Hamiltonian Control Research Articles

Stabilizing the coupled orbit–attitude dynamics of a rigid body in a J2 gravity field using Hamiltonian structure

The gravitationally coupled orbit–attitude dynamics of a rigid body in a J2 gravity field is a high-precision model for the spacecraft in the close proximity of a small spheroid celestial body, since the gravitational orbit–attitude coupling of the spacecraft is naturally taken into account in this model. A Hamiltonian structure-based feedback control law is proposed to stabilize relative equilibria of the rigid body in the J2 gravity field. The proposed stabilization control law is consisted of three parts: potential shaping, momentum control, and energy control. The potential shaping is used to modify the gravitational potential artificially so that the relative equilibrium is a minimum of the modified Hamiltonian. It is shown that the unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then the momentum control leads the motion to the same invariant manifold with the relative equilibrium, which is actually a momentum level set. The energy control introduces energy dissipation into the system and the motion will asymptotically converge to the minimum of the modified Hamiltonian on the invariant manifold, i.e., the relative equilibrium. The feasibility of the proposed stabilization control law is validated through numerical simulations. The control law can be implemented by the attitude control system and low-thrust engines onboard and the fuel consumption is reasonable. This Hamiltonian structure-based stabilization approach is applicable to non-canonical Hamiltonian systems commonly existing in the astrodynamics. Since the natural dynamical behaviors of the system is fully utilized, the proposed control law is very simple and is easy to implement autonomously with little computation in the space applications.

Time-optimal navigation through quantum wind

The quantum navigation problem of finding the time-optimal control Hamiltonian that transports a given initial state to a target state through quantum wind, that is, under the influence of external fields or potentials, is analyzed. By lifting the problem from the state space to the space of unitary gates realizing the required task, we are able to deduce the form of the solution to the problem by deriving a universal quantum speed limit. The expression thus obtained indicates that further simplifications of this apparently difficult problem are possible if we switch to the interaction picture of quantum mechanics. A complete solution to the navigation problem for an arbitrary quantum system is then obtained, and the behaviour of the solution is illustrated in the case of a two-level system.

A port-Hamiltonian formulation of a 2D boundary controlled acoustic system

This paper deals with the port Hamiltonian formulation of a 2D boundary controlled acoustic system. The system under consideration consits of an acoustic wave traveling in a tube equipped with a network of microphones/loudspeakers. The purpose of this smart skin is to damp the acoustic wave and reduce its effect at the output of the tube. It is first commented how the original 3D system can be reduced to a 2D system by considering symmetries. Then, the boundary port variables associated with the wave equation are parametrized in order to define a Dirac structure in two dimensions, compatible with the interconnection at the boundaries with the actuation system. The overall system (wave+actuators/sensors) is finally expressed as a port Hamiltonian control system and a first stabilizing distributed control law is proposed.

Controlling the coherence in a pure dephasing model for arbitrary prescibed time span

We present an open-loop unitary strategy to control the coherence in a pure dephasing model (related to the phase-flip channel) that is able to recover, for whatever prescribed time span, the initial coherence at the end of the control process. The strategy's key idea is to steer the quantum state to the subset of invariant states and keep it there the necessary time, using a fine tuned control Hamiltonian.

Model discrimination for dephasing two-level systems

The problem of model discriminability and parameter identifiability for dephasing two-level systems subject to Hamiltonian control is studied. Analytic solutions of the Bloch equations are used to derive explicit expressions for observables as functions of time for different models. This information is used to give criteria for model discrimination and parameter estimation based on simple experimental paradigms.

Construction of arbitrary robust one-qubit operations using planar geometry

We show how to construct an arbitrary robust one-qubit unitary operation with a control Hamiltonian of $A_x(t) \sigma_x + A_y(t) \sigma_y$, where $\sigma_i$ is a Pauli matrix and $A_i(t)$ is piecewise constant. Our method, based on planar geometry, admits a simple and intuitive interpretation. Furthermore, the total execution time and the number of elementary gates of the obtained sequence are comparable to those of the shortest known concatenated composite pulses.

A dynamic systems approach to fermions and their relation to spins

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental set-up. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given set-up can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds. PACS Codes: 03.67.Ac, 02.30.Yy, 75.10.Pq.

A Convergent Control Strategy for Quantum Systems

AbstractIn the interaction picture, a sufficient and necessary condition that guarantees the convergence of closed quantum control system is proposed in this paper. Theoretical derivation and the proof show that it is possible to achieve the convergence to the target state by constructing an observable operator in an energy function and selecting control Hamiltonians. Numerical simulation experiments on a four-level system verify the effectiveness of the proposed control strategy.

