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Unformatted text preview: Topic #12 16.31 Feedback Control Systems StateSpace Systems • Statespace model features • Controllability Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 12–1 Controllability • Definition: An LTI system is controllable if, for every x ( t ) and every finite T > , there exists an input function u ( t ) , < t ≤ T , such that the system state goes from x (0) = 0 to x ( T ) = x . – Starting at is not a special case – if we can get to any state in finite time from the origin, then we can get from any initial condition to that state in finite time as well. 8 • This definition of controllability is consistent with the notion we used before of being able to “inﬂuence” all the states in the system in the decoupled examples (page 11–3). – ROT: For those decoupled examples, if part of the state cannot be “inﬂuenced” by u ( t ) , then it would be impossible to move that part of the state from to x • Need only consider the forced solution to study controllability. x f ( t ) = t e A ( t − τ ) B u ( τ ) dτ – Change of variables τ 2 = t − τ , dτ = − dτ 2 gives a form that is a little easier to work with: x f ( t ) = t e Aτ 2 B u ( t − τ 2 ) dτ 2 8 This controllability from the origin is often called reachability . October 14, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 12–2 • Note that, regardless of the eigenstructure of A , the CayleyHamilton theorem gives e At = n − 1 i =0 A i α i ( t ) for some computable scalars α i ( t ) , so that x f ( t ) = n − 1 i =0 ( A i B ) t α i ( τ 2 ) u ( t − τ 2 ) dτ 2 = n − 1 i =0 ( A i B ) β i ( t ) for scalars β i ( t ) that depend on the input u ( τ ) , < τ ≤ t . • Result can be interpreted as meaning that the state x f ( t ) is a linear combination of the n vectors A i B . – All linear combinations of these n vectors is the range space of the matrix formed from the A i B column vectors: M c = B AB A 2 B ··· A n − 1 B • Definition: Range space of M c is controllable subspace of the system – If a state x c ( t ) is not in the range space of M c , it is not a linear combination of these columns ⇒ it is impossible for x f ( t ) to ever equal x c ( t ) – called uncontrollable state....
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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