Hamiltonian Control Research Articles

Dynamics in the controlled center manifolds by Hamiltonian structure-preserving stabilization

A two-dimensional hyperbolic Hamiltonian system can be linearly stabilized by a Hamiltonian structure-preserving controller. A linear symplectic transformation and the Lie series method can successfully normalize the expanded Hamiltonian function around a controlled stable equilibrium point, then the dynamics in the controlled center manifolds of which can be described by a Poincaré section. With the implement of the inverse transformation of the Lie series, the analytical results of the controlled manifolds can be obtained. Applying normalization and analytical calculation to planar solar sail three-body problem, we can get the normal form of the corresponding Hamiltonian function and trajectory around the chosen equilibrium point by analytical results. Finally, typical KAM theory is used to analyze the nonlinear stability of the controlled equilibrium point, and the stable region of the control gains are given by numerical calculation.

Dynamics and Control for Nonideal Solar Sails Around Artificial Lagrangian Points

Due to their lack of fuel consumption, solar sails have potential applications for deep space exploration. However, due to the optical degradation and deformation of such a sail, a nonideal model is preferable for investigating their dynamics and control when moving around an artificial Lagrangian point. An improved imperfectly reflecting sail model is constructed to compare with the ideal one. With a Hamiltonian structure-preserving controller adopted, the trajectory around an artificial Lagrangian point is generated for ideal solar sails. The original trajectory changes and migrates when the imperfectly reflecting sail model is used. Two improvement strategies are proposed to adjust the changed trajectory. One strategy is to adjust the input parameters of the actuator to compensate for the change in acceleration. The other strategy is to design a new controller to enhance the change and migration of the trajectories. The effectiveness of the two strategies for coping with imperfectly reflecting solar sails is verified via numerical simulations.

A deterministic method for PWR loading pattern optimization

A new deterministic approach to core loading optimization is presented. Selection of fuel enrichment and burnable poison (BP) arrangement in a pressurized water reactor (PWR) core are optimally determined subject to power peaking factor (PF) constraints. The deterministic approach is based on the method of Lagrange multipliers and the direct adjoining approach for the inequality power PF constraint. The optimality conditions are derived using calculus of variations resulting in the presence of jump conditions and a linear control Hamiltonian. The optimal control problems are addressed by simultaneously minimizing the Hamiltonian using the gradient method and performing Newton's method in different regions of the core to produce viable solutions for the controls. Without active control of the power profile in PWRs during depletion, our methodology is able to satisfy the power PF constraint at all times by proper loading selection at the beginning of cycle (BOC). This is achieved by promoting the proper burnup path forward in time during the calculation of the forward optimality conditions. This information is utilized in the adjoint burnup calculation, which acts to propagate the optimal control path back to the BOC during the calculation of the adjoint optimality conditions. The methodology has been developed into a new code DMCO (Deterministic Multi-Control Optimization) which can produce two-dimensional two-group depletion results in approximately seven minutes per optimization iteration on a personal computer. No application of heuristics is required and the code is capable of finding solutions from initial control distributions. A preliminary case study is presented for the Westinghouse AP600 first cycle fuel loading design which achieved an extended fuel cycle while satisfying power PF constraints.

Pulse design for multilevel systems by utilizing Lie transforms

We put forward a scheme to design pulses to manipulate multilevel systems with Lie transforms. A formula to reverse construct a control Hamiltonian is given and is applied in pulse design in the three- and four-level systems as examples. To demonstrate the validity of the scheme, we perform numerical simulations, which show the population transfers for cascaded three-level and $N$-type four-level Rydberg atoms can be completed successfully with high fidelities. Therefore, the scheme may benefit quantum information tasks based on multilevel systems.

Distributed Complementary Control Research of Wind Turbines in Two Offshore Wind Farms

In order to stabilize the fluctuation of wind power and maintain a stable power output, a complementary control idea is proposed. This idea aims to make the output power from two wind farms complement each other. This study proposes a distributed control strategy to solve the complementary control problem of wind turbines in two offshore wind farms on the basis of the Hamiltonian energy theory. The proposed control strategy not only ensures synchronization for wind turbines in the same farm but also keeps the combined output power of the two wind farms stable. First, through the Hamiltonian realization, the single-machine model of a wind turbine is transformed into a port-controlled Hamiltonian system with dissipation (PCHD). Subsequently, the Hamiltonian energy control law is developed on the basis of the energy-shaping method to adjust the Hamiltonian energy function. The complementary control of the two wind farms is designed to synchronize the wind turbines within an individual wind farm and keep the combined output of the two wind farms stable. Furthermore, the complementary control strategy is modified to address the communication delay between the two wind farms by incorporating time delay into the control problem. Finally, the effectiveness of the distributed complementary control has been verified via simulations.

