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Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance
Issued as Progress report, and Final report, Project no. E-21-67
The Separation Principle in Stochastic Control, Redux
Over the last 50 years a steady stream of accounts have been written on the
separation principle of stochastic control. Even in the context of the
linear-quadratic regulator in continuous time with Gaussian white noise, subtle
difficulties arise, unexpected by many, that are often overlooked. In this
paper we propose a new framework for establishing the separation principle.
This approach takes the viewpoint that stochastic systems are well-defined maps
between sample paths rather than stochastic processes per se and allows us to
extend the separation principle to systems driven by martingales with possible
jumps. While the approach is more in line with "real-life" engineering thinking
where signals travel around the feedback loop, it is unconventional from a
probabilistic point of view in that control laws for which the feedback
equations are satisfied almost surely, and not deterministically for every
sample path, are excluded.Comment: 23 pages, 6 figures, 2nd revision: added references, correction
Almost Sure Stabilization for Adaptive Controls of Regime-switching LQ Systems with A Hidden Markov Chain
This work is devoted to the almost sure stabilization of adaptive control
systems that involve an unknown Markov chain. The control system displays
continuous dynamics represented by differential equations and discrete events
given by a hidden Markov chain. Different from previous work on stabilization
of adaptive controlled systems with a hidden Markov chain, where average
criteria were considered, this work focuses on the almost sure stabilization or
sample path stabilization of the underlying processes. Under simple conditions,
it is shown that as long as the feedback controls have linear growth in the
continuous component, the resulting process is regular. Moreover, by
appropriate choice of the Lyapunov functions, it is shown that the adaptive
system is stabilizable almost surely. As a by-product, it is also established
that the controlled process is positive recurrent
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