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This article is a comprehensive study of a long-observed phenomenon of increase in the stability margin and so the rate of convergence of a class of linear systems due to time delay. We use the Lambert W function to determine: 1) in what systems the delay can lead to increase in the rate of convergence; 2) the exact range of time delay, for which the rate of convergence is greater than that of the delay-free system; and 3) an estimate on the value of the delay that leads to the maximum rate of convergence. For the special case when the system matrix eigenvalues are all negative real numbers, we expand our results to show that the rate of convergence in the presence of delay depends only on the eigenvalues with minimum and maximum real parts. Moreover, we determine the exact value of the maximum rate of convergence and the corresponding maximizing time delay. We demonstrate our results through a numerical example on the practical application in accelerating an agreement algorithm for networked systems by using a delayed feedback. [ABSTRACT FROM AUTHOR]

To achieve control objectives for extremely large-scale complex networks using standard methods is essentially intractable. In this article, a theory of the approximate control of complex network systems is proposed and developed by the use of graphon theory and the theory of infinite dimensional systems. First, graphon dynamical system models are formulated in an appropriate infinite dimensional space in order to represent arbitrary-size networks of linear dynamical systems, and to define the convergence of sequences of network systems with limits in the space. Exact controllability and approximate controllability of graphon dynamical systems are then investigated. Second, the minimum energy state-to-state control problem and the linear quadratic regulator problem for systems on complex networks are considered. The control problem for graphon limit systems is solved in each case and approximations are defined which yield control laws for the original control problems. Furthermore, convergence properties of the approximation schemes are established. A systematic control design methodology is developed within this framework. Finally, numerical examples of networks with randomly sampled weightings are presented to illustrate the effectiveness of the graphon control methodology. [ABSTRACT FROM AUTHOR]

This article considers the problem of designing physical watermark signals in order to optimally detect possible replay attack in a linear time-invariant system, under the assumption that the system parameters are unknown and need to be identified online. We first provide a replay attack model, where an adversary replays the previous sensor data in order to fool the system. A physical watermarking scheme, which leverages a random input as a watermark to detect the replay attack, is then introduced. The optimal watermark signal design problem is cast as an optimization problem, which aims to achieve the optimal trade-off between control performance and intrusion detection. An online watermarking design and system identification algorithm is provided to deal with systems with unknown parameters. We prove that the proposed algorithm converges to the optimal one and characterize the almost sure convergence rate. An industrial process example is provided to illustrate the effectiveness of the proposed strategy. [ABSTRACT FROM AUTHOR]

In this technical note, the tradeoff between the attack detectability and the performance degradation in stochastic cyber-physical systems is investigated. We consider a linear time-invariant system in which the attack detector performs a hypothesis test on the innovation of the Kalman filter to detect malicious tampering with the actuator signals. We adopt a notion of attack stealthiness to quantify the degree of stealth by limiting the maximum achievable exponents of both false alarm probability and detection probability below certain thresholds. And the conditions for any actuator attack to have a specific level of stealthiness are derived. Additionally, we characterize the upper bound of the performance degradation induced by attacks with a given extent of stealthiness that produces independent and identically distributed Gaussian innovations, and design the attack, which achieves the stated upper bound for right-invertible systems. Finally, our results are illustrated via numerical examples. [ABSTRACT FROM AUTHOR]

A time delay present in the observation represents a mathematical challenge in an output feedback stabilization for linear infinite-dimensional systems. It is well known that for a linear hyperbolic system, a stabilizing output feedback control may become unstable when the observation has a time delay. For the fixed time delay in observation, the problem for one-dimensional partial differential equations (PDEs) has been solved by the observer-based feedback in the time interval where the observation is available and the predictor where the observation is not available. However, the generalization to multidimensional PDE systems has been a long-standing unsolved problem. In this paper, we investigate the problem from operator point of view for abstract first-order equation setting of infinite-dimensional systems. We formulate the problem in the framework of the well-posed and regular linear systems and solve it in the operator form. The result is then applied to the stabilization of a multidimensional Schrödinger equation. [ABSTRACT FROM AUTHOR]

