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This article investigates the robust cooperative output regulation problem of heterogeneous uncertain linear multiagent systems with switching topologies and infinite distributed communication delays. A novel distributed observer is proposed for each agent to estimate the state of the so-called exosystem by taking consideration of both switching topologies and infinite distributed communication delays. A distributed output feedback controller is then developed based on the internal model principle and the estimated state without using the prior knowledge of communication delays. It is shown via the newly developed Lyapunov-like method that the robust cooperative output regulation problem can be solved under some mild assumptions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

This article considers the distributed resource allocation problem over a network of n agents with individual optimization functions and coupled resource constraints. The optimization variables are bounded for each agent. We propose an asynchronous distributed gradient descent based method, which is able to tackle time-varying communication networks and communication problems. The main ideas are twofold. First, the degree of freedom of the network is reduced to deal with the global constraint. Second, we utilize delay compensation to counteract the delay of asynchronous communication. We prove that the convergence rate is $O(\frac{1}{k})$ and analyze both the communication complexity and the time complexity. This method is applied into the parallel pump control problem in HVAC system and shows good numerical results. [ABSTRACT FROM AUTHOR]

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by dynamic state feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by static linear feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article. [ABSTRACT FROM AUTHOR]

For a family of genuinely nonlinear systems with delays in the input and state, global strong stabilization (GSS) in the sense of Kurzweil is shown to be possible by memoryless state and/or output feedback under two conditions: 1) The input delay is within an appropriate range although the state delays can be sufficiently large; 2) the time-delay bounding system is homogeneous of degree zero and has a lower triangular structure. The proof is based on the Lyapunov–Krasovskii functional theorem combined with the homogeneous domination philosophy. Using the emulation approach, we design memoryless homogeneous state and output feedback controllers, respectively, achieving GSS of the time-delay closed-loop systems with severe nonlinearity. Extensions to a wider class of time-delay nonlinear systems in the Hessenberg form are also given. Examples and counter-examples presented in this article illustrate not only the necessity of the homogeneous growth conditions but also the significance of the memoryless non-Lipschitz continuous feedback control strategies. [ABSTRACT FROM AUTHOR]

In this article, we provide several characterizations of integral input-to-state stability (iISS) of time-delay systems. These characterizations differ in the way the considered Lyapunov–Krasovskii functionals (LKFs) dissipate along the solutions of the system. This dissipation can involve the LKF itself, as in existing iISS characterizations, but can alternatively involve the instantaneous value of the solution’s norm (pointwise dissipation), the supremum norm of the state history (historywise dissipation), or a mix of the two (${\mathcal K}{\mathcal L}$ dissipation). We show that all of them guarantee iISS. By relying on a recent converse result by Y. Lin and Y. Wang, we show that most of them are also necessary for iISS. These relaxed dissipation rates simplify the iISS analysis of time-delay systems and contribute to uniforming iISS theory with that of input-free systems. Proofs rely on several results for time-delay systems that may be of interest on their own, including a novel characterization of global asymptotic stability and the fact that iISS is equivalent to global asymptotic stability of the input-free system plus a uniform bounded energy-bounded state property. [ABSTRACT FROM AUTHOR]

The increasing ease of obtaining and processing data together with the growth in system complexity has sparked the interest in moving from conventional model-based control design toward data-driven concepts. Since in many engineering applications time delays naturally arise and are often a source of instability, we contribute to the data-driven control field by introducing data-based formulas for state feedback control design in linear discrete-time time-delay systems with uncertain delays. With the proposed approach, the problems of system stabilization as well as of guaranteed cost and $H_{\infty }$ control design are treated in a unified manner. Extensions to determine the system delays and to ensure robustness in the event of noisy data are also provided. [ABSTRACT FROM AUTHOR]

