Feedback Operator Research Articles

Structured invariant subspace and decomposition of systems with time delays and uncertainties

AbstractThis article discusses invariant subspaces of a matrix with a given partition structure. The existence of a nontrivial structured invariant subspace is equivalent to the possibility of decomposing the associated system with multiple feedback blocks such that the feedback operators are subject to a given constraint. The formulation is especially useful in the stability analysis of time‐delay systems using the Lyapunov–Krasovskii functional approach where computational efficiency is essential in order to achieve accuracy for large scale systems. The set of all structured invariant subspaces are obtained (thus all possible decompositions are obtained as a result) for the coupled differential‐difference equations (DDE) associated with the DDE of retarded and neutral types, as well as systems with a time‐varying delay. It was shown that the known ad hoc methods of reducing the dimensions of delay channels can be considered as special cases of decomposition where one subsystem has trivial dynamics. The reduction of computational cost is demonstrated by a numerical example. For the general case, a recursive procedure is developed to obtain the set of all structured invariant subspaces. Based on this procedure, a method is presented to obtain a nontrivial structured invariant subspace that considers computational efficiency and increased possibility of terminating in a finite number of steps.

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of mathbb {R}^2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

A Min-Plus Algebra System Theory for Traffic Networks

In this article, we introduce a comprehensive system theory based on the min-plus algebra of 2×2 matrices of functions. This novel approach enables the algebraic construction of traffic networks and the analytical derivation of performance bounds for such networks. We use the term “traffic networks” or “congestion networks” to refer to networks where high densities of transported particles lead to flow drops, as commonly observed in road networks. Initially, we present a model for a segment or section of a link within the network and demonstrate that the dynamics can be expressed linearly within the min-plus algebra. Subsequently, we formulate the linear system using the min-plus algebra of 2×2 matrices of functions. By deriving the impulse response of the system, we establish its interpretation as a service guarantee, considering the traffic system as a server. Furthermore, we define a concatenation operator that allows for the combination of two segment systems, demonstrating that multiple segments can be algebraically linked to form a larger network. We also introduce a feedback operator within this system theory, enabling the modeling of closed systems. Lastly, we extend this theoretical framework to encompass two-dimensional systems, where nodes within the network are also taken into account in addition to the links. We present a model for a controlled node and provide insights into other potential two-dimensional models, along with directions for further extensions and research.

A distributed neurodynamic algorithm for sparse signal reconstruction via [formula omitted]-minimization

This paper proposes a distributed neurodynamic algorithm for sparse signal reconstruction by addressing ℓ1-minimization problems. Firstly, a ℓ1-minimization problem is transformed into a distributed model, drawing support from multi-agent consensus theory. Secondly, to address this distributed model, a novel distributed neurodynamic algorithm is proposed by employing derivative feedback and projection operator. It is proved that the proposed neurodynamic algorithm is globally convergent by utilizing the properties of projection operator and set-valued system. Moreover, compared with the existing distributed neurodynamic algorithms, the proposed neurodynamic algorithm is inverse-free and does not involve any matrix decomposition. Finally, some experimental results on sparse signal reconstruction indicate the effectiveness of the proposed neurodynamic algorithm.

Neural Network Algorithm With Reinforcement Learning for Parameters Extraction of Photovoltaic Models.

This research focuses on the application of artificial neural networks (ANNs) on parameters extraction of photovoltaic (PV) models. Extracting parameters of the PV models accurately is crucial to control and optimize PV systems. Although many algorithms have been proposed to address this issue, how to extract the parameters of the PV models accurately and reliably is still a great challenge. Neural network algorithm (NNA) is a recently reported metaheuristic algorithm. NNA is inspired by ANNs. Benefiting from the unique structure of ANNs, NNA shows excellent global search ability. However, NNA faces the challenge of slow convergence rate and local optima stagnation in solving complex optimization problems. This article presents an improved NNA, named neural network algorithm with reinforcement learning (RLNNA), for extracting parameters of the PV models. In RLNNA, three strategies, namely modification factor with reinforcement learning (RL), transfer operator with historical population, and feedback operator, are designed to overcome the challenge of NNA. To verify the performance of RLNNA, it is employed to extract the parameters of the three PV models. Experimental results show that RLNNA can extract the parameters of the considered PV models with higher accuracy and stronger stability compared with NNA and the other 12 powerful algorithms, which fully indicates the effectiveness of the improved strategies.

