Abstract
The details of the linear algebra which must be performed in order to generate state space equations from primitive equations of the linear constant coefficient ordinary differential mathematical model are presented. The applicability of the procedure is not dependent on any modeling philosophy and/or regimen. In addition to generating the state variable representation of the system, the procedure also generates equations which relate all other system internal variables to the system states and the system input variables. The treatment is complete and identifies all the operations which must be done in implementing a digital computer automation of the process. Some useful insights into the effects of ill-posed mathematical models and of modeling errors result from the treatment.
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References
Kron, G.,Tensor Analysis of Networks, John Wiley and Sons, New York, New York, 1939.
Kron, G.,Equivalent Circuits of Electric Circuits, John Wiley and Sons, New York, New York, 1951.
Murphy, G. J.,Basic Automatic Control Theory, 2nd Edition, Van Nostrand Book Company, New York, New York, 1966.
Mason, S. J., andZimmerman, H. J.,Electronic Circuits, Signals, and Systems, John Wiley and Sons, New York, New York, 1960.
Mason, S. J.,Feedback Theory: Further Properties of Signal Flow Graphs, Proceedings of the Institute of Radio Engineers, Vol. 44, pp. 920–926, 1956.
Mason, S. J.,Feedback Theory: Some Properties of Signal Flow Graphs, Proceedings of the Institute of Radio Engineers, Vol. 41, pp. 1144–1156, 1953.
Koenig, H. E., andBlackwell, W. A.,Electromechanical System Theory, McGraw-Hill Book Company, New York, New York, 1961.
Shearer, J. L., Murphy, A. T., andRichardson, H. H.,Introduction to System Dynamics, Addison-Wesley Press, Reading, Massachusetts, 1967.
Karnopp, D. C., andRosenburg, R. C.,System Dynamics: A Unified Approach, Wiley-Interscience, New York, New York, 1975.
Karnopp, D. C., andRosenburg, R. C.,Analysis and Simulation of Multiport Systems, MIT Press, Cambridge, Massachusetts, 1968.
Paynter, H. M.,Analysis and Design of Engineering Systems, MIT Press, Cambridge, Massachusetts, 1961.
Levin, H.,Introduction to Computer Analysis; ECAP for Electronic Technicians and Engineers, Prentice-Hall, Englewood Cliffs, New Jersey, 1970.
Rosenburg, R. R.,A User's Guide to ENPORT—4, Wiley-Interscience, New York, New York, 1974.
Leitmann, G.,The Calculus of Variations and Optimal Control Theory, Plenum Press, New York, New York, 1981.
Blackwell, C. C., andWilson, T. A.,Examples of Cases and Conditions Encountered in State Space Model Generation, University of Texas at Arlington, Department of Mechanical Engineering, Technical Report No. 84-010, 1984.
Jamshidi, M.,Large-Scale Systems: Modeling and Control, North-Holland Press, New York, New York, 1983.
Luenberger, D. G.,Introduction to Dynamic Systems: Theory, Models, and Applications, John Wiley and Sons, New York, New York, 1979.
Brogan, W. L.,Modern Control Theory, Quantum Publishers, New York, New York, 1974.
Karnopp, D. C.,Sensible State Variables (Letter), Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 98, Series G, p. 353, 1976.
Karnopp, D. C.,Resolution of a Paradox Involving Mechanical Gyrators and Transformers, Journal of the Franklin Institute, Vol. 291, pp. 211–217, 1971.
Lacy, J. W.,Automation of the Reduction and Solution of the Natural Equations of Linear Systems with Time-Varying Coefficients, The University of Texas at Arlington, MS Thesis, 1974.
Wood, T. J.,Automation of the Natural Equation Reduction of Constant Coefficient Linear Systems, University of Texas at Arlington, Department of Mechanical Engineering, MS Thesis, 1972.
Wilson, T. A.,Converting Natural Equations to State Space Format, University of Texas at Arlington, Mechanical Engineering Design Report CAD4390602TW, 1983.
Blackwell, C. C., andWilson, T. A.,A Simplified Procedure for Generating a State Space Representation from Primitive Equations (to appear).
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Communicated by G. Leitmann
The author is indebted to several reviewers and colleagues for valuable suggestions for improvement and for encouragement.
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Blackwell, C.C. On generating state space equations of a linear constant coefficient system. J Optim Theory Appl 49, 65–79 (1986). https://doi.org/10.1007/BF00939248
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DOI: https://doi.org/10.1007/BF00939248