Showing papers by "Ian R. Petersen published in 2021"

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

Abstract: 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. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and minimization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from 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. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov diffusion process in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different fourth-order cumulants, thus showing that the risk-sensitive criteria are not reducible to quadratic–exponential moments of classical Gaussian processes. The results of the paper are illustrated 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.

Design of a Discrete-Time Fault-Tolerant Quantum Filter and Fault Detector

Abstract: This paper solves the problem of discrete-time fault-tolerant quantum filtering for a class of laser–atom open quantum systems subject to the stochastic faults. We show that by using the discrete-time quantum measurements, optimal estimates of both the atomic observables and the classical fault process can be simultaneously determined in terms of recursive quantum stochastic difference equations. A dispersive interaction quantum system example is used to demonstrate the proposed filtering approach.

Distributed Formation Control Using Fuzzy Self-Tuning of Strictly Negative Imaginary Consensus Controllers in Aerial Robotics

Abstract: Wind gusts are significant barriers to the outdoor operations of networked multiple unmanned aerial vehicles (UAVs), which fly in close proximity to each other or around obstacles. As such, traditional control methods such as PID control may not perform adequately. Based on the strictly negative imaginary (SNI) systems theory, this article presents a novel decentralized and adaptive consensus-based formation control law that drives multiple UAVs to follow the desired formation in the presence of limited bandwidth for information exchange and dynamically changing environmental conditions. To be consistent with a decentralized approach, each UAV only measures its relative position with respect to its neighbors according to a fixed information graph. As a result, the required formation is obtained by maintaining the desired relative positions among UAVs. Moreover, to deal with the challenging dynamics of flight environments, we also employ a knowledge-based fuzzy inference system to automatically adjust the parameters of the SNI consensus controllers, leading to the development of a fast and robust adaption method. In this article, we conduct a stability analysis based on the SNI theorem and rigorously compare the performance of our controllers with respect to the performance of conventional PID controllers. The efficacy of the overall closed loop control system is highlighted in real-time flight tests.

Fuzzy Self-Tuning of Strictly Negative-Imaginary Controllers for Trajectory Tracking of a Quadcopter Unmanned Aerial Vehicle

Abstract: Robustness in the face of uncertainties is an integral part of designing a real-time control system. Based on negative imaginary (NI) systems theory, we design robust and adaptive control systems for accurate trajectory tracking of a quadcopter aerial vehicle. Considering the challenging dynamics of unmanned aerial vehicles, we employ knowledge-based fuzzy inference systems (FIS) to facilitate automatic tuning in our SNI controllers, leading to the development of adaptive SNI control systems. Unlike fixed-gain controllers that have no ability to adapt to the variations in environmental conditions or changes in the dynamics of the plant, our adaptive SNI controllers are able to perform self-tuning to constantly update their parameters. The concept of adaptive autopilots will enhance the ability of the closed-loop control systems to accommodate large uncertainties. To demonstrate their efficacy, we design and implement our adaptive SNI controllers in the three-position control loops of the AR.Drone quadcopter after conducting extensive computer simulations. We also perform a rigorous comparative study with respect to the performance of fixed-gain SNI controllers, fixed-gain NI systems, in addition to model-predictive-control systems, and proportional integral derivative (PID) control systems as our benchmarks. To complete the study, we conduct a stability analysis based on Kharitonov's Theorem.

Robust Cooperative Control of Networked Train Platoons: a Negative-Imaginary Systems’ Perspective

Abstract: Cooperative control for trains shows great potential in improving line utilization, passenger comfort and operation flexibility.However,the operation of train platoons faces great challenges induced by model uncertainties,unpredictable resistances and time-varying disturbances. To address these problems,a consensus-based robust cooperative control scheme is presented in this paper for both homogeneous and heterogeneous train platoons.Taking the physical connection between carriages into consideration,each train in the platoons is modeled as a negative imaginary(NI) system,and the NI property is rigorously proved in this paper.In the foundation of cooperative control theories of NI systems,robust strictly negative imaginary(SNI) controllers considering the network topology are utilized to track a predefined motion reference.The proposed controllers are robust to both mass uncertainties and external disturbances.Moreover,the line utilization for the railway system and the ride comfort for the passengers are improved using the proposed control schemes.Numerical simulations are given to showcase the effectiveness and robustness of the proposed controllers.

Negative Imaginary State Feedback Equivalence for Systems of Relative Degree One and Relative Degree Two.

Abstract: This paper presents necessary and sufficient conditions under which a linear system of relative degree either one or two is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show for a class of linear time-invariant strictly proper systems, that such a system can be rendered minimal and negative imaginary using full state feedback if and only if it is controllable and weakly minimum phase. A strongly strict negative imaginary (SSNI) state feedback equivalence result is also provided. The NI state feedback equivalence result is then applied in a robust stabilization problem for an uncertain system with a strictly negative imaginary (SNI) uncertainty.

