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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 16.31 121 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 inuence all the states in the system in the decoupled examples (page 113). ROT: For those decoupled examples, if part of the state cannot be inuenced 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 ) B u ( ) d Change of variables 2 = t , d = d 2 gives a form that is a little easier to work with: x f ( t ) = t e A 2 B u ( t 2 ) d 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 16.31 122 Note that, regardless of the eigenstructure of A , the CayleyHamilton theorem gives e At = n 1 i =0 A i i ( t ) for some computable scalars i ( t ) , so that x f ( t ) = n 1 i =0 ( A i B ) t i ( 2 ) u ( t 2 ) d 2 = n 1 i =0 ( A i B ) i ( t ) for scalars i ( t ) that depend on the input u ( ) , < 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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 Fall '07
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