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Ian R. Petersen

Publications and source records attributed to Ian R. Petersen.

At least 55 records · Page 3Linked to original sources
arXiv2024

Robust Quantum Control via a Model Predictive Control Strategy

This article presents a robust control strategy using Time-Optimal Model Predictive Control (TOMPC) for a two-level quantum system subject to bounded uncertainties. In this method, the control field is optimized over a finite horizon using a nominal quantum system as the reference and then the optimal control for the first time interval is applied and a projective measurement is implemented on the uncertain system. The new control field for the next time interval will be iteratively optimized based on the measurement result. We present theoretical results to guarantee the stability of the TOMPC algorithm. We also characterize the robustness and the convergence rate of the TOMPC strategy for the control of two-level systems. Numerical simulations further demonstrate that, in the presence of uncertainties, our quantum TOMPC algorithm enhances robustness and steers the state to the desired state with high fidelity. This work contributes to the progress of Model Predictive Control in quantum control and explores its potential in practical applications of quantum technology.

quant-ph↗
arXiv2024

Competitive Equilibrium in Microgrids With Dynamic Loads

In this paper, we consider microgrids that interconnect prosumers with distributed energy resources and dynamic loads. Prosumers are connected through the microgrid to trade energy and gain profit while respecting the network constraints. We establish a local energy market by defining a competitive equilibrium which balances energy and satisfies voltage constraints within the microgrid for all time. Using duality theory, we prove that under some convexity assumptions, a competitive equilibrium is equivalent to a social welfare maximization solution. Additionally, we show that a competitive equilibrium is equivalent to a Nash equilibrium of a standard game. In general, the energy price for each prosumer is different, leading to the concept of locational prices. We investigate a case under which all prosumers have the same locational prices. Additionally, we show that under some assumptions on the resource supply and network topology, locational prices decay to zero after a period of time, implying the available supply will be more than the demand required to stabilize the system. Finally, two numerical examples are provided to validate the results, one of which is a direct application of our results on electric vehicle charging control.

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arXiv2023

The Quantum Kalman Decomposition: A Gramian Matrix Approach

The Kalman canonical form for quantum linear systems was derived in \cite{ZGPG18}. The purpose of this paper is to present an alternative derivation by means of a Gramian matrix approach. Controllability and observability Gramian matrices are defined for linear quantum systems, which are used to characterize various subspaces. Based on these characterizations, real orthogonal and block symplectic coordinate transformation matrices are constructed to transform a given quantum linear system to the Kalman canonical form. An example is used to illustrate the main results.

quant-ph↗
arXiv2023

A Nonlinear Negative Imaginary Systems Framework with Actuator Saturation for Control of Electrical Power Systems

In the transition to net zero, it has been suggested that a massive expansion of the electric power grid will be required to support emerging renewable energy zones. In this paper, we propose the use of battery-based feedback control and nonlinear negative imaginary systems theory to reduce the need for such an expansion by enabling the more complete utilization of existing grid infrastructure. By constructing a novel Lur'e-Postnikov-like Lyapunov function, a stability result is developed for the feedback interconnection of a nonlinear negative imaginary system and a nonlinear negative imaginary controller. Additionally, a new class of nonlinear negative imaginary controllers is proposed to deal with actuator saturation. We show that in this control framework, the controller eventually leaves the saturation boundary, and the feedback system is locally stable in the sense of Lyapunov. This provides theoretical support for the application of battery-based control in electrical power systems. Validation through simulation results for single-machine-infinite-bus power systems supports our results. Our approach has the potential to enable a transmission line to operate at its maximum power capacity, as stability robustness is ensured by the use of a feedback controller.

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arXiv2023

Decoherence time control by interconnection for finite-level quantum memory systems

This paper is concerned with open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices for finite-level systems. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics according to a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This setting includes the zero-Hamiltonian isolated system dynamics as a particular case, where the system variables are constant in time, which makes them potentially applicable as a quantum memory. In a more realistic case of nonvanishing system-field coupling, we define a memory decoherence time when a mean-square deviation of the system variables from their initial values becomes relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the decoherence time maximization over the energy parameters of the system and obtain a condition under which the zero Hamiltonian provides a suboptimal solution. This optimization problem is also discussed for a direct energy coupling interconnection of such systems.

quant-ph↗
arXiv2023

Decoherence time in quantum harmonic oscillators as quantum memory systems

This paper is concerned with open quantum harmonic oscillators (OQHOs) described by linear quantum stochastic differential equations. This framework includes isolated oscillators with zero Hamiltonian, whose system variables remain unchanged (in the Heisenberg picture of quantum dynamics) over the course of time, making such systems potentially applicable as quantum memory devices. In a more realistic case of system-environment coupling, we define a memory decoherence horizon as a typical time for a mean-square deviation of the system variables from their initial values to become relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the maximization of the decoherence time over the energy and coupling matrices of the OQHO as a memory system in its storage phase and obtain a condition under which the zero Hamiltonian delivers a suboptimal solution. This optimization problem is also discussed for an interconnection of OQHOs.

