Frequency control following a contingency event is of vital concern in power system operations. Leveraging inverter-based resources, it is not hard to shape the center of inertia (COI) frequency nicely. However, under weak grid conditions, it becomes insufficient to solely shape the COI frequency since this aggregate signal fails to reveal the inter-area oscillations. In this manuscript, we advocate for foolproof fine-tuning rules for frequency shaping control (FS) based on a systematic analysis of damping ratio and decay rate of inter-area oscillations to simultaneously meet specified metrics for frequency security and oscillatory stability. To this end, building on a modal decomposition, we simplify the oscillation damping problem into a pole-placement task for a set of scalar subsystems, which can be efficiently solved by only investigating the root locus of a scalar subsystem associated with the main mode, while FS inherently guarantees a Nadir-less COI frequency response. Through our proposed root-locus-based oscillatory stability analysis, we derive closed-form expressions for the minimum damping ratio and decay rate among inter-area oscillations in terms of networked system and control parameters under FS. Moreover, we propose useful tuning guidelines for FS which need only simple calculations or visualized tuning to not only shape the COI frequency into a first-order response that converges to a steady-state value within the allowed range but also ensure a satisfactory damping ratio and decay rate of inter-area oscillations following disturbances. As for the common virtual inertia control (VI), although similar oscillatory stability analysis becomes intractable, one can still glean some insights via the root locus method. Numerical simulations validate the proposed tuning for FS as well as the superiority of FS over VI in exponential convergence rate.

It is known that the stability of a feedback interconnection of two linear time-invariant systems implies that the graphs of the open-loop systems are quadratically separated. This separation is defined by an object known as the multiplier. The theory of integral quadratic constraints shows that the converse also holds under certain conditions. This article establishes that if the feedback is robustly stable against certain structured uncertainty, then there always exists a multiplier that takes a corresponding form. In particular, if the feedback is robustly stable to certain gain-type uncertainty, then there exists a corresponding multiplier that is of phase-type, i.e., its diagonal blocks are zeros. These results build on the notion of phases of matrices and systems, which was recently introduced in the field of control. Similarly, if the feedback is robustly stable to certain phase-type uncertainty, then there exists a gain-type multiplier, i.e., its off-diagonal blocks are zeros. The results are meaningfully instructive in the search for a valid multiplier for establishing robust closed-loop stability, and cover the well-known small-gain and the recent small-phase theorems.

This article proposes a closed-loop symplectic regularized algorithm for constrained time-varying optimal control problems. Due to the limitations of symplectic methods in constrained time-varying systems, the optimal control problem is transformed and discretized into a discretized symplectic Runge-Kutta form by using variational integrator. Moreover, we derive first-order necessary conditions for the discrete optimal control problem and obtain a set of Euler-Lagrange (EL) equations. To solve the EL equations, we provide a forward-backward sweep iteration algorithm and analyze its error estimation along with regularization terms. Based on this iteration algorithm, a closed-loop symplectic regularized algorithm is proposed consisting of symplectic update and regularized iteration. To be specific, a sequence of quadratic programmings are leveraged in the forward stage of symplectic update to provide good initial values for the regularized iteration. Furthermore, an interior-point barrier function is applied to handle the constraints in the regularized iteration. The convergence analysis of the proposed algorithm is provided, and simulations are conducted to verify its effectiveness.

This article introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a single-loop cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between -pi and pi. The proposed theorem complements a cyclic version of the celebrated small gain theorem. In addition, a generalization of the proposed theorem is made via the use of angular scaling techniques for reducing conservatism.

In this paper, we formulate a Phi(infinity) synthesis problem for multivariable linear time-invariant (LTI) systems, i.e., controller synthesis with closed-loop phase constraints, which complements the H-infinity synthesis problem. A direct imitation of the solution of H-infinity synthesis would generally lead to complex-valued controllers for Phi(infinity) synthesis, which are not physically realizable. We derive a necessary and sufficient condition for solvability of the state feedback Phi(infinity) synthesis, formulated through a linear matrix inequality (LMI) subject to a rank condition. An algorithm is proposed to handle the difficulty caused by the rank condition. To overcome the limitation of the algorithm in lacking global convergence, we further derive a sufficient solvability condition with a clear geometric interpretation, expressed in terms of a real-coefficient LMI that ensures the resulting controller is real-valued.

