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This paper presents a mathematical framework for the modelling and analysis of networks of memristors, where we describe a memristor as a monotone relation between electric charge and magnetic flux. Using this framework, we show that the port behaviour of networks of monotone memristors can equivalently be described by a single monotone memristor, the so-called effective memristor. The behaviour of the effective memristor is influenced by the monotonicity properties of the single memristors and the structure of the network of memristors. An algorithm is provided to derive an explicit characterisation of the effective memristor of any network of memristors. In addition, explicit bounds on the effective memristance curve are derived in terms of the effective resistance of associated resistor networks. Finally, an explicit description of the port behaviour of series and parallel interconnections is derived and simulation examples are provided for two simple circuits.

This note establishes the characterization, existence and uniqueness of equi-normalized polytopic robust positively invariant sets for linear difference inclusions. The computation of this set results in a nonconvex optimization problem. Although this may be reformulated exactly as a mixed integer linear programme, we propose a more practical and tractable alternative in the form of a fixed-point iteration based on linear programming. Convergence of the algorithm is established.

Deep learning is a topic of considerable current interest. The availability of massive data collections and powerful software resources has led to an impressive amount of results in many application areas that reveal essential but hidden properties of the observations. System identification learns mathematical descriptions of dynamic systems from input–output data and can thus benefit from the advances of deep neural networks to enrich the possible range of models to choose from. For this reason, we provide a survey of deep learning from a system identification perspective. We cover a wide spectrum of topics to enable researchers to understand the methods, providing rigorous practical and theoretical insights into the benefits and challenges of using them. The main aim of the identified model is to predict new data from previous observations. This can be achieved with different deep learning-based modelling techniques and we discuss architectures commonly adopted in the literature, like feedforward, convolutional, and recurrent networks. Their parameters have to be estimated from past data to optimize the prediction performance. For this purpose, we discuss a specific set of first-order optimization tools that have emerged as efficient. The survey then draws connections to the well-studied area of kernel-based methods. They control the data fit by regularization terms that penalize models not in line with prior assumptions. We illustrate how to cast them in deep architectures to obtain deep kernel-based methods. The success of deep learning also resulted in surprising empirical observations, like the counter-intuitive behaviour of models with many parameters. We discuss the role of overparameterized models, including their connection to kernels, as well as implicit regularization mechanisms which affect generalization, specifically the interesting phenomena of benign overfitting and double-descent. Finally, we highlight numerical, computational and software aspects in the area with the help of applied examples.

This paper is concerned with a class of linear–quadratic stochastic large-population problems with partial information, where the individual agent only has access to a noisy observation process related to the state. The dynamics of each agent follows a linear stochastic differential equation driven by the individual noise, and all agents are coupled together via the control average term. By studying the associated mean-field game and using the backward separation principle with a state decomposition technique, the decentralized optimal control can be obtained in the open-loop form through a forward–backward stochastic differential equation with the conditional expectation. The optimal filtering equation is also provided. Thanks to the decoupling method, the decentralized optimal control can also be further presented as the feedback of state filtering via the Riccati equation. The explicit solution of the control average limit is given, and the consistency condition system is discussed. Moreover, the related $\varepsilon$-Nash equilibrium property is verified. To illustrate the good performance of theoretical results, an example in finance is studied.

Motivated by the linearized model of unstable burning in solid propellant rockets, this article addresses the adaptive event-triggered output feedback control of uncertain parabolic PDEs. First, we construct an adaptive identifier that consists of a gradient estimator, and then design a continuous-in-time controller. On this basis, we design a novel event-triggered output feedback controller and construct dynamic triggering conditions to assure the global asymptotic stability of the closed-loop system around the limit points. Furthermore, the parameter estimation is proven to converge to the true value when an additional constant input at the boundary is applied to the closed-loop system. Finally, simulation data verifies the effectiveness of the theoretical analysis.

