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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 Problem Set 2 Solutions 1 Problem 2.1 Consider the feedback system with external input r = r ( t ) , a causal linear time invariant forward loop system G with input u = u ( t ) , output v = v ( t ) , a ) − 1 / 2 e − t , where ¯ and impulse response g ( t ) = . 1 ( t ) + ( t + ¯ a is a parameter, → and a memoryless nonlinear feedback loop u ( t ) = r ( t ) + π ( v ( t )) , where π ( y ) = sin( y ) . It is customary to require well-posedness of such feedback models, r- u- G- v π ( y ) Figure 2.1: Feedback setup for Problem 2.1 which will usually mean existence and uniqueness of solutions v = v ( t ) , u = u ( t ) of system equations ⎬ t v ( t ) = . 1 u ( t ) + h ( t − δ ) u ( δ ) dδ, u ( t ) = r ( t ) + π ( v ( t )) on the time interval t ≤ [0 , ⊂ ) for every bounded input signal r = r ( t ) . 1 Version of October 8, 2003 2 (a) Show how Theorem 3.1 from the lecture notes can be used to prove well-posedness in the case when ¯ a > . In terms of the new signal variable y ( t ) = v ( t ) − . 1 π ( v ( t )) − . 1 r ( t ) system equations can be re-written as ⎬ t y ( t ) = h ( t − δ )[ r ( δ ) + ( y ( δ ) + . 1 r ( δ ))] dδ, where − t ( t + a ) − 1 / 2 e , t h ( t ) = → , otherwise, and : R ∞ R is the function which maps z ≤ R into π ( q ), with q being the solution of q − . 1 π ( q ) = z. Since π is continuously differentiable, and its derivative ranges in [ − 1 , 1], is con- tinuously differentiable as well, and its derivative ranges between 1 / 1 . 1 and 1 / . 9. For every constant T ≤ [0 , ⊂ ), the equation for y ( t ) with t T can be re-written → as ⎬ t y ( t ) = y ( T ) + a T ( y ( δ ) , δ, t ) dδ, T where a T (¯ y, δ, t ) = h ( t − δ )[ r ( δ ) + ( y ( δ ) + . 1 r ( δ ))] + h T ( t ) , ⎬ T h T ( t ) = h ˙ ( t − δ )[ r ( δ ) + ( y ( δ ) + . 1 r ( δ ))] dδ. When parameter a takes a positive value, function a = a T satisfies conditions of ¯ Theorem 3.1 with X = R n , ¯ a ) being a x = y ( T ), r = 1, and t = T , with K = K (¯ function of ¯ a = 0, and ≥ M = M T = M ( a )(1 + max y ( t ) ) ....
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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