This paper investigates the non-redundant parametrization of state feedback gains that assign a prescribed set of invariant factors to a controllable linear time-invariant system. While traditional approaches often rely on Jordan forms, we specify the desired closed-loop structure directly through a chain of monic polynomials and its associated block companion matrix. By exploiting a right-coprime factorization of the input-to-state transfer matrix, we derive a complete but redundant parametrization of all feedback gains achieving the assignment based on the Sylvester equation. In the resulting Sylvester coordinates, two parameter matrices generate the same feedback gain exactly when they differ by the right action of the invertible centralizer of the target block companion matrix. This identifies the non-redundant parameter space with the quotient of the Sylvester parameter space by this action. An explicit formula for the centralizer is derived and used to construct local normalizations for feedback computation. We also characterize the invariant-factor chains that maximize the dimension of the quotient parameter space for a given pole multiset and quantify the trade-off between design freedom and the maximal Jordan-block sizes of the closed-loop matrix. Numerical examples illustrate the theoretical results.
This paper revisits the control law derived with the backstepping approach through delay compensation for systems with distinct nonuniform input delays. As one of the main topics of the present paper, an interpretation of its control philosophy is provided through a very elementary viewpoint for better understanding and explanation of the control law. The course of such an interpretation, however, is associated with the use of some ‘implicit idealizing assumptions,’ and with regards to their use, the arguments are further extended to an elementary proof for the finite spectrum assignment property of the closed-loop system. It immediately implies the stability of the closed-loop system, but the arguments in the present paper are carried out through the treatment of the characteristic equation in the frequency-domain, and thus are in stark contrast with the time-domain stability proof with the Lyapunov functional used in the existing studies in which the control law was derived. In addition, the case when the state of the delay-free part of the system is not accessible and a state observer is introduced for it is further studied, and the combined use of the state-feedback-based control law and the state observer is studied. Through a simple modification of the frequency-domain proof for the closed-loop stability in the state feedback case, the separation principle in such a context is further established. A numerical example is also studied to confirm the robustness of the closed-loop system under the plant uncertainties including the mismatch in the input delays.
Extremum seeking control (ESC) is a model-free and adaptive scheme for optimizing steady-state performance of dynamic systems in real time, particularly when system models are unavailable or uncertain. Classical ESC relies on exciting the system with a sinusoidal dithering signal, thereby ensuring convergence to the optimum, provided that the dither excitation frequency remains well below the dominant system dynamics. However, this time-scale separation constraint limits convergence speed and robustness, especially in systems whose dominant dynamics begin at low frequencies. Enhanced ESC methods, such as proportional-integral schemes, phasor-based estimation techniques, and frequency-domain approaches, address the time-scale separation limitation. However, the understanding of how objective definitions with identical steady-state optima affect convergence remains limited. This paper analyzes ESC closed-loop convergence in dynamical systems for state-dependent and bilinear (state-input-dependent) objectives using a Wiener-type model framework. Although both objectives share equal steady-state optima, they exhibit fundamentally different convergence behavior when the dither frequency is chosen beyond the system’s dominant dynamics (fast time scale). New insights are obtained through sequential linearization combined with frequency-domain examinations and signal-based analyses. Furthermore, improvements to the classical ESC scheme for dynamical bilinear objectives are proposed to relax time-scale separation constraints, enabling higher dither frequencies and leading to consistent and accelerated convergence.
The paper studies a problem of absolute stabilizability for a system with nonlinearities/uncertainties described by some integral quadratic constraint with periodically time-varying coefficients of unknown period (unknown frequency). A lower bound on the frequency above which the system is absolutely stabilizable by a linear output feedback controller of H∞ type is obtained. The controller absolutely stabilizes the system for all unknown periods from some interval. The developed method is based on results of Yakubovich on absolute stability and linear periodic Hamiltonian systems.
Distributed estimation of algebraic connectivity is a key component in connectivity-aware coordination, but existing continuous-time estimators often require topology-dependent gain tuning. This paper develops a continuous-time estimator for the algebraic connectivity and Fiedler eigenspace of connected undirected graphs. The estimator combines Laplacian descent, Rayleigh-quotient feedback, average-mode rejection, and norm regulation; the global averages appearing in the reference dynamics are supplied in the distributed implementation by dynamic average consensus filters. The resulting local laws use only neighbor exchanges and do not require agents to know the network size or Laplacian spectral bounds when evaluating their dynamics or choosing estimator gains. When the algebraic connectivity is simple, a high-gain scaling of the consensus layer gives explicit ultimate bounds for the consensus disagreement, state error, and connectivity-estimation error, all decreasing inversely with the gain scaling. Numerical studies compare the estimator with representative power-iteration-based methods and illustrate its behavior on fixed and switching topologies.
