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Unformatted text preview: EL 625 Lecture 9 1 EL 625 Lecture 9 Observability of linear systems Definition of observability : A system is called (completely) ob servable if for any initial time, t , any initial state, x ( t ) can be de termined from observation of the output y e [ t , t ] over a finite time interval, with the input u e [ t , t ] known over the same interval. Condition for observability : Consider the system, ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) (1) The solution for this linear system is: x ( t ) = φ ( t, t ) x ( t ) + Z t t φ ( t, τ ) B ( τ ) u ( τ ) dτ (2) y ( t ) = C ( t ) φ ( t, t ) x ( t ) + Z t t C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) dτ + D ( t ) u ( t ) (3) Hence, C ( t ) φ ( t, t ) x ( t ) = y ( t ) Z t t C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) dτ D ( t ) u ( t )(4) Note that the right hand side of the above equation is completely known. Hence, we need to determine the conditions under which x ( t ) can be found if C ( t ) φ ( t, t ) x ( t ) is known. EL 625 Lecture 9 2 Let ˆ y ( t ) = y ( t ) Z t t C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) dτ D ( t ) u ( t ) (5) Then, ˆ y ( t ) is known over the time interval over which u ( t ) and y ( t ) are known. We can derive, " Z t t φ T ( τ, t ) C T ( τ ) C ( τ ) φ ( τ, t ) dτ # x ( t ) = Z t t φ T ( τ, t ) C T ( τ )ˆ y ( τ ) dτ (6) If the ‘Observability Grammian’, S ( t , t ) = 4 R t t φ T ( τ, t ) C T ( τ ) C ( τ ) φ ( τ, t ) dτ is nonsingular, then the above equation can be solved for x ( t ). system (1)observable ≡ S ( t , t ) nonsingular An equivalent condition for observability is: The system (1) is observable iff given any t , for every μ 6 = 0, the vector w ( τ ) = 4 C ( τ ) φ ( τ, t ) μ is not identically zero for all τ ≥ t . Using the same methods as in the derivation of a similar test for controllabity, the following sufficient condition for timevarying systems can be derived which does not require computation of the transition matrix. A timevarying continuous system is observable if rank (Γ * ( t )) = n EL 625 Lecture 9 3 for some t > t where Γ * = h M * ( t ) M * 1 ( t ) . . . M * n 1 ( t ) i (7) and M * k ( τ ) = dM * k 1 ( τ ) dτ + A T ( τ ) M * k 1 ( τ ) for k = 1 , 2 , . . . (8) with M * ( t ) = C T ( t ) (9) Also, a timevarying continuous system is observable if there does not exist a constant vector η 6 = such that η T Γ * ( t ) = T for all t > t . For a fixed analog system, a necessary and sufficient condition for observability can be derived as, A timeinvariant continuous system is observable iff rank C CA CA 2 ....
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This note was uploaded on 12/06/2010 for the course EL 625 taught by Professor Khorami during the Spring '10 term at NYU Poly.
 Spring '10
 khorami

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