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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 7 Solutions 1 Problem 7.1 A stable linear system with a relay feedback excitation is modeled by x ˙( t ) = Ax ( t ) + B sgn( Cx ( t )) , (7.1) where A is a Hurwitz matrix, B is a column matrix, C is a row matrix, and sgn( y ) denotes the sign nonlinearity ⎞ ⎠ 1 , y > , sgn( y ) = , y = , ⎧ − 1 , y < . For T > , a 2 Tperiodic solution x = x ( t ) of (7.1) is called a regular unimodal limit cycle if Cx ( t ) = − Cx ( t + T ) > for all t ≤ (0 , T ) , and CAx (0) >  CB  . (a) Derive a necessary and sufficient condition of exponential local stability of the regular unimodal limit cycle (assuming it exists and A, B, C, T are given). x ≤ R n such that C ¯ Let Y denote the set of all ¯ x = 0. Let x = x (0). By assumptions, Cx ( t ) > and Cx ( − t ) = Cx ( T − t + T ) = − Cx ( T − t ) < for t ≤ (0 , T ). Hence Cx (0) = Cx = 0, i.e. x ≤ Y . Let F : R × Y be defined by x ) = e At (¯ F ( t, ¯ x + A − 1 B ) − A − 1 B. 1 Version of November 12, 2003 2 By definition F ( β, ¯ x ) is the value at t = β of the solution z = z ( t ) of the ODE dz/dt = Az + B . Since F ( t, x ) > for t ≤ (0 , T ) and dF (0 , ¯ x + B ) C ( Ax + B ) > x ) = C ( A ¯ dt ¯ x ) > for all whenever x ≤ Y is suﬃciently close to x , we conclude that F ( t, ¯ t ≤ (0 , T ) and for all ¯ x ≤ Y suﬃciently close to x . On the other hand, dCF ( T, x ) = C ( Ax ( T ) + B ) = − CAx + CB < . dt Hence, by the implicit mapping theorem, for ¯ x ≤ Y suﬃciently close to x equation x ) = has a unique solution t ¯ = h (¯ CF ( t, ¯ x ) in a neigborhood of t = T . Consider the map S defined for x 1 ≤ Y in a neigborhood of x by S ( x 1 ) = F ( h ( x 1 ) , x 1 ). Essentially, S is the Poincare map associated with the periodic so lution x = x ( t ). Local exponential stability of the trajectory of x = x ( t ) is therefore equivalent to local exponential stability of the equilibrium x of S . The differential of S at x is the composition of e AT and the projection on Y parallel to Ax ( T ) + B = B − Ax . In other words, the differential of S has matrix S ˙ ( x ) = e AT − [ C ( B − Ax )] − 1 ( B − Ax ) Ce AT in the standard basis of R n . In order for the limit cycle...
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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
 AlexandreMegretski
 Dynamics

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