In this letter, we propose a neuron-inspired method for phase synchronization of identical nonlinear oscillators in a distributed manner. The proposed framework has a hierarchical structure. In the upper layer, a network of pulse-coupled neurons achieves synchronization and provides synchronized forcing signals to the lower layer. In the lower layer, pairs of neurons generate spike trains that emulate the forcing signals, which are then applied to nonlinear oscillators, inducing an entrainment. As a result, asymptotic phase synchronization is achieved without continuous communication over the network, continuous forcing, or measurements from the oscillators. Numerical examples with Van der Pol oscillators illustrate the proposed scheme.
This paper investigates the problem of distributed Nash equilibrium seeking with fixed-time convergence for players of high-order integrator dynamics. A novel distributed Nash equilibrium seeking algorithm with a two time-scale framework is proposed based on the leader-following consensus protocol and sliding mode theory. It is proved that the actions of the players under the proposed distributed algorithm converge to the Nash equilibrium in a fixed-time with its upper bound explicitly given. The effectiveness of the proposed distributed algorithm is demonstrated through a non-cooperative navigation example involving a heterogeneous swarm of unmanned vehicles of high-order integrator dynamics.
The paper considers the problem of escaping saddle points in distributed nonconvex optimization, which is a fundamental problem in applications such as the Internet of Things and distributed deep learning. We propose a novel distributed nonconvex algorithm named DisCub by combining the cubic-regularized Newton method with tracking techniques in a distributed setting. Furthermore, a communication-efficient variant named CeDisCub is explored with the aid of rank-1 approximation to alleviate the communication burden. Numerical simulations validate the superior saddle point avoidance of our algorithm compared to other advanced algorithms.
The three main classical criteria of absolute stability for Lurie systems with several nonlinearities are obtained with the use of the convolution theorem and without the use of the S-procedure.
This paper studies consensus to the origin for nonlinear multi-agent systems subject to unknown time-varying perturbations and communication delays, under a linear nominal drift, a square state-dependent input gain, and uncertain inter-agent couplings. For nominally stable agents, a distributed controller is designed via Lyapunov redesign around an extended perturbation envelope that absorbs, through a single causal scalar bound, both delayed self-perturbations and uncertain couplings. A Gronwall-type inequality then yields global exponential stabilisation to the origin and hence consensus with closed-form expressions for the convergence rate and the overshoot bound. The framework is further extended to open-loop unstable agents through a Control Lyapunov Function condition, recovering the same exponential guarantees via an augmented envelope evaluated directly at the agent states. Throughout, the control signal is continuous in time along every closed-loop trajectory, so the high-frequency switching inherent in discontinuous sliding-mode designs is avoided without sacrificing robustness against bounded time-varying delays. Numerical examples illustrate the theoretical results.
This paper develops new multivariable model reference adaptive control (MRAC) schemes for multi-input and multi-output (MIMO) plants, whose design and analysis are based on the Kalman–Yakubovich–Popov lemma. It demonstrates that such adaptive control schemes can be designed using either an output feedback, state feedback or partial-state feedback controller structure, based on a unified tracking error parametrization framework. The paper clarifies the conditions on the interactor matrix and high frequency gain matrix of a MIMO system, for the design of such an adaptive control scheme. It presents a broad collection of Kalman–Yakubovich–Popov lemma based adaptive control schemes and illustrates their features as compared with the gradient algorithm based adaptive control schemes.