This article focuses on the energy management problem of microgrids with battery energy storage systems. The primary objective of this work is to develop a distributed algorithm over time-varying networks to real-time regulate the power output of dispatchable generators and energy storage devices, thereby achieving a balance between supply and demand. To accomplish this, the offline time-coupled optimization problem is initially relaxed to a time-average form and subsequently decoupled into a time-independent problem through the use of the Lyapunov optimization technique. An analytical framework is provided to demonstrate that the battery energy constraints can be satisfied by appropriately selecting parameters. Subsequently, a consensus-based distributed algorithm is formulated over time-varying communication topologies, with its linear convergence proven using the small-gain theorem. Finally, case studies are presented to validate the theoretical results.

We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.

This article addresses the distributed economic dispatch (ED) issue of microgrids. The primary objective of this study is to derive a distributed optimization algorithm with a compressed communication scheme over directed networks. Specifically, the algorithm aims to solve the ED problem, where the total power generation of distributed energy resources (DERs) is dispatched to meet the overall demand at the minimum operational cost under DER capacity constraints. To improve communication efficiency, a novel data compressed transmission mechanism is introduced into the consensus-based distributed algorithm by constructing estimator-like equations. Furthermore, by resorting to the property of matrix norms and system theory, a sufficient condition is derived to ensure that the proposed algorithm linearly converge to the optimal solution under arbitrary compression rate. This condition explicitly depends on the communication topologies and the algorithm parameters but is independent of the compression rate. Finally, simulated examples are provided to validate the theoretical claims and demonstrate the performance of the proposed algorithm.

In this paper, we systematically study a variant of the scaled relative graph (SRG), referred to as the θ-symmetric SRG, and apply it to the stability analysis of cactus networks. Compared with the previous SRG definition, the θ-symmetric SRG enables the characterization of phase lead and lag behaviors, and serves as a more natural multivariable extension of the classical Nyquist plot. We first analyze the gain and phase aspects of θ-symmetric SRG separately and build a connection between θ-segmental phase and a norm minimization problem. This connection makes it possible to compute θ-segmental phase via semidefinite programming. We further derive the submultiplicative and subadditive properties of θ-symmetric SRG. These algebraic properties are crucial to determine the nonsingularity of product-type and sum-type return difference matrices, which topologically correspond to the cyclic and parallel-feedback extreme cases of cactus networks. By taking the cyclic interconnection as the fundamental starting point, we establish necessary and sufficient conditions for its robust stability. Integrating this with the parallel case, we synthesize a unified stability framework for general multi-loop cactus networks. The θ-symmetric SRG framework is less conservative and provides a more intuitive geometric interpretation of system behaviors compared with some existing approaches. Several examples are included to demonstrate the effectiveness of the proposed methods.

In this paper, we show that the small phase condition is both sufficient and necessary to ensure the feedback stability when the interconnected systems are symmetric. Such symmetric systems arise in diverse applications. The key lies in that, for a complex symmetric and semi-sectorial matrix, the transformation matrix in its generalized sectorial decomposition can be taken to be real. Such a result fills the gap of phase based necessary condition for the feedback stability of symmetric systems, and serves as a counterpart of the necessity result for small gain condition. Moreover, we explore the necessity of small phase condition for general asymmetric systems. Some insightful results are presented, which help to clarify the main challenge in the general case.

