Abstract
The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.
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References
Truesdell, C., Elements of Continuum Mechanics. Berlin-Heidelberg-New York: Springer 1966.
Truesdell, C., Rational Thermodynamics. New York: McGraw-Hill 1969.
Onat, E. T., The Notion of State and Its Implications in Thermodynamics of Inelastic Solids, pp. 292–314 in: Proc. 1966 IUTAM Symposium, H. Parkus & L. I. Sedov, Ed. Berlin-Heidelberg-New York: Springer 1968.
Onat, E. T., Representation of Inelastic Mechanical Behavior by Means of State Variables, pp. 213–224 in: Proc. 1968 IUTAM Symposium, B. A. Boley, Ed. Berlin-Heidelberg-New York: Springer 1970.
Kalman, R. E., P. L. Falb, & M. A. Arbib, Topics in Mathematical System Theory. New York: McGraw-Hill 1969.
Desoer, C. A., Notes for a Second Course on Linear Systems. New York: Nostrand Reinhold 1970.
Zadeh, L. A., The Concepts of System, Aggregate, and State in System Theory, in: System Theory, L. A. Zadeh & E. Polak, Ed. New York: McGraw-Hill 1969.
Youla, D. C., The synthesis of linear dynamical systems from prescribed weighting patterns. J. SIAM Appl. Math. 14, No. 3, 527–549 (1966).
Balakrishnan, A. V., Foundations of the state-space theory of continuous systems I. J. Computer and System Sci. 1, 91–116 (1967).
Brockett, R. W., Finite Dimentional Linear Systesm. New York: Wiley 1970.
Willems, J. C., The construction of Lyapunov functions for input-output stable systems. SIAM J. Control 9, No. 1, 105–134 (1971).
Baker, R. A., & A. R. Bergen, Lyapunov stability and Lyapunov functions of infinite dimensional systems. IEEE Trans. Automatic Control AC-14, 325–334 (1969).
Estrada, R. F., & C. A. Desoer, Passivity and stability of systems with a state representation. Int. J. Control 13, No. 1, 1–26 (1971).
Breuer, S., & E. T. Onat, On recoverable work in linear visocelasticity. ZAMP 15, No. 1, 12–21 (1964).
Breuer, S., & E. T. Onat, On the determination of free energy in linear viscoelastic solids. ZAMP 15, No. 1, 184–191 (1964).
Youla, D. C., L. J. Castriota, & H. J. Carlin, Bounded real scattering matrices and the foundations of linear passive network theory. IRE Trans. on Circuit Theory CT-6, 102–124 (1959).
Zames, G., On the input-output stability of time-varying nonlinear feedback systems. Part I: Conditions derived using concepts of loop gain, conicity, and positivity; Part II: Conditions involving circles in the frequency plane and sector nonlinearities. IEEE Trans. Automatic Control AC-11, 228–238, 465–476 (1966).
Willems, J. C., The Analysis of Feedback Systems. Cambridge, Mass.: The M.I.T. Press 1971.
Meixner, J., On the theory of linear passive systems. Arch. Rational Mech. Anal. 17, 278–296 (1964).
Carlin, H. J., Network theory without circuit elements. Proc. IEEE 55, No. 4, 482–497 (1967).
Gruber, M., & J. L. Willems, On a Generalization of the Circle Criterion. Proc. of the Fourth Annual Allerton Conf. on Circuit and System Theory, pp. 827–835, 1966.
Freedman, M., & G. Zames, Logarithmic variation criteria for the stability of systems with time-varying gains. SIAM J. on Control 6, 487–507 (1968).
Willems, J. L., Stability Theory of Dynamical Systems. Nelson, 1970.
Yoshizawa, T., The Stability Theory by Liapunov's Second Method. Math. Soc. Japan, 1966.
La Salle, J. P., An Invariance Principle in the Theory of Stability, pp. 277–286, in: Differential Equations and Dynamical Systems, J. K. Hale & J. P. La Salle, Ed. New York: Academic Press 1967.
Brockett, R. W., Path Integrals, Liapunov Functions and Quadratic Minimization. Proc. 4th Annual Allerton Conf. on Circuit and System Theory, Monticello, Ill., pp. 685–698, 1966.
Coleman, B. D., Thermodynamics of materials with memory. Arch. Rational Mech. Anal. 17, 1–6 (1964).
Gurtin, M., On the thermodynamics of materials with memory. Arch. Rational Mech. Anal. 28, 40–50 (1968).
Popov, V. M., Hyperstability and optimality of automatic systems with several control functions. Rev. Roumaine Sci. Tech. Electrotechn. et Energ. 9, No. 4, 629–690 (1964).
Kalman, R. E., Lyapunov functions for the problem of Lur'e in automatic control. Proc. Nat. Acad. Sci. U.S.A. 49, 201–205 (1963).
Newcomb, R. W., Linear Multiport Synthesis. New York: McGraw-Hill 1966.
Meixner, J., On the Foundation of Thermodynamics of Processes, pp. 37–47 in: A Critical Review of Thermodynamics, E. B. Stuart, B. Gal-Or & A. J. Brainard, Ed. Mono Book Corp. 1970.
Day, W. A., Thermodynamics based on a work axiom. Arch. Rational Mech. Anal. 31, 1–34 (1968).
Brayton, R. K., & J. K. Moser, A theory of nonlinear networks. Quart. Appl. Math. 22, 1–33, 81–104 (1964).
Day, W. A., A theory of thermodynamics for materials with memory. Arch. Rational Mech. Anal. 34, 86–96 (1969).
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Communicated by C. Truesdell
This research was supported in part by the National Science Foundation under Grant No. GK-25781 and in part by the U.K. Science Research Council. This paper was prepared while the author was a Senior Visiting Fellow at the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge, Cambridge, England.
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Willems, J.C. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal. 45, 321–351 (1972). https://doi.org/10.1007/BF00276493
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DOI: https://doi.org/10.1007/BF00276493