Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.     

Stochastic model order reduction in randomly parametered linear dynamical systems

This study focuses on the development of reduced order models for stochastic analysis of complex large ordered linear dynamical systems with parametric uncertainties, with an aim to reduce the computational costs without compromising on the accuracy of the solution. Here, a twin approach to model order reduction is adopted. A reduction in the state space dimension is first achieved through system equivalent reduction expansion process which involves linear transformations that couple the effects of state space truncation in conjunction with normal mode approximations. These developments are subsequently extended to the stochastic case by projecting the uncertain parameters into the Hilbert subspace and obtaining a solution of the random eigenvalue problem using polynomial chaos expansion. Reduction in the stochastic dimension is achieved by retaining only the dominant stochastic modes in the basis space. The proposed developments enable building surrogate models for complex large ordered stochastically parametered dynamical systems which lead to accurate predictions at significantly reduced computational costs.     

Exact model matching of generalized state space systems

The problem of exact model matching for generalized state space (GSS) systems via pure proportional state and output feedback is studied. The following two major issues are resolved here for the first time: The necessary and sufficient conditions for the problem to have a solution and the general analytical expressions for the exact modelmatching controller matrices. The important case of left invertible systems is treated separately wherein simple solutions are established for the above two major issues and results on structural properties of the closed-loop system are reported. Known results on model matching of regular systems are derived as a special case of the GSS systems results, thus unifying the solution of the exact model-matching problem of regular and singular systems.The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

Stationary Analysis of a Fluid Queue Driven by Some Countable State Space Markov Chain

Motivated by queueing systems playing a key role in the performance evaluation of telecommunication networks, we analyze in this paper the stationary behavior of a fluid queue, when the instantaneous input rate is driven by a continuous-time Markov chain with finite or infinite state space. In the case of an infinite state space and for particular classes of Markov chains with a countable state space, such as quasi birth and death processes or Markov chains of the G/M/1 type, we develop an algorithm to compute the stationary probability distribution function of the buffer level in the fluid queue. This algorithm relies on simple recurrence relations satisfied by key characteristics of an auxiliary queueing system with normalized input rates.      

Coherent states for a generalized oscillator with a finite-dimensional Hilbert space

A construction of oscillator-like systems connected with a given set of orthogonal polynomials and coherent states for such systems developed by the authors is extended to the case of systems with a finite-dimensional state space. As an example, we consider a generalized oscillator connected with Krawtchouk polynomials. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 75–99.     

一类四阶偏微分多智能体系统的一致性控制

研究一类四阶偏微分多智能体系统的一致性控制问题,该类系统中的每个智能体是由四阶偏微分方程构建而成.针对系统的特点,通过构建合适空间上的Lyapunov泛函,得到分布式反馈控制律.当该反馈控制律作用于系统时,系统状态变量的一致性误差于L2(0,l)×L2(0,l)空间内收敛到零.最后,通过仿真算例验证了方法的有效性.     

A least squares identification algorithm for a state space model with multi-state delays

A parameter estimator is presented for a state space model with time delay based on the given input–output data. The basic idea is to expand the state equations and to eliminate some state variables, and to substitute the state equation into the output equation to obtain the identification model which contains the information vector and parameter vector. A least squares algorithm is developed to estimate the system parameter vectors. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithm.     

On a class of strongly stabilizable systems of neutral type

We consider the strong stabilizability problem for a delayed system of neutral type. For simplicity the case of one delay in state is studied. We separate a class of such systems and give a constructive solution of the problem in this case, without the derivative of the localized delayed state. Our results are based on an abstract theorem on the strong stabilizability of contractive systems in Hilbert space. An illustrating example is also given.     

Application of a Near-Optimal Reinforcement Learning Controller to a Robotics Problem in Manufacturing: A Hybrid Approach

Optimization theory provides a framework for determining the best decisions or actions with respect to some mathematical model of a process. This paper focuses on learning to act in a near-optimal manner through reinforcement learning for problems that either have no model or the model is too complex. One approach to solving this class of problems is via approximate dynamic programming. The application of these methods are established primarily for the case of discrete state and action spaces. In this paper we develop efficient methods of learning which act in complex systems with continuous state and action spaces. Monte-Carlo approaches are employed to estimate function values in an iterative, incremental procedure. Derivative-free line search methods are used to obtain a near-optimal action in the continuous action space for a discrete subset of the state space. This near-optimal control policy is then extended to the entire continuous state space via a fuzzy additive model. To compensate for approximation errors, a modified procedure for perturbing the generated control policy is developed. Convergence results under moderate assumptions and stopping criteria are established.     

On a class of operators on Hilbert space with applications to factorization and systems theory

A class of operators is defined in a Hilbert resolution space setting that offers a new perspective on problems of causal invertibility, special factorization, and the theory of quadratic cost optimization problems for dynamical systems. The major results include an extension of the special factorization to a class of noncompact operators and the definition of an abstract state space. These results are then used to obtain an optimal feedback solution to an abstract linear regular-quadratic cost problem.     