This work proposes a decentralized event-triggered control law with a dwell time for a class of semi-linear time-varying delay reaction–diffusion systems (RDSs). The time-varying delay RDS, modeled as a semi-linear parabolic partial differential equation, achieves exponential stabilization through linear local averaged state feedback. To be specific, a finite number of distributed sensors are configured to measure the locally averaged state information, and a decentralized event-triggered mechanism (ETM) is designed. Each triggering mechanism corresponding to a sensor can decide independently whether to transmit the measurement signal. Compared with centralized ETMs, the proposed scheme enables timely acquisition of critical local information, thereby improving transmission efficiency while preserving closed-loop performance. In addition, incorporating a dwell time effectively prevents Zeno behavior. Furthermore, a closed-loop system is established by introducing a shape function to integrate the local control signal. Combined with the inequality technique, some sufficient exponential stability criteria for the closed-loop system are constructed. Moreover, centralized ETMs with a dwell time are included to verify the advantages of the proposed decentralized ETM. Finally, numerical examples based on a catalytic reaction model and a reaction–diffusion neural network model are presented to demonstrate the effectiveness and practical applicability of the proposed method.
Recently, a condition has been established to ensure that stochastic systems remain within a predefined safety set at all times with a predetermined worst-case probability p, utilizing the framework of stochastic zeroing control barrier functions (ZCBFs). However, when the control input set is subject to bounded constraints, it is not guaranteed that the controlled stochastic systems will maintain their presence within the predefined safety set with the specified worst-case probability p, as the boundedness of the control inputs may limit the feasible control actions. In this study, by formulating a stochastic ZCBF that accounts for input constraints, we propose two methods to identify an inner safety set wherein the controlled stochastic systems are assured safety with the worst-case probability p. The first approach derives an implicit representation of the stochastic ZCBF with input constraints, thereby extending analogous results from deterministic systems to the stochastic domain. Notably, in the absence of stochastic disturbances, the structure of the corresponding inner safety set aligns with that established for deterministic controlled systems under input constraints. The second approach involves the explicit construction of a stochastic ZCBF with input constraints, accompanied by a design procedure for a continuous state-feedback controller. Two illustrative examples are presented to demonstrate the validity and effectiveness of the proposed methods.
Dynamic weighted average consensus, which enables a network of agents to collaboratively track the weighted average of their time-varying local reference signals, is a fundamental primitive in multi-agent systems due to its wide range of applications. However, conventional algorithms, including the special case of dynamic average consensus, require explicit exchange of state information, rendering them vulnerable to external eavesdropping attacks and thereby compromising the privacy of local reference signals. To address this, we propose a novel privacy-preserving algorithm based on an agent decomposition mechanism in which each agent is virtually decomposed into two coupled sub-agents: a public α-sub-agent responsible for inter-agent communication and a private β-sub-agent operating exclusively internally. By exploiting the inherent robustness of weighted consensus dynamics, our algorithm ensures convergence to the dynamic weighted average with an ultimately bounded tracking error while providing rigorous privacy preservation. Based on the property of observational indistinguishability, we rigorously prove that an external eavesdropper with full knowledge of the network topology, algorithm design, and all communicated messages cannot infer the true reference signals. Compared to existing privacy-preserving methods, many of which either fail to protect against eavesdropping, impose restrictive assumptions on reference signals, or rely on cooperation from neighboring agents to satisfy sum-conservation constraints, our algorithm guarantees privacy without such limitations and achieves a superior balance between privacy preservation and convergence accuracy. Numerical simulations confirm the effectiveness of our algorithm.
This paper presents a novel framework for the formation and trajectory tracking of multi-agent systems in time-varying tasks. The proposed approach leverages a time-varying diffeomorphism to simplify the control design, decoupling the complexity of reaching a desired trajectory from adhering to motion profiles along it. This framework ensures aggregation and stability in a virtual reference frame, which is then mapped to the real frame to achieve the desired dynamic behavior. The method is demonstrated on agents modeled as double integrators, showcasing its scalability and adaptability to various time-varying scenarios. Numerical results validate the effectiveness of the approach in achieving desired formations and adapting to evolving environments, while maintaining proper performance and predictable behavior. The proposed strategy offers a promising tool for dynamic applications such as surveillance, monitoring, and coordinated motion planning.
This paper addresses the problem of hyperexponential stabilization of a chain of integrators using time-varying feedback when the state is not fully available for measurement. The proposed approach makes use of the delayed output measurement to reconstruct the state and requires only a finite available time-varying history of a scalar output instead of full state availability. We employ a Lyapunov-based small-gain framework and a forced comparison inequality for a running maximum to provide sufficient conditions that ensure hyperexponential stability of the closed-loop system without using explicit observer dynamics. An illustrative example shows the effectiveness of the method.