Stabilizing a class of mixed states for stochastic quantum systems via switching control

The global stabilization of a class of mixed states for finite dimensional stochastic quantum systems with degenerate measurement operator is investigated in this paper. We construct a measurement operator and control Hamiltonian that make the target state one of the system equilibria. Based on the proposed Lyapunov function, a control law is designed following Lyapunov’s method to steer system state to the target mixed state from an initial state in the convergence domain, which is obtained through the analyses of invariant set based on LaSella’s invariance principle. When the initial state isn’t in the convergence domain, a constant control is used to steer the system state so that it enters the convergence domain in finite time. The constant control and the control designed by Lyapunov method compose a switching control strategy, which can steer system state to the target mixed state from any arbitrary state in the state space, i.e., the target mixed state is globally stable under the switching control. The convergence of the switching control is proved based on state sample trajectories. Moreover, the numerical experiments on a three dimensional stochastic quantum system are performed to demonstrate the effectiveness of switching control.

Fast state transfer in a Λ-system: a shortcut-to-adiabaticity approach to robust and resource optimized control

We propose an efficient strategy to find optimal control functions for state-to-state quantum control problems. Our procedure first chooses an input state trajectory, that can realize the desired transformation by adiabatic variation of the system Hamiltonian. The shortcut-to-adiabaticity formalism then provides a control Hamiltonian that realizes the reference trajectory exactly but on a finite time scale. As the final state is achieved with certainty, we define a cost functional that incorporates the resource requirements and a perturbative expression for robustness. We optimize this functional by systematically varying the reference trajectory. We demonstrate the method by application to population transfer in a laser driven three-level Λ-system, where we find solutions that are fast and robust against perturbations while maintaining a low peak laser power.

Multi-objective optimization in quantum parameter estimation

We investigate quantum parameter estimation based on linear and Kerr-type nonlinear controls in an open quantum system, and consider the dissipation rate as an unknown parameter. We show that while the precision of parameter estimation is improved, it usually introduces a significant deformation to the system state. Moreover, we propose a multi-objective model to optimize the two conflicting objectives: (1) maximizing the Fisher information, improving the parameter estimation precision, and (2) minimizing the deformation of the system state, which maintains its fidelity. Finally, simulations of a simplified {\epsilon}-constrained model demonstrate the feasibility of the Hamiltonian control in improving the precision of the quantum parameter estimation.

Port-Controlled Hamiltonian and Sliding Mode Control of Gantry Robot Based on Induction Motor Drives

A port-controlled Hamiltonian (PCH) control approach is presented to solve the position tracking problem of gantry robot based on induction motor (IM) drives. First, a robot model is established. Second, a PCH controller is designed to realize accurate position tracking of a gantry robot. For IM drives, it is convenient to choose a direct torque control strategy based on the sliding mode control, which overcomes the higher ripples of torque and flux. Third, a voltage reconstruction technique is introduced to calculate the stator voltage of the IM, which replaces the stator voltage measurement of the IM. Finally, the load torque observer is developed to estimate an unknown load torque. The asymptotic stability of the robot system is proved by the Lyapunov stability theory. Simulation results indicate that the system has excellent position tracking performances and load disturbance attenuation ability.

Hybrid Tracking Control of 2-DOF SCARA Robot via Port-Controlled Hamiltonian and Backstepping

In this paper, we propose a hybrid coordinated control method based on port-controlled Hamiltonian and backstepping to improve the position tracking performance for two degree of freedom SCARA robot. The port-controlled Hamiltonian (PCH) control is designated to ensure the stability of the system, and the backstepping control targets to improve the response speed of the system. Exponential function is used as a coordination function to achieve the coordinated control strategy to adapt to the position tracking control of 2-DOF SCARA robot. This hybrid coordination control system not only realizes a quick tracking control, but also improves the steady-state performance of the output signal. The simulation results show that when the external interference exists in the mechanical system of a 2-DOF SCARA robot, the hybrid tracking control system takes on the advantages of both methods, which shows good dynamic performance, good steady-state performance, and strong resistance to external interference.