This article aims to reveal an interesting phenomenon about the time delays (TDs) in control systems: stability preservation with over large intermittently or unbounded variable TDs. That is, given a system asymptotically stable for bounded TDs, it could remain to be stable when the TDs are enhanced to have peak values over large intermittently or unbounded. The investigation is carried out by theoretical analysis plus case studies with the stochastic system models. The stability of systems is the core issue; the explicit involvement of the TDs in the stability criteria is the premise for them being concerned. To these ends, a series of lemmas are established preliminarily, a new type stability theorem with its practical version is established in succession, and a stability criterion with the TDs as explicit parameters is induced. The Lyapunov function method is applied for less restriction and conservativeness. The mechanism and the approach for the emergence of the phenomenon under probe are analyzed. Implementation is considered for the linear system model. Typical variable TDs are constructed for illustrations theoretically. Finally, the obtained results are verified with numerical experiments. [ABSTRACT FROM AUTHOR]

A novel framework is introduced for the supervisory control (SC) of timed discrete event systems based on Time Petri nets. The method encompasses both logical (markings to reach or avoid) and temporal specifications (arrival and departure times in specific markings). It relies on the construction of a partial forward reachability graph of the modified state class graph type and the formulation of integer linear programming problems to establish suitable firing time intervals (FTIs) for the controllable transitions. For each enabled controllable transition, the SC algorithm provides the largest FTI that that the specifications are met, irrespectively of the firing times of the uncontrollable transitions. [ABSTRACT FROM AUTHOR]

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the Bell polynomials, the generalized Laguerre polynomials, the Lah numbers, and a suitable polynomial-based Vandermonde matrix. The results can be used to design a new controller—with time-varying gains—ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway et al. 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE–ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of time-varying kernels. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results. [ABSTRACT FROM AUTHOR]

A special class of retarded and neutral time delay systems is considered. These are plants with internal delayed feedback, and they may have finitely many or infinitely many unstable poles. Stabilizing controllers are obtained from a particular interpolation. A parametrization of all stabilizing integral-action controllers is obtained. Examples are given to illustrate this simple design procedure and its robustness properties for various uncertainties. [ABSTRACT FROM AUTHOR]

We investigate state estimation and safe controller synthesis for networked discrete-event systems (DESs), where supervisors send control decisions to plants via communication channels subject to communication delays. Previous works on state estimation of networked DES are based on the open-loop system without utilizing the knowledge of the control policy. In this article, we propose a new approach for online estimation and control of networked DES with control delays. We first propose a new state estimation algorithm for the closed-loop system utilizing the information of control decision history. The proposed state estimation algorithm can be implemented recursively upon the occurrence of each new observable event. Then we investigate how to predict the effect of control delays in order to calculate a control decision online at each instant. We show that the proposed online supervisor can be updated effectively and the resulting closed-loop behavior is safe. Furthermore, we compare the proposed online supervisor with the predictive supervisor proposed in the literature and show that our proposed online supervisor is more permissive than predictive supervisor in the sense of language inclusion. [ABSTRACT FROM AUTHOR]

In this article, the framework of network aggregative game (NAG) is extended by considering the interaction of players in a dynamic environment whose model is not known to them. The cost function of each player depends on its own strategy, the aggregate strategy of its neighbor players in the network, and also the state of the environment. The state of the environment evolves with a dynamics that is affected by the strategies of players as an input to the model. The players of NAG are modeled as myopic selfish agents who observe the decision value of their neighbors and the state of the environment with some time delay and, then, adjust their decisions using a projected subgradient rule. The main contribution of this article is to provide conditions in which the strategies of myopic agents converge to the unique fixed-point Nash equilibrium point of the proposed dynamic NAG. Furthermore, as a case study, the framework is applied to a wireless network by modeling the service providers as the agents and evolution of service selection by users as the dynamics of the environment. [ABSTRACT FROM AUTHOR]

We study transportation networks controlled by dynamic feedback tolls. We focus on a multiscale model, whereby the dynamics of the traffic flows are intertwined with those of the routing choices. The latter are influenced by the current traffic state of the network as well as by dynamic tolls controlled in feedback by the system planner. We prove that a class of decentralized monotone flow-dependent tolls allows for globally stabilizing the transportation network around a generalized Wardrop equilibrium. In particular, our results imply that using decentralized marginal cost tolls, stability of the dynamic transportation network is guaranteed around the social optimum traffic assignment. This is particularly remarkable as such dynamic feedback tolls can be computed in a fully local way without the need for any global information about the network structure, its state, or the exogenous network loads. Through numerical simulations, we also compare the performance of such decentralized dynamic feedback marginal cost tolls with constant offline (and centrally) optimized tolls both in the asymptotic and in the transient regime, and we investigate their robustness to information delays. [ABSTRACT FROM AUTHOR]