quant-ph↗
arXiv2023

Negative Imaginary Control Using Hybrid Integrator-Gain Systems: Application to MEMS Nanopositioner

In this paper, we propose a new approach to address the control problem for negative imaginary (NI) systems by using hybrid integrator-gain systems (HIGS). We investigate the single HIGS of its original form and its two variations, including a multi-HIGS and the serial cascade of two HIGS. A single HIGS is shown to be a nonlinear negative imaginary system, and so is the multi-HIGS and the cascade of two HIGS. We show that these three types of HIGS can be used as controllers to asymptotically stabilize linear NI systems. The results of this paper are then illustrated in a real-world experiment where a 2-DOF microelectromechanical system nanopositioner is stabilized by a multi-HIGS.

eess.SY↗
arXiv2023

Design and Stability of Angle based Feedback Control in Power Systems: A Negative-Imaginary Approach

This paper considers a power transmission network characterized by interconnected nonlinear swing dynamics on generator buses. At the steady state, frequencies across different buses synchronize to a common nominal value such as $50$Hz or $60$Hz, and power flows on transmission lines are within steady-state envelopes. We assume that fast measurements of generator rotor angles are available. Our approach to frequency and angle control centers on equipping generator buses with large-scale batteries that are controllable on a fast timescale. We link angle based feedback linearization control with negative-imaginary systems theory. Angle based feedback controllers are designed using large-scale batteries as actuators and can be implemented in a distributed manner incorporating local information. Our analysis demonstrates the internal stability of the interconnection between the power transmission network and the angle based feedback controllers. This internal stability underscores the benefits of achieving frequency synchronization and preserving steady-state power flows within network envelopes through the use of feedback controllers. Our approach will enable transmission lines to be operated at maximum power capacity since stability robustness is ensured by the use of feedback controllers rather than conservative criteria such as the equal area criterion. By means of numerical simulations, we illustrate our results.

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arXiv2023

Transactive Multi-Agent Systems over Flow Networks

This paper presented insights into the implementation of transactive multi-agent systems over flow networks where local resources are decentralized. Agents have local resource demand and supply, and are interconnected through a flow network to support the sharing of local resources while respecting restricted sharing/flow capacity. We first establish a competitive market with a pricing mechanism that internalizes flow capacity constraints into agents' private decisions. We then demonstrate through duality theory that competitive equilibrium and social welfare equilibrium exist and agree under convexity assumptions, indicating the efficiency of the pricing mechanism. Additionally, a new social acceptance sharing problem is defined to investigate homogeneous pricing when the optimal sharing prices at all agents under competitive equilibrium are always equal for social acceptance. A conceptual computation method is proposed, prescribing a class of socially admissible utility functions to solve the social acceptance problem. A special case of linear-quadratic multi-agent systems over undirected star graphs is provided as a pedagogical example of how to explicitly prescribe socially admissible utility functions. Finally, extensive experiments are provided to validate the results.

cs.MA↗
arXiv2023

Quadratic-exponential coherent feedback control of linear quantum stochastic systems

This paper considers a risk-sensitive optimal control problem for a field-mediated interconnection of a quantum plant with a coherent (measurement-free) quantum controller. The plant and the controller are multimode open quantum harmonic oscillators governed by linear quantum stochastic differential equations, which are coupled to each other and driven by multichannel quantum Wiener processes modelling the external bosonic fields. The control objective is to internally stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional which penalizes the plant variables and the controller output. We obtain first-order necessary conditions of optimality for this problem by computing the partial Frechet derivatives of the cost functional with respect to the energy and coupling matrices of the controller in frequency domain and state space. An infinitesimal equivalence between the risk-sensitive and weighted coherent quantum LQG control problems is also established. In addition to variational methods, we employ spectral factorizations and infinite cascades of auxiliary classical systems. Their truncations are applicable to numerical optimization algorithms (such as the gradient descent) for coherent quantum risk-sensitive feedback synthesis.

math.OC↗
arXiv2023

Two-step feedback preparation of entanglement for qubit systems with time delay

Quantum entanglement plays a fundamental role in quantum computation and quantum communication. Feedback control has been widely used in stochastic quantum systems to generate given entangled states since it has good robustness, where the time required to compute filter states and conduct filter based control usually cannot be ignored in many practical applications. This paper designed two control strategies based on the Lyapunov method to prepare a class of entangled states for qubit systems with a constant delay time. The first one is bang bang like control strategy, which has a simple form with switching between a constant value and zero, the stability of which is proved. Another control strategy is switching Lyapunov control, where a constant delay time is introduced in the filter-based feedback control law to compensate for the computation time. Numerical results on a two qubit system illustrate the effectiveness of these two proposed control strategies.