Distributional linear quadratic regulator (LQR) is a new framework that integrates the distributional reinforcement learning and classical LQR, which offers a new way to study the random return instead of the expected cost. Unlike iterative approximation using dynamic programming in the DRL, a closed-form expression for the random return can be exactly characterized in the distributional LQR, which is defined over infinitely many random variables. In recent work [1, 2], it has been shown that this random return can be well approximated by a finite number of random variables, which we call truncated random return. In this paper, we study the truncated random return in the distributional LQR. We show that the truncated random return can be naturally expressed in the quadratic form. We develop a sufficient condition for the positive definiteness of the block symmetric matrix in the quadratic form and provide the lower and upper bounds on the eigenvalues of this matrix. We further show that in this case, the truncated random return follows a positively weighted non-central chi-square distribution if the random disturbances admits Gaussian, and its cumulative distribution function is log-concave if the probability density function of the random disturbances is log-concave.

Time delays in information exchanges, often arising from real-time task scheduling or communication congestion, have been extensively studied in control systems. Recent advances suggest that random delays can be effectively modeled as stochastic multiplicative uncertainty, enabling the characterization of system stability via mean-square criteria and the application of robust control techniques. This paper contributes to this body of knowledge by confirming, for the first time, that the long-standing law for state feedback stabilization—namely, that the total channel capacity should exceed the topological entropy of the open-loop plant—remains valid for random delay channels, at least in the context of a first-order system with one-step delay. Unlike prior work that focused on minimum-phase mean channels, our study lifts this restriction, offering a more general perspective, which may provide more insights for further research in networked control systems with random delays.

This study first introduces the frequency-wise phases of n-port linear time-invariant networks based on recently defined phases of complex matrices. Such a phase characterization can be used to quantify the well-known notion of passivity for networks. Further, a class of matrix operations induced by fairly common n-port network connections is examined. The intrinsic phase properties of networks under such connections are preserved. Concretely, a scalable phase-preserving criterion is proposed, which involves only the phase properties of individual subnetworks, under the matrix operations featured by connections. This criterion ensures that the phase range of the integrated network can be verified effectively and that the scalability of the analyses can be maintained. In addition, the inverse operations of the considered connections, that is, network subtractions with correspondences are examined. With the known phase ranges of the integrated network and one of its subnetworks, the maximal allowable phase range of the remaining subnetwork can also be determined explicitly in a unified form for all types of subtractions. Finally, we extend the phase-preserving properties from the aforementioned connections to more general matrix operations defined using a certain indefinite inner product.

In this paper, we study the problem of designing a uniform controller for heterogeneous multi-agent systems to achieve consensus, taking into account transient performance requirement characterized by convergence rate and damping. Two main issues are addressed: 1) Under what conditions the problem is solvable? 2) When the problem is solvable, how to design such a uniform controller? To answer these questions, we define a measure of diversity of the agents through simultaneous phase alignment of a set of matrices, and define a measure of interaction quality using the essential phase of the Laplacian matrix of a graph. The main finding of the paper is a critical trade-off among the diversity of the agents, the interaction quality among them, and the desired damping performance that constitutes the solvability condition. We also propose a method to design the controller when the condition is satisfied. The analysis of departure angles of multivariable root loci plays a useful role in our study.

This article addresses the privacy-preserving energy management problem of battery energy storage systems (BESSs). An autonomous privacy-preserving distributed optimization (APPDO) scheme is developed over time-varying networks with the aim of regulating the power output of local BESS to fulfill the total load demand at the minimum economic cost under battery capacity constraints without privacy leakage. To this end, a linearly convergent distributed algorithm is proposed by combining the gradient descent algorithm with leaderless and leader-following consensus schemes. This algorithm is applicable to both islanded and grid-connected modes of BESSs. Furthermore, a novel privacy-preserving approach is constructed by injecting well-designed perturbation sequences into the data exchanged between neighboring nodes, making it effective against malicious eavesdroppers. Furthermore, a comprehensive analysis framework is established to evaluate the convergence, optimality, and privacy-preserving performance of the APPDO algorithm. Finally, numerical studies are conducted to demonstrate the effectiveness of the developed APPDO scheme.