A linear delayed vector equation $\dot{x}(t)=\sum _{k=1}^m A_k(t)x(h_{k}(t)), \quad t\in [0,\infty)$ is investigated, where $x=(x_1,\ldots,x_n)^T$ is an unknown vector function. The system is considered in the most general setting and under weak assumptions about the entries of matrices $A_k$ and delays $h_{k}$. The main result on uniform exponential stability is universal in the sense that it generates a set of $2^m-1$ independent explicit statements (that can depend on all delays) on uniform exponential stability. The advantages over the existing results are demonstrated. The main tools employed by this article include the Bohl–Perron method, a priori estimates of solutions, and transformations of differential equations. [ABSTRACT FROM AUTHOR]

This article studies the robust cooperative output regulation problem of heterogeneous uncertain linear multiagent systems under unbounded distributed transmission delays. A novel distributed observer is first proposed to estimate the state of an exosystem in the presence of unbounded distributed transmission delays. Then, two novel distributed controllers, one based on state feedback and the other based on output feedback, are further developed without any prior knowledge of the unbounded transmission delays. It is shown that the resulting closed-loop multiagent systems can achieve robust cooperative output regulation. It is also shown that the better transient performance is achieved with our proposed controllers in contrast to many existing low-gain controllers, which are proposed to deal with both bounded and unbounded transmission delays. Our results can be directly applied to solve cooperative output regulation problems of multiagent systems with bounded distributed or constant transmission delays. Furthermore, our results can also be directly applied to solve leader–following consensus problems of multiagent systems with unbounded distributed, bounded distributed, or constant transmission delays. Finally, a simulation example is provided to illustrate the effectiveness of the proposed controllers. [ABSTRACT FROM AUTHOR]

The optimal linear estimation problem is studied for multirate sampling systems with multiple random measurement time delays (TDs). The considered multirate sampling scheme is that the sensor uniformly samples at a slow rate and the state uniformly updates at a fast rate. Known stochastic variable sequences obeying Bernoulli distributions are adopted to depict random measurement TDs, including missing measurements as a special case. First, using a state iterating method, the original system with multirate sampling and delayed measurements is transformed into a state-space model with single-rate sampling and delay-free measurements at measurement sampling (MS) instants. Then, by utilizing projection theory, a nonaugmented recursive optimal linear state filter is presented based on the established model in the linear minimum variance sense, where the estimators for the process noise are involved. Furthermore, the state estimator at state update instants is achieved through filtering or prediction based on the filter at MS instants. Finally, the centralized fusion estimator and the distributed covariance intersection fusion estimator are proposed for the multisensor case. Simulation research on a vehicle suspension system verifies the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In this article, we study delay robustness of PID controllers in stabilizing systems containing uncertain delays. We consider second-order systems and seek analytical characterization and exact computation of the PID delay margin, where by PID delay margin we mean the maximal range of delay values within which the system can be robustly stabilized by a PID controller. Our primary contribution is threefold. First, we show that the delay margin achieved by PID control coincides with that by PD controllers. Second, we show that the PID delay margin can be computed efficiently by solving a pseudoconcave unimodal problem, i.e., a univariate optimization problem that admits a unique maximum and, hence, is a convex optimization problem in one variable. Finally, we demonstrate analytically the tradeoff between achieving delay margin and tracking performance, showing that for several canonical performance criteria, integral control reduces the delay margin. These results lend useful insights into the PID control of delay systems, and useful guidelines in the tuning and analytical design of PID controllers. [ABSTRACT FROM AUTHOR]

