Bilinear Stochastic Systems Research Articles

Maximum likelihood based multi‐innovation stochastic gradient identification algorithms for bilinear stochastic systems with ARMA noise

SummaryThis paper considers the parameter estimation problem for bilinear stochastic systems with autoregressive moving average (ARMA) noise using the stochastic gradient method. First, the identification model is derived by eliminating the state variables. Based on the obtained identification model, a multi‐innovation generalized extended stochastic gradient (MI‐GESG) algorithm is proposed using the multi‐innovation identification theory. Furthermore, to enhance the parameter estimation accuracy, a maximum likelihood based MI‐GESG (ML‐MI‐GESG) algorithm is developed by using the maximum likelihood identification principle. Finally, an illustrative simulation example is provided to testify the proposed algorithms. The simulation results show the effectiveness of the proposed algorithms for identifying bilinear systems.

Fault diagnosis and fault‐tolerant control for bilinear stochastic distribution systems with actuator fault

SummaryIn the paper, the problem of fault diagnosis and fault‐tolerant control for the bilinear stochastic distribution system with actuator fault is studied. Firstly, based on a grain processing device, a bilinear stochastic distribution system model is established. Then, a new unknown input observer is designed to diagnose the fault, and a model‐predicted fault‐tolerant controller is designed for the system on the basis of the estimated fault information, in order to keep the system stable or still maintain the control performance after the fault occurs. Finally, the validity of the proposed algorithm is verified by the grain processing process.

Filtering-based maximum likelihood hierarchical recursive identification algorithms for bilinear stochastic systems

Filtering-based maximum likelihood hierarchical recursive identification algorithms for bilinear stochastic systems

Order Selection for Stochastic Bilinear Systems*

Identification of bilinear systems subject to white inputs is studied. Assuming bilinearity we use the crosscovariances between the output and the higher order Hermite polynomials of the input for estimating the coefficients of the Hermite series expansion of the output. The parameters of the bilinear model are obtained via a balanced factorization of the appropriately constructed Hankel matrix. The skewness of the singular values of the estimated Hankel matrix is applied for testing the order selection for the bilinear model.

Model predictive control design for constrained Markov jump bilinear stochastic systems with an application in finance

In this study, we propose a solution to the model predictive control problem for a class of constrained discrete-time bilinear stochastic systems consisting of two coupled subsystems with Markov jumps. The first one includes a bilinear term in the state variables of the second subsystem and the input, whereas the second subsystem is described by a Markov switching vector autoregressive model. Furthermore, hard constraints imposed on the input manipulated variables. The results obtained are applied to the dynamic investment portfolio selection problem for a financial market with serially dependent returns and switching modes, subject to hard constraints on trading amounts. Our approach is tested on a real dataset from the New York Stock Exchange and the Russian Stock Exchange MOEX.

Two-stage gradient-based iterative algorithm for bilinear stochastic systems over the moving data window

For the bilinear stochastic system, the difficulty of identification lies in the product of the state vector and input in the system. This paper studies the iterative estimation of the parameters and states for the bilinear state-space systems in the observer canonical form. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. Therefore, this paper proposes a state filter (SF) for the bilinear systems based on the extremum principle. By means of the hierarchical principle, we decompose the identification model into two sub-identification models by introducing two fictitious output variables. Then an SF two-stage gradient-based iterative algorithm is proposed to achieve the combined parameter and state estimation according to the gradient search. For the purpose of improving the identification performance, an SF two-stage moving data window gradient-based iterative algorithm is derived by increasing the data utilization. The numerical example demonstrates the validity of the proposed algorithms.

A new type of singular perturbation approximation for stochastic bilinear systems

Model order reduction (MOR) techniques are often used to reduce the order of spatially discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure has already been extended to bilinear equations, an important subclass of nonlinear systems. The choice of Gramians in Al-Baiyat and Bettayeb (In: Proceedings of the 32nd IEEE conference on decision and control, 1993) is the most frequently used approach. A balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on this choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend SPA to stochastic systems with bilinear drift and linear diffusion term. However, we propose a slightly modified reduced order model in comparison to previous work and choose a different reachability Gramian. Based on this new approach, an L^2-error bound is proved for SPA which is the main result of this paper. This bound is new even for deterministic bilinear systems.

