This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The problem is to find a stabilizing measurement-free quantum controller for a quantum plant so as to minimize a mean square cost for the fully quantum closed-loop system. The plant and controller are open quantum systems interconnected through bosonic quantum fields. In comparison with the observation-actuation structure of classical controllers, coherent quantum feedback is less invasive to the quantum dynamics. The plant and controller variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions in the form of quadratic constraints on the state-space matrices, reflecting the CCR preservation in time. The symmetry of the problem is taken into account by introducing equivalence classes of coherent quantum controllers generated by symplectic similarity transformations. We discuss a modified gradient flow, which is concerned with norm-balanced realizations of controllers. A line-search gradient descent algorithm with adaptive stepsize selection is proposed for the numerical solution of the CQLQG control problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing coherent quantum controller, thus taking the PR constraints into account. A convergence analysis of the algorithm is presented. Numerical examples of designing locally optimal CQLQG controllers are provided in order to demonstrate the algorithm performance.
This paper is concerned with open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by quantum stochastic differential equations. The latter are driven by quantum Wiener processes which represent external boson fields. The system-field coupling operators are linear functions of the system variables. The Hamiltonian consists of a nominal quadratic function of the system variables and an uncertain perturbation which is represented in a Weyl quantization form. Assuming that the nominal linear quantum system is stable, we develop sufficient conditions on the perturbation of the Hamiltonian which guarantee robust mean square stability of the perturbed system. Examples are given to illustrate these results for a class of Hamiltonian perturbations in the form of trigonometric polynomials of the system variables. Index Termsopen quantum stochastic system, Hamiltonian perturbation, Weyl quantization, robust mean square stability. I. INTRODUCTIONThe quantum mechanical concept of quantization is concerned with assigning quantum observables to classical variables (and functions thereof). Weyl's proposal for the development of a general quantization scheme was introduced in 1927 (see for example, [23, Section IV.14]) soon after the invention of quantum mechanics. An important feature of the Weyl association is that it treats quantum dynamic variables equally and leads to correct marginal distributions for them [1, Chapter 8]. The Weyl quantization scheme employs Fourier transforms and is known to be a convenient and, in many respects, satisfactory procedure for quantization [1], [3], [24]. In addition to providing a mathematical formalism, this scheme also offers an interpretation of quantum mechanical phenomena, thus leading to a better understanding of their physical aspects [1], [24]. The aim of the present paper is to use the Weyl quantization for the modeling of perturbations of Hamiltonians for a class of open quantum systems and the robust stability analysis based on this description of uncertainty.A wide range of open quantum systems, which interact with their environment, can be modelled by using the apparatus of quantum stochastic differential equations (QSDEs) [7], [14]. In this framework, which follows the Heisenberg picture of quantum dynamics, a quantum noise is introduced in order to represent the surroundings as a heat bath of external fields acting on a boson Fock space [14]. The QSDE approach to open quantum systems is employed by the quantum dissipative systems theory [8] which addresses robust stability issues.The robustness of various classes of perturbed open quantum systems, modelled by QSDEs, has been addressed in the literature using dissipativity theory and different notions of stability (see for example [15], [16], [20], [21]). In particular, robust mean square stability with respect to a class of perturbations of Hamiltonians has been studied in [15] and its applications have been presented in [17], [18]. In these papers, the classical and qua...
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