Bilinear Stochastic Systems Research Articles

Parametrization of all Linear Compensators for Discrete-time Stochastic Bilinear Systems

For discrete-time stochastic bilinear systems, this paper presents a characterization of all state covariances assignable by linear controller schemes and a parametrization of all controllers that achieve a desired covariance. These results indirectly provide the parametrization of all linear fixed-order compensators which are mean square stabilizing for this class of systems. The paper also includes robustificauon of the derived compensators

Constrained variance control with state feedback for bilinear stochastic discrete systems

Abstract The aim of this paper is to develop a methodology to find the feedback gain matrices that achieve the specified state covariance which satisfies the variance constraints for bilinear stochastic discrete systems. Moreover, a simple example is given to show the usefulness and applicability of the present method.

Dynamic covariance controllers for perturbed bilinear stochastic systems

Abstract Two objectives are reached in this paper. First, we find the set of dynamic covariance controllers such that the Root‐Mean‐Squared (RMS) value (performance requirements) for each system state of the nominal bilinear stochastic system is less than or equal to some prespecified values set by these controllers. Secondly, when modeling errors are present in the plant parameters, we seek the set of admissible perturbations such that the RMS value for each state of the uncertain closed‐loop system will still be less than or equal to some prespecified values.

Full and reduced-order observer design for discrete stochastic bilinear systems

The author considers a general discrete-time stochastic bilinear system model and derives the mean square optimal linear unbiased observer equations to be used in reconstructing a prespecified linear combination of state variables based on noisy output measurements. >

Approximation of Upper Lyapunov Exponents of Bilinear Stochastic Differential Systems

A bilinear Stochastic Differential System is considered whose solution starting from x is denoted by $(X(t,x))$; it is supposed that its (upper) Lyapunov exponent $\lambda $ exists. The purpose of the paper is to propose an efficient algorithm to approximate $\lambda $.It is shown that, for Markov chains $(\bar X_p^h (x),p \in \mathbb{N})$ defined by several approximation schemes, the Lyapunov exponents $\bar \lambda ^h $ are well defined and can be computed; estimates are given in terms of the discretization step h of the theoretical error $| {\lambda - \bar \lambda ^h } |$.Then these results are applied to an engineering problem: the stability of the motion of helicopter rotor blades in a turbulent wind.

Estimation and control of stochastic bilinear systems with prescribed degree of stability

The design is described of exponentially data-weighting state estimators and feedback controllers with a prescribed degree of stability for stochastic bilinear (stochastic-parameter) discrete-time systems. Not only stability but also stability robustness in the presence of deterministic and stochastic parameter perturbations/modelling uncertainties are considered. The existence is shown of robustness properties similar to those encountered in the theory of linear deterministic systems.

Constrained variance design for bilinear stochastic continuous systems

Recently, a theory for the control of covariance has been introduced for linear continuous systems. The theory was motivated mainly by the fact that performance requirements of many engineering control problems are naturally described in terms of the root-mean-squared (RMS) values of the system states or outputs. However, the primary purpose of this paper is to deal with the theory of covariance control for a general class of bilinear stochastic continuous systems. Moreover, on the basis of the theory of the state covariance assignment, this paper will address a methodology which deals more directly with these RMS values.

Adaptive suboptimal filtering of bilinear systems

The optimal filter (minimum mean square error) of discrete bilinear stochastic systems with output feedback is studied here. The sequential filter of bilinear systems is derived for suboptimal adaptive estimation of the unknown a priori state and observation-noise statistics simultaneously with the bilinear system state. The unbiased estimations of state-noise variance Q and observation-noise variance R are obtained under some usual conditions. For on-line operation, this paper gives the recursive form of this adaptive suboptimal filter (ASF). Computer simulations show that ASF approximates the minimum mean square error (MSE) filter very well, and ASF provides improved state estimates at little computing expense

On the controller design for perturbed bilinear stochastic systems.

The performance requirements of many engineering control problems are naturally described in terms of Root-Mean-Squared (RMS) values of the system states or outputs. The primary purpose of this paper is twofold : the first is to find the set of state feedback gains such that the RMS value for each system state of the nominal bilinear stochastic systems is less than or equal to some prespecified values. And the second is to render the closed-loop system insensitlve to modeling errors in the plant parameters. Of much significance is that the RMS design specifications will still be satisfied for all perturbations of the model parameters withn a certain set.

Adaptive Control of Multidimensional Discrete Bilinear Systems

Adaptive regulators for controlling multidimensional stochastic bilinear systems with uncertain and changing within a wide range of parameters are considered. The methods of synthesis for adaptive re-gulation laws are proposed, under which a closed loop system has a ge-neralized minimum-variance. The controlled object in this case can be nonminimum-phase. The algorithms of the adaptive control law taking ac-count of uncontrolled coordinate disturbances are discribed.