Controlling several atoms in a cavity

We treat control of several two-level atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinite-dimensional system algebras. Hence we address problems arising with infinite-dimensional Lie algebras and those of unbounded operators. For the models considered, these problems can be solved by splitting the set of control Hamiltonians into two subsets: the first obeys an Abelian symmetry and can be treated in terms of infinite-dimensional Lie algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry, and its discussion introduces new arguments. Yet, full controllability can be achieved in a strong sense: e.g., in a time dependent Jaynes–Cummings model we show that, by tuning coupling constants appropriately, every unitary of the coupled system (atoms and cavity) can be approximated with arbitrarily small error.

Hamiltonian quantum simulation with bounded-strength controls

We propose dynamical control schemes for Hamiltonian simulation in many-body quantum systems that avoid instantaneous control operations and rely solely on realistic bounded-strength control Hamiltonians. Each simulation protocol consists of periodic repetitions of a basic control block, constructed as a modification of an ‘Eulerian decoupling cycle,’ that would otherwise implement a trivial (zero) target Hamiltonian. For an open quantum system coupled to an uncontrollable environment, our approach may be employed to engineer an effective evolution that simulates a target Hamiltonian on the system while suppressing unwanted decoherence to the leading order, thereby allowing for dynamically corrected simulation. We present illustrative applications to both closed- and open-system simulation settings, with emphasis on simulation of non-local (two-body) Hamiltonians using only local (one-body) controls. In particular, we provide simulation schemes applicable to Heisenberg-coupled spin chains exposed to general linear decoherence, and show how to simulate Kitaevʼs honeycomb lattice Hamiltonian starting from Ising-coupled qubits, as potentially relevant to the dynamical generation of a topologically protected quantum memory. Additional implications for quantum information processing are discussed.

Steady-state entanglement by engineered quasi-local Markovian dissipation

We characterize time-independent Markovian dynamics that drive a finite-dimensional multipartite quantum system into a target (pure) entangled steady state, subject to physical locality constraints. New control schemes are introduced in situations where the desired stabilization task {\em cannot} be attained solely based on quasi-local dissipative means, as considered in previous analysis. The new schemes either allow for Hamiltonian control or, if the latter is not an option, suitably restrict the set of admissible initial states. In both cases, we provide explicit algorithms for constructing a Markovian master equation that achieves the intended objective and show how this genuinely extends the manifold of stabilizable states. In particular, we present dissipative quasi-local control protocols for deterministically engineering multipartite GHZ ``cat'' states and W states on $n$ qubits. For GHZ states, we show that no scalable procedure exists for achieving stabilization from arbitrary initial states, whereas this is possible for a target W state by a suitable combination of a two-body Hamiltonian and dissipators. Interestingly, for both entanglement classes, we show that quasi-local stabilization may be {\em scalably} achieved conditional to initialization of the system in a large, appropriately chosen subspace.

The generalized Hamiltonian model for the shafting transient analysis of the hydro turbine generating sets

Traditional rotor dynamics mainly focuses on the steady- state behavior of the rotor and shafting. However, for systems such as hydro turbine generating sets (HTGS) where the control and regulation is frequently applied, the shafting safety and stabilization in transient state is then a key factor. The shafting transient state inevitably involves multiparameter domain, multifield coupling, and coupling dynamics. In this paper, the relative value form of the Lagrange function and its equations have been established by defining the base value system of the shafting. Taking the rotation angle and the angular speed of the shafting as a link, the shafting lateral vibration and generator equations are integrated into the framework of generalized Hamiltonian system. The generalized Hamiltonian control model is thus established. To make the model more general, additional forces of the shafting are taken as the input excitation in proposed model. The control system of the HTGS can be easily connected with the shafting model to form the whole simulation system of the HTGS. It is expected that this study will build a foundation for the coupling dynamics theory using the generalized Hamiltonian theory to investigate coupling dynamic mechanism among the shafting vibration, transient of hydro turbine generating sets, and additional forces of the shafting.

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.

Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity

The convergence of closed quantum systems in the degenerate cases to the arbitrary desired target state by using the quantum Lyapunov control based on the average value of an imaginary mechanical quantity is studied in this paper. On the basis of the existing methods which can only ensure the pure states and single-control Hamiltonian systems converge toward a set, we propose a control laws design to make the multi-control Hamiltonian systems, which can also converge from the arbitrary initial state to the arbitrary target state of the quantum systems whose internal Hamiltonian are not strongly regular or/and control Hamiltonians are not full connected. The degenerate cases' problems solved in this paper widely exist in the practical quantum systems, so it has the great significance in quantum systems control. People can make use of those conditions obtained to design a convergent controller for the quantum control system, which can instruct the experimental scientists to obtain high successful probability to the actual quantum systems control. This research work establishes a completed quantum Lyapunov control theory in closed quantum systems. The convergence of the control system is proved. How to make the convergence conditions to be satisfied is proved or analyzed. Finally, numerical simulations for a three level system in the degenrate case transfering from an initial eigenstate to a target superposition state are studied.

Quantum control with noisy fields: computational complexity versus sensitivity to noise

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise, which has to be suppressed to retain controllability. Can one design control fields such that the effect of noise is negligible on the time-scale of the transformation? This question is intimately related to the fundamental problem of a connection between the computational complexity of the control problem and the sensitivity of the controlled system to noise. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases with the dimension of the Hilbert space representation of the algebra. We find two types of control tasks, easy and hard. Easy tasks are characterized by a small variance of the evolving state with respect to the operators of the control operators. They are relatively immune to noise and the control field is easy to find. Hard tasks have a large variance, are sensitive to noise and the control field is hard to find. The influence of noise increases with the size of the system, which is measured by the scaling factor N of the largest weight of the representation. For fixed time and control field the ability to control degrades as O(N) for easy tasks and as O(N2) for hard tasks. As a consequence, even in the most favorable estimate, for large quantum systems, generic noise in the controls dominates for a typical class of target transformations, i.e. complete controllability is destroyed by noise.

Speeding up and slowing down the relaxation of a qubit by optimal control

We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As by-products of our analysis, we find that (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control can not speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control, it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.

Research on a Nonlinear Control Strategy for Three-Phase Voltage Sources PWM Rectifier with Resistive and Inductive Load

The mathematical model of three-phase voltage sources Pulse Width Modulation (PWM) rectifier is nonlinear, in view of the traditional linear control strategys weak disturbance-rejection ability with resistive and inductive load, a new passivity-based control strategy was proposed according to passivity-based control theory. Energy shaping method based on PCHD (Port Control Hamiltonian with Dissipation) model and the IDA-PBC (Interconnection and Damping Assignment Passivity Based Control) control algorithm is adopted to design passivity-based controller, which is able to make the energy function have the minimum value when rectifier is at the desired point, thus improving the stability and the load disturbance-rejection ability. Simulation results show that passivity-based control method can make this system possess the high-performance of robustness and dynamic.

Suboptimal Control Strategies for Finite-Time Nonlinear Processes with Input Constraints

Novel techniques for the optimization and control of finite-time processes in real-time are pursued. These are developed in the framework of the Hamiltonian optimal control. Two methods are designed. The first one constructs the reference control trajectory as an approximation of the optimal control via the Riccati equations in an adaptive fashion based on the solutions of a set of partial differential equations called the α and β matrices. These allow calculating the Riccati gain for a range of the duration of the process T and the final penalization S. The second method introduces input constraints to the general optimization formulation. The notions of linear matrix inequalities allow us to recuperate the Riccati gain as in the first method, but using an infinite horizon optimization method. Finally, the performance of the proposed strategies is illustrated through numerical simulations applied to a batch reactor and a penicillin fed-batch reactor.

Tailoring many-body entanglement through local control

We construct optimal time-local control pulses based on a multipartite entanglement measure as target functional. The underlying control Hamiltonians are derived in a purely algebraic fashion, and the resulting pulses drive a composite quantum system rapidly into that highly entangled state which can be created most efficiently for a given interaction mechanism, and which bears entanglement that is robust against decoherence. Moreover, it is shown that the control scheme is insensitive to experimental imperfections in first order.

Port-Controlled Hamiltonian Control of IM Based on Back-to-Back Converter

A method of port-controlled Hamiltonian (PCH) systems energy-shaping control is presented for the four-quadrant control problem of squirrel cage induction motor (IM) drive system. First, the PCH models of IM drive system are developed. Then, the PCH controllers are designed based on the control theory of closed-loop state error PCH system. The coordinated control of the grid-side and the motor-side are applied in Matlab/simulink, which the motor-side is started after the grid-side is achieved up to stable. The controlled system can be achieved that DC bus voltage controllable, unity power factor, energy bidirectional flow, IM run in the four-quadrant, etc.