Optimal feedback controls of stochastic linear quadratic control problems in infinite dimensions with random coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [22] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of an optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the C0-group assumption of unbounded linear operator A in [22] and using the contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

Existence, uniqueness, and stabilization results for parabolic variational inequalities

In this paper, we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau–Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.

Centralized and Collective Neurodynamic Optimization Approaches for Sparse Signal Reconstruction via L₁-Minimization.

This article develops several centralized and collective neurodynamic approaches for sparse signal reconstruction by solving the L1 -minimization problem. First, two centralized neurodynamic approaches are designed based on the augmented Lagrange method and the Lagrange method with derivative feedback and projection operator. Then, the optimality and global convergence of them are derived. In addition, considering that the collective neurodynamic approaches have the function of information protection and distributed information processing, first, under mild conditions, we transform the L1 -minimization problem into two network optimization problems. Later, two collective neurodynamic approaches based on the above centralized neurodynamic approaches and multiagent consensus theory are proposed to address the obtained network optimization problems. As far as we know, this is the first attempt to use the collective neurodynamic approaches to deal with the L1 -minimization problem in a distributed manner. Finally, several comparative experiments on sparse signal and image reconstruction demonstrate that our proposed centralized and collective neurodynamic approaches are efficient and effective.

Interior and H∞ feedback stabilization for sabra shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations that exhibit at most resemblance with turbulent flows. One of the mathematically well‐studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model, namely, feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus, only finitely many modes of the shell model would suffice to stabilize the system. Later, we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max‐min control problem. We first prove the stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic Riccati equation. Using the same Riccati operator, we establish asymptotic stability of the nonlinear system.

Using mobile sensor density to approximate state feedback controllers for a class of PDEs

This work is concerned with the use of mobile sensors to approximate and replace the full state feedback controller by static output feedback controllers for a class of PDEs. Assuming the feedback operator associated with the full-state feedback controller admits a kernel representation, the proposed optimization aims to approximate the inner product of the kernel and the full state by a finite sum of weighted scalar outputs provided by the mobile sensors. When the full state feedback operator is time-dependent thus rendering its associated kernel time-varying, the approximation results in moving sensors with time-varying static gains. To calculate the velocity of the mobile sensors within the spatial domain the time-varying kernel is set equal to the sensor density and thus the solution to an associated advection PDE reveals the velocity field of the sensor network. To obtain the speed of the finite number of sensors, a domain decomposition based on a modification of the Centroidal Voronoi Tessellations (µ-CVT) is used to decompose the kernel into a finite number of cells, each of which contains a single sensor. A subsequent application of the µ-CVT on the velocity field provides the individual sensor speeds. The nature of this µ-CVT ensures collision avoidance by the very structure of the kernel decomposition into non-intersecting cells. Numerical simulations are provided to highlight the proposed sensor guidance.

An End-to-End Stochastic Action and Visual Estimation System Towards Autonomous Teleoperation

Teleoperation systems have been getting significant attention from many application areas for decades. However, classical teleoperation systems suffer from problems such as lack of natural feedback, latency, and inefficient operator throughput. Researchers attempted to address these issues by performing some of the teleoperation sub-tasks autonomously whenever requested by the operator. Nevertheless, these systems still need the operator to see the need for autonomous actions and initiate these actions manually, which is demanding for the operators. This paper proposes a novel end-to-end Stochastic Assistive Teleoperation System (SATS) that always stays in the loop, automatically detects applicable actions with probabilities, and produces visual scene estimations for each of these actions, which results in increased operator efficiency and throughput. We introduce several methods that combine ideas from Markov processes and recurrent neural networks to stochastically predict future action sequences and scene configurations with tractable algorithms. Experiments performed with a group of operators on real and simulative teleoperation environments show that operators issue a considerably smaller number of commands compared to alternative methods. We also showed that the operators can manipulate multiple robots simultaneously using our technique, which boosts the operator throughput even further. We provide supplementary video material that demonstrates SATS in action.

Learning an Optimal Feedback Operator Semiglobally Stabilizing Semilinear Parabolic Equations

Stabilizing feedback operators are presented which depend only on the orthogonal projection of the state onto the finite-dimensional control space. A class of monotone feedback operators mapping the finite-dimensional control space into itself is considered. The special case of the scaled identity operator is included. Conditions are given on the set of actuators and on the magnitude of the monotonicity, which guarantee the semiglobal stabilizing property of the feedback for a class of semilinear parabolic-like equations. Subsequently an optimal feedback control minimizing the quadratic energy cost is computed by a deep neural network, exploiting the fact that the feedback depends only on a finite dimensional component of the state. Numerical simulations demonstrate the stabilizing performance of explicitly scaled orthogonal projection feedbacks, and of deep neural network feedbacks.

Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

<p style='text-indent:20px;'>This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.</p>

Improving parameter precision of estimation by no-knowledge feedback control

We investigate a scheme to improve the precision of parameter estimation using no-knowledge measurement-based feedback control. The scheme shows that combination of no-knowledge measurement-based feedback control and quantum weak measurement is the optimal way to suppress decoherence. Our results indicate that compared with knowledge quantum feedback control, under the same weak measurement condition, the precision of parameter estimation can be improved more effectively with no knowledge quantum feedback control. Further, based on numerical simulation, we find that the feedback operator is chosen as [Formula: see text] (or both [Formula: see text] and [Formula: see text], which can protect quantum Fisher information (QFI) for a long time.

Locally optimal measurement-based quantum feedback with application to multiqubit entanglement generation

We present a general approach to measurement-based quantum feedback that employs proportional and quantum state-based (PaQS) feedback components to obtain locally optimal protocols. To demonstrate the power of the method, we first show that it reproduces many known feedback protocols, and then apply it to generation of multipartite entanglement with an emphasis on remote entanglement, which requires spatially local feedback Hamiltonians. The symmetry of both measurement and feedback operators is found to be essential for construction of effective protocols. We show that under perfect measurement efficiency, entangled states can be reached with fidelity approaching unity under non-Markovian feedback control protocols, while Markovian protocols resulting from optimizing the feedback unitaries on ensemble averaged states still yield fidelities above 94%. Application of the PaQS approach to generation of N-qubit W, general Dicke and GHZ states shows that such entangled states can be efficiently generated with high fidelity, for up to N = 100 in some cases.

Well-posedness and exponential stability of boundary control systems with dynamic boundary conditions

The well-posedness of abstract boundary control systems with dynamic boundary conditions (i.e., boundary conditions evolving according to an operator semigroup acting on the boundary space) is established . Here, the boundary feedback operator is unbounded which makes the investigation more interesting in many applications. The positivity of such problem is well studied. By exploiting the positivity, we find a characterization of exponential stability. Our approach is based on the feedback theory of infinite dimensional linear systems. Finally, these results are applied to coupled systems consisting of a partial differential equation (PDE) and an ordinary differential equation (ODE) and to diffusion processes in networks with dynamic ramification nodes.

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.

Path Planning of Pattern Transfer Based on Dual-Operator and a Dual-Population Ant Colony Algorithm for Digital Mask Projection Lithography.

In the process of digital micromirror device (DMD) digital mask projection lithography, the lithography efficiency will be enhanced greatly by path planning of pattern transfer. This paper proposes a new dual operator and dual population ant colony (DODPACO) algorithm. Firstly, load operators and feedback operators are used to update the local and global pheromones in the white ant colony, and the feedback operator is used in the yellow ant colony. The concept of information entropy is used to regulate the number of yellow and white ant colonies adaptively. Secondly, take eight groups of large-scale data in TSPLIB as examples to compare with two classical ACO and six improved ACO algorithms; the results show that the DODPACO algorithm is superior in solving large-scale events in terms of solution quality and convergence speed. Thirdly, take PCB production as an example to verify the time saved after path planning; the DODPACO algorithm is used for path planning, which saves 34.3% of time compared with no path planning, and is about 1% shorter than the suboptimal algorithm. The DODPACO algorithm is applicable to the path planning of pattern transfer in DMD digital mask projection lithography and other digital mask lithography.

Event-Triggered Control of Infinite-Dimensional Systems

This paper addresses the problem of event-triggered control for infinite-dimensional systems. We employ event-triggering mechanisms that compare the plant state and the error of the control input induced by the event-triggered implementation. Under the assumption that feedback operators are compact, a strictly positive lower bound on the interevent times can be guaranteed. We show that if the threshold of the event-triggering mechanisms is sufficiently small, then the event-triggered control system with a bounded control operator and a compact feedback operator is exponentially stable. For infinite-dimensional systems with unbounded control operators, we employ two event-triggering mechanisms that are based on system decomposition and periodic event-triggering, respectively, and then analyze the exponential stability of the closed-loop system under each event-triggering mechanism.

Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.