Showing papers by "Ian R. Petersen published in 2018"

Optimal Cooperative Guidance Law for Salvo Attack: An MPC-Based Consensus Perspective

Abstract: In this paper, a model-predictive-control (MPC)-based cooperative guidance law is presented to perform a salvo attack against a stationary target, which guarantees that multiple missiles hit the target simultaneously. Inspired by the concept of consensus in the multiagent systems, the impact time coordination is reformulated as a consensus on the ranges and range rates of the missiles, leading to a solution without explicitly exploiting the time-to-go or its estimate. By the virtue of an MPC framework, constraints on the normal acceleration and field-of-view are also addressed, and the finite-horizon optimality is guaranteed. Three numerical simulation cases are provided to demonstrate the effectiveness of the proposed approach.

A Quantum Hamiltonian Identification Algorithm: Computational Complexity and Error Analysis

Abstract: Quantum Hamiltonian identification (QHI) is important for characterizing the dynamics of quantum systems, calibrating quantum devices, and achieving precise quantum control. In this paper, an effective two-step optimization (TSO) QHI algorithm is developed within the framework of quantum process tomography. In the identification method, different probe states are input into quantum systems and the output states are estimated using the quantum state tomography protocol via linear regression estimation. The time-independent system Hamiltonian is reconstructed based on the experimental data for the output states. The Hamiltonian identification method has computational complexity $O(d^6)$ , where $d$ is the dimension of the system Hamiltonian. An error upper bound $O(\frac{d^3}{\sqrt{N}})$ is also established, where $N$ is the resource number for the tomography of each output state, and several numerical examples demonstrate the effectiveness of the proposed TSO Hamiltonian identification method.

A Survey of Methods Used to Control Piezoelectric Tube Scanners in High‐Speed AFM Imaging

Abstract: In most nanotechnology applications, speed and precision are important requirements for obtaining good topographical maps of material surfaces using atomic force microscopes (AFMs), many o...

The Kalman Decomposition for Linear Quantum Systems

Abstract: This paper studies the Kalman decomposition for linear quantum systems. Contrary to the classical case, the coordinate transformation used for the decomposition must belong to a specific class of transformations as a consequence of the laws of quantum mechanics. We propose a construction method for such transformations that put the system in a Kalman canonical form. Furthermore, we uncover an interesting structure for the obtained decomposition. In the case of passive systems, it is shown that there exist only controllable/observable and uncontrollable/unobservable subsystems. In the general case, controllable/unobservable and uncontrollable/observable subsystems may also be present, but their respective system variables must be conjugate variables of each other. This decomposition naturally exposes decoherence-free modes, quantum-nondemolition modes, quantum-mechanics-free subsystems, and back-action evasion measurements in the quantum system, which are useful resources for quantum information processing, and quantum measurements. The theory developed is applied to physical examples.

Extending Negative Imaginary Systems Theory to Nonlinear Systems

Abstract: In this paper, we extend the negative imaginary systems framework to nonlinear systems using Lyapunov and dissipativity theories. This enables us to extend some existing results from linear negative imaginary systems theory to nonlinear systems. In particular, asymptotic stability of a positive feedback interconnection of nonlinear negative imaginary systems has been established under suitable assumptions.

Time-Varying Formation Control of a Collaborative Multi-Agent System Using Negative-Imaginary Systems Theory.

Abstract: The movement of cooperative robots in a densely cluttered environment may not be possible if the formation type is invariant. Hence, we investigate a new method for time-varying formation control for a group of heterogeneous autonomous vehicles, which may include Unmanned Ground Vehicles (UGV) and Unmanned Aerial Vehicles (UAV). We have extended a Negative-Imaginary (NI) consensus control approach to switch the formation shape of the robots whilst only using the relative distance between agents and between agents and obstacles. All agents can automatically create a new safe formation to overcome obstacles based on a novel geometric method, then restore the prototype formation once the obstacles are cleared. Furthermore, we improve the position consensus at sharp corners by achieving yaw consensus between robots. Simulation and experimental results are then analyzed to validate the feasibility of our proposed approach.

Risk-sensitive performance criteria and robustness of quantum systems with a relative entropy description of state uncertainty.

Abstract: This paper considers links between the original risk-sensitive performance criterion for quantum control systems and its recent quadratic-exponential counterpart We discuss a connection between the minimization of these cost functionals and robustness with respect to uncertainty in system-environment quantum states whose deviation from a nominal state is described in terms of the quantum relative entropy These relations are similar to those in minimax LQG control for classical systems The results of the paper can be of use in providing a rational choice of the risk-sensitivity parameter in the context of robust quantum control with entropy theoretic quantification of statistical uncertainty in the system-field state

A Modified Frequency Domain Condition for the Physical Realizability of Linear Quantum Stochastic Systems

Abstract: This paper is concerned with a modified version of the frequency domain physical realizability (PR) condition for linear quantum systems. We consider open quantum systems whose dynamic variables satisfy the canonical commutation relations of an open quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to physical quantum systems, these QSDEs must satisfy PR conditions. We provide a relatively simple proof that the PR condition is equivalent to the frequency domain $(J,J)$ -unitarity of the input–output transfer function and orthogonality of the feedthrough matrix of the system without the technical spectral assumptions required in previous work. We also show that the poles and transmission zeros associated with the transfer function of PR linear quantum systems are the mirror reflections of each other about the imaginary axis. An example is provided to illustrate the results.

Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

Abstract: This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicable to recursive computation of such moments

Quantum Filtering for a Qubit System Subject to Classical Disturbances

Abstract: In this paper, we consider the filtering problem for a hybrid system where a quantum qubit system is disturbed by a classical signal. The quantum filtering theory, which is based on quantum probability theory, can not be directly applied to a hybrid system where a classical stochastic process is also needed in describing the system dynamics. An optical cavity system is employed to model the classical disturbance. By designing the parameters of the auxiliary cavity system, the expectation of the quadrature operator of the cavity shares the same dynamics with the classical signal. With this correspondence guaranteed, one can obtain the real time expectation of the classical signal. The quantum concatenation product is adopted to describe the quantum system which contains both the qubit subsystem and the cavity subsystem. A stochastic master equation, which provides estimates for the quantum state and the classical signal, is given. To reduce the computational complexity, the quantum extended Kalman filter is also applied to this system.

Adaptive Target Scheme for Learning Control of Quantum Systems

Abstract: Developing efficient algorithms is an important task in the learning control of quantum systems, since most learning control problems for quantum systems involve a heavy requirement for computational resources. In this paper, we employ a learning control algorithm with an adaptive target state developed in the chemical physics community for several classes of quantum control problems. For these problems applied to some new quantum control tasks, we further demonstrate that the algorithm using an adaptive target state can be more efficient than traditional learning control algorithms using a fixed target state. In the algorithm, the target state is updated according to the renormalized fragmentary yield in the desired region (or subspace) throughout the learning iterations. The adaptive target scheme is applied to three significant quantum control tasks, including the slow collision of a sodium cation and an iodine anion, the orientation of a LiH molecule, and population transfer between subspaces. Shaped laser pulses are obtained using the learning control algorithm, and numerical results are presented to demonstrate the advantages of the adaptive target scheme over the algorithm using a fixed target state. The adaptive target scheme is especially useful for learning control problems of quantum systems where the target state is not unique or known.

Optimal identification of non-Markovian environments for spin chains.

Abstract: Correlations of an environment are crucial for the dynamics of non-Markovian quantum systems, which may not be known in advance. In this paper, we propose a gradient algorithm for identifying the correlations in terms of time-varying damping rate functions in a time-convolution-less master equation for spin chains. By measuring time trace observables of the system, the identification procedure can be formulated as an optimization problem. The gradient algorithm is designed based on a calculation of the derivative of an objective function with respect to the damping rate functions, whose effectiveness is shown in a comparison to a differential approach for a two-qubit spin chain.

A Systems Theory Approach to the Synthesis of Minimum Noise Phase-Insensitive Quantum Amplifiers

Abstract: We present a systems theory approach to the proof of a result bounding the required level of added quantum noise in a phase-insensitive quantum amplifier We also present a synthesis procedure for constructing a quantum optical phase-insensitive quantum amplifier which adds the minimum level of quantum noise and achieves a required gain and bandwidth This synthesis procedure is based on a singularly perturbed quantum system and leads to an amplifier involving two squeezers and two beamsplitters

A derivation of moment evolution equations for linear open quantum systems

Abstract: Given a linear open quantum system which is described by a Lindblad master equation, we detail the calculation of the moment evolution equations from this master equation. We stress that the moment evolution equations are well-known, but their explicit derivation from the master equation cannot be found in the literature to the best of our knowledge, and so we provide this derivation for the interested reader.

Planar Cooperative Extremum Seeking with Guaranteed Convergence Using A Three-Robot Formation

Abstract: In this paper, a combined formation acquisition and cooperative extremum seeking control scheme is proposed for a team of three robots moving on a plane. The extremum seeking task is to find the maximizer of an unknown two-dimensional function on the plane. The function represents the signal strength field due to a source located at the maximizer, and is assumed to be locally concave around the maximizer and monotonically decreasing in distance to the source location. Taylor expansions of the field function at the location of a particular lead robot and the maximizer are used together with a gradient estimator based on signal strength measurements of the robots to design and analyze the proposed control scheme. The proposed scheme is proven to exponentially and simultaneously (i) acquire the specified geometric formation and (ii) drive the lead robot to a specified neighbourhood disk around the maximizer, whose radius depends on the specified desired formation size as well as the norm bounds of the Hessian of the field function. The performance of the proposed control scheme is evaluated using a set of simulation experiments.

Negative Imaginary State Feedback Control with a Prescribed Degree of Stability

Abstract: This paper presents a method for the synthesis of negative imaginary closed-loop systems with a prescribed degree of stability under the assumption of full state feedback. The approach extends existing work by using a perturbation method to ensure a closed-loop system that has both the negative imaginary property and a prescribed degree of stability. This approach involves the real Schur decomposition of a matrix followed by the solution to two Lyapunov equations which provides computational advantages over alternate state feedback synthesis techniques. Also, some counterexamples are presented which clarify the perturbation properties of strictly negative imaginary systems. Finally, an illustrative example demonstrates the approach.

