An Overview of FiniteTime Stability
 Peter Dorato
 … show all 1 hide
Summary
Finitetime stability (FTS) is a concept that was first introduced in the 1950s. The FTS concept differs from classical stability in two important ways. First, it deals with systems whose operation is limited to a fixed finite interval of time. Second, FTS requires prescribed bounds on system variables. For systems that are known to operate only over a finite interval of time and whenever, from practical considerations, the systems’ variables must lie within specific bounds, FTS is the only meaningful definition of stability. This overview will first present a short history of the development of the concept of FTS. Then it will present some important analysis and design results for linear, nonlinear, and stochastic systems. Finally some applications of the theory will be presented.
 Amato F, Ariola M, Abdallah C, Cosentino C (2002) Application of finitetime stability concepts to control of ATM networks. In: 40th Allerton Conf. on Communication, Control and Computers, Allerton, IL
 Amato F, Ariola M, Abdallah C, Dorato P (1999a) Dynamic output feedback finitetime control of LTI systems subject to parametric uncertainties and disturbances. In: European Control Conference, Karlsruhe, Germany
 Amato F, Ariola M, Abdallah C, Dorato P (1999b) Finitetime control for uncertain linear systems with disturbance inputs. 1776–1780. In: Proc. American Control Conf., San Diego, CA
 Amato F, Ariola M, Cosentino C, Abdallah C, Dorato P (2003) Necessary and sufficient conditions for finitetime stability of linear systems. 4452–4456. In: Proc. American Control Conf., Denver, CO
 Amato F, Ariola M, Dorato P (1998) Robust finitetime stabilization of linear systems depending on parameter uncertainties. 1207–1208. In: Proc. IEEE Conf. on Decision and Control, Tampa, FL
 Amato F, Ariola M, Dorato P (2001) Finitetime control of linear systems subject to parametric uncertainties and disturbances. Automatica 37:1459–1463 CrossRef
 Bhat S, Bernstein D (1998) Continuous finitetime stabilization of the translational and rotational double integrators. IEEE Trans. Automat. Contr. 43:678–682 CrossRef
 ChzhanSyIn (1959a) On stability of motion for a finite interval of time. Journal of Applied Math. and Mechanics (PMM) 23:333–344 CrossRef
 ChzhanSyIn (1959b) On estimates of solutions of systems of differential equations, accumulation of perturbation, and stability of motion during a finite time interval. Journal of Applied Math. and Mechanics 23:920–933 CrossRef
 D’Angelo H (1970) Timevarying systems: analysis and design. Allyn and Bacon, Boston, MA
 Dorato P (1961a) Shorttime stability in linear timevarying systems. 83–87. In: IRE International Convention Record, Part IV
 Dorato P (1961b) Shorttime stability in linear timevarying systems, PhD thesis, Polytechnic Institute of Brooklyn
 Dorato P (1961c) Shorttime stability. IRE Trans. Automat. Contr. 6:86
 Dorato P (1967) Comment on finitetime stability under perturbing forces and on product spaces. IEEE Trans. Automat. Contr. 12:340 CrossRef
 Dorato P (2000) Quantified multivariate polynomial inequalities. IEEE Control Systems Magazine 20:48–58 CrossRef
 Dorato P, Abdallah C, Famularo D (1997) Robust finitetime stability design via linear matrix inequalities. In: Proc. IEEE Conf. on Decision and Control, San Diego, CA
 Garrard W (1969) Finitetime stability in control system synthesis. 21–31. In: Proc. 4th IFAC Congress, Warsaw, Poland
 Garrard W (1972) Further results on the synthesis of finitetime stable systems. IEEE Trans. Automat. Contr. 17:142–144 CrossRef
 Grujic L (1973) On practical stability. Int. J. Control 17:881–887
 Grujic L (1975) Uniform practical and finitetime stability of largescale systems. Int. J. Systems Sci. 6:181–195
 Grujic L (1976) Finitetime adaptive control. In: Proc. 1976 JACC, Purdue University
 Grujic L (1977) Finite time noninertial adaptive control. AIAA Journal 15:354–359
 Gunderson R (1967a) Qualitative solution behavior on a finite time interval, PhD thesis, University of Alabama
 Gunderson R (1967b) On stability over a finite interval. IEEE Trans. Automat. Contr. AC12:634–635 CrossRef
 Hahn W (1963) Theory and applications of Liapunov’s direct method. PrenticeHall, Englewood Cliffs, NJ
 Haimo V (1986) Finite time controllers. SIAM J. Control and Optimization 24:760–770 CrossRef
 Hallam T, Komkov V (1969) Application of Liapunov’s functions to finite time stability. Revue Roumaine de Mathematiques Pure et Appliquees 14:495–501
