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observability_2009_11_09_01_2up

observability_2009_11_09_01_2up - 14 1 Observability S Lall...

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14 - 1 Observability S. Lall, Stanford 2009.11.09.01 14. Observability Estimating the initial state Observability Example: observability Least-squares observers The observability ellipsoid Infinite time Computing observability Example: one mass attached to two springs Estimating other states Example: estimating other states 14 - 2 Observability S. Lall, Stanford 2009.11.09.01 The Key Points of This Section once we know u (0) , . . . , u ( T 1) and y (0) , . . . , y ( T 1) , we just have a linear equation relating the data to the initial state x (0) so we can use least squares to estimate it and we can compute the estimation ellipsoid which tells how sensitive the estimate is to errors the infinite-time case is easy to compute via a Lyapunov equation which gives a practical measure of observability
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14 - 3 Observability S. Lall, Stanford 2009.11.09.01 Estimating the Initial State discrete-time system x ( t + 1) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) which means y (0) y (1) y (2) . . . y ( T 1) = D CB D CAB CB D . . . . . . CA T 2 B CA T 3 B . . . CB D u (0) u (1) u (2) . . . u ( T 1) + C CA CA 2 . . . CA T 1 x (0) we’d like to estimate x (0) ; ask estimation questions given u (0) , . . . , u ( T 1) and y (0) , . . . , y ( T 1) , find x (0) find all x (0) consistent with measured data if there is no exactly consistent x (0) , find an approximate one 14 - 4 Observability S. Lall, Stanford 2009.11.09.01 estimating the initial state this is just y (0) y (1) y (2) . . . y ( T 1) = P T u (0) u (1) u (2) . . . u ( T 1) + J T x (0) where P T = D CB D CAB CB D .
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