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

Vladimirov, Igor G.; Petersen, Ian R.

doi:10.3182/20110828-6-it-1002.01617

Cited by 25 publications

(76 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, if the additional quantum noise ω is effectively eliminated from the state dynamics of the quantum controller by letting b = 0, then (25) implies that c = 0, and hence, the matrix A in (11) becomes block lower triangular. In this case, the closed-loop system in (10) cannot be internally stable if A is not Hurwitz.…”

Section: Quantum Controllermentioning

confidence: 99%

“…1. By using a quadratic cost adopted in quantum control settings [16], [25] from classical LQG control [10], (1), (3) and is influenced by the environment through the quantum Wiener processes w, ω.…”

Section: Quantum Controllermentioning

confidence: 99%

“…A numerical procedure for finding suboptimal controllers for this problem was proposed in [16], and algebraic equations for the optimal CQLQG controller were obtained in [25]. Despite the previous results, the CQLQG control problem does not lend itself to an "elegant" solution (for example, in the form of decoupled Riccati equations as in the classical case [10]) and remains a subject of research.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we develop an algorithm for the numerical solution of the CQLQG control problem by using the gradient descent method and the Hamiltonian parameterization of PR quantum controllers [25]. The latter is a different technique to handle the PR constraints by reformulating the CQLQG control problem in an unconstrained fashion.…”

Section: Introductionmentioning

confidence: 99%

“…We obtain ordinary differential equations (ODEs) for the gradient descent in the Hilbert space of matrix-valued parameters. For this purpose, Fréchet differentiation is used together with related algebraic techniques [3], [14], [22], [23], [25] to employ the analytic structure of the LQG cost as a composite function of the matrix-valued variables, involving Lyapunov equations. The advantage of the proposed approach is that, at intermediate steps, it produces stabilizing PR quantum controllers which can be used as gradually improving suboptimal solutions of the CQLQG control problem, and a locally optimal solution (if it exists) is achieved asymptotically by moving along negative gradient directions with a suitable choice of stepsizes.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A gradient descent approach to optimal coherent quantum LQG controller design

Sichani

Vladimirov

Petersen

2015

2015 American Control Conference (ACC)

Self Cite

View full text Add to dashboard Cite

This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic 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, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented. A numerical example of a locally optimal CQLQG controller design is provided to demonstrate the algorithm performance.

show abstract

Section: Quantum Controllermentioning

confidence: 99%

Section: Quantum Controllermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A gradient descent approach to optimal coherent quantum LQG controller design

Sichani

Vladimirov

Petersen

2015

2015 American Control Conference (ACC)

Self Cite

View full text Add to dashboard Cite

show abstract

Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

2018

Self Cite

View full text Add to dashboard Cite

This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. An integro-differential equation is obtained for the time evolution of this quadratic-exponential functional, which is compared with the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. We also outline further approximations of the cost functional, based on higher-order cumulants and their growth rates, together with large deviations estimates. For comparison, an auxiliary classical Gaussian Markov diffusion process is considered in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different higher-order moments relevant to the risk-sensitive criteria. The results of the paper are also demonstrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals.Keywords Linear quantum stochastic system · Gaussian quantum state · Risk-sensitive quantum control Mathematics Subject Classification (2010) 81S25 · 81S05 · 81S22 · 81P16 · 81P40 · 81Q93 · 81Q10 · 60G15

show abstract

Time averaged consensus in a direct coupled coherent quantum observer network

Petersen

2017

Control Theory Technol.

Self Cite

View full text Add to dashboard Cite

This paper considers the problem of constructing a direct coupling quantum observer for a closed linear quantum system. The proposed distributed observer consists of a network of quantum harmonic oscillators and it is shown that the observer network converges to a consensus in a time averaged sense in which each element of the observer estimates the specified output of the quantum plant. An example and simulations are included to illustrate the properties of the observer network.

show abstract

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

Cited by 25 publications

References 4 publications

A gradient descent approach to optimal coherent quantum LQG controller design

A gradient descent approach to optimal coherent quantum LQG controller design

Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

Time averaged consensus in a direct coupled coherent quantum observer network

Contact Info

Product

Resources

About