Non-linear filtering for bilinear stochastic differential systems: A Stratonovich perspective

Bilinear stochastic differential equations have found applications to model turbulence in autonomous systems as well as switching uncertainty in non-linear dynamic circuits. In signal processing and control literature, bilinear stochastic differential equations are ubiquitous, since they capture non-linear qualitative characteristics of dynamic systems as well as offer closed-form solutions. The novelties of the paper are two: we weave bilinear filtering for the Stratonovich stochasticity. Then this paper unfolds the usefulness of bilinear filtering for switched dynamic systems. First, the Stratonovich stochasticity is embedded into a vector ‘bilinear’ time-varying stochastic differential equations. Then, coupled non-linear filtering equations are achieved. Finally, the non-linear filtering results are applied to an appealing bilinear stochastic Ćuk converter circuit. This paper also encompasses a system of coupled bilinear filtering equations for the vector input Brownian motion case. This paper brings the notions of systems theory, that is, bilinearity, Stratonovich stochasticity, non-linear filtering techniques and switched electrical networks together. Numerical simulation results are presented to demonstrate that the proposed bilinear filter can achieve much better and accurate filtering performance than the conventional Extended Kalman Filter (EKF).

The Auxiliary Model Based Hierarchical Estimation Algorithms for Bilinear Stochastic Systems with Colored Noises

This paper considers the parameter identification for a class of nonlinear stochastic systems with colored noise. An input-output representation is derived by eliminating the state variables in the bilinear system. Based on the obtained identification model, a recursive generalized extend least squares algorithm is proposed by using the auxiliary model identification idea. Moreover, a two-stage recursive generalized extended least squares algorithm is presented to reduce the computational burden by using the hierarchical identification principle and the auxiliary model identification idea, respectively. A stochastic gradient identification algorithm is proposed for comparison. The simulation results show that the proposed algorithms have a good performance in estimating the parameters of the bilinear systems with colored noises.

A Stochastic Optimal Regulator for a Class of Nonlinear Systems

This work investigates an optimal control problem for a class of stochastic differential bilinear systems, affected by a persistent disturbance provided by a nonlinear stochastic exogenous system (nonlinear drift and multiplicative state noise). The optimal control problem aims at minimizing the average value of a standard quadratic-cost functional on a finite horizon. It has been supposed that neither the state of the system nor the state of the exosystem is directly measurable (incomplete information case). The approach is based on the Carleman embedding, which allows to approximate the nonlinear stochastic exosystem in the form of a bilinear system (linear drift and multiplicative noise) with respect to an extended state that includes the state Kronecker powers up to a chosen degree. This way the stochastic optimal control problem may be restated in a bilinear setting and the optimal solution is provided among all the affine transformations of the measurements. The present work is a nontrivial extension of previous work of the authors, where the Carleman approach was exploited in a framework where only additive noises had been conceived for the state and for the exosystem. Numerical simulations support theoretical results by showing the improvements in the regulator performances by increasing the order of the approximation.

Energy estimates and model order reduction for stochastic bilinear systems

ABSTRACT In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretised systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied, the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an -error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.

Asynchronous control of discrete-time stochastic bilinear systems with Markovian switchings

This paper is concerned with the problem of asynchronous control for a class of discrete-time Markov systems with multiplicative stochastic white noises. Based on a stability analysis scheme developed from mode-dependent Lyapunov function method, we first derive testable conditions in linear matrix inequality (LMI) setting to ensure the robust stability of the closed-loop system. We then recast the proposed stability conditions into equivalent forms that are later utilised to design a multi-mode asynchronous state-feedback controller (ASFC) that makes the closed-loop system stable. An extension to the case of deficient mode information (i.e. transition rates of the system and the controller are not fully accessible) is also presented. Finally, a model of networked control with DC devices is given to demonstrate the efficacy of the proposed design scheme.

Free-endpoint optimal control of inhomogeneous bilinear ensemble systems

Optimal control of bilinear systems has been a well-studied subject in the areas of mathematical and computational optimal control. However, effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped. These burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience. In this work, we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free-endpoint conditions. The central idea is to represent the bilinear ensemble system at each iteration as a time-varying linear ensemble system, and then solve it in an iterative manner. We analyze convergence of the iterative procedure and discuss optimality of the convergent solutions. The method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes. We demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control.

