Continuous-time Feedback Research Articles

Stabilization of impulsive hybrid stochastic differential equations with Lévy noise by feedback control based on discrete-time state observations

In this paper, we investigate the problem of the mean-square exponential stabilization for a class of unstable impulsive hybrid stochastic differential equations with Lévy noise (IHSDEs-LN) via feedback control based on discrete-time state observations. Our results show that if feedback control of continuous-time observations can stabilize the controlled system in the sense of mean-square exponential stability, then feedback control of discrete-time state observations can also stabilize the controlled system. And there exists an upper bound on the interval between adjacent observation moments that can be accurately computed. Based on this new theory, we first design continuous-time feedback control for unstable IHSDEs-LN to achieve the stabilization problem, and then improve the continuous-time feedback control to discrete-time state observations feedback control.

Semi-global exponential stabilization of stochastic nonlinear systems with distributed delay under discrete-time state observations and application to circuit systems

This paper is aimed at studying the semi-global exponential stabilization problem for stochastic nonlinear systems with distributed delay under discrete-time state observations. An extended definition of horizontal translation of the segment function along the investigated systems is given, related to the stochastic disturbance and Brownian motion. Basically, the right and upper Dini’s derivative of the constructed functional is presented. Combining this with the global exponential stability of the stochastic nonlinear systems with distributed delay under continuous-time feedback control, the Lyapunov method and Halanay inequality, the maximum observation interval and some stabilization criteria to make the investigated systems under discrete-time state observations reach semi-global exponential stabilization are provided. The results indicate that the semi-global exponential stabilization is dependent upon stochastic disturbance, initial condition range, and maximum observation interval. Finally, an application to the circuit systems is discussed and the involved sufficient conditions for the circuit systems to keep semi-global stabilization are also proposed. Besides, the numerical simulation results confirm the effectiveness of the theoretical results.

Optimal Output Consensus of Heterogeneous Linear Multiagent Systems Over Weight-Unbalanced Directed Networks.

This article investigates the distributed optimal output consensus problem of heterogeneous linear multiagent systems over weight-unbalanced directed networks. A novel distributed continuous-time state feedback controller is proposed to steer the outputs of all the agents to converge to the optimal solution of the global cost function. Under the standard condition that the unbalanced digraph is strongly connected and the local cost functions are strongly convex with global Lipschitz gradients, the exponential convergence of the closed-loop multiagent system is established. Then, the proposed state feedback control law is extended to an observer-based output feedback setting. Two examples are finally provided to illustrate the effectiveness of the proposed control schemes.

Soft-Reset Control With Max-of-Quadratics Lyapunov Certificates

We further develop a novel control paradigm encoding the core behavior of a reset control system within a continuous-time feedback, described by a differential inclusion. A tuning knob allows adjusting how close the continuous-time feedback resembles the behavior of the underlying reset controller. The Lyapunov conditions that we derive naturally lead to a sum-of-squares formulation allowing for the design of max-of-quadratics Lyapunov certificates associated with polynomial multipliers. A few application examples confirm the practical effectiveness of the proposed solutions.

Understanding a Class of Decentralized and Federated Optimization Algorithms: A Multirate Feedback Control Perspective

Distributed algorithms have been playing an increasingly important role in many applications such as machine learning, signal processing, and control. Significant research efforts have been devoted to developing and analyzing new algorithms for various applications. In this work, we provide a fresh perspective to understand, analyze, and design distributed optimization algorithms. Through the lens of multirate feedback control, we show that a wide class of distributed algorithms, including popular decentralized/federated schemes, can be viewed as discretizing a certain continuous-time feedback control system, possibly with multiple sampling rates, such as decentralized gradient descent, gradient tracking, and federated averaging. This key observation not only allows us to develop a generic framework to analyze the convergence of the entire algorithm class, but, more importantly, it also leads to an interesting way of designing new distributed algorithms. We develop the theory behind our framework and provide examples to highlight how the framework can be used in practice.

Partial stability of nonlinear dissipative feedback systems

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

On Semi-Global Exponential Stability Under Sampling for Locally Lipschitz Time-Delay Systems

In this article, the stability preservation problem for locally Lipschitz nonlinear time-delay systems, by emulation of continuous-time dynamic output feedback controllers, is studied. Sufficient Lyapunov-like conditions are provided such that, by suitably fast sampling, the Euler emulation of continuous-time dynamic output feedback controllers ensures the semiglobal exponential stability of the related sampled-data closed-loop system. If other emulation schemes than the Euler one are used, practical semiglobal exponential stability is guaranteed, with arbitrarily small final target ball, by suitably fast sampling. The intersampling system behavior, as well as time-varying sampling intervals, is taken into account. In the provided results, the delay-free case is included as a special case. The practical applicability of the provided results is shown through the study of a nonlinear chemical reactor system with recycle and a glucose–insulin system.