This note considers the distributed optimal coordination (DOC) problem for heterogeneous linear multiagent systems. The local gradients are locally Lipschitz and the local convexity constants are unknown. A control law is proposed to drive the states of all agents to the optimal coordination that minimizes a global objective function. By exploring certain features of the invariant projection of the Laplacian matrix, the global asymptotic convergence is guaranteed utilizing only local interaction. The proposed control law is then extended with event-triggered communication schemes, which removes the requirement for continuous communications. Under the event-triggered control law, it is proved that no Zeno behavior is exhibited and the global asymptotic convergence is preserved. The proposed control laws are fully distributed, in the sense that the control design only uses the information in the connected neighborhood. Furthermore, to achieve the DOC for linear multiagent systems with unmeasurable states, an observer-based event-triggered control law is proposed. A simulation example is given to validate the proposed control laws. [ABSTRACT FROM AUTHOR]

In this paper, a numerically robust solver for least-square problems with bounded variables (BVLS) is presented for applications including, but not limited to, model predictive control (MPC). The proposed BVLS algorithm solves the problem efficiently by employing a recursive QR factorization method based on Gram–Schmidt orthogonalization. A reorthogonalization procedure that iteratively refines the QR factors provides numerical robustness for the described primal active-set method, which solves a system of linear equations in each of its iterations via recursive updates. The performance of the proposed BVLS solver, which is implemented in C without external software libraries, is compared in terms of computational efficiency against state-of-the-art quadratic programming solvers for small- to medium-sized random BVLS problems and a typical example of embedded linear MPC application. The numerical tests demonstrate that the solver performs very well even when solving ill-conditioned problems in single-precision floating-point arithmetic. [ABSTRACT FROM AUTHOR]

To solve a time-varying overdetermined problem, a novel varying-parameter convergent-differential neural network (VP-CDNN) is proposed, designed, and discussed. Specifically, a vector-error function is first defined. Second, according to neural dynamic design method, an implicit-dynamic equation with a time-varying parameter is derived. Mathematics analysis and theoretical proof verify that the VP-CDNN can obtain the least-squares solution with a super exponential convergence rate. In addition, it is also proved that VP-CDNN can restrain the noise efficiently. Simulations among the VP-CDNN, gradient-based recurrent neural network and zeroing neural network verify that the VP-CDNN has faster speed, higher accuracy, and stronger robustness. At last, applications to data fitting and system identification further verify the high effectiveness and efficiency of the VP-CDNN. [ABSTRACT FROM AUTHOR]

Periodic orbits often describe desired state trajectories of dynamical systems in various engineering applications. Stability analysis of periodic solutions lays a foundation for control design to achieve convergence to a prescribed orbit. Here we consider a class of perturbed nonlinear systems with fast and slow dynamics and develop a novel averaging method for analyzing the local exponential orbital stability of a periodic solution. A framework is then proposed for feedback control design to stabilize a natural oscillation of an uncertain nonlinear system using a synchronous adaptive oscillator. The idea is applied to linear mechanical systems and a design theory is established. In particular, we propose a controller based on the Andronov–Hopf oscillator with additional adaptation mechanisms for estimating the unknown natural frequency and damping parameters. We prove that, with sufficiently slow adaptation, the estimated parameters locally converge to their true values and entrainment to the natural oscillation is achieved as part of an orbitally stable limit cycle. Numerical examples demonstrate that adaptation and convergence can in fact be fast. [ABSTRACT FROM AUTHOR]

This paper proposes an approximate/adaptive optimal control (AOC) design for completely unknown continuous-time linear time invariant systems, without requiring the restrictive persistence of excitation (PE) condition for parameter convergence. The proposed AOC algorithm utilizes two layers of filtering—the first layer filters strategically eliminate the need for state derivative information, while the second layer filters provide suitable algebraic relations for iteratively obtaining the optimal policy under a milder online-verifiable initial excitation assumption. Unlike previous AOC algorithms, the proposed method does not require finite window integrals, intelligent data-storage, and the restrictive PE assumption. Further, the proposed method relaxes the sufficient condition required for obtaining successive stabilizing control policies. The intermediate policies are proved to be stabilizing and converging to the optimal policy. Simulation results validate the efficacy of the proposed adaptive/approximate linear quadratic regulator algorithm. [ABSTRACT FROM AUTHOR]

