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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 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 timedependent 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 mdimensional timevarying 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 ∗ R n . The reachable set for a given T > the ( Ulocally) reachable set R U (¯ x ,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 ⇒∀ R n ⇒∀ R m and x ( t ) ∗ U for all t ∗ [0 ,T ]. Our task is to find conditions under which R U (¯ x ,T ) is guaranteed to contain a neig borhood of some point in R n , or, alternatively, conditions which guarante that R U (¯ x ,T ) has an empty interior. In particular, when ¯ x is a controlled equilibrium of (12.1), i.e. x , ¯ u ∗ R m , complete local controllability of (12.1) at ¯ a (¯ u ) = for some ¯ x means that for every φ > and T > there exists > such that R U (¯ x ) for every x,T ) B (¯ ¯ x ), where U = B α (¯ x ∗ B (¯ x ) 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 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 ,T ] , u : [0 ,T ] be a bounded solution of (12.1). The standard ⇒∀ R n ⇒∀ R m linearization of (12.1) around the solution ( x ( · ) ,u ( · )) describes the dependency of small state increments x ( t ) = x ( t ) − x ( 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 ( t ) ,u = u ( t ) x = x ( t ) ,u = u ( t ) are bounded measureable matrixvalued functions of time. Let us call system (12.2) controllable on time interval [0 ,T ] if for every ¯ ¯ 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) = ¯ satisfies x ( T ) = T . The following simple criterion of controllability x x is well known from the linear system theory....
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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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