 Heinen J, Wu S (1969) Further results concerning finitetime stability. IEEE Trans. Automat. Contr. AC14:211–212 CrossRef
 Kamenkov G (1953) On stability of motion over a finite interval of time [in Russian]. Journal of Applied Math. and Mechanics (PMM) 17:529–540
 Kamenkov G, Lebedev A (1954) Remarks on the paper on stability in finite time interval [in Russian]. Journal of Applied Math. and Mechanics (PMM) 18:512
 Kayande A (1971) A theorem on contractive stability. SIAM J. Appl. Math. 21:601–604 CrossRef
 Kayande A, Wong J (1968) Finite time stability and comparison principles. Proc. Cambridge Philosophical Society 64:749–756 CrossRef
 Kushner H (1967) Stochastic stability and control. Academic Press, New York, NY
 Kushner H (1966) Finitetime stochastic stability and the analysis of tracking systems. IEEE Trans. Automat. Contr. AC11:219–227 CrossRef
 Lakshmikantham V, Leela S, Martynyuk A (1990) Practical stability of nonlinear systems. World Scientific, Singapore
 Lam L, Weiss L (1974) Finite time stability with respect to timevarying sets. J. Franklin Inst. 298:425–421
 LaSalle J, Lefschetz S (1961) Stability by Liapunov’s direct method. Academic Press, New York, NY
 Lebedev A (1954a) The problem of stability in a finite interval of time [in Russian]. Journal of Applied Math. and Mechanics (PMM) 18:75–94
 Lebedev A (1954b) On stability of motion during a given interval of time [in Russian]. Journal of Applied Math. and Mechanics (PMM) 18:139–148
 Mastellone S (2004) Finitetime stability of nonlinear networked control systems, Master’s thesis, University of New Mexico
 Michel A (1970) Quantitative analysis of simple and interconnected systems: Stability, boundedness, and trajectory behavior. IEEE Trans. Circuit Theory CT17:292–301 CrossRef
 Michel A, Porter D (1972) Practical stability and finitetime stability of discontinuous systems. IEEE Trans. Circuit Theory CT19:123–129
 Michel A, Wu S (1969) Stability of discretetime systems over a finite interval of time. Int. J. Control 9:679–694
 Richards J (1983) Analysis of periodically timevarying systems. SpringerVerlag, Berlin
 San Filippo F, Dorato P (1974) Shorttime parameter optimization with flight control applications. Automatica 10:425–430 CrossRef
 San Fillipo F (1973) Short time optimization of parametrically disturbed linear control system, PhD thesis, Polytechnic Institute of Brooklyn
 Van Mellaert L (1967) Inclusionprobabilityoptimal control, PhD thesis, Polytechnic Institute of Brooklyn
 Van Mellaert L, Dorato P (1972) Numerical solution of an optimal control problem with a probability criterion. IEEE Trans. Automat. Contr. AC17:543–546 CrossRef
 Watson J, Stubberud A (1967) Stability of systems operating in a finite time interval. IEEE Trans. Automat. Contr. AC12:116 CrossRef
 Weiss L (1969) On uniform and nonuniform finite time stability. IEEE Trans. Automat. Contr. AC14:313–314 CrossRef
 Weiss L (1968) Converse theorems for finitetime stability. SIAM J. Appl. Math. 16:1319–1324 CrossRef
 Weiss L, Infante E (1967) Finite time stability under perturbing forces and on product spaces. IEEE Trans. Automat. Contr. AC12:54–59 CrossRef
 Weiss L, Infante E (1965) On the stability of systems defined over a finite time interval. Proc. of the National Academy of Sciences 54:440–448
 Wonham W (1970) Random differential equations in control theory In: BharuchaReid A (ed), Probabilistic methods in applied mathematics, Volume 2, 132–208. Academic Press, New York
 Title
 An Overview of FiniteTime Stability
 Book Title
 Current Trends in Nonlinear Systems and Control
 Book Subtitle
 In Honor of Petar Kokotović and Turi Nicosia
 Book Part
 Part II
 Pages
 pp 185194
 Copyright
 2006
 DOI
 10.1007/0817644709_10
 Print ISBN
 9780817643836
 Online ISBN
 9780817644703
 Series Title
 Systems and Control: Foundations & Applications
 Publisher
 Birkhäuser Boston
 Copyright Holder
 Birkhäuser Boston
 Additional Links
 Topics
 eBook Packages
 Editors

 Laura Menini ^{(12)}
 Luca Zaccarian ^{(12)}
 Chaouki T. Abdallah ^{(13)}
 Editor Affiliations

 12. Dipartimento di Informatica Sistemi e Produzione, Università di Roma “Tor Vergata”
 13. Department of Electrical and Computer Engineering, University of New Mexico
 Authors

 Peter Dorato ^{(14)}
 Author Affiliations

 14. Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM, 87131, USA
Continue reading...
To view the rest of this content please follow the download PDF link above.