Continuous-Time Systems With Perfect State Measurements
In this chapter, we develop the continuous-time counterparts of the results presented in Chapter 3. The disturbance attenuation problem will be the one formulated in Chapter 1, through (1.9)-(1.10), under only perfect state measurements. This will include the closed-loop perfect state (CLPS), sampled-data perfect state (SDPS), and delayed perfect state (DPS) information patterns, which were introduced in Chapter 2 (Section 2.3).
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- 1.Note that if the game is completely decoupled in terms of the effects of the controls of Player 1 and Player 2 on the state vector as well as the cost function (such as matrices A and Q being block-diagonal and the range spaces of BB’ and DD’ having empty intersection), then we would have an equality in (4.17); this, however, is not a problem of real interest to us, especially in the context of the disturbance attenuation problem to be studied in the next section.Google Scholar
- 2.This is, in fact, all we need for the disturbance attenuation problem, since our interest there is primarily in the boundedness of the upper value, and the construction of only minimizer’s (Player l’s) saddle-point policy, as the limit of the feedback policies obtained from finite-horizon games.Google Scholar
- 3.For a similar argument used in the discrete time, see the proof of Theorem 3.8. Note that with ry = oo, the ARE (4.29) becomes the standard ARE that arises in linear-quadratic regulator theory (, ), which is known to admit a unique positive definite solution under the given conditions.Google Scholar