassi and H. K. Khalil: A separation principle for the control of a class of nonlinear systems. IEEE Trans. Automat. Control 46 (2001), 742-746. · Zbl 1055.93064 · doi:10.1109/9.920793 |
| [3] | D. Bestle and M. Zeitz: Canonical form observer design for non-linear time-variable systems. Internat. J. Control 38 (1983), 419-431. · Zbl 0521.93012 · doi:10.1080/00207178308933084 |
| [4] | R. S. Bucy and P. D. Joseph: Filtering for Stochastic Processes with Applications to Guidance. Interscience Publishers, New York 1968. · Zbl 0174.21903 |
| [5] | V. Černý and J. Hrušák: Non-linear observer design method based on dissipation normal form. Kybernetika 41 (2005), 59-74. · Zbl 1249.93021 |
| [6] | V. Černý and J. Hrušák: Comparing frequency domain, optimal and asymptotic filtering: a tutorial. Internat. J. Control and Intelligent Systems 34 (2006), 136-142. · Zbl 1172.93406 |
| [7] | V. Černý, D. Mayer, and J. Hrušák: Generalized Tellegen principle and physical correctness of system representations. J. Systemics, Cybernetics and Informatics 4 (2006), 38-42. |
| [8] | F. Esfandiari and H. K. Khalil: Output feedback stabilization of fully linearizable systems. Internat. J. Control 56 (1992), 1007-1037. · Zbl 0762.93069 · doi:10.1080/00207179208934355 |
| [9] | J. P. Gauthier and G. Bornard: Observability for any \(u(t)\) of a class of nonlinear systems. IEEE Trans. Automat. Control 26 (1981), 922-926. · Zbl 0553.93014 · doi:10.1109/TAC.1981.1102743 |
| [10] | M. S. Ghausi and K. R. Laker: Modern Filter Design. Prentice Hall, Englewood Cliffs, New Jersey 1981. |
| [11] | P. Glendinning: Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press, New York 1994. · Zbl 0808.34001 · doi:10.1017/CBO9780511626296 |
| [12] | A. Glumineau, C. H. Moog, and F. Plestan: New algebro-geometric conditions for the linearization by input-output injection. IEEE Trans. Automat. Control 41 (1996), 598-603. · Zbl 0851.93018 · doi:10.1109/9.489283 |
| [13] | R. Hermann and A. J. Krener: Nonlinear controllability and observability. IEEE Trans. Automat. Control 22 (1977), 728-740. · Zbl 0396.93015 · doi:10.1109/TAC.1977.1101601 |
| [14] | J. Hrušák: Anwendung der Äquivalenz bei Stabilitätsprüfung, Tagung über die Regelungstheorie. Mathematisches Forschungsinstitut, Stuttgart 1969. |
| [15] | J. Hrušák and V. Černý: Non-linear and signal energy optimal asymptotic filter design. J. Systemics, Cybernetics and Informatics 1 (2003), 55-62. |
| [16] | E. C. Ifeachor and B. W. Jervis: Digital Signal Processing: A Practical Approach. Addison Wesley, Wokingham 1993. |
| [17] | A. H. Jazwinski: Stochastic Processes and Filtering Theory. Academic Press, New York 1970. · Zbl 0203.50101 |
| [18] | R. E. Kalman and J. E. Bertram: Control system analysis and design via the second method of Lyapunov: I. continuous-time systems, II. discrete-time systems. ASME J. Basic Engrg. 82 (1960), 371-393, 394-400. |
| [19] | R. E. Kalman and R. S. Bucy: New results in linear filtering and prediction theory. ASME J. Basic Engrg. 83 (1961), 95-108. |
| [20] | H. Keller: Non-linear observer design by transformation into a generalized observer canonical form. Internat. J. Control 46 (1987), 1915-1930. · Zbl 0634.93012 · doi:10.1080/00207178708934024 |
| [21] | H. Kimura: Generalized Schwarz form and lattice-ladder realizations of digital filters. IEEE Trans. Circuits Systems 32 (1985), 1130-1139. · Zbl 0579.94024 · doi:10.1109/TCS.1985.1085647 |
| [22] | A. J. Krener and A. Isidori: Linearization by output injection and nonlinear observers. Systems Control Lett. 3 (1983), 47-52. · Zbl 0524.93030 · doi:10.1016/0167-6911(83)90037-3 |
| [23] | A. J. Krener and W. Respondek: Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23 (1985), 197-216. · Zbl 0569.93035 · doi:10.1137/0323016 |
| [24] | D. G. Luenberger: An introduction to observers. IEEE Trans. Automat. Control 16 (1971), 596-602. |
| [25] | A. Muszynska: Rotordynamics. Taylor & Francis, London 2005. · Zbl 1089.70001 |
| [26] | A. W. Oppenheim and R. W. Schafer: Digital Signal Processing. Prentice Hall, Englewood Cliffs, New Jersey 1975. · Zbl 0369.94002 |
| [27] | M. R. Patel, F. Fallside, and P. C. Parks: A new proof of the Routh and Hurwitz criterion by the second method of Lyapunov with application to optimum transfer functions. IEEE Trans. Automat. Control 9 (1963), 319-322. |
| [28] | P. Penfield, S. Spence, and S. Dunker: Tellegen’s Theorem and Electrical Networks. MIT Press, Cambridge, Mass. 1970. |
| [29] | T. Ph. Proychev and R. L. Mishkov: Transformation of nonlinear systems in observer canonical form with reduced dependency on derivatives of the input. Automatica 29 (1993), 495-498. · Zbl 0772.93017 · doi:10.1016/0005-1098(93)90145-J |
| [30] | J. W. Rayleigh: The Theory of Sound. Dover Publications, New York 1945. · Zbl 0061.45904 |
| [31] | H. R. Schwarz: Ein Verfahren zur Stabilitätsfrage bei Matrizen Eigenwertproblemen. Z. Angew. Math. Phys. 7 (1956), 473-500. · Zbl 0073.33901 |
| [32] | S. W. Smith: The Scientist and Engineer’s Guide to Digital Signal Processing. California Technical Publishing, San Diego 1999. |
| [33] | J. C. Willems: Dissipative dynamical systems - Part I: General theory. Arch. Rational Mechanics and Analysis 45 (1972), 321-351. · Zbl 0252.93002 |
| [34] | M. Zeitz: Observability canonical (phase-variable) form for non-linear time-variable systems. Internat. J. Control 15 (1984), 949-958. · Zbl 0546.93011 · doi:10.1080/00207728408926614 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
