The riccati equation and its bounds

✍ Scribed by R.S. Bucy

• Book ID: 104148080
• Publisher: Elsevier Science
• Year: 1972
• Tongue: English
• Weight: 330 KB
• Volume: 6
• Category: Article
• ISSN: 0022-0000
• DOI: 10.1016/s0022-0000(72)80026-6

✦ Synopsis

In the asymptotic theory of the Riccati equation, certain a priori bounds are of importance. The upper bound is of the form W -1 + _dlC with W the observability matrix, C the controllability matrix, and A 1 a constant. In [3,4,6] we asserted that A 1 could be taken to be 1, this is not the case in general, although in certain cases where the Sensor is in the positive eigenfunction space of an operator related to the controllability matrix--see Theorem 2.2--it is. In this paper we find an explicit bound for A 1 dependent only on the information rate and control energy rate and state dimensions. Further, a simple counterexample is given which shows A 1 may be greater than 1.

In order to effectively compute this example, we developed explicit expressions for the infinite lag error variance for stationary problems as well as a very simple expression for the limit solution of the autonomous Riccati equation as the initial matrix gets large. The results all are given in terms of the equilibrium solution to a dual Riccati equation.