Dynamics and steady-state analysis of a consumer-resource model

This paper is concerned with the dynamics of a consumer-resource reaction diffusion model in the heterogeneous environment, proposed by Zhang et al. (2017). We use the comparison principle to improve the ultimate bounds step by step, and show that the unique steady state is globally asymptotically stable if the resources are fully limited uniformly in space and consumer population abundance is homogeneous in space.     

An impulse control problem of a production model with interruptions to follow stochastic demand

The paper deals with the stochastic optimal intervention problem which arises in a production & storage system involving identical items. The requests for items arrive at random and the production of an item can be interrupted during production to meet the corresponding demand. The operational costs considered are due to the stock/backlog, running costs and set up costs associated to interruptions and re-initializations. The process presents distinct behaviour on each of two disjoint identical subsets of the state space, and the state process can only be transferred from one subset to the other by interventions associated to interruptions/re-initializations. A characterization is given in terms of piecewise deterministic Markov process, which explores the aforementioned structure, and a method of solution with assured convergence, that does not require any special initialization, is provided.Additionally, we demonstrate that under conditions on the data, the optimal policy is to produce the item completely in a certain region of the state space of low stock level.     

Algorithm of the resolving of a boundary-value problem for a nonlinear controlled system and its numerical modeling

An algorithm to construct a differentiable control function guaranteeing the transfer nonlinear stationary systems of ordinary differential equations from the initial state to a given final state of the phase space such that restrictions for the control are satisfied is proposed. The proposed algorithm is convenient for numerical implementation and is applicable to a broad class of systems. A sufficient condition of the existence of a desired transfer function is constructed. A certain practical problem is considered and simulated numerically by means of the presented method.     

Noise-induced basin hopping in a gearbox model

We investigate some dynamical effects of adding a certain amount of noise in a theoretical model for rattling in single-stage gearbox systems with a backlash, consisting of two wheels with a sinusoidal driving. The parameter intervals we are dealing with show an extremely involved attraction basin structure in phase space. One of the observable effects of noise is basin hopping, or the switching between basins of different attractors. We characterize this effect and its relation to the presence of chaotic transients.     

Noise-induced basin hopping in a gearbox model

We investigate some dynamical effects of adding a certain amount of noise in a theoretical model for rattling in single-stage gearbox systems with a backlash, consisting of two wheels with a sinusoidal driving. The parameter intervals we are dealing with show an extremely involved attraction basin structure in phase space. One of the observable effects of noise is basin hopping, or the switching between basins of different attractors. We characterize this effect and its relation to the presence of chaotic transients.     

Effect of a Concentrated Normal Force on the Surface of an Orthotropic Half Space

An analytic solution is given for the stressed-deformed state of an orthotropic half space acted on by a normal concentrated force applied to its boundary. It is shown that the traditional choice of a general representation of the displacements in a gradient form, where the potential functions are in the form of double Fourier integrals with unknown densities, reduces the problem to solving two systems of linear algebraic equations – homogeneous and inhomogeneous. A solution obtained by taking a limiting transition is given for the case of an isotropic half space.     

Systems on universe spaces

The paper studies the class of analytic input/state/output systems for which the vector fields and the observation functions are only partially defined. The state space of the system is a universe space — a newly defined concept. As an example, quotient systems and observability are studied in detail. Observability is defined in a new way, consistent with the universe approach.     

Relay control of an evolution process in a Banach space in a neighborhood of an unstable stationary regime

We study a controlled evolution process whose states are characterized by points of a real Banach space and which is described by a family of semigroups in a neighborhood of an unstable stationary regime. The goal is to find control to keep the state of the process however long in the neighborhood. We find conditions under which this control is possible, and prove an existence theorem. Also, we apply the theorem to the processes described by systems of parabolic differential equations.     

Solving a Local Boundary Value Problem for a Nonlinear Nonstationary System in the Class of Feedback Controls

An algorithm convenient for numerical implementation is proposed for constructing differentiable control functions that transfer a wide class of nonlinear nonstationary systems of ordinary differential equations from an initial state to a given point of the phase space. Constructive sufficient conditions imposed on the right-hand side of the controlled system are obtained under which this transfer is possible. The control of a robotic manipulator is considered, and its numerical simulation is performed.     

Stability of hybrid stochastic delay systems whose discrete components have a large state space: A two-time-scale approach

This work is concerned with stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain has a large state space and is subject to both fast and slow movements. Under simple conditions, we demonstrate that if the limit systems are pth-moment exponentially stable, then the original systems are pth-moment exponentially stable in an appropriate sense. In addition, the exponential stability is also investigated. Moreover, stability in distribution is obtained for such hybrid systems.