In this technical note, a distributed algorithm is proposed for multiagent networks to achieve a least squares solution of a system of linear equations, in which each agent only knows part of the overall equations and communicates only with its nearby neighbors. The proposed algorithm is discrete time but does not involve small or time-varying step sizes. Given that the network is fixed, connected, and undirected, the proposed algorithm enables all agents in the network to achieve exponentially fast the same least squares solution; this is validated by simulations. [ABSTRACT FROM AUTHOR]

An adaptive algorithm solving the on-line parameter estimation problem for a broad class of linear systems is proposed. The approach can be applied to systems with delay, distributed-parameter systems, fractional-order systems, and others that are stable or stabilized by linear feedback. The proposed scheme can be applied to simultaneously track sufficiently slow changes in process gains, delays, time constants, diffusivity, and other parameters. The proposed method is gradient-based, and it yields a relatively efficient numerical implementation. Convergence and robustness of the algorithm are investigated through Lyapunov analysis, yielding explicit convergence conditions that generalize the well-known “persistence of excitation” and identifiability requirements arising in conventional adaptive estimation. The method is illustrated by several examples. [ABSTRACT FROM AUTHOR]

For linear systems in the observer canonical form, we introduce a state observer with time-varying gains that tend to infinity as time approaches a prescribed convergence time. The observer is shown to exhibit fixed-time stability with an arbitrary convergence time, which is prescribed by the user irrespective of initial conditions. The output estimation error injection terms are also shown to remain uniformly bounded and converge to zero at the prescribed time. [ABSTRACT FROM AUTHOR]

Model reference adaptive control is a well-understood and popular method used in the case when the plant's constant parameters are unknown and must be identified. The related literature is very rich and there exist many proofs of stability and convergence. Lyapunov functions for such systems, having the property that the total derivative is negative definite, are, however, very scarce. In this note, we use the Mazenc construction to design a simple strict Lyapunov function in a rather intuitive manner, based on a first-choice function whose derivative is negative semidefinite. Furthermore, we provide, for the first time in the literature, a Lyapunov function for a popular passivity-based adaptive controller for Lagrangian systems. [ABSTRACT FROM AUTHOR]

This paper studies linear stochastic approximation (SA) algorithms and their application to multiagent systems in engineering and sociology. As main contribution, we provide necessary and sufficient conditions for convergence of linear SA algorithms to a deterministic or random final vector. We also characterize the system convergence rate, when the system is convergent. Moreover, differing from non-negative gain functions in traditional SA algorithms, this paper considers also the case when the gain functions are allowed to take arbitrary real numbers. Using our general treatment, we provide necessary and sufficient conditions to reach consensus and group consensus for first-order discrete-time multiagent system over random signed networks and with state-dependent noise. Finally, we extend our results to the setting of multidimensional linear SA algorithms and characterize the behavior of the multidimensional Friedkin–Johnsen model over random interaction networks. [ABSTRACT FROM AUTHOR]

In this paper, the problem of convergence for the optimal linear estimator of discrete-time linear systems with multiplicative and time-correlated additive measurement noises is studied. By defining a new random vector that consists of the innovation, error, and part of the noise in the new measurement obtained from the measurement differencing method, we obtain convergence conditions of the optimal linear estimator by equivalently studying the convergence of the expectation of a random matrix where the random matrix is the product of the new vector and its transpose. It is also shown that the state error covariance matrix of the optimal linear estimator converges to a unique fixed point under appropriate conditions and, moreover, this fixed point can be obtained by solving a set of matrix equations. [ABSTRACT FROM AUTHOR]

This technical note considers a consensus problem of high-order multiagent systems with antagonistic interactions and communication noises. The interaction network associated with the multiagent system is modeled by a signed graph (called coopetition network) and the agent dynamics is described by a general linear system. A novel stochastic-approximation based control strategy is designed for each agent by using the relative state information from its neighbors. Additionally, convergence of a consensus error system is analyzed by the stochastic stability theory. Finally, the effectiveness of our results is demonstrated by simulation. [ABSTRACT FROM AUTHOR]

