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The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales as $O(1/N) $ for both QST and QDT, where $N $ denotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples., Comment: 36 pages, 6 figures

In this paper, we develop an online optimization algorithm for solving a class of nonconvex optimization problems with a linearly varying optimal point. The global convergence of the algorithm is guaranteed using the circle criterion for the class of functions whose gradient is bounded within a sector. Also, we show that the corresponding Lur\'e-type nonlinear system involves a double integrator, which demonstrates its ability to track a linearly varying optimal point with zero steady-state error. The algorithm is applied to solving a time-of-arrival based localization problem with constant velocity and the results show that the algorithm is able to estimate the source location with zero steady-state error., Comment: 8 pages, 7 figures, submitted to 2025 American Control Conference

In this paper, we quantify voltage and current phasor-based measurement requirements for the unique estimation of the electric grid topology and admittance parameters. Our approach is underpinned by the concept of a rigidity matrix that has been extensively studied in graph rigidity theory. Specifically, we show that the rank of the rigidity matrix is the same as that of a voltage coefficient matrix in a corresponding electric power system. Accordingly, we show that there is a minimum number of measurements required to uniquely estimate the admittance matrix and corresponding grid topology. By means of a numerical example on the IEEE 4-node radial network, we demonstrate that our approach is suitable for applications in electric power grids.

This paper is concerned with open quantum harmonic oscillators with position-momentum system variables, whose internal dynamics and interaction with the environment are governed by linear quantum stochastic differential equations. A recently proposed approach to such systems as Heisenberg picture quantum memories exploits their ability to approximately retain initial conditions over a decoherence horizon. Using the quantum memory decoherence time defined previously in terms of a fidelity threshold on a weighted mean-square deviation of the system variables from their initial values, we apply this approach to a partially isolated subsystem of the oscillator, which is not directly affected by the external fields. The partial isolation leads to an appropriate system decomposition and a qualitatively different short-horizon asymptotic behaviour of the deviation, which yields a longer decoherence time in the high-fidelity limit. The resulting approximate decoherence time maximization over the energy parameters for improving the quantum memory performance is discussed for a coherent feedback interconnection of such systems., Comment: 9 pages, 3 figures, submitted to ANZCC 2025

The noncommutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which, in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. This paper aims to study this structure by investigating the zeros and poles of linear quantum systems. Firstly, it is shown that -s_0 is a transmission zero if and only if s_0 is a pole of the transfer function, and -s_0 is an invariant zero if and only if s_0 is an eigenvalue of the A-matrix, of a linear quantum system. Moreover, s_0 is an output-decoupling zero if and only if -s_0 is an input-decoupling zero. Secondly, based on these zero-pole correspondences, we show that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Two types of stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated; in particular, the fundamental tradeoff between ideal input squeezing and system robustness is studied on the basis of system sensitivity analysis., Comment: 16 pages, 2 figures, comments are welcome. arXiv admin note: text overlap with arXiv:2408.03177

We propose a discrete-time integral resonant control (IRC) approach for negative imaginary (NI) systems, which overcomes several limitations of continuous-time IRC. We show that a discrete-time IRC has a step-advanced negative imaginary property. A zero-order hold-sampled NI system can be asymptotically stabilized using a discrete-time IRC with suitable parameters. A hardware experiment is conducted where a high-speed flexure-guided nanopositioner is efficiently damped using the proposed discrete-time IRC with the discrete-time controller being implemented in FPGA hardware at the sampling rate of 1.25 MHz., Comment: 10 pages, 10 figures

In the transition to achieving net zero emissions, it has been suggested that a substantial expansion of electric power grids will be necessary to support emerging renewable energy zones. In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Lur\'e-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time., Comment: 8 pages, 2 figures, 26th International Symposium on Mathematical Theory of Networks and Systems

This paper proposes a unified methodology for voltage regulation, frequency synchronization, and rotor angle control in power transmission systems considering a one-axis generator model with time-varying voltages. First, we formulate an output consensus problem with a passivity and negative-imaginary (NI) based control framework. We establish output consensus results for both networked passive systems and networked NI systems. Next, we apply the output consensus problem by controlling large-scale batteries co-located with synchronous generators -- using real-time voltage phasor measurements. By controlling the battery storage systems so as to dispatch real and reactive power, we enable simultaneous control of voltage, frequency, and power angle differences across a transmission network. Validation through numerical simulations on a four-area transmission network confirms the robustness of our unified control framework., Comment: 8 pages, 7 figures, the 63rd IEEE Conference on Decision and Control. arXiv admin note: text overlap with arXiv:2406.01206