quant-ph↗
arXiv2023

Fault-tolerant $H^\infty$ control for optical parametric oscillators with pumping fluctuations

Optical Parametric Oscillators (OPOs) have wide applications in quantum optics for generating squeezed states and developing advanced technologies. When the phase or/and the amplitude of the pumping field for an OPO have fluctuations due to fault signals, time-varying uncertainties will be introduced in the dynamic parameters of the system. In this paper, we investigate how to design a fault-tolerant $H^\infty$ controller for an OPO with a disturbance input and time-varying uncertainties, which can achieve the required $H^\infty$ performance of the quantum system. We apply robust $H^\infty$ control theory to a quantum system, and design a passive controller and an active controller based on the solutions to two Riccati equations. The passive controller has a simple structure and is easy to be implemented by using only passive optical components, while the active quantum controller may achieve improved performance. The control performance of the proposed two controllers and one controller that was designed without consideration of system uncertainties is compared by numerical simulations in a specific OPO, and the results show that the designed controllers work effectively for fluctuations in both the phase and amplitude of the pumping field.

quant-ph↗
arXiv2023

Whiplash Gradient Descent Dynamics

In this paper, we propose the Whiplash Inertial Gradient dynamics, a closed-loop optimization method that utilises gradient information, to find the minima of a cost function in finite-dimensional settings. We introduce the symplectic asymptotic convergence analysis for the Whiplash system for convex functions. We also introduce relaxation sequences to explain the non-classical nature of the algorithm and an exploring heuristic variant of the Whiplash algorithm to escape saddle points, deterministically. We study the algorithm's performance for various costs and provide a practical methodology for analyzing convergence rates using integral constraint bounds and a novel Lyapunov rate method. Our results demonstrate polynomial and exponential rates of convergence for quadratic cost functions.

math.OC↗
arXiv2023

On the regularization and optimization in quantum detector tomography

Quantum detector tomography (QDT) is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we utilize regularization to improve the QDT accuracy whenever the probe states are informationally complete or informationally incomplete. In the informationally complete scenario, without regularization, we optimize the resource (probe state) distribution by converting it to a semidefinite programming problem. Then in both the informationally complete and informationally incomplete scenarios, we discuss different regularization forms and prove the mean squared error scales as $ O(\frac{1}{N}) $ or tends to a constant with $ N $ state copies under the static assumption. We also characterize the ideal best regularization for the identifiable parameters, accounting for both the informationally complete and informationally incomplete scenarios. Numerical examples demonstrate the effectiveness of different regularization forms and a quantum optical experiment test shows that a suitable regularization form can reach a reduced mean squared error.

quant-ph↗
arXiv2023

Strictly negative imaginary state feedback control for relative degree two systems

In this paper, we present a strictly negative imaginary state feedback control methodology for relative degree two negative imaginary systems such as flexible structures with collocated sensors and actuators. We show that by augmenting a nominal model with a destabilising PID controller, we can apply state feedback control to the system, resulting in a strictly negative imaginary nominal closed loop system with a prescribed degree of stability. Moreover, the overall system is guaranteed to be internally stable in the face of unmodelled spillover dynamics.

math.OC↗
arXiv2023

Nonlinear Negative Imaginary Systems with Switching

In this paper, we extend nonlinear negative imaginary (NI) systems theory to switched systems. Switched nonlinear NI systems and switched nonlinear output strictly negative imaginary (OSNI) systems are defined. We show that the interconnection of two switched nonlinear NI systems is still switched nonlinear NI. The interconnection of a switched nonlinear NI system and a switched nonlinear OSNI system is asymptotically stable under some assumptions. This stability result is then illustrated using a numerical example.

eess.SY↗
arXiv2023

A Coherent LQG approach to Quantum Equalization

We propose a method to design a suboptimal, coherent quantum LQG controller to solve a quantum equalization problem. Our method involves reformulating the problem as a control problem and then designing a classical LQG controller and implementing it as a quantum system. Illustrative examples are included which demonstrate the algorithm for both active and passive systems, i.e., systems where the dynamics are described in terms of both position and momentum operators and systems with dynamics in terms of annihilation operators only.

quant-ph↗
arXiv2023

A negative imaginary approach to hybrid integrator-gain system control

In this paper, we show that a hybrid integrator-gain system (HIGS) is a nonlinear negative imaginary (NNI) system. We prove that the positive feedback interconnection of a linear negative imaginary (NI) system and a HIGS is asymptotically stable. We apply the HIGS to a MEMS nanopositioner, as an example of a linear NI system, in a single-input single-output framework. We analyze the stability and the performance of the closed-loop interconnection in both time and frequency domains through simulations and demonstrate the applicability of HIGS as an NNI controller to a linear NI system.

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