Feedback representations of critical controls for well-posed linear systems
✍ Scribed by Olof J. Staffans
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 223 KB
- Volume
- 8
- Category
- Article
- ISSN
- 1049-8923
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✦ Synopsis
This is the first part in a three part study of the suboptimal full information H problem for a well-posed linear system with input space º, state space H, and output space ½. We define a cost function Q(x , u)" R> 1y(s), Jy(s)2 7 ds, where y3¸ (R>; ½) is the output of the system with initial state x 3H and control u3¸
(R>; º), and J is a self-adjoint operator on ½. The cost function Q is quadratic in x and u, and we suppose (in the stable case) that the second derivative of Q( x, u) with respect to u is non-singular. This implies that, for each x 3H, there is a unique critical control u such that the derivative of Q(x , u) with respect to u vanishes at u"u . We show that u can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)-inner numerator; here S is a particular self-adjoint operator on º. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace.