lect8 - EL 625 Lecture 8 1 EL 625 Lecture 8 Controllability...

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EL 625 Lecture 8 1 EL 625 Lecture 8 Controllability and observability of linear systems Definition of controllability : A system is called completely con- trollable if for any pair of arbitrary initial and final states, an input can be found to transfer the system from the given initial state to the given final state in finite time, i.e. starting from any initial time t 0 and any initial state x ( t 0 ), any final state x * can be reached in finite time using some input, u e [ t 0 , t ]. Example : The system in the figure below is not controllable. - - Z - - Z - u ( t ) x 1 ( t ) x 2 ( t ) ˙ x 1 = u ˙ x 2 = u (1) x 1 ( t ) = x 1 (0) + Z t 0 u ( τ )
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EL 625 Lecture 8 2 x 2 ( t ) = x 2 (0) + Z t 0 u ( τ ) (2) = x 2 ( t ) = x 1 ( t ) + x 2 (0) - x 1 (0) (3) Thus, if x 1 (0) = x 2 (0), then, x 1 ( t ) = x 2 ( t ) for all time t and it is not possible to reach a state for which x 1 6 = x 2 . Thus, the set of reachable states is the line in the state space passing through the origin and making an angle of 45 o with the x 1 axis as shown in the figure below. - 6 x 1 x 2 x 1 = x 2 In general, for arbitrary initial states x 1 (0) and x 2 (0), the set of reachable states is a line passing through ( x 1 (0), x 2 (0)) and parallel to the line, x 1 = x 2 . Thus, the state space has been foliated into a set of parallel lines. Given any initial state lying on one of these lines, we can find a
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EL 625 Lecture 8 3 control input to transfer the state to any other point on the same line, but, it is impossible to transfer the state to a point lying outside this line. This system is not controllable. Note however that though, the system is not controllable, there exist initial and final states such that a control can be found to transfer the initial state to the final state. Controllability to (and from) the origin : Consider the sys- tem, ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) (4) The solution for this linear system is x ( t ) = φ ( t, t 0 ) x ( t 0 ) + Z t t 0 φ ( t, τ ) B ( τ ) u ( τ ) (5) Thus, x ( t ) - φ ( t, t 0 ) x ( t 0 ) = Z t t 0 φ ( t, τ ) B ( τ ) u ( τ ) (6) The righthand side of this equation is the state that would be reached at time t starting from the origin at time t 0 with input u e [ t 0 , t ]. Thus, the ability to go from an arbitrary initial state to an
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EL 625 Lecture 8 4 arbitrary final state is equivalent to the ability to go from the zero state to an arbitrary final state. Complete controllability Controllability from the origin Consider a system which is controllable to the origin i.e. starting from any initial state, a control can be found to transfer the state to the origin. 0
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