Quadratic-exponential functionals of Gaussian quantum processes.

Abstract: This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

Fault-tolerant Coherent H_infinity Control for Linear Quantum Systems

Abstract: Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this paper is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance and H_infinity disturbance attenuation performance. This paper first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault tolerant H_infinity control problem to the dissipation properties and the solutions of Riccati differential equations, an H_infinity controller for the quantum system is then designed by solving a set of linear matrix inequalities (LMIs). In particular, an algorithm is employed to introduce additional quantum inputs and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. Also, we propose a real application of the developed fault-tolerant control strategy to quantum optical systems. A linear quantum system example from quantum optics, where the amplitude of the pumping field r

Social Shaping of Competitive Equilibriums for Resilient Multi-Agent Systems

Abstract: In this paper, we study multi-agent systems with decentralized resource allocations. Agents have local demand and resource supply, and are interconnected through a network designed to support sharing of the local resource; and the network has no external resource supply. It is known from classical welfare economics theory that by pricing the flow of resource, balance between the demand and supply is possible. Agents decide on the consumed resource, and perhaps further the traded resource as well, to maximize their payoffs considering both the utility of the consumption, and the income from the trading. When the network supply and demand are balanced, a competitive equilibrium is achieved if all agents maximize their individual payoffs, and a social welfare equilibrium is achieved if the total agent utilities are maximized. First, we consider multi-agent systems with static local allocations, and prove from duality theory that under general convexity assumptions, the competitive equilibrium and the social welfare equilibrium exist and agree. Compared to similar results in the literature based on KKT arguments, duality theory provides a direct way for connecting the two notions and for a more general (e.g. nonsmooth) class of utility functions. Next, we show that the agent utility functions can be prescribed in a family of socially admissible functions, under which the resource price at the competitive equilibrium is kept below a threshold. Finally, we extend the study to dynamical multi-agent systems where agents are associated with dynamical states from linear processes, and we prove that the dynamic the competitive equilibrium and social welfare equilibrium continue to exist and coincide with each other.

Two-Stage Estimation for Quantum Detector Tomography: Error Analysis, Numerical and Experimental Results

Abstract: Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, a novel quantum detector tomography method is proposed. First, a series of different probe states are used to generate measurement data. Then, using constrained linear regression estimation, a stage-1 estimation of the detector is obtained. Finally, the positive semidefinite requirement is added to guarantee a physical stage-2 estimation. This Two-stage Estimation (TSE) method has computational complexity $O(nd^{2}M)$ , where $n$ is the number of $d$ -dimensional detector matrices and $M$ is the number of different probe states. An error upper bound is established, and optimization on the coherent probe states is investigated. We perform simulation and a quantum optical experiment to testify the effectiveness of the TSE method.

Necessary and Sufficient Conditions for State Feedback Equivalence to Negative Imaginary Systems.

Abstract: In this paper, we present necessary and sufficient conditions under which a linear time-invariant (LTI) system is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show that a minimal LTI strictly proper system can be rendered NI using full state feedback if and only if it can be output transformed into a system, which has relative degree less than or equal to two and is weakly minimum phase. We also considered the problems of state feedback equivalence to output strictly negative imaginary systems and strongly strict negative imaginary systems. Then we apply the NI state feedback equivalence result to robustly stabilize an uncertain system with strictly negative imaginary uncertainty. An example is provided to illustrate the proposed results, for the purpose of stabilizing an uncertain system.

Velocity-Free Attitude Control of Quadrotors: A Nonlinear Negative Imaginary Approach.

Abstract: In this paper, we propose a new approach to the attitude control of quadrotors, by which angular velocity measurements or a model-based observer reconstructing the angular velocity are not needed. The proposed approach is based on recent stability results obtained for nonlinear negative imaginary systems. In specific, through an inner-outer loop method, we establish the nonlinear negative imaginary property of the quadrotor rotational subsystem. Then, a strictly negative imaginary controller is synthesized using the nonlinear negative imaginary results. This guarantees the robust asymptotic stability of the attitude of the quadrotor in the face of modeling uncertainties and external disturbances. First simulation results underline the effectiveness of the proposed attitude control approach are presented.