This article is concerned with the design of a feedback control based on past states in order to make a given unstable hybrid stochastic differential equation (SDE) to be stable in distribution (stabilization in distribution). This is the first article in this direction. Under the global Lipschitz condition on the coefficients of the given unstable hybrid SDE, we will show that the stabilization in distribution can be achieved by linear delay feedback controls. In particular, we discuss how to design feedback controls in two structure cases: state feedback and output injection. [ABSTRACT FROM AUTHOR]

This article studies the event-triggered control problem of general nonlinear systems with time delay. A novel event-triggering scheme is presented with two tunable design parameters, based on a Lyapunov functional result for the input-to-state stability of time-delay systems. The proposed event-triggered control algorithm guarantees the resulting closed-loop systems to be globally asymptotically stable, uniformly bounded, and/or globally attractive for different choices of these parameters. Sufficient conditions on the parameters are derived to exclude Zeno behavior. Two illustrative examples are studied to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

This article studies fixed-time stabilization (FxTS) of a general controllable linear system with an input delay $\tau$. It is shown that such a problem is not solvable if the prescribed convergence time $T_{\tau }$ is smaller than $2\tau$. For $T_{\tau }\geq 3\tau$ , a solution based on linear periodic delayed feedback (PDF) without any distributed delay is established. For $T_{\tau }>2\tau$ , a solution based on linear predictor-based PDF containing a distributed delay is proposed. For both cases, the gains of the PDF can be chosen as continuous, continuously differentiable, and even smooth, in the sense of infinitely many times differentiable. If only an output signal is available for feedback, two classes of linear observers with periodic coefficients are designed, so that their states converge to the current and future states of the system at a prescribed finite time, respectively. With the observed current and future states, FxTS can also be achieved by using, respectively, the PDF and observer-based PDF. A linear periodic feedback (without delay) is also established to solve the FxTS problem of linear systems with both instantaneous and delayed controls, which cannot be stabilized by any constant instantaneous feedback in certain cases. Two numerical examples verify the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this article, we study the digital implementation of derivative-dependent control for consensus of stochastic multiagent systems. The consensus controllers that depend on the output and its derivatives are approximated as delayed sampled-data controllers. First, we consider the $n$ th-order stochastic multiagent systems. Second, we consider PID control of the second-order stochastic multiagent systems. For the consensus analysis, we propose novel Lyapunov functionals to derive linear matrix inequalities that allow us to find an admissible sampling period. The efficiency of the presented approach is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This article studies the problem of when a globally asymptotically stabilizable nonlinear system by smooth feedback is tolerable with respect to input delay. We illustrate, by means of theoretic and practical examples, some fundamental limitations including: 1) A globally exponentially stabilizable (GES) nonlinear system may not guarantee global asymptotic stabilizability (GAS) of the nonlinear system even with an arbitrarily small input delay; 2) GAS cannot even ensure semiglobal asymptotic stabilizability of the nonlinear system with a small input delay. To overcome these obstacles, we introduce the notion of semiglobal input delay tolerance (SGIDT) and present a semiglobal control framework for the asymptotic analysis/synthesis of input delay tolerance of multi-input–multi-output (MIMO) nonlinear systems under smooth feedback. With the aid of Razumikhin theorem and the converse Lyapunov theorem on global asymptotic local exponential stability (GALES), it is proved that in the case of state feedback, GALES does ensure the SGIDT of a MIMO nonlinear system. In the case of output feedback, it is further proved that GALES and uniform observability imply the SGIDT of the nonlinear system. [ABSTRACT FROM AUTHOR]

Unlike the situation with gain and phase margins in robust stabilization, the problem to determine an exact maximum delay margin is still an open problem, although extensive work has been done to establish upper and lower bounds. The problem is that the corresponding constraints in the Nyquist plot are frequency dependent, and encircling the point $s=-1$ has to be done at sufficiently low frequencies, as the possibility to do so closes at higher frequencies. In this article, we present a new method for determining a sharper lower bound by introducing a frequency-dependent shift. The problem of finding such a bound simultaneously with gain and phase margin constraints is also considered. In all these problems, we take an analytic interpolation approach. [ABSTRACT FROM AUTHOR]

