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lec2_6243_2003 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 2: Differential Equations As System Models 1 Ordinary differential requations (ODE) are the most frequently used tool for modeling continuous-time nonlinear dynamical systems. This section presens results on existence of solutions for ODE models, which, in a systems context, translate into ways of proving well-posedness of interconnections. 2.1 ODE models and their solutions Ordinary differential equations are used to describe responses of a dynamical system to all possible inputs and initial conditions. Equations which do not have a solution for some valid inputs and initial conditions do not define system’s behavior completely, and, hence, are inappropriate for use in analysis and design. This is the reason a special attention is paid in this lecture to the general question of existence of solution of differential equation. 2.1.1 ODE and their solutions An ordinary differential equation on a subset Z R n × R is defined by a function a : Z ∈� R n . Let T be a non-empty convex subset of R (i.e. T can be a single point set, or an open, closed, or semi-open interval in R ). A function x : T ∈� R n is called a solution of the ODE x ˙( t ) = a ( x ( t ) , t ) (2.1) if ( x ( t ) , t ) Z for all t T , and t 2 x ( t 2 ) x ( t 1 ) = a ( x ( t ) , t ) dt t 1 , t 2 T. (2.2) t 1 1 Version of September 10, 2003
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2 The variable t is usually referred to as the “time”. Note the use of an integral form in the formal definition (2.2): it assumes that the function t ∈� a ( x ( t ) , t ) is integrable on T , but does not require x = x ( t ) to be differentiable at any particular point, which turns out to be convenient for working with discontinuous input signals, such as steps, rectangular impulses, etc. Example 2.1 Let sgn denote the “sign” function sgn : R { 0 , 1 , 1 } defined by 1 , y > 0 , sgn( y ) = 0 , y = 0 , 1 , y < 0 . The notation x ˙ = sgn( x ) , (2.3) which can be thought of as representing the action of an on/off negative feedback (or describing behavior of velocity subject to dry friction), refers to a differential equation defined as above with n = 1, Z = R × R (since sgn( x ) is defined for all real x , and no restrictions on x or the time variable are explicitly imposed in (2.3)), and a ( x, t ) = sgn( x ). It can be verified 2 that all solutions of (2.3) have the form x ( t ) = max { c t, 0 } or x ( t ) = min { t c, 0 } , where c is an arbitrary real constant. These solutions are not differentiable at the critical “stopping moment” t = c . 2.1.2 Standard ODE system models Ordinary differential equations can be used in many ways for modeling of dynamical systems. The notion of a standard ODE system model describes the most straightforward way of doing this.
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