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Unformatted text preview: Chapter 7 Kalman Filter In this chapter, we study Kalman Filtering, which is a celebrated optimal observer/filter/prediction design method. In many ways, Kalman filter is the dual of LQ control. We shall develop Kalman filter in this way. Recall that for time invariant systems, the design method for state feedback can be used to design a state estimator (observer). Here, the State Estimator design problem is to choose L in: x = A x + Bu + L ( y ( t ) C x ) x = ( A LC ) x so that the observer error dynamics is stable. This is ensured if A LC is stable. The related State feedback Problem is to choose K in x = A T x + C T u ; u = Kx x = ( A T C T K ) x. A T C T K is stable. By choosing L = K T for the observer, the observer is ensured to be stable. Sine the K obtained by LQ optimal (infinite horizon) control design is stabilizing as long as some stabilizability and detectability conditions are satisfied, L = K T can be used as a stabilizing observer gain as well. The questions we need to ask are: How is it done - in the steady state case, and in the time varying (or finite horizon) case? In what way is the resulting observer optimal? 7.1 Steady State Kalman Filter Consider first the Infinite Horizon LQ Control : x = Ax + Bu where R is positive definite, Q = Q T 2 Q 1 2 with ( A,B ) stabilizable, ( A,Q 1 2 ) detectable. We can solve for the positive (semi-) definite P in the ARE: A T P + P A P BR- 1 B T P + Q = 0 . (7.1) 123 124 c circlecopyrt Perry Y.Li e.g. via the Hamiltonian matrix approach (using its eigenvectors) or by simply integrating the Riccati equation backwards in time. We can get the optimal feedback gain: u = Kx where K = R- 1 B T P Because ( A,B ) and ( A,Q 1 2 ) are stabilizable and detectable, A BK has eigenvalues on the open left half plane. The stable observer problem is to find L in: x = A x + Bu + L ( y C x ) so that A LC is stable. Let us solve LQ control problem for the dual problem: x = A T x + C T u (7.2) Transform the Algebraic Riccati Equation: C T B , A T A . AP + P A T P C T R- 1 CP + Q = 0 . Stabilizing feedback gain K for (7.2) is: K = L T = R- 1 CP L = P C T R- 1 This generates the observer: x = A x + Bu + P C T R- 1 ( y C x ) with observer error dynamics: x = ( A P C T R- 1 C ) x which will be stable, as long as 1. ( A T ,C T ) is stabilizable. This is the same as ( A,C ) being detectable. 2. ( A T ,Q 1 2 ) is detectable. This is the same as ( A,Q 1 2 ) being stabilizable. This design can be easily achieved by merely solving the Algebraic Riccati Equation (ARE). Mat- labs ARE.m command can be used. Questions we need to ask are: In what way is this observer optimal?...
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This note was uploaded on 02/07/2012 for the course ME 8281 taught by Professor Staff during the Fall '08 term at Minnesota.
- Fall '08