lect9 - EL 625 Lecture 9 1 EL 625 Lecture 9 Observability...

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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 0 , any initial state, x ( t 0 ) can be de- termined from observation of the output y e [ t 0 , t ] over a finite time interval, with the input u e [ t 0 , 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 0 ) x ( t 0 ) + Z t t 0 φ ( t, τ ) B ( τ ) u ( τ ) (2) y ( t ) = C ( t ) φ ( t, t 0 ) x ( t 0 ) + Z t t 0 C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) + D ( t ) u ( t ) (3) Hence, C ( t ) φ ( t, t 0 ) x ( t 0 ) = y ( t ) - Z t t 0 C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) - 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 0 ) can be found if C ( t ) φ ( t, t 0 ) x ( t 0 ) is known.
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EL 625 Lecture 9 2 Let ˆ y ( t ) = y ( t ) - Z t t 0 C ( t ) φ ( t, τ ) B ( τ ) u ( τ ) - 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 0 φ T ( τ, t 0 ) C T ( τ ) C ( τ ) φ ( τ, t 0 ) # x ( t 0 ) = Z t t 0 φ T ( τ, t 0 ) C T ( τ y ( τ ) (6) If the ‘Observability Grammian’, S ( t 0 , t ) = 4 R t t 0 φ T ( τ, t 0 ) C T ( τ ) C ( τ ) φ ( τ, t 0 ) is nonsingular, then the above equation can be solved for x ( t 0 ). system (1)observable S ( t 0 , t ) nonsingular An equivalent condition for observability is: The system (1) is observable iff given any t 0 , for every μ 6 = 0, the vector w ( τ ) = 4 C ( τ ) φ ( τ, t 0 ) μ is not identically zero for all τ t 0 . Using the same methods as in the derivation of a similar test for controllabity, the following sufficient condition for time-varying systems can be derived which does not require computation of the transition matrix. A time-varying continuous system is observable if rank * ( t )) = n
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EL 625 Lecture 9 3 for some t > t 0 where Γ * = h M * 0 ( t ) M * 1 ( t ) . . . M * n - 1 ( t ) i (7) and M * k ( τ ) = dM * k - 1 ( τ ) + A T ( τ ) M * k - 1 ( τ ) for k = 1 , 2 , . . . (8) with M * 0 ( t ) = C T ( t ) (9) Also, a time-varying continuous system is observable if there does not exist a constant vector η 6 = 0 such that η T Γ * ( t ) = 0 T for all t > t 0 . For a fixed analog system, a necessary and sufficient condition for observability can be derived as, A time-invariant continuous system is observable iff rank C CA CA 2 . . . CA n - 1 = n (10) The condition for observability of a linear fixed system can also be derived as below. ˙ x = A x + B u
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EL 625 Lecture 9 4 y = C x (11) ˙ y = CA x + CB u (12) ¨ y = CA 2 x + CAB u + CB ˙ u (13) . . . y ( n - 1) = CA n - 1 x + CA n - 2 B u + CA n - 3 B ˙ u + . . . + CAB u ( n - 3) + CB u ( n - 2) (14) y ˙ y . . . y ( n - 1) = C CA CA 2 .
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