On Stochastic Bilinear Systems

: Given a mean-square continuous stochastic vector process y with stationary increments and a rational spectral density Phi such that Phi(infinity) is finite an nonsingular, consider the problem of finding all minimal Gauss-Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal Stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families of external realizations defined on the same probability space and totally ordered with respect to state covariances. (Author)

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

Event-triggered [formula omitted] stabilization for networked stochastic systems with multiplicative noise and network-induced delays

SummaryThe aim of this survey paper is to sketch the problems and results of the qualitative theory of stochastic dynamical systems (SDS) and to give reference to papers where details can be found. Here an SDS is an ordinary differential equation ẋ = f (x,ξ) with a random noise process ξ in the r.h.s. and random initial conditions x(o) = xo. Qualitative theory studies the general nature of a solution in the entire time interval, e.g. recurrence and stability properties. Both the white noise and the real (i.e. non-white) noise case are covered. Emphasis is given to nonlinear systems (including multiplicative noise linear systems) and to stationary and Markovian noise. Standing reference is ARNOLD and KLIEMANN [7].

Structure and Entropy for Positive-Definite Toeplitz Kernels on Free Semigroups

The solutions of linear fuzzy stochastic differential systems

On minimality in the partial realization problem

In this paper, we show how to design a perfect sampling algorithm for stochastic Free-Choice Petri nets by backward coupling. For Markovian event graphs, the simulation time can be greatly reduced by using extremal initial states, namely blocking marking, although such nets do not exhibit any natural monotonicity property. Another approach for perfect simulation of non-Markovian event graphs is based on a (max,plus) representation of the system and the theory of (max,plus) stochastic systems. Next, we show how to extend this approach to one-bounded free choice nets to the expense of keeping all states. Finally, experimental runs show that the (max,plus) approach needs a larger simulation time than the Markovian approach.

Existing results in averaging theory for deterministic and stochastic systems are reviewed. In the stochastic case, the system structures considered include an ODE, which is either forced by a stochastic perturbation process or by the state of a stochastic differential equation. The history of averaging methods is briefly reviewed and the restrictions imposed in the existing stochastic averaging theory are discussed.

Partial realization of symmetric matrix rational functions

Visual sensing and adaptive control were combined in aluminum alloy pulse gas tungsten arc welding(GTAW) to control the penetration. The property of aluminum alloy GTAW was analyzed firstly. Stochastic system theory was used in the welding process, and the stochastic system model between the welding parameters and welding pool geometric parameters was established. Then real-time identification of model parameters was realized. Based on minimum variance adjustment, the adaptive controller for dominating weld formation was designed. The experiments show that adaptive control is an effective method for weld penetration in aluminum alloy GTAW.

This paper is concerned with an application of the Kalman filter to integrated navigation systems. It is shown that this filter is well adapted to resetting the dead‐reckoning position by means of a radio‐hyperbolic system or of a Doppler sonar. The algorithm can also be used for resetting the dead‐reckoning position by means of a satellite navigation system.The first part is briefly concerned with stochastic processes and system theory. The canonical equations for navigation problems are written. The second part is devoted to the Kalman filter. The third part discusses the application of the algorithm to navigation problems and presents some simulation results.

Systems and Control is now a mature subject with a deep mathematical theory and a powerful engineering methodology; its roots lie in antiquity but it emerged in the form of the Watt governor more than two centuries ago as an enabling technology of the Industrial Revolution, and its development since then involves critical contributions in a broad range of domains stretching from the Space Race to the economically vital area of industrial process control. However, the current natural and designed environment presents a vast array of control problems generated by systems of immense complexity which are often populated by subsystems possessing both discrete and continuous states and which, moreover, display so-called intelligent behaviour. A sample would have to include (i) the need to stabilize and control in a distributed manner the huge communication, information, transportation and economic networks of the modern world, (ii) the control of systems such as agricultural, fisheries and ecological systems with the objective of achieving sustainability, (iii) the management of the current integration of distributed and intrinsically stochastic renewable energy sources into deregulated and immensely expanded transmission networks serving users of all scales, and, finally, the most difficult problem area of all, (iv) the effective stabilization of the planet's climate through political and economic intervention based on system theoretic concepts and grounded in the physical sciences. Clearly the problem areas above demand a range of responses from various disciplines, but they raise at least two specific, interlinked challenges of a basic nature for Systems and Control: 1. To create fundamental theory and effective methodology to analyze and control systems with both discrete and continuous states and dynamics. Despite two decades of advances, Hybrid System theory is still in an early stage of development, for instance continuous-to-discrete state aggregation still lacks an adequate theory and so hierarchical control is in its infancy. Furthermore, work taking inspiration from the life sciences, such as gene controlled and self-healing systems, is limited to say the least. 2. To create adequate methods for the analysis and decentralized control of networked systems of huge complexity with social characteristics. While there has been progress in the development of game theory for stochastic dynamical systems, in particular Mean Field Game (MFG) theory, many challenges lie ahead, two examples being (i) relating MFG methods to the currently emerging information based control theory wherein the capacity of each of the feedback channels in a network plays a critical role, and (ii) the development of a dynamical game theory of the formation and stability of coalitions. In addition, the state aggregation problem introduced in the first challenge area also naturally falls under this second heading.

Fractional stochastic differential equations with memory effects are fundamental in modeling phenomena across physics, biology, and finance, where long‐range dependencies and random fluctuations coexist, yet their stability analysis under non‐Lipschitz conditions remains a significant challenge, particularly for systems involving Riemann–Liouville fractional operators with stochastic perturbations. To address this challenge, this article establishes sufficient conditions for the existence, uniqueness, and Ulam–Hyers–Rassias (UHR) stability of mild solutions to nonlinear Riemann–Liouville fractional stochastic differential equations (RL‐FSDEs) driven by additive white noise. Through the application of a generalized Banach contraction principle within a carefully constructed weighted metric space, our methodology relaxes the restrictive Lipschitz constraints commonly found in existing literature, thereby significantly broadening the class of admissible nonlinearities. The UHR stability is rigorously established in the mean‐square sense by deriving explicit bounds that systematically relate approximate solutions to exact trajectories. Our theoretical findings are substantiated through four illustrative examples demonstrating that solutions remain consistently close to approximate trajectories under random fluctuations, thereby significantly advancing the stability theory of stochastic fractional systems and providing a robust analytical framework for applications in control theory, financial modeling, and engineering systems governed by memory effects and stochastic noise.

In this paper, we study the synchronization of identical nonlinear systems over large-scale network with uncertainty in the interconnections. We consider a special class of nonlinear systems which have a dissipative nonlinearity and the stability of such systems can be analyzed using absolute stability theory tools like the Positive Real Lemma, Bounded Real Lemma and dissipativity theory. We extend this analysis to the stochastic setting over a network where the interconnection weights are drive by Wiener process with given mean and variance. To capture the stability of the synchronized state, we study the notion of mean square stability from stochastic stability theory and formulate a network size-independent sufficient condition based on the theory of stochastic dissipative systems. We also compute a heuristic margin of synchronization for the networked systems to indicate the tolerance to stochastic uncertainty in interconnection links.