Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II


We address the matter of steering the state of a linear stochastic system to a prescribed distribution over a finite horizon with minimum energy, and the matter to keep up the state at a stationary distribution over an infinite horizon with minimum power. For each problems the control and Gaussian noise channels are allowed to be distinct, thereby, putting the results of this paper outside of the scope of previous work both in likelihood and in control. The special case where the disturbance and control enter through the identical channels has been addressed in the first half of this work that was presented as Half I. Herein, we have a tendency to present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. We have a tendency to then address the question of feasibility for each problems. For the finite-horizon case, provided the system is controllable, we have a tendency to prove that while not any restriction on the directionality of the stochastic disturbance it is always potential to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it's not forever doable to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It's shown that covariances of admissible stationary Gaussian distributions are characterised by a certain Lyapunov-like equation and, of course, they coincide with the class of stationary state covariances that can be attained by a appropriate stationary coloured noise as input. We finally address the question of the way to compute appropriate controls numerically. We have a tendency to present another to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial - articles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.

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