Feedback Law Research Articles

Feedback-based quantum algorithms for ground state preparation

The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024

Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

Sufficient conditions on input-to-stabilisation of time-varying semi-linear systems with disturbances in Hilbert spaces

In this paper, we investigate the problem of input-to-state stabilisation of time-varying semi-linear systems with disturbance in Hilbert spaces. An original adaptive input-to-state stabilisation strategy to derive this exceptional problem under the nonlinearities modelling is proposed using the Lyapunov method. Since all the states are not measurable, a Luenberger observer is designed to estimate unmeasurable states for input-to-state stabilisation. Thus, in the design procedure, first, a Luenberger observer is designed, and then the adaptive output feedback controller is constructed via the estimated states and adaptation law. Moreover, we give a sufficient condition, in terms of a perturbation term, to guarantee the input-to-state stability of the time-varying semi-linear systems under a stabilising nonlinear feedback law. An example of a controlled reaction-diffusion equation is given.

Hybrid feedback control of PMSG-based WECS with multilevel flying-capacitor inverter enhanced by fractional order extremum seeking

This paper proposes an original hybrid control strategy for a three-phase multilevel flying capacitor inverter (FCI) used in wind energy conversion system (WECS). A state feedback law, computed using a Lyapunov function and an invariance principle, is designed to manage grid current control and capacitor voltage balancing, guaranteeing the asymptotic stability of the closed-loop system. Additionally, a fractional-order extremum seeking control (FOESC) algorithm for Maximum Power Point Tracking (MPPT) is introduced, which optimizes power capture by adjusting turbine speed from the DC side without requiring a detailed system model (model-free approach). The use of fractional-order calculus leads to improved convergence and more seamless performance while preserving the robust characteristics of the system. The proposed control strategies are validated through simulation, demonstrating improved transient response and stability under varying wind conditions and parameter uncertainties. These results highlight the potential of advanced control algorithms to enhance the efficiency and reliability of wind energy systems, contributing to global efforts to achieve sustainable energy goals.

L2-gain analysis and fault-tolerant control of nonlinear discrete time-delay saturated switched systems with memory state feedback

For nonlinear discrete time-delay saturated switched systems with memory feedback, the problem of actuator failure with external disturbances is studied. First, we propose a sufficient condition for the existence of a fault-tolerant state feedback law with memory by using the switched Lyapunov function (SLF) strategy, which ensures that the closed-loop system has a maximum tolerant disturbance capacity by solving the optimization problem. Then, the minimum upper bound on the [Formula: see text]-gain is estimated by solving the optimization problem. Finally, the reliability of the proposed method is verified by examples.

La traduction, un processus évolutif vu en rétroaction par la théorie de construction de niche

Translation, an Evolving Process seen in Retroaction by Niche Construction Theory This article delves into the translation process as a complex, self-organised system, viewed through the lens of niche construction theory. It highlights three key aspects: the modification of translation’s environmental conditions, the selection pressures acting on translation as it shapes its environment, and the evolutionary responses observed within the translation process. Firstly, it underlines how translation modifies its own environmental conditions through feedback loops between translators and their practices. Drawing on the insights of Gideon Toury and Kobus Marais, this perspective portrays translation as a dynamic and retroactive system. For example, the semantic evolution of the term “translate” itself reflects the dynamic interaction between translators and their environments. This model illustrates how translation continually shapes and redefines its context, resulting in a continuously evolving discipline. Additionally, the article explores the selection pressures that impact translation, akin to natural selection, where translation serves as a cultural and linguistic vector undergoing processes of variation, adaptation, and heredity. It examines how translation adapts to changing environments while preserving knowledge. Moreover, the article examines evolutionary responses observable in the translation process, such as computer-assisted translation, machine translation, and translation memories. These examples underscore translation’s capacity for self-observation, self-organisation, and self-creation (autopoiesis), reflecting its inherent complexity and adaptability. In conclusion, the article draws parallels between the translation process and biological niche construction, particularly focusing on the feedback law. It proposes a conceptual framework for understanding the factors contributing to the translation environment, emphasizing the complexity and multiplicity inherent in translation processes. This niche construction theory provides insights into the dynamic nature of translation and its reciprocal relationship with its environment. As technology, particularly AI, advances, the translation niche may evolve into a cognitive niche, enhancing our ability to observe and control the translation process. Thus, this study advocates for an interdisciplinary approach that bridges sociological and biological perspectives, paving the way for future research in the evolving landscape of translation studies..

