MIT16_30F10_lec10

MIT16_30F10_lec10 - Topic #10 16.30/31 Feedback Control...

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Topic #10 16.30/31 Feedback Control Systems State-Space Systems State-space model features Controllability
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Fall 2010 16.30/31 10–1 Controllability Defnition: An LTI system is controllable if, for every x ( t ) and every Fnite T > 0 , there exists an input function u ( t ) , 0 < t T , such that the system state goes from x (0) = 0 to x ( T ) = x . Starting at 0 is not a special case if we can get to any state in Fnite time from the origin, then we can get from any initial condition to that state in Fnite time as well. 1 This deFnition 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 9– ?? ). ROT: ±or 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 0 to x Need only consider the forced solution to study controllability. t x f ( t ) = e A ( t τ ) B u ( τ ) 0 Change of variables τ 2 = t τ , = 2 gives a form that is a little easier to work with: t x f ( t ) = e 2 B u ( t τ 2 ) 2 0 Assume system has m inputs. 1 This controllability from the origin is often called reachability . October 13, 2010
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± ² ± ² Fall 2010 16.30/31 10–2 Note that, regardless of the eigenstructure of A , the Cayley-Hamilton theorem gives n 1 e At = A i α i ( t ) i =0 for some computable scalars α i ( t ) , so that n 1 ³ t n 1 x f ( t ) = ( A i B ) α i ( τ 2 ) u ( t τ 2 ) 2 = ( A i B ) β i ( t ) 0 i =0 i =0 for coefficients β i ( t ) that depend on the input u ( τ ) , 0 < τ t . Result can be interpreted as meaning that the state x f ( t ) is a linear combination of the nm vectors A i B (with m inputs).
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MIT16_30F10_lec10 - Topic #10 16.30/31 Feedback Control...

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