Planar Cooperative Extremum Seeking with Guaranteed Convergence Using A Three-Robot Formation.

Abstract: In this paper, a combined formation acquisition and cooperative extremum seeking control scheme is proposed for a team of three robots moving on a plane The extremum seeking task is to find the maximizer of an unknown two-dimensional function on the plane The function represents the signal strength field due to a source located at maximizer, and is assumed to be locally concave around maximizer and monotonically decreasing in distance to the source location Taylor expansions of the field function at the location of a particular lead robot and the maximizer are used together with a gradient estimator based on signal strength measurements of the robots to design and analyze the proposed control scheme The proposed scheme is proven to exponentially and simultaneously (i) acquire the specified geometric formation and (ii) drive the lead robot to a specified neighborhood disk around maximizer, whose radius depends on the specified desired formation size as well as the norm bounds of the Hessian of the field function The performance of the proposed control scheme is evaluated using a set of simulation experiments

Direct coupling coherent quantum observers with discounted mean square performance criteria and penalized back-action.

Abstract: This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant-observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples.

Controller Synthesis to Render a Closed Loop Transfer Function Strongly Strictly Negative Imaginary

Abstract: In this paper, we show that there exists an alternative transformation from the class of negative imaginary to the class of positive real systems. We use this to offer a solution to the problem of designing a controller such that the closed loop is strongly strictly negative imaginary and its associated linear fractional interconnection is internally stable.

A Systems Theory Approach to the Synthesis of Minimum Noise Phase-Insensitive Quantum Amplifiers

Abstract: We present a systems theory approach to the proof of a result bounding the required level of added quantum noise in a phase-insensitive quantum amplifier. We also present a synthesis procedure for constructing a quantum optical phase-insensitive quantum amplifier which adds the minimum level of quantum noise and achieves a required gain and bandwidth. This synthesis procedure is based on a singularly perturbed quantum system and leads to an amplifier involving two squeezers and two beamsplitters.

Structural characterization of linear quantum systems with application to back-action evading measurement

Abstract: The purpose of this paper is to study the structure of quantum linear systems in terms of their Kalman canonical form, which was proposed in a recent paper \cite{ZGPG18}. The spectral structure of quantum linear systems is explored, which indicates that a quantum linear system is both controllable and observable provided that it is Hurwitz stable. A new parameterization method for quantum linear systems is proposed. This parameterization is designed for the Kalman canonical form directly. Consequently, the parameters involved are in a blockwise form in correspondence with the blockwise structure of the Kalman canonical form. This parameter structure can be used to simplify various quantum control design problems. For example, necessary and sufficient conditions for the realization of quantum back-action evading (BAE) measurements are given in terms of these new parameters. Due to their blockwise nature, a small number of parameters are required for realizing quantum BAE measurements.

Negative Imaginary State Feedback Control with a Prescribed Degree of Stability

Abstract: This paper presents a method for the synthesis of negative imaginary closed-loop systems with a prescribed degree of stability under the assumption of full state feedback. The approach extends existing work by using a perturbation method to ensure a closed-loop system that has both the negative imaginary property and a prescribed degree of stability. This approach involves the real Schur decomposition of a matrix followed by the solution to two Lyapunov equations which provides computational advantages over alternate state feedback synthesis techniques. Also, some counterexamples are presented which clarify the perturbation properties of strictly negative imaginary systems.

Distributed Obstacle and Multi-Robot Collision Avoidance in Uncertain Environments.

Abstract: This paper tackles the distributed leader-follower (L-F) control problem for heterogeneous mobile robots in unknown environments requiring obstacle avoidance, inter-robot collision avoidance, and reliable robot communications. To prevent an inter-robot collision, we employ a virtual propulsive force between robots. For obstacle avoidance, we present a novel distributed Negative-Imaginary (NI) variant formation tracking control approach and a dynamic network topology methodology which allows the formation to change its shape and the robot to switch their roles. In the case of communication or sensor loss, a UAV, controlled by a Strictly-Negative-Imaginary (SNI) controller with good wind resistance characteristics, is utilized to track the position of the UGV formation using its camera. Simulations and indoor experiments have been conducted to validate the proposed methods.

Dissipative Linear Stochastic Hamiltonian Systems

Abstract: This paper is concerned with stochastic Hamiltonian systems which model a class of open dynamical systems subject to random external forces. Their dynamics are governed by Ito stochastic differential equations whose structure is specified by a Hamiltonian, viscous damping parameters and system-environment coupling functions. We consider energy balance relations for such systems with an emphasis on linear stochastic Hamiltonian (LSH) systems with quadratic Hamiltonians and linear coupling. For LSH systems, we also discuss stability conditions, the structure of the invariant measure and its relation with stochastic versions of the virial theorem. Using Lyapunov functions, organised as deformed Hamiltonians, dissipation relations are also considered for LSH systems driven by statistically uncertain external forces. An application of these results to feedback connections of LSH systems is outlined.