Local exponential stabilisation of the delayed Fisher's equation

Exponential stabilisation is undoubtedly an essential property to study for any control system. This property is often difficult to establish for nonlinear distributed systems and even more difficult if the state of such a system is subject to a time-delay. In this context, this paper is devoted to the study of the well-posedness and exponential stabilisation problems for the one-dimensional delayed Fisher's partial differential equation (PDE) posed in a bounded interval, with homogeneous Dirichlet condition at the left boundary and controlled at the right boundary by Dirichlet condition. Based on the infinite dimensional backstepping method for the linear and undelayed corresponding case, a feedback law is built. Under this feedback, we prove that the closed-loop system is well-posed in L 2 ( 0 , 1 ) and locally exponentially stable in the topology of the L 2 ( 0 , 1 ) -norm by the use of semigroup theory relating to differential difference equations and Lyapunov–Razumikhin analysis, respectively. A numerical example that reflects the effectiveness of the proposed approach is given.

Feedback Control for Directional Rendezvous Using Constant-Magnitude Low Thrust

Feedback control for close-range orbital rendezvous, which permits choosing the direction of the final approach, is beneficial for numerous space missions. However, developing such a feedback controller for satellites operating with constant-magnitude low thrust without introducing chattering is nontrivial due to the underactuated nature of the dynamics. This paper develops a feedback control law for rendezvous with a target on a nearly circular orbit, assuming that the chaser satellite utilizes constant-magnitude low thrust. Based on the Clohessy–Wiltshire (CW) model, a sliding surface is chosen on which the states approach the origin. A feedback law driving given initial states to the sliding surface is developed first. Close to the origin, a bang-bang type controller is then employed, which drives the state errors to zero in finite time while allowing to choose the final approach direction along the R-, V-, or H-bar. Solutions to the state trajectories are obtained in closed form, except for the case of the R-bar approach, which is proven to be finite-time stable. It is shown that the higher-order dynamics neglected in the CW equations can be handled through dynamic inversion combined with a hopping maneuver.

Adaptive sliding mode boundary control of a perturbed diffusion process

SummaryThis paper proposes a sliding‐mode‐based adaptive boundary control law for stabilizing a class of uncertain diffusion processes affected by a matched disturbance. The matched disturbance is assumed to be uniformly bounded along with its time derivative, whereas the corresponding upper bounding constants are not known. This motivates the use of adaptive control strategies. In addition, the spatially‐varying diffusion coefficient is also uncertain. To achieve asymptotic stability of the plant origin in the ‐sense in the presence of the disturbance, a discontinuous boundary feedback law is proposed where the gain of the discontinuous control term is adjusted according to a gradient‐based adaptation law. A constructive Lyapunov analysis supports the stability properties of the considered closed‐loop system, yielding sufficient convergence conditions in terms of suitable inequalities involving the controllers' tuning parameters. Simulation results are presented to corroborate the theoretical findings.

Overcoming local extrema in torque-actuated source seeking using the divergence theorem and delay

We study the problem of seeking the source of a scalar signal with a force- and torque-controlled unicycle. The vehicle does not have the capability to measure or track its position. The only information about the vehicle’s position relative to the source is given by measurements of the signal. We propose a feedback law that induces a uniform forward motion and an almost uniform rotational motion. This in turn leads to an almost circular motion of the unicycle; i.e., a motion along the boundary of a disk. An application of the divergence theorem reveals that the measurements of the signal along the boundary of the disk provide access to the gradient of an averaged signal, where the average is taken over the entire disk. Such a local average of the sensed signal can eliminate undesired local extrema, which in turn is beneficial for the goal of reaching a global extremum. We prove practical asymptotic stability for the closed-loop system with respect to extrema of the averaged signal. Our theoretical analysis takes L∞-measurement errors of the signal into account. We compare the performance of the proposed control law to another source seeking method in the presence of local extrema through numerical simulations.