This paper focuses on the design of an asynchronous dual solver suitable for model predictive control (MPC) applications. The proposed solver relies on a state-of-the-art variance reduction (VR) scheme, previously used in the context of proximal stochastic gradient methods (Prox-SVRG) and on the alternating minimization algorithm (AMA). The resultant algorithm, a stochastic AMA with VR (SVR-AMA), shows geometric convergence (in the expectation) to a suboptimal solution of the MPC problem and, compared to other state-of-the-art dual asynchronous algorithms, allows one to tune the probability of the asynchronous updates to improve the quality of the estimates. Two novel accelerated versions of the Prox-SVRG (and, by duality, of SVR-AMA) are also provided. We apply the proposed algorithm to a specific class of splitting methods, that is, the decomposition along the length of the prediction horizon. Numerical results on the longitudinal control problem of an Airbus passenger aircraft show the benefits that we can gain in terms of computation time when using our proposed solver with an adaptive probability distribution. [ABSTRACT FROM AUTHOR]

This paper proposes adaptive iterative learning control (ILC) schemes for continuous-time parametric nonlinear systems with iteration lengths that randomly vary. As opposed to the existing ILC works that feature nonuniform trial lengths, this paper is applicable to nonlinear systems that do not satisfy the globally Lipschitz continuous condition. In addition, this paper introduces a novel composite energy function based on newly defined virtual tracking error information for proving the asymptotical convergence. Both an original update algorithm and a projection-based update algorithm for estimating the unknown parameters are proposed. Extensions to cases with unknown input gains, iteration-varying tracking references, nonparametric uncertainty, high-order nonlinear systems, and multi-input-multi-output systems are all elaborated upon. Illustrative simulations are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Linear discriminant analysis (LDA) is a well-known technique for linear classification, feature extraction, and dimension reduction. To improve the accuracy of LDA under the high dimension low sample size (HDLSS) settings, shrunken estimators, such as Graphical Lasso, can be used to strike a balance between biases and variances. Although the estimator with induced sparsity obtains a faster convergence rate, however, the introduced bias may also degrade the performance. In this paper, we theoretically analyze how the sparsity and the convergence rate of the precision matrix (also known as inverse covariance matrix) estimator would affect the classification accuracy by proposing an analytic model on the upper bound of an LDA misclassification rate. Guided by the model, we propose a novel classifier, $\mathcal {DBSDA}$ , which improves classification accuracy through debiasing. Theoretical analysis shows that $\mathcal {DBSDA}$ possesses a reduced upper bound of misclassification rate and better asymptotic properties than sparse LDA (SDA). We conduct experiments on both synthetic datasets and real application datasets to confirm the correctness of our theoretical analysis and demonstrate the superiority of $\mathcal {DBSDA}$ over LDA, SDA, and other downstream competitors under HDLSS settings. [ABSTRACT FROM AUTHOR]

This paper provides an extension of the observability rank condition to nonlinear systems driven by both known and unknown inputs. In particular, the systems here considered are characterized by dynamics that are nonlinear in the state and linear in the inputs and characterized by a single unknown input (or disturbance) and multiple known inputs. Additionally, it is assumed that the unknown input is a differentiable function of time (up to a given order). The goal of the paper is not to design new observers for these systems but to provide the analytic condition in order to check the weak local observability of the state. This condition is simple and can be easily and automatically applied to the nonlinear systems mentioned above, independently of their complexity and type of nonlinearity. In particular, the complexity of the analytic condition is comparable to the complexity of the standard method to check the state observability in the case without unknown inputs (i.e., the observability rank condition). This is a fundamental practical (and unexpected) advantage. As for the observability rank condition, the proposed analytic condition is based on the computation of the observable codistribution. Similarly to the case of only known inputs, the observable codistribution is obtained by recursively computing the Lie derivatives of the outputs along the vector fields that characterize the dynamics. However, in correspondence of the unknown input, the corresponding vector field must be suitably rescaled. Additionally, the Lie derivatives of the outputs must also be computed along a new set of vector fields that are obtained by recursively performing suitable Lie bracketing of the vector fields that define the dynamics. In practice, the entire observable codistribution is obtained by a very simple recursive algorithm. Finally, it is shown that the recursive algorithm converges in a finite number of steps, and the criterion to establish that the convergence has been reached is provided. The proposed analytic extension of the observability rank condition is illustrated by checking the weak local observability of several nonlinear systems driven by known inputs and a single disturbance. [ABSTRACT FROM AUTHOR]