Multi-time scale control and optimization via averaging and singular perturbation theory: From ODEs to hybrid dynamical systems

Multi-time scale techniques based on singular perturbations and averaging theory are among the most powerful tools developed for the synthesis and analysis of feedback control algorithms. This paper introduces some of the recent advances in singular perturbation theory and averaging theory for continuous-time dynamical systems modeled as ordinary differential equations (ODEs), as well as for hybrid dynamical systems that combine continuous-time dynamics and discrete-time dynamics. Novel multi-time scale analytical tools based on higher-order averaging and singular perturbation theory are also discussed and illustrated via different examples. In the context of hybrid dynamical systems, a class of sufficient Lyapunov-based conditions for global stability results are also presented. The analytical tools are illustrated through various new architectures and algorithms within the context of adaptive and extremum-seeking systems. These tools are suitable for the study of model-free optimization and stabilization problems that require the synergistic use of continuous-time and discrete-time feedback. The paper aims to acquaint the reader with a range of modern tools for studying multi-time scale phenomena in optimization and control systems, providing some guidelines for future research in this field.

Output Feedback Stabilization of Nonlinear Strict-feedback Networked Control Systems

Design of output feedback semiglobally practically input-to-state (SPIS) stabilizing controllers for nonlinear strict-feedback networked systems is considered. We first design continuous-time state feedback globally asymptotically (GA) stabilizing controllers and discrete-time reduced-order observers and combine the emulation of state feedback controllers and the designed observers to construct discrete-time output feedback controllers. Then we show that the designed controllers SPIS stabilize nonlinear strict-feedback networked control systems when transmission intervals are sufficiently small and communication networks satisfy suitable assumptions. A numerical example is also given to illustrate the efficiency of proposed design of controllers.

A framework for distributed control via dynamic periodic event-triggering mechanisms

Dynamic event-triggered mechanisms (ETMs) have shown potential in further reducing the number of events in comparison to their static counterparts while delivering similar system performance. In this paper, we provide a framework to design dynamic periodic event-triggered controllers for multi-agent systems (MASs) with nonlinear dynamics. A preliminary version of this work on single-agent systems was studied in Dhullipalla et al. (2020). The design methodology adopts an emulation-based technique that assumes the existence of a continuous-time state feedback controller that stabilizes the MAS. The dynamic ETMs are constructed based on an agent’s ability or inability to sense states (or relative states) of fellow agents in the network. To illustrate the design methodologies, two case studies on nonlinear MASs (with Lipschitz and one-sided Lipschitz dynamics), interacting over undirected and directed communication networks, are presented. Finally, the results of the case studies are demonstrated via numerical examples.

Uniqueness of solutions and linearized stability for impulsive differential equations with state-dependent delay

We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds.

Safety-critical dynamic event-triggered control of nonlinear systems

This paper is concerned with the problem of safety-critical dynamic event-triggered control (ETC) for nonlinear systems. A novel safety-critical dynamic ETC strategy is proposed to simultaneously guarantee event-triggered safety and stability based on the union of input-to-state safe barrier function (ISSf-BF) and input-to-state stable Lyapunov function (ISS-LF). Also, an appropriate dynamic event-triggered mechanism (DETM) is constructed, and Zeno phenomenon of ETC in the safety-critical control framework is definitely excluded based on the DETM constructed. Compared with the existing safety-critical control designed in the framework of continuous-time feedback control, the proposed strategy constructs safety-critical event-triggered controller and an DETM, which greatly alleviates the unnecessary waste of communication and computation resources. Finally, three examples are presented to demonstrate the effectiveness of the proposed design strategy.

Constrained Continuous-Time Dynamics for Linear Model Predictive Control

In this paper, we consider the use of a continuous-time dynamic feedback controller for linear model predictive control (MPC). The controller incorporates both integral-action and anti-windup conditioning and ensures optimality through online emulation of the solution trajectory of an underlying MPC problem. We establish closed-loop stability when the controller is interconnected with an open-loop stable linear-time invariant system using the passivity argument and the invariance principle. We include a computational example to illustrate the effectiveness and the constraint handling capability of the proposed controller.

Sampled-data and event-triggered boundary control of a class of reaction–diffusion PDEs with collocated sensing and actuation

This paper provides observer-based sampled-data and event-triggered boundary control strategies for a class of reaction–diffusion PDEs with collocated sensing and Robin actuation. Infinite-dimensional backstepping design is used as the underlying control approach. It is shown that the continuous-time output feedback boundary control applied in a sample-and-hold fashion ensures global closed-loop exponential stability, provided that the sampling period is sufficiently small. Further, robustness to perturbations of the sampling schedule is guaranteed. For the event-triggered implementation of the continuous-time controller, a dynamic triggering condition is utilized. The triggering condition determines the time instants at which the control input needs to be updated. Under the observer-based event-triggered boundary control, it is shown that there is a minimal dwell-time between two triggering instants independent of initial conditions. Further, the global exponential convergence of the closed-loop system to the equilibrium point is established. A simulation example is provided to validate the theoretical results.