Homogeneous Hamiltonian Control Systems Part I: Geometric Formulation

Contact geometry has been successfully employed for the geometric formulation and control of systems containing thermodynamic components. In this paper we elaborate on the geometric theory of symplectization of contact manifolds in order to lift contact control systems to Hamiltonian control systems with a Hamiltonian that is homogeneous in the co-state variables. This provides a new view on contact control systems as used in thermodynamics, and offers possibilities for unifying the theories of contact control systems, Hamiltonian input-output systems and port-Hamiltonian systems.

Homogeneous Hamiltonian Control Systems Part II: Application to thermodynamic systems

In this paper, we give several examples of the formulation of the dynamics of open physical systems in terms of homogeneous Hamiltonian control systems. Therefore we have systematically used Gibbs-Massieu equations in the entropy form for the definition of the Lagrangian submanifold associated with the thermodynamical properties of the system. Accordingly we have used the total entropy function of the system to define the generating function of this Lagrange submanifold and have used the canonical lifting of systems of balance equations of the remaining independent extensive variables. In the first part we consider two examples of thermodynamic systems: two cells exchanging a heat flow and the Continuous Stirred Tank Reactor (CSTR). In the second part, we consider an elementary dissipative mechanical, the mass-spring-damper system and then embed it into the gas-piston system.

Hybrid control of 2-DOF joint robot based on Port-Controlled Hamiltonian and PD algorithm

Aiming at the unsatisfactory performance of Two Degree-of-Freedom joint robot trajectory tracking control system which is controlled by one method, a hybrid control method that integrates Port-Controlled Hamiltonian and Proportional Differential algorithm is designed. The Port-Controlled Hamiltonian control is used to ensure the stability of the system, and Proportional Differential control is used to improve the response speed of the system. Exponential function is used as the hybrid function to achieve the coordinated control strategy, and to adapt to the error disturbance of Two Degree-of-Freedom joint robot. The control system not only realizes the fast-tracking control, but also raises the output signal of the robot to a higher precision. The simulation results show that when the modelling error exists in the mechanical system of Two Degree-of-Freedom joint robot, the hybrid control trajectory tracking system takes on both advantages of the two methods. It has good dynamic performance and steady-state performance. In addition, it can eliminate the error quickly.

Hamiltonian methods of modeling and control of AC microgrids with spinning machines and inverters

This paper presents a novel approach to the modeling and control of AC microgrids that contain spinning machines, power electronic inverters and energy storage devices. The inverters in the system can adjust their frequencies and power angles very quickly, so the modeling focuses on establishing a common reference frequency and angle in the microgrid based on the spinning machines. From this dynamic model, nonlinear Hamiltonian surface shaping and power flow control method is applied and shown to stabilize. From this approach the energy flow in the system is used to show the energy storage device requirements and limitations for the system. This paper first describes the model for a single bus AC microgrid with a Hamiltonian control, then extends this model and control to a more general class of multiple bus AC microgrids. Simulation results demonstrate the efficacy of the approach in stabilizing and optimization of the microgrid.

Influence of localization transition on dynamical properties for an extended Aubry–André–Harper model

We show the localization transition and its influence on two dynamical processes of an extended Aubry–André–Harper model with incommensurate on-site and hopping potentials. After specifying an extended Aubry–André–Harper model, we check the localization transition for all the eigenstates and eigenenergy band splitting behavior versus a system parameter. To check the influence of localization transition on dynamical processes, firstly, the slowly pumping of edge states are examined. The adiabatic pumping is suppressed in the localized region. Then by quantum Lyapunov control method in terms of different control Hamiltonians, we prepare an edge-localized state in the nonlocal region. The control effect is suppressed in the localized region compared to that in the nonlocal region. In the dynamical processes, the system acts as conductor for the excitation in the nonlocal region and insulator in the localized region. Then we adopt the occupation imbalance between even and odd sites and entropy to indicate the localization transition further. Finally, the experimental schemes based on cold atoms trapped in quasiperiodic optical lattice and coupled optical waveguide arrays are suggested.

