Controllability of non-linear stochastic systems

Complete controllability of a semi-linear stochastic system assuming controllability of the associated linear sytem is studied. It is also shown that a non-linear stochastic system is locally null controllable provided that the corrsponding linearized system is controllable.     

Approximate analysis of non-linear stochastic systems

In this paper a method is developed whereby the statistical characteristics of the response of non-linear stochastic systems are calculated with good accuracy. The method consists of the formulation and solution of the differential equations for the statistical characteristics of the processes under consideration. The development is based on the theory of Markov processes. In particular, use is made of Ito's differential rule to obtain a basic result which is used for the formulation of the differential equations. For the solution of the differential equations the method employs an approximate analytic representation of the probability density function of a random process. The representation is in the form of a finite Edgeworth (asymptotic) expansion. The method is quite general and yields accurate results. The treatment is illustrated by examples.     

Controllability of non-linear impulsive stochastic systems

In this article, we consider the finite-dimensional dynamical control systems described by non-linear impulsive stochastic differential equations. Sufficient conditions for the complete and approximate controllability of non-linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable.     

Closed-loop control of stochastic non-linear systems

A new approach for optimum control of stochastic non-linear systems is developed from practical engineering assumptions. Systems amenable to this new approach include optimum guidance and navigation systems for space and terrestrial vehicles, optimum closed-loop process controllers and optimum controllers for systems with unknown parameters. The classical quadratic synthesis approach—optimization of a deterministic cost and perturbation estimation and control about that solution—is shown to give 24 per cent more cost and 97 per cent more mean-squared terminal error than the combined optimization approach presented for a sample problem involving control of a first-order system with an unknown time constant. A similar improvement in terminal error is shown for an atmospheric entry problem. Furthermore, the optimum controller automatically designs the best controller to minimize the effects of the unknown parameter without artificial augmentation of the cost function as is done in the sensitivity theory approach. The solution is obtained by expansion of the cost function in a power series around a deterministic trajectory with the assumption of linear perturbation estimation and control about that trajectory. Optimization of the expanded cost function gives necessary conditions dependent on the covariance matrices and the deterministic portion of the cost. When the necessary conditions are solved, a set of open-loop controls, perturbation controller gains, and perturbation estimator gains are obtained that can be precomputed and implemented into the system.     

Modelling and analysis of non-linear time series

The modelling of non-linear time series is reviewed and new results are introduced by employing some ideas from the identification of non-linear control systems. Both global and local conditions for stationary and invertibility are established for the general non-linear time-series model and it is shown how these results provide a framework for time-series estimators. Methods of computing multistep-ahead predictions are studied and the usefulness of polynomial models is discussed.     

Using the C language to approximate non-linear stochastic systems

Recognizing the growing popularity and demand for scientific or numerical libraries in the C computer language, and the shortage of systematic methods for the increasingly important non-linear stochastic systems, this article attempts to present a method to approximate the solution of systems governed by non-linear stochastic differential equations and the corresponding algorithm and computer code in the C language. It was found that the C language offers promising possibilities for the stochastic systems modeler. It is hoped the use of C for the analysis of these type of problems will be encouraged.     

Exact and approximate response of non-linear stochastic systems

The method of finding the exact solution and a new approximation method for a class of non-linear stochastic systems is presented. In both methods a non-linear transformation of the original system is applied. The idea of this method is presented for a non-linear system with stochastic parametric excitations. If the transformed system is a linear one then one can find an exact solution, otherwise the cumulant neglect closure technique is applied to this new system. The resuts obtained are illustrated by a numerical example.     

Design of decoupled non-linear multivariable stochastic control systems

A technique is presented for decoupling a non-linear system with input disturbance, The class of matrices which decouple the nominal system is first obtained. Then, the free parameters in the feedback matrix are selected to minimize the effect of disturbances on the output by minimizing a performance index which involves the input-output error magnitude. An example is presented to illustrate the approach.     

Inversion of non-linear stochastic models for the purpose of parameter estimation

Prediction error and maximum likelihood estimation of non-linear stochastic models requires inversion of the model, a step which may require substantial efforts, either in terms of manual calculations or through the use of software capable of symbolic computations. In this paper we show that model inversion can be easily implemented in numerical software such as, e.g. Simulink and Matrix X, by means of a feedback connection based on the model. It is further shown how the gradients, used for the optimization of the cost function, can be generated by a linear time-varying feedback system associated with the non-linear model. In addition, we derive sufficient conditions for the existence of a stable causal inverse as well as sufficient conditions for the initial transient to decay. These conditions are given in terms of properties for a linear time-varying system associated with the non-linear model. The method is illustrated on numerical examples.     