We present a switching control strategy based on Lyapunov control for arbitrary state transitions in open qubit systems. With coherent vector representation, we propose a switching control strategy, which can prevent the state of the qubit from entering invariant sets and singular value sets, effectively driving the system ultimately to a sufficiently small neighborhood of target states. In comparison to existing works, this control strategy relaxes the strict constraints on system models imposed by special target states. Furthermore, we identify conditions under which the open qubit system achieves finite-time stability (FTS) and finite-time contractive stability (FTCS), respectively. This represents a critical improvement in quantum state transitions, especially considering the asymptotic stability of arbitrary target states is unattainable in open quantum systems. The effectiveness of our proposed method is convincingly demonstrated through its application in a qubit system affected by various types of decoherence, including amplitude, dephasing and polarization decoherence., Comment: 12 pages, 7 figures

A hybrid integrator-gain system (HIGS) is a control element that switches between an integrator and a gain, which overcomes some inherent limitations of linear controllers. In this paper, we consider using discrete-time HIGS controllers for the digital control of negative imaginary (NI) systems. We show that the discrete-time HIGS themselves are step-advanced negative imaginary systems. For a minimal linear NI system, there always exists a HIGS controller that can asymptotically stablize it. An illustrative example is provided, where we use the proposed HIGS control method to stabilize a discrete-time mass-spring system., Comment: To appear in the 2024 European Control Conference. 7 pages, 3 figures

We introduce a hybrid control system called a hybrid integrator-gain system (HIGS) based integral resonant controller (IRC) to stabilize negative imaginary (NI) systems. A HIGS-based IRC has a similar structure to an IRC, with the integrator replaced by a HIGS. We show that a HIGS-based IRC is an NI system. Also, for a SISO NI system with a minimal realization, we show there exists a HIGS-based IRC such that their closed-loop interconnection is asymptotically stable. Also, we propose a proportional-integral-double-integral resonant controller and a HIGS-based proportional-integral-double-integral resonant controller, and we show that both of them can be applied to asymptotically stabilize an NI system. An example is provided to illustrate the proposed results., Comment: 9 pages, 9 figures. The 63rd IEEE Conference on Decision and Control (CDC 2024)

Quantum process tomography is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose a two-stage solution for both trace-preserving and non-trace-preserving quantum process tomography. Utilizing a tensor structure, our algorithm exhibits a computational complexity of $O(MLd^2)$ where $d$ is the dimension of the quantum system and $ M $, $ L $ represent the numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number. Numerical examples and testing on IBM quantum devices are presented to demonstrate the performance and efficiency of our algorithm., Comment: 41 pages, 7 figures

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., Comment: 22 pages, 3 figures

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.

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., Comment: 22 pages, 2 figures, submitted for publication. Comments are welcome

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case., Comment: 8 pages, 4 figures. 26th International Symposium on Mathematical Theory of Networks and 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., Comment: 8 pages, 5 figures, European Control Conference

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., Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:2310.17232

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., Comment: 10 pages, 3 figures

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., Comment: 13 pages, 9 figures. Accepted for publication as a Full Paper in the IEEE Transactions on Control Systems Technology (TCST)

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., Comment: 8 pages, 7 figures

One of the fundamental tasks in information theory is the compression of information. To achieve this in the quantum domain, quantum autoencoders that aim to compress quantum states to low-dimensional ones have been proposed. When taking a pure state as the reference state, there exists an upper bound for the encoding fidelity. This bound limits the compression rate for high-rank states that have high entropy. To overcome the entropy inconsistency between the initial states and the reconstructed states, we allow the reference state to be a mixed state. A new cost function that combines the encoding fidelity and the quantum mutual information is proposed for compressing general input states. In particular, we consider the reference states to be a mixture of maximally mixed states and pure states. To achieve efficient compression for different states, two strategies for setting the ratio of mixedness (in the mixture of maximally mixed states and pure states) are provided based on prior knowledge about quantum states or observations obtained from the training process. Numerical results on thermal states of the transverse-field Ising model, Werner states, and maximally mixed states blended with pure states illustrate the effectiveness of the proposed method. In addition, quantum autoencoders using mixed reference states are experimentally implemented on IBM Quantum devices to compress and reconstruct thermal states and Werner states.

