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lec1_6243_2003

lec1_6243_2003 - Massachusetts Institute of Technology...

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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 1: Input/Output and State-Space Models 1 This lecture presents some basic definitions and simple examples on nonlinear dynam- ical systems modeling. 1.1 Behavioral Models. The most general (though rarely the most convenient) way to define a system is by using a behavioral input/output model. 1.1.1 What is a signal? In these lectures, a signal is a locally integrable function z : R + ≤ R k , where R + denotes the set of all non-negative real numbers. The notion of “local integrability” comes from the Lebesque measure theory, and means simply that the function can be safely and meaningfully integrated over finite intervals. Generalized functions, such as the delta function ( t ), are not allowed. The argument t → R + of a signal function will be referred to as “time” (which it usually is). Example 1.1 Function z = z ( · ) defined by t − . 9 sgn( cos (1 /t )) for t > , z ( t ) = for t = 1 Version of September 3, 2003 2 is a valid signal, while 1 /t for t > , z ( t ) = for t = and z ( t ) = ˙ ( t ) are not. The definition above formally covers the so-called continuous time (CT) signals. Dis- crete time (DT) signals can be represented within this framework as special CT signals....
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