Modelling IEEE 802.11 CSMA/CA RTS/CTS with stochastic bigraphs with sharing

Stochastic bigraphical reactive systems (SBRS) is a recent formalism for modelling systems that evolve in time and space. However, the underlying spatial model is based on sets of trees and thus cannot represent spatial locations that are shared among several entities in a simple or intuitive way. We adopt an extension of the formalism, SBRS with sharing, in which the topology is modelled by a directed acyclic graph structure. We give an overview of SBRS with sharing, we extend it with rule priorities, and then use it to develop a model of the 802.11 CSMA/CA RTS/CTS protocol with exponential backoff, for an arbitrary network topology with possibly overlapping signals. The model uses sharing to model overlapping connectedness areas, instantaneous prioritised rules for deterministic computations, and stochastic rules with exponential reaction rates to model constant and uniformly distributed timeouts and constant transmission times. Equivalence classes of model states modulo instantaneous reactions yield states in a CTMC that can be analysed using the model checker PRISM. We illustrate the model on a simple example wireless network with three overlapping signals and we present some example quantitative properties.     

Feedback stabilizability of non-linear stochastic systems with state-dependent noise

This paper deals with non-linear stochastic systems with state-dependent noise. Based on the solution of the stochastic algebraic Riccati equation, sufficient conditions are derived which assure the existence of state feedback control laws such that the closed-loop system is locally and globally stable in probability.     

Modelling solitary waves of a fifth-order non-linear wave equation

A numerical solution of a fifth-order non-linear dispersive wave equation is set up using collocation of seventh-order B-spline interpolation functions over finite elements. A linear stability analysis shows that this numerical scheme, based on a Crank–Nicolson approximation in time, is unconditionally stable. The method is used to model the behaviour of solitary waves.     

Modelling and control of non-minimal non-linear realisations of tensegrity systems

The dynamics of tensegrity systems have their simplest mathematical form when the realisation is non-minimal. In this article, the simplicity of this structure is exploited to design controls. A Lyapunov-based approach creates a non-linear feedback control law exploring the tools of linear algebra due to the linearity of the model on a new set of transformed control variables. Explicit solutions are given for different choices of norm minimisation of the instantaneous control. The theory is illustrated through numerical simulations of simple tensegrity structures.     

State-dependent parameter modelling and identification of stochastic non-linear sampled-data systems

State-dependent parameter representations of stochastic non-linear sampled-data systems are studied. Velocity-based linearization is used to construct state-dependent parameter models which have a nominally linear structure but whose parameters can be characterized as functions of past outputs and inputs. For stochastic systems state-dependent parameter ARMAX (quasi-ARMAX) representations are obtained. The models are identified from input–output data using feedforward neural networks to represent the model parameters as functions of past inputs and outputs. Simulated examples are presented to illustrate the usefulness of the proposed approach for the modelling and identification of non-linear stochastic sampled-data systems.     

Exponential stability of switched stochastic delay systems with non-linear uncertainties

This article considers the robust exponential stability of uncertain switched stochastic systems with time-delay. Both almost sure (sample) stability and stability in mean square are investigated. Based on Lyapunov functional methods and linear matrix inequality techniques, new criteria for exponential robust stability of switched stochastic delay systems with non-linear uncertainties are derived in terms of linear matrix inequalities and average dwell-time conditions. Numerical examples are also given to illustrate the results.     

Identification of non-linear stochastic systems by state dependent parameter estimation

The paper outlines how improved estimates of time variable parameters in models of stochastic dynamic systems can be obtained using recursive filtering and fixed interval smoothing techniques, with the associated hyper-parameters optimized by maximum likelihood based on prediction error decomposition. It then shows how, by exploiting special data re-ordering and back-fitting procedures, similar recursive parameter estimation techniques can be utilized to estimate much more rapid State Dependent Parameter (SDP) variations. In this manner, it is possible to identify and estimate a widely applicable class of nonlinear stochastic systems, as illustrated by several examples that include simulated and real data from chaotic processes. Finally, the paper points out that such SDP models can form the basis for new methods of signal processing, automatic control and state estimation for nonlinear stochastic systems.     

Existence and stability behaviour of a non-linear perturbed stochastic system

The aim of the present paper is to study a non-linear perturbed integral equation of the form for t?0 and ω?Ω the supporting set of the probability measure space (Omega;, F, P). Sufficiant conditions are given for the existence and uniqueness of a random solution, a. mean square continuous stochastic process, and the stability behaviour of the system. Stability in the mean, almost surely stable, and exponentially stable; in the moan of the above system are studied.     

Diagnosis of unknown parametric faults in non-linear stochastic dynamical systems

A new detection and isolation scheme for unknown parametric faults in non-linear stochastic systems is presented that is particularly suited for small parametric changes. The proposed residual is the moving angle between two stochastic processes: the error derived from the reference model and the expected error in case a certain fault has occurred. Since the fault is unknown, fault classes are defined. Most of the faults that belong to such a fault class can be detected and isolated by just testing one representative of this fault class. As the moving angle itself is a stochastic process, an estimator is designed and characterised. Conditions for the detectability and isolability of fault classes are given for this estimator based on hypotheses tests. Finally, the theoretical results are confirmed by simulations and Monte Carlo tests.     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.