The $\mathcal {H}_2$ norm of an exponentially stable system described by delay differential algebraic equations (DDAEs) might be infinite due to the existence of hidden feedthrough terms and, as shown in this article, it might become infinite as a result of infinitesimal changes to the delay parameters. We introduce the notion of strong $\mathcal {H}_2$ norm of semi-explicit DDAEs, a robustified measure that considers delay perturbations, and we analyze its properties. We derive necessary and sufficient finiteness criteria for the strong $\mathcal {H}_2$ norm in terms of a finite number of equalities involving multidimensional powers of a finite set of matrices. As the main contribution, we present a strengthened, sufficient, condition for finiteness of the strong $\mathcal {H}_2$ norm, along with an algorithm for checking it, which has significantly better scalability properties in terms of both the system dimension and the number of delays. We show that the satisfaction of the novel condition is equivalent to the existence of a simultaneous block triangularization of the matrices of the delay difference equation associated with the DDAE. The latter is instrumental to a novel regularization procedure that allows us to transform the DDAE to a neutral-type system with the same transfer matrix, without any need for differentiation of inputs or outputs. This transformation enables computing of the strong $\mathcal {H}_2$ norm using Lyapunov matrices. Finally, we show by a counterexample that the strengthened condition is, in general, not necessary, inducing open problems, but we also list several classes of DDAEs for which it is necessary and sufficient. [ABSTRACT FROM AUTHOR]

In this article, we propose an observer for a class of Lipschitz nonlinear systems affected by time-varying and known measurement delays, which is an improvement of the one presented by Cacace et al., 2014. Under the assumption that the delay function is piecewise continuous and differentiable, we prove that exponential convergence to zero of the observation error can be achieved with any desired decay rate, by suitably tuning a gain vector. The delay bound achieved with the observer proposed here is less conservative than the one obtained by Cacace et al., 2014, as confirmed by numerical tests. For the sake of brevity, in this article, only one-step observers are considered. However, a cascade observer can be arranged to cope with arbitrarily long delays. [ABSTRACT FROM AUTHOR]

A novel control law is proposed to attenuate the influence of bounded disturbances and measurement noises for plants with vector output and sector-bounded nonlinearities. The control law is based on the estimation of measurement noises. Differently from the existing results, the ultimate bound of the closed-loop system depends only on one component of the noise vector (as well as on the disturbance). The proposed control law is extended to systems with uncertain input and output delays. The input-to-state stability conditions are given in terms of matrix inequalities. The efficiency and advantages of the results over the existing methods are demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

An output robust adaptive control is designed for a class of Lipschitz nonlinear systems under assumption that the measurements are available with a constant bias and the state equations linearly parameterized by unknown parameters and external disturbances. A dynamic state reconstruction (synthesis of an observer) is avoided by using delayed values of the output in the feedback and adaptation laws. The analysis of robust stability for the resulted time-delay system is performed by using the Lyapunov–Krasovskii approach. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities. This research is motivated by a nonlinear pendulum control problem, and the efficacy of the developed control is demonstrated on this application through experiments. [ABSTRACT FROM AUTHOR]

In this article, we study constrained online convex optimization (OCO) in the presence of feedback delays, where a decision maker chooses sequential actions without knowing the loss functions and constraint functions a priori. The loss/constraint functions vary with time and their feedback information is revealed to the decision maker with delays, which arise in many applications. We first consider the scenario of delayed function feedback, in which the complete information of the loss/constraint functions is revealed to the decision maker with delays. We develop a modified online saddle point algorithm suitable for constrained OCO with feedback delays. Sublinear regret and sublinear constraint violation bounds are established for the algorithm in terms of the delays. In practice, the complete information (functional forms) of the loss/constraint functions may not be revealed to the decision maker. Thus, we further examine the scenario of delayed bandit feedback, where only the values of the loss/constraint functions at two random points close to the chosen action are revealed to the decision maker with delays. A delayed version of the bandit online saddle point algorithm is proposed by utilizing stochastic gradient estimates of the loss/constraint functions based on delayed bandit feedback. We also establish sublinear regret and sublinear constraint violation bounds for this bandit optimization algorithm in terms of the delays. Finally, numerical results for online quadratically constrained quadratic programs are presented to corroborate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]