Bounded Motions near Contact Binary Asteroids by Hamiltonian Structure-Preserving Control

This paper is devoted to stabilizing motions near a special type of equilibrium point of contact binary asteroids by Hamiltonian structure-preserving control. Aiming at noncollinear equilibrium points of the asteroid 1996 HW1, a Hamiltonian structure-preserving controller is built up using stable and unstable manifolds. Compared with traditional Hamiltonian structure-preserving controllers, it is capable of stabilizing the motion near a special kind of equilibrium point only using stable or unstable manifolds. The analytical solution of controlled motions is approximated by a nonlinear series using the method of variation of parameters. An optimization strategy is proposed including the numerical optimization of control gains by maximum-likelihood and the global optimization of initial deviation analytically. To achieve collision avoidance, the concept of an artificial potential function is introduced to the controller, which still has the capability to change equilibrium points’ topological type to elliptic without breaking the system’s Hamiltonian structure.

Speeding up adiabatic passage by adding Lyapunov control

We propose a scheme to speed up adiabatic passage by using Lyapunov control theory. This is a good choice to solve the problem that may emerge in Berry's transitionless quantum driving [M. V. Berry, J. Phys. A 42, 365303 (2009)]. That is, the extra couplings in the counterdiabatic driving Hamiltonian can be avoided by choosing the available control Hamiltonian in an actual physical system. As examples, we shorten the evolution time of adiabatic population transfer in a three-level system and the entanglement generation in a cavity quantum electrodynamics system. Moreover, the occupation of an intermediate state can be sharply suppressed by properly choosing the control Hamiltonian in the three-level system. The scheme can also be generalized to a complex system where the exact expressions of adiabatic eigenstates are difficult to obtain.

Feedback stabilization of N-dimensional stochastic quantum systems based on bang-bang control

For an N-dimensional quantum system under the influence of continuous measurement, this paper presents a switching control scheme where the control law is of bang-bang type and achieves asymptotic preparation of an arbitrarily given eigenstate of a non-degenerate and degenerate measurement operator, respectively. In the switching control strategy, we divide the state space into two parts: a set containing a target state, and its complementary set. By analyzing the stability of the stochastic system model under consideration, we design a constant control law and give some conditions that the control Hamiltonian satisfies so that the system trajectories in the complementary set converge to the set which contains the target state. Further, for the case of a non-degenerate measurement operator, we show that the system trajectories in the set containing the target state will automatically converge to the target state via quantum continuous measurement theory; while for the case of a degenerate measurement operator, the corresponding system trajectories will also converge to the target state via the construction of the control Hamiltonians. The convergence of the whole closed-loop systems under the cases of a non-degenerate and a degenerate measurement operator is strictly proved. The effectiveness of the proposed switching control scheme is verified by the simulation experiments on a finite-dimensional angular momentum system and a two-qubit system.

Superadiabatic driving of a three-level quantum system

We study superadiabatic quantum control of a three-level quantum system whose energy spectrum exhibits multiple avoided crossings. In particular, we investigate the possibility of treating the full control task in terms of independent two-level Landau-Zener problems. We first show that the time profiles of the elements of the full control Hamiltonian are characterized by peaks centered around the crossing times. These peaks decay algebraically for large times. In principle, such a power-law scaling invalidates the hypothesis of perfect separability. Nonetheless, we address the problem from a pragmatic point of view by studying the fidelity obtained through separate control as a function of the intercrossing separation. This procedure may be a good approach to achieve approximate adiabatic driving of a specific instantaneous eigenstate in realistic implementations.

Fast and Robust Quantum Information Transfer in Annular and Radial Superconducting Networks

AbstractIn this paper, we propose a protocol to achieve fast and robustness quantum information transfer (QIT) in annular and radial superconducting networks, where each quantum node is composed of a superconducting quantum interference device (SQUID) inside a coplanar waveguide resonator (CPWR). The process is based on reversely constructing time‐dependent control Hamiltonian by designing evolution operator. With the protocol, the maximal population of lossy intermediate states and the amplitudes of pulses can be easily controlled by two corresponding control parameters. Therefore, one can design feasible pulses for QIT with great flexibility. Besides, the speed of the QIT here is much faster compared with that with adiabatic QIT. Moreover, numerical simulations show that the protocol still possesses high fidelity when lossy factors and imperfect operations are taken into account. Therefore, the protocol may provide a useful way to manipulate quantum information networks.