A comparative study on how neural networks enhance quantum state tomography

Abstract: Quantum state tomography aiming at reconstructing the density matrix of a quantum state plays an important role in various emerging quantum technologies. Inspired by the intuition that machine learning has favorable robustness and generalization, we propose a deep neural networks based quantum state tomography (DNN-QST) approach, that can be applied to three cases, including few measurement copies and incomplete measurements as well as noisy measurements. Numerical results demonstrate that DNN-QST exhibits a great potential to achieve high fidelity for quantum state tomography with limited measurement resources and can achieve improved estimation when tomographic measurements suffer from noise. In addition, the results for 2-qubit states from quantum optical devices demonstrate the generalization of DNN-QST and its robustness against possible error in the experimental devices

Voltage Stability Studies for Distribution Networks: Assessing Load Dynamics.

Abstract: This paper presents an investigation into load dynamics that potentially cause voltage instability or collapse in distribution networks. Through phasor-based, time domain simulations of a dynamic load (DL) model from the literature, we show that the load dynamics alone do not cause voltage instability or collapse. By comparing the DL model to a benchmark model, we identify an important limitation with the DL model. We characterise this limitation and recommend that future work use load models with physical state variables. By investigating when and how load dynamics cause voltage instability, we are well-positioned to develop systems to control and maintain voltage stability in distribution networks.

Structural Controllability on Graphs for Drifted Bilinear Systems over Lie Groups

Abstract: In this paper, we study graphical conditions for structural controllability and accessibility of drifted bilinear systems over Lie groups. We consider a bilinear control system with drift and controlled terms that evolves over the special orthogonal group, the general linear group, and the special unitary group. Zero patterns are prescribed for the drift and controlled dynamics with respect to a set of base elements in the corresponding Lie algebra. The drift dynamics must respect a rigid zero-pattern in the sense that the drift takes values as a linear combination of base elements with strictly non-zero coefficients; the controlled dynamics are allowed to follow a free zero pattern with potentially zero coefficients in the configuration of the controlled term by linear combination of the controlled base elements. First of all, for such bilinear systems over the special orthogonal group or the special unitary group, the zero patterns are shown to be associated with two undirected or directed graphs whose connectivity and connected components ensure structural controllability/accessibility. Next, for bilinear systems over the special unitary group, we introduce two edge-colored graphs associated with the drift and controlled zero patterns, and prove structural controllability conditions related to connectivity and the number of edges of a particular color.

Velocity-free Attitude Control of Quadrotors: A Nonlinear Negative Imaginary Approach

Abstract: In this paper, we propose a new approach to the attitude control of quadrotors, by which angular velocity measurements or a model-based observer reconstructing the angular velocity are not needed. The proposed approach is based on recent stability results obtained for nonlinear negative imaginary systems. In specific, by constructing the respective storage functions, we establish the nonlinear negative imaginary properties of the whole quadrotor system and the quadrotor rotational subsystem. Then, an inner-outer loop method will be implemented to synthesize a strictly negative imaginary controller. This guarantees the robust asymptotic stability of the attitude of the quadrotor about its reference signal in the face of modeling uncertainties and external disturbances.

Social Shaping for Transactive Energy Systems

Abstract: This paper considers the problem of shaping agent utility functions in a transactive energy system to ensure the optimal energy price at a competitive equilibrium is always socially acceptable, that is, below a prescribed threshold. Agents in a distributed energy system aim to maximize their individual payoffs, as a combination of the utility of energy consumption and the income/expenditure from energy exchange. The utility function of each agent is parameterized by individual preference vectors, with the overall system operating at competitive equilibriums. We show the social shaping problem of the proposed transactive energy system is conceptually captured by a set decision problem. The set of agent preferences that guarantees a socially acceptable price is characterized by an implicit algebraic equation for strictly concave and continuously differentiable utility functions. We also present two analytical solutions where tight ranges for the coefficients of linear-quadratic utilities and piece-wise linear utilities are established under which optimal pricing is proven to be always socially acceptable.

A Closed Loop Gradient Descent Algorithm applied to Rosenbrock's function.

Abstract: We introduce a novel adaptive damping technique for an inertial gradient system which finds application as a gradient descent algorithm for unconstrained optimisation. In an example using the non-convex Rosenbrock's function, we show an improvement on existing momentum-based gradient optimisation methods. Also using Lyapunov stability analysis, we demonstrate the performance of the continuous-time version of the algorithm. Using numerical simulations, we consider the performance of its discrete-time counterpart obtained by using the symplectic Euler method of discretisation.

Towards Online Optimization for Power Grids

Abstract: In this note, we discuss potential advantages in extending distributed optimization frameworks to enhance support for power grid operators managing an influx of online sequential decisions. First, we review the state-of-the-art distributed optimization frameworks for electric power systems, and explain how distributed algorithms deliver scalable solutions. Next, we introduce key concepts and paradigms for online optimization, and present a distributed online optimization framework highlighting important performance characteristics. Finally, we discuss the connection and difference between offline and online distributed optimization, showcasing the suitability of such optimization techniques for power grid applications.