lec12_6243_2003

lec12_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 12: Local Controllability 1 In this lecture, nonlinear ODE models with an input are considered. Partial answers to the general controllability question (which states can be reached in given time from a given state by selecting appropriate time-dependent control action) are presented. More precisely, we consider systems described by x ˙( t ) = a ( x ( t ) , u ( t )) , (12.1) where a : R n × R m is a given continuously differentiable function, and u = u ( t ) ⇒∀ R n is an m -dimensional time-varying input to be chosen to steer the solution x = x ( t ) in a desired direction. Let U be an open subset of R n , ¯ x 0 R n . The reachable set for a given T > 0 the ( U -locally) reachable set R U x 0 , T ) is defined as the set of all x ( T ) where x : [0 , T ] , u : [0 , T ] is a bounded solution of (12.1) such that x (0) = ¯ x 0 ⇒∀ R n ⇒∀ R m and x ( t ) U for all t [0 , T ]. Our task is to find conditions under which R U x 0 , T ) is guaranteed to contain a neig- borhood of some point in R n , or, alternatively, conditions which guarante that R U x 0 , T ) has an empty interior. In particular, when ¯ x 0 is a controlled equilibrium of (12.1), i.e. x 0 , ¯ u 0 R m , complete local controllability of (12.1) at ¯ a u 0 ) = 0 for some ¯ x 0 means that for every φ > 0 and T > 0 there exists > 0 such that R U x 0 ) for every x, T ) B ¯ x 0 ), where U = B α x B x 0 ) and x ) = { ¯ x B r x 1 R n : x 1 ¯ | ¯ | √ r } denotes the ball of radius r centered at ¯ x . 1 Version of October 31, 2003
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2 12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations. 12.1.1 Controllability of linearized system Let x 0 : [0 , T ] , u 0 : [0 , T ] be a bounded solution of (12.1). The standard ⇒∀ R n ⇒∀ R m linearization of (12.1) around the solution ( x 0 ( · ) , u 0 ( · )) describes the dependency of small state increments x ( t ) = x ( t ) x 0 ( t ) + o ( x ( t )) on small input increments u ( t ) = u ( t ) u ( t ): ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , (12.2) where da da A ( t ) = , B ( t ) = (12.3) dx du x = x 0 ( t ) ,u = u 0 ( t ) x = x 0 ( t ) ,u = u 0 ( t ) are bounded measureable matrix-valued functions of time. Let us call system (12.2) controllable on time interval [0 , T ] if for every ¯ 0 ¯ T R n x , x there exists a bounded measureable function u : [0 , T ] such that the solution of ⇒∀ R m ¯ (12.2) with x (0) = ¯ 0 satisfies x ( T ) = T . The following simple criterion of controllability x x is well known from the linear system theory. Theorem 12.1 System (12.2) is controllable on interval [0 , T ] if and only if the matrix T W c = M ( t ) 1 B ( t ) B ( t ) ( M ( t ) ) 1 dt 0 is positive definite, where M = M ( t ) is the evolution matrix of (12.2), defined by ˙ M ( t ) = A ( t ) M ( t ) , M (0) = I.
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