CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. III Stability Theory - Peter C. Müller

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1 STABILITY THEORY Peter C. Müller University of Wuertal, Germany Keywords: Asymtotic stability, Eonential stability, Linearization, Linear systems, Lyaunov equation, Lyaunov function, Lyaunov stability, Nonlinear systems, Stability in the first aroimation, Uniform stability Contents 1. Introduction 2. Linearization: Stability in the First Aroimation 3. The Direct Method of Lyaunov 3.1. Nonlinear Systems 3.2. Linear Systems Glossary Bibliograhy Biograhical Sketch Summary Stability lays an imortant role in the theory of dynamical systems and control. It characterizes the roerty of an unerturbed trajectory that all erturbed trajectories starting nearby stay nearby: small erturbations cause only small changes in the system behavior. The most imortant concet of stability has bee introduced by the Russian mathematician A. M. Lyaunov in Based on his famous work a general Lyaunov theory has been develoed to investigate the stability behavior of general dynamical systems. Here, the basic ideas and results of this theory are resented. 1. Introduction The roerty of stability is very imortant for the behavior of dynamical systems. Intuitively, stability can be understood as the requirement that small erturbations of the system cause only small changes of the system behavior. As erturbations eternal ecitations or changes of the initial conditions can be considered as well. The develoment of stability concets always tried to meet these intuitive ideas. But for a correct formulation of the stability roblem eact definitions and criteria are required. While for linear time-invariant systems indeendent investigations were erformed resulting in algebraic and geometric criteria, the stability analysis of nonlinear dynamical systems is still based on the famous work of the Russian mathematician A. M. Lyaunov ( ) who resented a book on the general roblem of the stability of motion in Lyaunov s stability theory shows the comleity of the stability roblem requiring a very recise discussion of the roblem. The following stability analysis is based on a state sace model of nonlinear dynamical systems Encycloedia of Life Suort Systems (EOLSS)

2 () t = a( (),) t t (1) or () t = a( (),) t t + B( (),) t t u() t, (2) y() t = c ( (),) t t + D ( (),) t t u () t. (3) For the free system (1) an uncontrolled nonlinear system or a closed-loo controlled system is considered. For system (2, 3) the inut-outut behavior is considered where u can be interreted as eternal disturbances or as control inuts which have to be designed suitably; the outut y reresents usually the measurements. For linear systems we can seak about the stability of the system in general, but for nonlinear systems we have to distinguish more recisely if the stability of an equilibrium oint, a trajectory, a limit cycle or an arbitrary attractor is considered. If a articular solution () t = () t is assumed, if necessary with a suitable inut function u() t = u () t, satisfying (1) or (2, 3) resectively, then the equations can be reformulated with resect to the deviations from () t, u () t : () t = () t + () t, This leads to u() t = u () t + u () t. (4) () t = a( (),) t t, a0 (,) t = 0 (5) with a ( (),) t t = a ( () t + (),) t t a ( (),) t t (6) or, a0 (,) t = 0, (7) y() t = c ( (),) t t + D ( (),) t t u () t, c0 (,) t = 0 (8) with a ( (),) t t = a ( (),) t t + B ( +,) t B (,) t u () t, (9) B ( ( t), t) = B ( +, t), (10) c ( (),) t t = c ( +,) t c (,) t + D ( +,) t D (,) t u () t, (11) Encycloedia of Life Suort Systems (EOLSS)

3 D ( ( t), t) = D ( +, t), (12) y() t = y() t y () t, = + y () t c(,) t D(,) t u () t. (13) By the transformation (4) the general stability investigation for the articular solution () t is attributed to the stability analysis of the equilibrium oint = 0 of the system (5) of (7, 8). But it has to be acceted that a time-invariant system (1) or (2, 3) is transformed in a time-variant system (5) or (7, 8) for time-varying trajectories () t. Only secial cases such as constant oeration oints () t = 0 lead to time-invariant systems (5) or (7, 8) if the original systems are time invariant, too. In the following the descritions (5) or (7, 8) are assumed. By that agreement the bars on the vectors in (5) and (7, 8) are droed without being confused. 2. Linearization: Stability in the First Aroimation In many alications a feedback control is designed to stabilize an equilibrium oint (or a articular motion). In case of additional disturbances only small deviations () t from the equilibrium oint are allowed. As long as the nonlinearity functions are continuously differentiable in a neighborhood of the desired equilibrium oint, then the system behavior may be aroimated by the linearized equations () t = A() t () t (14) or () t = A() t () t + B() t u() t, (15) y () t = C() t () t + D() t u() t. (16) The system matrices are designed as A : a (, t) = T = 0 (, ), B: = B( 0, t), : = ct C, D: = D( 0, t) (17) T = 0 where A, C are Jacobian matrices evaluated at = 0. Very often Eqs. (15, 16) are the starting oint of the stability analysis or the control design. This established rocedure is justified by the method of first aroimation. Theorem 1: If the linearized system (14) is eonentially stable (cf. (20)) and a (, t) A( t) lim = 0 (18) 0 holds, then the equilibrium oint = 0 of the nonlinear system (5) is eonentially Encycloedia of Life Suort Systems (EOLSS)