Optimization, guidance, and control of low-thrust transfers from the Lunar Gateway to low lunar orbit

The Lunar Gateway will represent a primary space system useful for the Artemis program, Earth–Moon transportation, and deep space exploration. It is expected to serve as a staging location and logistic outpost on the way to the lunar surface. This study focuses on low-thrust transfer dynamics, from the Near-Rectilinear Halo Orbit traveled by Gateway to a specified Low-altitude Lunar Orbit (LLO). More specifically, this research addresses two closely-related problems: (i) determination of the minimum-time low-thrust trajectory and (ii) design, implementation, and testing of a guidance and control architecture, for a space vehicle that travels from Gateway to LLO. Orbit dynamics is described in terms of modified equinoctial elements, with the inclusion of all the relevant perturbations, in the context of a high-fidelity multibody ephemeris model. The minimum-time trajectory from Gateway to a specified lunar orbit is detected through an indirect heuristic approach, which uses the analytical conditions arising in optimal control theory in conjunction with a heuristic technique. However, future missions will pursue a growing level of autonomy, and this circumstance implies the mandatory design and implementation of an efficient feedback guidance scheme, capable of compensating for nonnominal flight conditions. This research proposes nonlinear orbit control as a viable and effective option for autonomous explicit guidance of low-thrust transfers from Gateway to LLO. This approach allows defining a feedback law that enjoys quasi-global stability properties without requiring any offline reference trajectory. The overall spacecraft dynamics is modeled and simulated, including attitude control and actuation. The latter is demanded to an array of reaction wheels, arranged in a pyramidal configuration. Guidance, attitude control, and actuation are implemented in an iterative scheme. Monte Carlo simulations demonstrate that the guidance and control architecture at hand is effective in nonnominal flight conditions, i.e. with random starting point from Gateway as well as in case of temporary unavailability of the propulsion system. The numerical results also point out that only a modest propellant penalty is associated with the use of feedback guidance and control in comparison to the minimum-time optimal trajectory.

Global practical stabilization of the double integrator system with an imperfect sensor and subject to a bounded disturbance

This paper is concerned with global practical stabilization of the double integrator system with an imperfect sensor and subject to an additive bounded output disturbance. The imperfect sensor nonlinearity possesses the nonlinear characteristics of saturation and dead zone. Because of the presence of output dead zone and the additive disturbance, the states cannot be expected to driven into an arbitrarily small neighborhood of the origin. To solve the global practical stabilization problem, we proposes a low gain-based linear dynamic output feedback law, under which the first state enters and remains in a bounded set whose size is depended on the bound of disturbance and the range of dead zone and the second state enters and remains in a pre-specified arbitrarily small set, both in finite time. Simulation results illustrate the effectiveness of our proposed control method.

An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

SIMULATION OF TRANSIENT PROCESSES IN THE STABILIZATION SYSTEM OF DARRIEUS WIND TURBINE ROTOR ROTATIONS

In the work, the simulation of transient processes of the rotor speed system stabilization of vertical-axis wind turbines, controlled by changes of the blade length, is carried out. It is proved that real roots of the characteristic equation of a closed system with the proportional-integral law of feedback leads to a significantly longer length of transient processes than with complex-connected roots.

On the boundary stabilization of the KdV–KdV system with time-dependent delay

The boundary stabilization problem of the Boussinesq KdV–KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato’s approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV–KdV system decays exponentially by employing the Lyapunov method.