On adaptive control of stochastic bilinear systems

The minimum variance control law for bilinear systems with known parameters is shown to yield in most cases controls with infinite variance; this calls into question the use of the so-called bilinear self-tuning regulators. An adaptive weighted minimum variance controller based upon the cost with weighted control efforts is suggested for first-order bilinear systems and is shown to yield boundedness of the closed-loop system variables under a certain condition on the parameter estimate.

Realization of bilinear stochastic systems

This note considers the problem of realization of bilinear stochastic systems (BLSR). First an explicit statement for the BLSR problem is presented. Next a realization algorithm is developed. In this algorithm the state vector is picked as a basis in the subspace obtained by projecting an appropriately defined past vector onto an appropriately defined future vector. Also, the realization algorithm involves solving a matrix nonlinear equation which is akin to the algebraic Riccati equation, except for one additional term.

On discrete stochastic bilinear systems stability

Mean square stability conditions for discrete-time bilinear systems operating in a stochastic environment are given in this paper. Only independence and wide sense stationarity are required for the second order disturbance sequences involved, thus dismissing ergodicity and zero-mean assumptions. Stochastic stability conditions are derived by using a deterministic stability result for a class of separable nonlinear dynamical systems evolving in a Banach space.

Mean square stability conditions for discrete stochastic bilinear systems

Necessary and sufficient conditions for mean square stability are proved for the following class of nonlinear dynamical systems: finite-dimensional bilinear models, evolving in discrete-time, and driven by random sequences. The stochastic environment under consideration is characterized only by independence, wide sense stationarity, and second-order properties. Thus, we do not assume random sequences to be Gaussian, zero-mean, or ergodic. The probability distributions involved are allowed to be arbitrary and unknown. Limiting state moments are given in terms of the model parameters and disturbances moments.

Identification of a class of stochastic bilinear systems†

Abstract This paper considers the identification of a class of stochastic bilinear systems in which the transfer matrix in state equation is the sum of a constant matrix A and a random matrix S(i), and observation equation is linear. Estimations of covariance matrices of the random disturbances, including the state noise, observation noise and random part in transfer matrix, associated with this system are also presented. Strong consistency of the estimations is also established.

Control of Coupled Bilinear Stochastic Systems

A brief survey is presented on the state-of-the-art of bilinear, stochastic, control systems. A new contribution includes the optimal control of a generalized form of coupled, bilinear, stochastic, control systems with partial state information. While the solution generally requires the solution of an infinite number of moment equations, it is shown that if the unknown initial states are conditionally Gaussian with respect to the “measurable” ones, then under certain general conditions, the solution is possible. In fact, the solution involves a generalization of the familiar Riccati equation.

Identification of discrete-time stochastic bilinear systems

This paper considers the problem of identifying unknown parameters appearing in discrete time-varying bilinear models. The system is assumed to operate in a stochastic environment, where the input disturbance and the noise observation have unknown probability distribution. Identifiability conditions are investigated in the light of a state covariance analysis for stochastic bilinear systems. By assuming that the only accessible process is supplied by the noisy observation sequence, a recursive identification procedure is proposed which is suitable for on-line applications. Stochastic approximation algorithms are used for identifying time-invariant bilinear systems.

A separation principle for a class of bilinear stochastic systems

A separation principle for a class of bilinear stochastic systems

The use of harmonic analysis in suboptimal estimator design

The state estimation problem for bilinear stochastic systems evolving on the spheres Sn, the special orthogonal groups SO(n) , and certain other compact Lie groups and homogeneous spaces is considered. The problem is motivated by some applications involving rotational processes in three dimensions. The theory of harmonic analysis on compact Lie groups is used to define assumed density approximations which result in implementable suboptimal estimators for the state of the bilinear system. The results of Monte Carlo simulations are reported; these indicate that simple filters designed by these techniques perform well as compared to other filters.

Analysis of bilinear noise models in circuits and devices

There are a number of applications in which linear noise models are inappropriate. In this paper, the use of bilinear noise models in circuits and devices is considered. Several physical problems are studied in this framework. These include circuits involving varying parameters (such as variable resistance circuits constructed using field-effect transistors), the effect of switching jitter on sampled data system performance and communication systems involving voltage-controlled oscillators and phase-lock loops. In addition, several types of analytical techniques for stochastic bilinear systems are considered. Specifically, the moment equations of Brockett for bilinear systems driven by white noise are discussed, and closed-form expressions for certain bilinear systems (those that evolve on abelian or solvable Lie groups) driven by white or colored noise are derived. In addition, an approximate statistical technique involving the use of harmonic expansions is described.