Dissipation Reduction and Information-to-Measurement Conversion in DNA Pulling Experiments with Feedback Protocols

Information-to-energy conversion with feedback measurement stands as one of the most intriguing aspects of the thermodynamics of information in the nanoscale. To date, experiments have focused on feedback protocols for work extraction. Here we address the novel case of dissipation reduction in non-equilibrium systems with feedback. We perform pulling experiments on DNA hairpins with optical tweezers, with a general feedback protocol based on multiple measurements that includes either discrete-time or continuous-time feedback. While feedback can reduce dissipation, it remains unanswered whether it also improves free energy determination (information-to-measurement conversion). We define thermodynamic information {\Upsilon} as the natural logarithm of the feedback efficacy, a quantitative measure of the efficiency of information-to-energy and information-to-measurement conversion in feedback protocols. We find that discrete and continuous-time feedback reduces dissipation by roughly kBT{\Upsilon} without improvement in free energy determination. Remarkably, a feedback strategy (defined as a correlated sequence of feedback protocols) further reduces dissipation, enhancing information-to-measurement efficiency. Our study underlines the role of temporal correlations to develop feedback strategies for efficient information-to-energy conversion in small systems.

Efficient stability analysis approaches for nonlinear weakly-hard real-time control systems

This article considers the stability analysis of nonlinear control systems with failures in the feedback loop that satisfy weakly-hard real-time (WHRT) constraints. Such WHRT constraints are window-based guarantees like for example, that the feedback loop is at least closed for n times in each window of m sampling times. WHRT constraints can, e.g., describe the packet loss properties of a communication network. We consider two different actuation strategies to deal with failures in the feedback loop: holding the last received input or setting the input to zero if no new input arrives. We present stability analysis procedures for both strategies. The procedures are based on the emulation of a sampled-data controller from continuous-time feedback. For the hold strategy, we use a graph description for the WHRT constraints and a modified version of the Viterbi algorithm to bound a Lyapunov function candidate. For the zero strategy, we employ a bound on the evolution of the Lyapunov function candidate for zero input. For both strategies, efficient stability analysis procedures based on non-monotonic Lyapunov functions are derived. In addition, we demonstrate how switched controllers, that switch depending on the past dropout sequence, can be used for the hold strategy. Such switched controllers can significantly increase the maximum allowable sampling period for the hold strategy. The proposed approaches are illustrated with a numerical example.

Global Stability and Exponential Decay of Processes in Nonlinear Feedback Systems with Different Fractional Orders

The global stability of continuous-time multi-input multi-output nonlinear feedback systems with different fractional orders and interval matrices of positive linear parts is investigated. New sufficient conditions for the global stability of this class of positive nonlinear systems are established. Sufficient conditions for the exponential decay of processes in fractional nonlinear systems are given. Procedures for computation of a gain matrix characterizing the class of nonlinear elements are proposed and illustrated by examples.

Optimal control of port-Hamiltonian systems: A continuous-time learning approach

In this paper, we propose a continuous-time adaptive feedback controller for the optimal control of input-state-output port-Hamiltonian systems with respect to general Lagrangian performance indices. The proposed control law implements an online learning procedure which uses the Hamiltonian of the system as an initial value function candidate. The continuous-time learning of the value function is achieved by means of a certain Lagrange multiplier that allows to evaluate the optimality of the current solution. In particular, constructive conditions for stabilizing initial value function candidates are stated and asymptotic stability of the closed-loop equilibrium is proven. Simulations of an exemplary nonlinear optimal control problem demonstrate the performance of the controller resulting from the proposed online learning procedure.

Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices

Abstract The global stability of continuous-time fractional orders nonlinear feedback systems with positive linear parts and interval state matrices is investigated. New sufficient conditions for the global stability of this class of positive feedback nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple example.

Switched-Observer-Based Event-Triggered Adaptive Fuzzy Funnel Control for Switched Nonlinear Systems

This article addresses the event-triggered output-feedback funnel tracking control problem for a class of switched nonlinear systems. A novel switched-observer-based event-triggered adaptive fuzzy funnel control approach for solving the problem is proposed by exploiting the average dwell time method and backstepping. To estimate unmeasurable system states, a switched fuzzy observer and switched update laws for individual subsystems are designed. Compared with the existing funnel controllers designed in the framework of continuous-time feedback control, the proposed approach constructs event-triggered funnel controllers of subsystems and a switching event-triggered mechanism (ETM), which greatly alleviates the unnecessary waste of communication and computation resources, and, without any known limitation on maximum asynchronous time, handles the issue of asynchronous switching between the candidate controllers of subsystems and subsystems. Also, a strictly positive lower bound on interevent time is ensured by introducing a piecewise constant variable into the ETM, which overcomes the difficulty of switched measurement error being discontinuous. It is guaranteed that the tracking error evolves within a prespecified performance funnel. Finally, two examples including a mass-spring-damper system are presented for illustrating the validity of the developed approach.