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.

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., Comment: 29 pages, 3 figures

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.

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.

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., Comment: 8 pages, IFAC 2023. arXiv admin note: text overlap with arXiv:2303.06861

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., Comment: 6 pages, 5 figures

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., Comment: 7 pages, 4 figures. Full archive version for the paper of the same title to appear in the proceedings of IFAC World Congress 2023

This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associated with the classical separation principle. This leads to a quadratic constraint on the controller gain matrices, which is formulated in the framework of a swapping transformation for the conjugate positions and momenta in the canonical representation of the controller variables. For the class of coherent quantum controllers with the Luenberger dynamics, we obtain first-order necessary conditions of optimality in the form of algebraic equations, involving a matrix-valued Lagrange multiplier., Comment: 8 pages, 1 figure

In this paper, we consider dynamic multi-agent systems (MAS) for decentralized resource allocation. The MAS operates at a competitive equilibrium to ensure supply and demand are balanced. First, we investigate the MAS over a finite horizon. The utility functions of agents are parameterized to incorporate individual preferences. We shape individual preferences through a set of utility functions to guarantee the resource price at a competitive equilibrium remains socially acceptable, i.e., the price is upper-bounded by an affordability threshold. We show this problem is solvable at the conceptual level. Next, we consider quadratic MAS and formulate the associated social shaping problem as a multi-agent linear quadratic regulator (LQR) problem which enables us to propose explicit utility sets using quadratic programming and dynamic programming. Then, a numerical algorithm is presented for calculating a tight range of the preference function parameters which guarantees a socially accepted price. We investigate the properties of a competitive equilibrium over an infinite horizon. Considering general utility functions, we show that under feasibility assumptions, any competitive equilibrium maximizes the social welfare. Then, we prove that for sufficiently small initial conditions, the social welfare maximization solution constitutes a competitive equilibrium with zero price. We also prove for general feasible initial conditions, there exists a time instant after which the optimal price, corresponding to a competitive equilibrium, becomes zero. Finally, we specifically focus on quadratic MAS and propose explicit results., Comment: arXiv admin note: substantial text overlap with arXiv:2209.04621

This paper considers a class of open quantum systems with an algebraic structure of dynamic variables, including the Pauli matrices for finite-level systems as a particular case. 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 in the form of a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This quasilinearity leads to a tractable evolution of the two-point commutator matrix of the system variables (and their multi-point mixed moments in the case of vacuum input fields) involving time-ordered operator exponentials. The resulting exponential decay in the two-point commutation relations is a manifestation of quantum decoherence, caused by the dissipative system-field interaction and making the system lose specific unitary dynamics features which it would have in isolation from the environment. We quantify the decoherence in terms of the rate of the commutation relations decay and apply system theoretic and matrix analytic techniques, such as algebraic Lyapunov inequalities and spectrum perturbation results, to the study of the asymptotic behaviour of the related Lyapunov exponents in the presence of a small scaling parameter in the system-field coupling. These findings are illustrated for finite-level quantum systems (and their interconnections through a direct energy coupling) with multichannel external fields and the Pauli matrices as internal variables., Comment: 41 pages, 1 figure

This paper studies self-sustained dynamic multiagent systems (MAS) for decentralized resource allocation operating at a competitive equilibrium over a finite horizon. The utility of resource consumption, along with the income from resource exchange, forms each agent's payoff which is aimed to be maximized. Each utility function is parameterized by individual preferences which can be designed by agents independently. By shaping these preferences and proposing a set of utility functions, we can guarantee that the optimal resource price at the competitive equilibrium always remains socially acceptable, i.e., it never violates a given threshold that indicates affordability. First, we show this problem is solvable at the conceptual level under some convexity assumptions. Then, as a benchmark case, we consider quadratic MAS and formulate the associated social shaping problem as a multi-agent LQR problem which enables us to propose explicit utility sets using quadratic programming and dynamic programming. Finally, a numerical algorithm is presented for calculating the range of the preference function parameters which guarantee a socially accepted price. Some illustrative examples are given to examine the effectiveness of the proposed methods.

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., Comment: This paper was presented at the 61st IEEE Conference on Decision and Control (CDC), 2022. A short version was published in the proceedings of the conference

This paper gives a J-spectral factorization condition for the implementation of a strictly proper transfer function matrix as a physically realizable quantum system using only direct feedthrough quantum noise. A necessary frequency response condition is also presented. Examples are included to illustrate the main results.

