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arXiv:1010.3125 · Systems & Control Letters

A Quasi-separation Principle and Newton-like Scheme for Coherent Quantum LQG Control

Igor G. Vladimirov, Ian R. Petersen

Published Jan 1, 2011• 39 citations
arXivHTMLPDF
DOI
10.1016/j.sysconle.2013.03.004

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arXiv:1105.1574•Australian Control Conference•May 9, 2011•12 citations
This paper is concerned with linear-quadratic-Gaussian (LQG) control for a field-mediated feedback connection of a plant and a coherent (measurement-free) controller. The plant and controller are multimode open quantum harmonic oscillators governed by linear quantum stochastic differential equations. The control objective is to make the closed-loop system internally stable and to minimize the infinite-horizon quadratic cost for the plant variables and the controller output subject to quantum physical realizability (PR) constraints. This coherent quantum LQG (CQLQG) control problem, which has been of active research interest for over ten years, does not admit a solution in the form of separation principle and independent Riccati equations known for its classical counterpart. We apply variational techniques to a family of discounted CQLQG control problems parameterized by an effective time horizon. This yields a homotopy algorithm, which is initialized with a PR (but not necessarily stabilizing) controller and aims at a locally optimal stabilizing controller for the original problem in the limit.
The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control problem for time-varying quantum plants governed by linear quantum stochastic differential equations over a bounded time interval. A controller is sought among quantum linear systems satisfying physical realizability (PR) conditions. The latter describe the dynamic equivalence of the system to an open quantum harmonic oscillator and relate its state-space matrices to the free Hamiltonian, coupling and scattering operators of the oscillator. Using the Hamiltonian parameterization of PR controllers, the CQLQG problem is recast into an optimal control problem for a deterministic system governed by a differential Lyapunov equation. The state of this subsidiary system is the symmetric part of the quantum covariance matrix of the plant-controller state vector. The resulting covariance control problem is treated using dynamic programming and Pontryagin's minimum principle. The associated Hamilton-Jacobi-Bellman equation for the minimum cost function involves Frechet differentiation with respect to matrix-valued variables. The gain matrices of the CQLQG optimal controller are shown to satisfy a quasi-separation property as a weaker quantum counterpart of the filtering/control decomposition of classical LQG controllers.