Realization Theory for a Class of Stochastic Bilinear Systems

This paper presents a complete realization theory for a class of discrete-time, bilinear systems with observed stochastic inputs, which we call generalized bilinear systems . This class of systems includes subclasses of bilinear systems, linear parameter varying (LPV) systems, and jump-Markov linear systems. We present necessary and sufficient conditions for the existence of a realization of generalized bilinear systems, along with a characterization of minimality in terms of rank conditions. We also formulate a realization algorithm and a minimization algorithm and we show that minimality can be checked algorithmically.

Output Feedback Stabilization for a Class of Multi-Variable Bilinear Stochastic Systems With Stochastic Coupling Attenuation

In this technical note, stochastic coupling attenuation is investigated for a class of multi-variable bilinear stochastic systems and a novel output feedback $m$ -block backstepping controller with linear estimator is designed, where gradient descent optimization is used to tune the design parameters of the controller. It has been shown that the trajectories of the closed-loop stochastic systems are bounded in probability sense and the stochastic coupling of the system outputs can be effectively attenuated by the proposed control algorithm. Moreover, the stability of the stochastic systems is analyzed and the effectiveness of the proposed method has been demonstrated using a simulated example.

Recursive Least Squares and Multi-innovation Gradient Estimation Algorithms for Bilinear Stochastic Systems

Bilinear systems are a special class of nonlinear systems. Some systems can be described by using bilinear models. This paper considers the parameter identification problems of bilinear stochastic systems. The difficulty of identification is that the model structure of the bilinear systems includes the products of the states and inputs. To this point, this paper gives the input---output representation of the bilinear systems through eliminating the state variables in the model and derives a least squares algorithm and a multi-innovation stochastic gradient algorithm for identifying the parameters of bilinear systems based on the least squares principle and the multi-innovation identification theory. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

A half-wave rectifier circuit as a bilinear stochastic differential system

Abstract Bilinear stochastic differential systems have found striking applications in diverse fields, e.g. autonomous systems under noise influence, population models, theoretical biology and switched electrical networks. Because of their surprising applications, this paper develops the closed-form solution for a Stratonovich time-varying bilinear Stochastic Differential Equation (SDE). Subsequently, the analytic result of this paper is applied to a simple, but an appealing half-wave rectifier electronic circuit. Furthermore, we derive the mean and variance expressions of the state satisfying the rectifier SDE. The stochastic differential rule in combination with the concept of characteristic function of the Gaussian stochastic process plays the key role to develop the theory of this paper. The results of this paper will be useful to accomplish the noise analysis of bilinear systems arising from systems and control literature.

Simple spectral bounds for sums of certain Kronecker products

New bounds are derived for the eigenvalues of sums of Kronecker products of square matrices by relating the corresponding matrix expressions to the covariance structure of suitable bi-linear stochastic systems in discrete and continuous time.

Some new results on time-varying Stratonovich and Itô bilinear stochastic systems

The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term, where the input is a random input and the state is the output of dynamical systems. Stochastically influenced bilinear systems are described via bilinear stochastic differential equations. In this paper, first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itô counterpart. Second, the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit. The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well. The theory of this paper hinges on the ‘Stratonovich calculus’, conversion of the Stratonovich integral into the Itô integral and characteristic function of the vector Brownian motion. The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.

M-measurements Inde¯nite Linear Quadratic Optimal Control for Bilinear Stochastic Systems with Multiplicative Noises

摘要: 研究了带有乘性噪声和受扰动观测的离散时间随机系统不定线性二次(Linear quadratic, LQ) 最优输出反馈控制问题. 对此类问题而言,二次成本函数的加权矩阵不定号,并且最优控制具有对偶效果.为在最优性和计算复杂度间 进行折衷,本文采用了一种M量测反馈控制设计方法.基于动态规划方法,将未来的测量结合到当前控制 计算当中的M量测反馈控制可以通过倒向求解一类与原系统维数相同的广义差分Riccati方程(Generalized difference Riccati equation, GDRE)得到.仿真结果 表明本文提出的算法与目前普遍采用的确定等价性方法相比具有优越性. 关键词: 双线性随机系统 / 不定LQ控制 / M量测反馈 / 乘性噪声