This paper investigates the optimal tracking performance of a discrete-time linear time-invariant (LTI) multi-input and single-output (MISO) plant responding to a step reference signal, in the presence of temporally correlated multiplicative uncertainty. By temporally correlated, we mean the uncertainty has a practical structure of finite impulse response (FIR) and certain dynamical first and second moments, which includes multiplicative white noises and significant network-induced uncertainties. A two-degree-of-freedom (2DOF) controller is adopted and the tracking performance is measured by the expected energy of the tracking error. By a projection lemma, the criterion of achievability of asymptotic tracking is proposed in an innovative form, which explicitly characterizes how the plant properties (i.e., the potentially repeated unstable output poles, nonminimum phase output zeros, and relative degree) and the uncertainty property (captured by a rational function) may affect the performance limitation. It turns out that, when the achievability condition holds, the minimal achievable tracking performance of the closed-loop system with the uncertainty is proportional to the tracking performance limit of the system without uncertainty, made worse by a quantity related to the inverse of the largest stability margin of the closed-loop system against the uncertainty. In addition, some well-known criteria are reproduced by applying the result to the systems with random packet dropout or multiplicative white uncertainty. Several simulations are also conducted to validate the results. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and

Data-enabled predictive control (DeePC) has been extensively studied for its ability to achieve safe control of multiconstrained unknown systems without requiring an explicit system model. Traditional DeePC constructs a Hankel matrix using historical input-output data of an unknown system to replace the system model, enabling receding horizon predictive control. This paper proposes an online data-updated DeePC framework, which combines real-time data with historical data to construct a Mosaic Hankel Matrix online, addressing the issue of data unreliability caused by external system state variations or improper data collection. Furthermore, an adaptive prediction horizon strategy is designed subject to control frequency constraints, and the penalty formulation of slack variables is improved using a sigmoid function, achieving a balance between control efficiency and optimization performance. Finally, quadrotor trajectory tracking experiments were conducted on a ROS+PX4-based software-in-the-loop (SITL) simulation platform to validate the proposed approach.

In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the θ-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of θ-symmetric SRG is established, enabling it to handle cyclic interconnections for which conventional graph separation methods are not applicable. By integrating both gain and refined phase information, the θ-symmetric SRG provides a unified graphical characterization of the system, which better captures system properties and yields less conservative results. In the scalar case, the θ-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise θ-symmetric SRG is more suitable than the standard SRG as a multi-input multi-output extension of the classical Nyquist plot. Illustrative examples are included to demonstrate the effectiveness of the θ-symmetric SRG.

In this paper, we study the state-space characterization of multi-input multi-output linear time-invariant systems. A sectored real lemma is developed for phase-bounded systems, serving as a counterpart of bounded real lemma and an extension of positive real lemma. Moreover, we propose a mixed bounded/sectored real lemma that integrates the gain and phase information, thereby enhancing its applicability to practical systems. All results are formulated in a novel phase-related terminology, which is equivalent to the LMI statement but provides a distinct conceptual perspective on phase.

Existing studies on event-triggered output tracking control have focused only on addressing steady errors while neglecting transmit errors, which leads to the obtained results in a non-optimal implementation fashion. To solve this problem, this paper adopts a reinforcement learning algorithm to investigate the optimal output tracking control of heterogeneous multi-agent systems with a novel event-triggered mechanism. First, different from existing model-based predictor method for each agent, a novel edge-based predictor using the relative state information is proposed to estimate the relative state information among agents during the time interval between two adjacent triggering instants. Then, the predicted relative state is put forwarded to design event-triggered distributed observer to provide the state estimation of the leader’s information, and a novel event-triggered condition based on the control input signal is developed. As a result, the proposed edge-based distributed observer method not only avoids continuous communication among followers, the leader and its children, and Zeno behavior, but also the explicit control input signal can be protected from the view of privacy protection. Second, the state feedback control policy under a reinforcement learning method is considered to achieve the model-based optimal output tracking control, where the optimal control policy is learned by solving the Bellman equation iteratively. Beside, the model-free optimal output tracking control is also achieved by verifying the rank condition based on the collected system data without relying on accurate system dynamics. It is shown that the proposed algorithm ensure the model-free optimal output tracking control without continuous communication and prior system knowledge. Finally, the effectiveness of the proposed theoretical algorithm is verified using a simulation example.