This paper is concerned with security controller design of unknown networked systems under aperiodic denial-of-service (DoS) attacks, using only noise data but no model knowledge. First, a novel attack parameter-dependent stability criterion of linear networked systems under a class of time-constraint DoS attacks is proposed by using DoS attack parameter-dependent time-varying Lyapunov function method, where the considered system model, the state-feedback gain, and the lower and upper bounds of sleeping/active periods of DoS attack signal are known in advance. Based on this model-based stability condition and by combining tools from data-driven control theory, robust control theory, and switched system approach to security control, a new data-based stability criterion of all linear networked control systems (NCSs) which are consistent with the measured data and the assumed noise bound in the presence of DoS attacks is derived in terms of linear matrix inequalities. Based on this data-dependent parametrization, the data-driven security state-feedback controllers are designed correspondingly. Our control method guarantees the exponential stability properties robustly for all linear systems consistent with the measured data despite the presence of DoS attacks. As a byproduct, the proposed method embeds existing approaches for event-triggered control (ETC) into a general data-based event-triggered security control framework, which can be extended to co-design of data-based robust controller and event-triggering mechanism for uncertain NCSs under DoS attacks. Finally, the efficiency and superiority of the proposed methodology are verified through a numerical example.

This paper presents weighted stochastic Riccati (WSR) equations for designing multiple types of controllers for linear stochastic systems. The system matrices are independent and identically distributed (i.i.d.) to represent noise in the systems. While the stochasticity can invoke unpredictable control results, it is essentially difficult to design controllers for systems with i.i.d. matrices because the controllers can be solutions to non-algebraic equations. Although an existing method has tackled this difficulty, the method has not realized the generality because it relies on the special form of cost functions for risk-sensitive linear (RSL) control. Furthermore, designing controllers over an infinite-horizon remains challenging because many iterations of solving nonlinear optimization is needed. To overcome these problems, the proposed WSR equations employ a weighted expectation of stochastic equations. Solutions to the WSR equations provide multiple types of controllers characterized by the weight, which contain stochastic optimal and RSL controllers. Two approaches calculating simple recursive formulas are proposed to solve the WSR equations without solving the nonlinear optimization. Moreover, designing the weight yields a novel controller termed the robust RSL controller that has both a risk-sensitive policy and robustness to randomness occurring in stochastic controller design.

We study a multi-agent resilient consensus problem, where some agents are of the Byzantine type and try to prevent the normal ones from reaching consensus. In our setting, normal agents communicate with each other asynchronously over multi-hop relay channels with delays. To solve this asynchronous Byzantine consensus problem, we develop the multi-hop weighted mean subsequence reduced (MW-MSR) algorithm. The main contribution is that we characterize a tight graph condition for our algorithm to achieve Byzantine consensus, which is expressed in the novel notion of strictly robust graphs. We show that the multi-hop communication is effective for enhancing the network’s resilience against Byzantine agents. As a result, we also obtain novel conditions for resilient consensus under the malicious attack model, which are tighter than those known in the literature. Furthermore, the proposed algorithm can be viewed as a generalization of the conventional flooding-based algorithms, with less computational complexity. Lastly, we provide numerical examples to show the effectiveness of the proposed algorithm.

The famous discretized Lyapunov functional method of K. Gu employing the functionals of general structure with piecewise linear matrix kernels is known to deliver effective stability conditions in the form of linear matrix inequalities (LMIs). In parallel, the role of the delay Lyapunov matrix for linear time-invariant systems with delay was recently revealed. In Gomez et al. (2019), it was shown that the positive definiteness of a beautiful block matrix which involves the delay Lyapunov matrix values at several discretization points of the delay interval constitutes a necessary and sufficient condition for the exponential stability. The only drawback is that the dimension of the block matrix turns out to be very high in practice. In this study, we significantly reduce the dimension by combining the delay Lyapunov matrix framework with the discretized Lyapunov functional method. The component of the latter method that pertains to the discretization of the functional derivative is replaced with bounding the difference between the values of the functional possessing a prescribed derivative and its discretized counterpart. The key breakthrough lies in the fact that the structure of the block matrix is kept the same as in Gomez et al. (2019). Numerical examples show the superiority of our method in many cases compared to the other techniques known in the literature.

This note is concerned with the stability analysis problem of stochastic neutral-type time-delay systems with multiple delays. A less restrictive constraint is imposed to ensure that the difference operator (which appears on the left hand side of the stochastic neutral-type time-delay systems) possesses the property that “exponentially converging input implies exponentially converging state”. Such a constraint is necessary and sufficient for the strong stability of the difference equation associated with the system in the deterministic setting, and thus cannot be further relaxed.