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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 4: Analysis Based On Continuity 1 This lecture presents several techniques of qualitative systems analysis based on what is frequently called topological arguments , i.e. on the arguments relying on continuity of functions involved. 4.1 Analysis using general topology arguments This section covers results which do not rely specifically on the shape of the state space, and thus remain valid for very general classes of systems. We will start by proving gener- alizations of theorems from the previous lecture to the case of discrete-time autonomous systems. 4.1.1 Attractor of an asymptotically stable equilibrium Consider an autonomous time invariant discrete time system governed by equation x ( t + 1) = f ( x ( t )) , x ( t ) ⊂ X, t = , 1 , 2 , . . . , (4.1) where X is a given subset of R n , f : X ∞ X is a given function. Remember that f is called continuous if f ( x k ) f ( x ) as k → whenever x k , x ⊂ X are such that x k x as k → ). In particular, this means that every function defined on a finite set X is continuous. One important source of discrete time models is discretization of differential equations. R n Assume that function a : ∞ R n is such that solutions of the ODE x ˙( t ) = a ( x ( t )) , (4.2) 1 Version of September 17, 2003 2 with x (0) = x exist and are unique on the time interval t ⊂ [0 , 1] for all ¯ ¯ x ⊂ R n . Then discrete time system (4.1) with f (¯) = x (1 , ¯) describes the evolution of continuous time x x system (4.2) at discrete time samples. In particular, if a is continuous then so is f . Let us call a point in the closure of X locally attractive for system (4.1) if there exists d > such that x ( t ) ¯ ¯ x as t → for every x = x ( t ) satisfying (4.1) with | x (0) − x | < d . Note that locally attractive points are not necessarily equilibria, and, even if they are, they are not necessarily asymptotically stable equilibria. x ⊂ R n the set A = A (¯ x ⊂ X in (4.1) which define a For ¯ x ) of all initial conditions ¯ solution x ( t ) converging to ¯ x x as t → is called the attractor of ¯ ....
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This note was uploaded on 11/07/2011 for the course AERO 16.36 taught by Professor Alexandremegretski during the Spring '09 term at MIT.
- Spring '09