You are here : Control System Design  Index  Book Contents  Appendix D  Section D.5 D. Properties of ContinuousTime Riccati Equations

(D.5.7) 
Substituting (D.5.5) into (D.5.4) gives
Using integration by parts, we then obtain
Finally, using (22.10.5) and (D.5.6), we obtain
Hence, squaring and taking mathematical expectations, we obtain
(upon using (D.5.3),
(22.10.3) and (22.10.4) ) the following:
The last term in (D.5.11) is zero if . Thus, we see that the design of the optimal linear filter can be achieved by minimizing
where satisfies the reversetime equations (D.5.5) and (D.5.6).
We recognize the set of equations formed by (D.5.5), (D.5.6), and (D.5.12) as a standard linear regulator problem, provided that the connections shown in Table D.1 are made.
Table D.1: Duality in quadratic regulators and filters
Finally, by using the (dual) optimal control results presented earlier, we see that the optimal filter is given by
where
and satisfies the dual form of (D.0.1), (22.4.18):
Substituting (D.5.14) into (D.5.5), (D.5.6) we see that
(D.5.21) 
We see that is the output of a linear homogeneous equation. Let , and define as the state transition matrix from for the timevarying system having equal to . Then
Hence, the optimal filter satisfies
where
We then observe that (D.5.24) is actually the solution of
the following state space (optimal
filter).
We see that the final solution depends on only through (D.5.27). Thus, as predicted, (D.5.25), (D.5.26) can be used to generate an optimal estimate of any linear combination of states.
Of course, the optimal filter (D.5.25) is identical to that given in (22.10.23)
All of the properties of the optimal filter follow by analogy from the (dual) optimal linear regulator. In particular, we observe that (D.5.16) and (D.5.17) are a CTDRE and its boundary condition, respectively. The only difference is that, in the optimalfilter case, this equation has to be solved forward in time. Also, (D.5.16) has an associated CTARE, given by
Thus, the existence, uniqueness, and properties of stabilizing solutions for (D.5.16) and (D.5.28) satisfy the same conditions as the corresponding Riccati equations for the optimal regulator.