Based on a nonlifted iterative dynamic linearization formulation, a novel data-driven higher order optimal iterative learning control (DDHOILC) is proposed for a class of nonlinear repetitive discrete-time systems. By using the historical data, additional tracking errors and control inputs in previous iterations are used to enhance the online control performance. From the online data, additional control inputs of previous time instants within the current iteration are utilized to improve transient response. The data-driven property of the proposed method implies that no model information except for the I/O data is utilized. The computational complexity is reduced by avoiding matrix inverse operation in the proposed DDHOILC approach due to the nonlifted linear formulation of the original model. The asymptotic convergence is proved rigorously. Furthermore, the convergence property is analyzed and evaluated via three performance indexes. By elaborately selecting the higher order factors, the higher order learning control law outperforms the lower order one in terms of convergence performance. Simulation results verify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We consider a version of the pole assignment problem for linear discrete time-varying systems with linear state feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have an assignable Lyapunov spectrum if and only if the original system is uniformly completely controllable. We also provide an example to show that uniform complete controllability is not a necessary condition for assignability of the Lyapunov spectrum of the original system. [ABSTRACT FROM AUTHOR]

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, which are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr {L}_2$ -gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables. [ABSTRACT FROM AUTHOR]

This paper deals with the stabilization of linear systems with process noise under packet drops between the sensor and the controller. Our aim is to ensure exponential convergence of the second moment of the plant state to a given bound in finite time. Motivated by considerations about the efficient use of the available resources, we adopt an event-triggering approach to design the transmission policy. In our design, the sensor's decision to transmit or not is based on an online evaluation of whether the control objective may be violated in the future. The resulting event-triggering policy is hence specifically tailored to the control objective. We formally establish that the proposed event-triggering policy meets the desired objective and quantify its efficiency by providing an upper bound on the fraction of expected number of transmissions in an infinite time interval. Simulations for scalar and vector systems illustrate the results. [ABSTRACT FROM AUTHOR]

We solve the problem of stabilizing a general class of linear first-order hyperbolic systems using actuation at both boundaries of the spatial domain. We design a novel Fredholm transformation similarly to backstepping approaches to derive a boundary controller and a boundary observer enabling stabilization by output feedback. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence to zero. [ABSTRACT FROM AUTHOR]

The problem of convergent fault estimation for a class of linear systems with faults and disturbances is addressed for the first time in this technical note. A differential-equation-based convergence theorem of mean sequence of estimates is proposed, which provides an iterative mean estimation technique that has not been fully reported yet in the literature. Based on the technique, a convergent fault estimation strategy for the linear system in consideration is designed, and the convergence of mean sequence of fault estimates is proved theoretically. An iterative algorithm is given subsequently to compute the fault estimates. Numerical simulation results demonstrate the effectiveness and merits of the proposed method. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the performance analysis and synthesis of distributed system throttlers (DST). A throttler is a mechanism that limits the flow rate of incoming metrics, e.g., byte per second, network bandwidth usage, capacity, traffic, etc. This can be used to protect a service's backend/clients from getting overloaded, or to reduce the effects of uncertainties in demand for shared services. We study performance deterioration of DSTs subject to demand uncertainty. We then consider network synthesis problems that aim to improve the performance of noisy DSTs via communication link modifications as well as server update cycle modifications. [ABSTRACT FROM PUBLISHER]

Sufficient conditions for the existence and convergence to zero of numeric approximations to solutions of asymptotically stable homogeneous systems are obtained for the explicit and implicit Euler integration schemes. It is shown that the explicit Euler method has certain drawbacks for the global approximation of homogeneous systems with nonzero degrees, whereas the implicit Euler scheme ensures convergence of the approximating solutions to zero. Properties of absolute and relative errors of the respective discretizations are investigated. [ABSTRACT FROM AUTHOR]

In the paper “ISS with respect to boundary disturbances for 1-D parabolic PDEs” (IEEE Transactions on Automatic Control, vol. 61, pp. 3712–3724, 2016), input-to-state stability properties are established for 1-D spatially varying parabolic partial differential equations (PDEs) under certain assumptions, imposed on eigenvalues and eigenfunctions of an associated Sturm–Liouville operator. A key assumption on the absolute convergence of an associated Fourier series, composed of the normalized eigenfunctions and inverse eigenvalues of the Sturm–Liouville operator, is analyzed in the present note. General properties of the Sturm–Liouville operator are carried out to demonstrate that such a key assumption becomes redundant for the underlying PDEs with sign-definite sufficiently smooth coefficients. [ABSTRACT FROM PUBLISHER]