Berthing proximity maneuvers and trajectory planning of a space manipulator via Kane’s formulation

On-orbit servicing is one of the most challenging tasks of the recent space era. Space Manipulator Systems (SMS) can represent viable technological solutions for several space operations, such as in-situ refueling, repairing of operating spacecraft, or debris grappling. Regardless of the specific task, SMS must approach the target object and then establish a physical connection with it using a single or more robotic arms. This work considers a space vehicle equipped with a pair of two-link robotic arms, with the intent of efficiently designing and simulating the overall spacecraft dynamics until contact takes place. The links that compose each arm are modeled as rigid bodies, assuming relatively slow dynamics. Instead, flexibility is introduced in the model of the rotary joints. The overall dynamics is investigated using Kane’s methodology, which is general, highly systematic, and joins the advantages of both the Newton/Euler and the Lagrange formulation. Both SMS and the target, assumed as noncooperative, are placed in low Earth orbit, in close proximity. Trajectory planning of the robotic arm is investigated, while considering the effect of the arm motion on the overall attitude dynamics. Precise attitude maneuvering is mandatory in order that grappling be successfully completed. To do this, a quaternion-based, globally stable attitude feedback law is designed, implemented, and tested. Actuation is demanded to an arrangement of control momentum gyroscopes. These devices are modeled with the inclusion of gyroscopic coupling effects, and are driven by means of suitable steering laws, for the purpose of pursuing the desired attitude control action. Accurate modeling of SMS, with also the inclusion of disturbances due to joint flexibility, in conjunction with the use of suitable feedback attitude control and steering laws for both the robotic arms and the actuation devices, allows obtaining the desired variables, i.e. (i) the motor torques (for steering both the joints and the attitude devices) and (ii) the constraint actions (forces and torques). A Monte Carlo analysis points out effectiveness of the control architecture proposed in this study in the challenging mission scenario of interest.

Adaptive observer and control of spatiotemporal delayed neural fields

An adaptive observer is proposed to estimate the synaptic distribution between neurons asymptotically from the measurement of a part of the neuronal activity and a delayed neural field evolution model. The convergence of the observer is proved under a persistency of excitation condition. Then, the observer is used to derive a feedback law ensuring asymptotic stabilization of the neural fields. Finally, the feedback law is modified to ensure simultaneously practical stabilization of the neural fields and asymptotic convergence of the observer under additional restrictions on the system. Numerical simulations confirm the relevance of the approach.

Global inverse optimality for a class of recurrent neural networks with multiple proportional delays

This paper formulates two novel theoretical designs of input-to-state stabilizing control for a class of recurrent neural networks with multiple proportional delays. The analysis tool developed in this paper is based on Lyapunov function and inverse optimality method, which does not require solving Hamilton-Jacobi-Bellman equations. Two inverse optimal feedback laws are constructed via the dimensions of state and input, which ensure the input-state stability for the considered system. When the dimensions of state and input are different, we establish a scalar function and give one of the control laws by Sontag's formula. Furthermore, the designs of inverse optimal control reach both global inverse optimality and global asymptotic stability of the system for some meaningful cost functional. Four numerical examples are provided to show the effectiveness of the inverse optimal control.

Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs

Abstract We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${\mathcal O}(h^{2}+\tau )$ for its solution, and conclude from it a rate of almost ${\mathcal O}(h^{2}+\tau )$ resp. ${\mathcal O}(h^{2}+\tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.

Stabilization of age‐structured chemostat hyperbolic PDE with actuator dynamics

AbstractFor population systems modeled by age‐structured hyperbolic partial differential equations (PDEs), we redesign the existing feedback laws, designed under the assumption that the dilution input is directly actuated, to the more realistic case where dilution is governed by actuation dynamics (modeled simply by an integrator). In addition to the standard constraint that the population density must remain positive, the dilution dynamics introduce constraints of not only positivity of dilution, but possibly of given positive lower and upper bounds on dilution. We present several designs, of varying complexity, and with various measurement requirements, which not only ensure global asymptotic (and local exponential) stabilization of a desired positive population density profile from all positive initial conditions, but do so without violating the constraints on the dilution state. To develop the results, we exploit the relation between first‐order hyperbolic PDEs and an equivalent representation in which a scalar input‐driven mode is decoupled from input‐free infinite‐dimensional internal dynamics represented by an integral delay system.