4 stable as well. Theorem 2: If the linearized system (15) is stabilized eonentially by a linear static or dynamic outut feedback of (16) and c (, t) C( t) lim = 0 0 (19) holds additionally to (18), then the equilibrium oint = 0 of the nonlinear system (7, 8) is eonentially stabilized by the same feedback. The requirements (18, 19) mean that the deviations between the nonlinear and the linearized system are small of higher than first order. Additionally, we have to be aware of the effect that stability of the linearized system is a global roerty while for the nonlinear system the stability of the equilibrium oint = 0 is a local roerty in its neighborhood. Eonential stability is only guaranteed in a domain of attraction which may be small. A control design according to Theorem 2 should also make this domain as large as ossible. The notion of eonential stability of = 0 means the requirement that any trajectory () t for arbitrary initial conditions (0) = 0 in a small but full neighborhood of = 0 satisfies the estimate β () t α e t (20) 0 for certain ositive constants α, β. For time-invariant system matrices A( t ) = A the system (14) is eonentially stable if and only if the conditions Re λ ( A ) < 0, i= 1,, n (21) i hold where λi ( A ) are the eigenvalues of A. The test for (21) consists either in the calculation of the eigenvalues λ i or in the alication of one of the stability criteria for linear time-invariant systems. For linear time-variant systems the roof of eonential stability is much more difficult. Sometimes there is a seculation with resect to the frozen eigenvalues λ i ( t), i = 1,, n, [ ] det λi ( t) I A ( t) = 0, (22) such that a condition Re λi ( t) δ < 0 would guarantee eonential stability. But this seculation is definitely wrong which is shown by a simle counter-eamle. Eamle 1: Consider the linear eriodic system Encycloedia of Life Suort Systems (EOLSS)

5 = cos 3t + ( 1 3sin t), = ( 1+ sin3t) cos3t+ (23) The frozen eigenvalues are constant: λi = λ2 =. Nevertheless the system (23) is 2 eonentially unstable because of the solution t 3 2t 3 t 3 2t 3 1() t = e cos t 10 + e sin t, 20 2() t = e sin t 10 + e cos t. (24) The stability analysis of time-variant systems is difficult in general. In the secial case of linear eriodic systems Floquet s theory can be alied, which offers a systematic numerical aroach for the stability check at least Bibliograhy TO ACCESS ALL THE 12 PAGES OF THIS CHAPTER, Click here Barbashin E. A. and Krasoviskii N. N. (1952). On the Stability of a Motion in the Large, Doklady Akad. Nauk SSSR 86, Cesari L. (1963). Asymtotic Behavior and Stability Problems in Ordinary Differential Equations, 2nd Edition, Sringer, Berlin-Göttingen-Heidelberg. Hahn W. (1967). Stability of Motion, Sringer, Berlin-Heidelberg-New York. LaSalle J. and Lefschetz S. (1961). Stability by Liaunov s Direct Method with Alications, Academic Press, New York-London. Lyaunov A. M. (1892,1907, 1992). Problème Générale de la stabilité de mouvement. Ann. Fac. Sci. Toulouse 9 (1907), [French translation of the 1892 Russian original ublished in Comm. Soc. Math. Charkow. English translation: The General Problem of the Stability of Motion. Int. J. Control 55 (1992), , and as book by Taylor & Francis, London 1992.] Magnus K. (1959). Zur Entwicklung des Stabilitätsbegriffes in der Mechnik (On the develoment of the concet of stability in mechanics), Naturwiss. 56, Malkin I. G. (1952, 1959, 1966). Theory of Stability of Motion [in Russian]. Moscow [German Translation: Theorie der Stabilität einer Bewegung. R. Oldenbourg, München English translation: 2nd Edition, Nauka,Moscow 1966.] Müller P. C. (1977). Stabilität und Matrizen [Stability and Matrices, in German]. Sringer, Berlin- Heidelberg-New York. Rouche N., Habets P. and Laloy M. (1977). Stability Theory by Liaunov s Direct Method. Sringer, New York-Heidelberg-Berlin. Encycloedia of Life Suort Systems (EOLSS)

6 Biograhical Sketch Peter C. Müller, born in 1940 at Stuttgart, Germany. He got degree of "Dilom-Mathematiker in 1965 from the Technical University of Stuttgart Profession: Institute of Mechanics, Technical University of München 1970 Dr. rer. nat. ("Time-otimal alignment of inertial latforms") 1974 Habilitation in Mechanics ("Stability and matrices") 1978 Professor of Mechanics Since 1981 University of Wuertal, Professor of Safety Control Engineering Dean of the Deartment of Safety Engineering Since 1987 Head of the Institute of Robotics Vice Rector of the University of Wuertal Membershi: GAMM, VDI, IEEE Co-editor: Archive of Alied Mechanics Mathematical and Comuter Modelling of Dynamical Systems ZAMM" : Selection Committee of the Aleander von Humboldt Foundation : Committee for Postgraduate Research Grous of the German Research Council (DFG) Research: Control of Mechanical Systems, Nonlinear Control Systems, Descritor Systems, Robotics and Mechatronics, Fault Detection and Suervisory Control Publications: About 275 journal articles, book chaters and conference ublications Books: Secial Problems of Gyrodynamics, Linear Vibrations (with W. O. Schiehlen), Forced Linear Vibrations (with W. O. Schiehlen), Stability and Matrices Encycloedia of Life Suort Systems (EOLSS)