This paper is concerned with multimode open quantum harmonic oscillators (OQHOs), described by linear quantum stochastic differential equations with multichannel external bosonic fields. We consider the exponentially fast decay in the two-point commutator matrix of the system variables as a manifestation of quantum decoherence. Such dissipative effects are caused by the interaction of the system with its environment and lead to a loss of specific features of the unitary evolution which the system would have in the case of isolated dynamics. These features are exploited as nonclassical resources in quantum computation and quantum information processing technologies. A system-theoretic definition of decoherence time in terms of the commutator matrix decay is discussed, and an upper bound for it is provided using algebraic Lyapunov inequalities. Employing spectrum perturbation techniques, we investigate the asymptotic behaviour of a related Lyapunov exponent for the oscillator when the system-field coupling is specified by a small coupling strength parameter and a given coupling shape matrix. The invariant quantum state of the system, driven by vacuum fields, in the weak-coupling limit is also studied. We illustrate the results for one- and two-mode oscillators with multichannel external fields and outline their application to a decoherence control problem for a feedback interconnection of OQHOs., Comment: 21 pages, 2 figures

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., Comment: 19 pages, 10 figures

In this paper, we investigate the necessary and sufficient conditions under which a class of nonlinear systems are state feedback equivalent to nonlinear negative imaginary (NI) systems with positive definite storage functions. The nonlinear systems of interest have a normal form of relative degree less than or equal to two. The nonlinearity of the system is restricted with respect to a subset of the state variables, which are the state variables that have external dynamics. Under mild assumptions, such systems are state feedback equivalent to nonlinear NI systems and nonlinear output strictly negative imaginary (OSNI) systems if and only if they are weakly minimum phase. Such a state feedback control approach can also asymptotically stabilize the systems in question against nonlinear OSNI system uncertainties. A numerical example is provided to show the process of the state feedback equivalence control and stabilization of uncertain systems., Comment: 8 pages, 2 figures

This paper is concerned with linear stochastic systems whose output is a stationary Gaussian random process related by an integral operator to a standard Wiener process at the input. We consider a performance criterion which involves the trace of an analytic function of the spectral density of the output process. This class of "covariance-analytic" cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the presence of the "cost-shaping" analytic function, the performance criterion is related to higher-order Hardy-Schatten norms of the system transfer function. These norms have links with the asymptotic properties of cumulants of finite-horizon quadratic functionals of the system output and satisfy variational inequalities pertaining to system robustness to statistically uncertain inputs. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of annihilation and creation operators in quantum mechanics. The resulting computational procedure involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared $\mathcal{H}_2$-norm of an auxiliary cascaded system. These results are also compared with an alternative approach which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations., Comment: 33 pages, 5 figures, to be submitted to Mathematics of Control, Signals, and Systems

Quadrotors are one of the popular unmanned aerial vehicles (UAVs) due to their versatility and simple design. However, the tuning of gains for quadrotor flight controllers can be laborious, and accurately stable control of trajectories can be difficult to maintain under exogenous disturbances and uncertain system parameters. This paper introduces a novel robust and adaptive control synthesis methodology for a quadrotor robot's attitude and altitude stabilization. The developed method is based on the fuzzy reinforcement learning and Strictly Negative Imaginary (SNI) property. The first stage of our control approach is to transform a nonlinear quadrotor system into an equivalent Negative-Imaginary (NI) linear model by means of the feedback linearization (FL) technique. The second phase is to design a control scheme that adapts online the Strictly Negative Imaginary (SNI) controller gains via fuzzy Q-learning, inspired by biological learning. The proposed controller does not require any prior training. The performance of the designed controller is compared with that of a fixed-gain SNI controller, a fuzzy-SNI controller, and a conventional PID controller in a series of numerical simulations. Furthermore, the stability of the proposed controller and the adaptive laws are proofed using the NI theorem.

This paper provides a state feedback stabilization approach for nonlinear systems of relative degree less than or equal to two by rendering them nonlinear negative imaginary (NI) systems. Conditions are provided under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly negative imaginary (OSNI) system. Roughly speaking, an affine nonlinear system that has a normal form with relative degree less than or equal to two, after possible output transformation, can be rendered nonlinear NI and nonlinear OSNI. In addition, if the internal dynamics of the normal form are input-to-state stable, then there exists a state feedback input that stabilizes the system. This stabilization result is then extended to achieve stability for systems with a nonlinear NI uncertainty., Comment: 10 pages, 2 figures

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., Comment: Shorter version published in Asian Journal of Control, Special Edition, 2023

In this letter we revisit the famous heavy ball method and study its global convergence for a class of non-convex problems with sector-bounded gradient. We characterize the parameters that render the method globally convergent and yield the best $R$-convergence factor. We show that for this family of functions, this convergence factor is superior to the factor obtained from the triple momentum method., Comment: 6 pages, 4 figures, to appear in CSS Letters

This paper is concerned with networks of identical linear quantum stochastic systems which interact with each other and external bosonic fields in a translation invariant fashion. The systems are associated with sites of a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). The block Toeplitz coefficients of these QSDEs are specified by the energy and coupling matrices which quantify the Hamiltonian and coupling operators for the component systems. We discuss the invariant Gaussian quantum state of the network when it satisfies a stability condition and is driven by statistically independent vacuum fields. A quadratic-exponential functional (QEF) is considered as a risk-sensitive performance criterion for a finite fragment of the network over a bounded time interval. This functional involves a quadratic function of dynamic variables of the component systems with a block Toeplitz weighting matrix. Assuming the invariant state, we study the spatio-temporal asymptotic rate of the QEF per unit time and per lattice site in the thermodynamic limit of unboundedly growing time horizons and fragments of the lattice. A spatio-temporal frequency-domain formula is obtained for the QEF rate in terms of two spectral functions associated with the real and imaginary parts of the invariant quantum covariance kernel of the network variables. A homotopy method and asymptotic expansions for evaluating the QEF rate are also discussed., Comment: 30 pages, 3 figures, to be submitted to Infinite Dimensional Analysis, Quantum Probability and Related Topics

This paper is concerned with infinite-horizon growth rates of quadratic-exponential functionals (QEFs) for linear quantum stochastic systems driven by multichannel bosonic fields. Such risk-sensitive performance criteria impose an exponential penalty on the integral of a quadratic function of the system variables, and their minimization improves robustness properties of the system with respect to quantum statistical uncertainties and makes its behaviour more conservative in terms of tail distributions. We use a frequency-domain representation of the QEF growth rate for the invariant Gaussian quantum state of the system with vacuum input fields in order to compute it in state space. The QEF rate is related to a similar functional for a classical stationary Gaussian random process generated by an infinite cascade of linear systems. A truncation of this shaping filter allows the QEF rate to be computed with any accuracy by solving a recurrent sequence of algebraic Lyapunov equations together with an algebraic Riccati equation. The state-space computation of the QEF rate and its comparison with the frequency-domain results are demonstrated by a numerical example for an open quantum harmonic oscillator., Comment: 29 pages, 7 figures, to be submitted to the Journal of the Franklin Institute

The development of estimation and control theories for quantum systems is a fundamental task for practical quantum technology. This vision article presents a brief introduction to challenging problems and potential opportunities in the emerging areas of quantum estimation, control and learning. The topics cover quantum state estimation, quantum parameter identification, quantum filtering, quantum open-loop control, quantum feedback control, machine learning for estimation and control of quantum systems, and quantum machine learning., Comment: 11 pages, vision paper from the perspective of systems and control

Negative imaginary (NI) systems theory is a well-established system theoretic framework for analysis and design of linear-time-invariant (LTI) control systems. In this paper, we aim to generalize negative imaginary systems theory to a class of nonlinear systems. Based on the time domain interpretation of the NI property for LTI systems, a formal definition in terms of a dissipativity with an appropriate work rate will be used to define the nonlinear negative imaginary (NNI) property for a general nonlinear system. Mechanical systems with force actuators and position sensors are nonlinear negative imaginary according to this new definition. Using Lyapunov stability theory, we seek to establish a nonlinear generalization of the NI robust stability result for positive feedback interconnections of NNI systems. An example of a nonlinear mass-spring-damper system with force as input and displacement of the mass as output will be presented to illustrate the applicability of the NNI stability result. Furthermore, the case of NI systems with free motion will be investigated in the nonlinear domain based on the dissipativity framework of NNI systems.

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, which are applied to three measurement-constrained 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.

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., Comment: 11 pages