Stabilization of switched linear stochastic dynamical systems under limited mode information

This paper derives conditions under which co-diagonalization of the controllability and observability Gramians can be performed in a physically consistent manner for the class of linear quantum stochastic systems that are common in the fields of quantum optics and related areas like quantum nano- and optomechanical systems. Such a class is essentially a quantum analogue of classical (non-quantum) linear stochastic systems. However, unlike their classical counterparts, the laws of quantum mechanics impose nonlinear equality constraints on the system matrices of linear quantum stochastic systems, the so-called physical realizability constraints. Therefore, a meaningful model reduction procedure must yield a physically realizable reduced model satisfying these constraints. Previous work has shown that model reduction by subsystem truncation preserves physical realizability, but did not propose a method for truncating a subsystem. One candidate approach for this truncation is a quantum adaptation of the well-known balanced truncation method for classical linear time-invariant systems. It is shown in this paper that balanced realization for linear quantum stochastic systems is only possible under a strongly restrictive necessary and sufficient condition. However, the paper also derives less restrictive necessary and sufficient conditions for other realizations with simultaneously diagonal controllability and observability Gramians, and introduces the notion of a quasi-balanced realization. An example of the application of the results is provided to demonstrate quasi-balanced truncation in the linear quantum stochastic setting.

This paper presents a preliminary algorithm for determining the orders of a linear stochastic dynamic system solely from the observation of the system output. The scalar transfer function of the system contains both poles and zeros, however, no a priori information is available about the numbers and values of these poles and zeros. The driving input of this system is assumed to be white. Such a system is called a mixed type process, since it is a combination of the autoregressive (all-pole) and the moving-average (all zero) processes. The objective of this algorithm is thus to determine the number of poles (order of the autoregressive portion of the process) and the number of zeros (order of the moving-average portion of the process) of the mixed type process. The algorithm suggested in this paper is based on a two stage process: first take those output correlations which follow a repetitive pattern to determine the order of the autoregressive portion, then proceed to estimate the autoregressive parameters and lower the index of the correlation until the significant deviation from the repetitive pattern is observed, the order of the moving-average then can be determined.

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems, we have designed an optimal controller which guarantees the exponential stability of the system. Actually, we employed Lyapunov function approach and the stochastic algebraic Riccati equation (SARE) to have shown the robustness of the linear quadratic(LQ) optimal control law. And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed-loop systems are given.

This paper discusses the optimal preview control problem for a class of linear continuous stochastic control systems in the infinite horizon, based on the augmented error system method. Firstly, an assistant system is designed and the state equation is translated to the assistant system. Then, an integrator is introduced to construct a stochastic augmented error system. As a result, the tracking problem is converted to a regulation problem. Secondly, the optimal regulator is solved based on dynamic programming principle for the stochastic system, and the optimal preview controller of the original system is obtained. Compared with the finite horizon, we simplify the performance index. We also study the stability of the stochastic augmented error system and design the observer for the original stochastic system. Finally, the simulation example shows the effectiveness of the conclusion in this paper.

Linear discrete time stochastic dynamical systems with switching parameters may represent a wide class of physical processes. Given the fact that, for non linear systems, the conditional density has no finite parametrization, optimal filters for this class of models are generally infinite dimensional. An exact recursive hybrid filter form has been recently found by Elliott and al [3]. The purpose of this note is to present a new simple approximation scheme of this exact version, which avoids the exponential growth memory problem of the original form. This new numerical scheme is shown to be computationnally efficient. Its implementation is discussed and a comparison of its performances with the Interacting Multiple Models (IMM) algorithm [2] shows its efficiency.

Due to the informational constraints, the computation and implementation of the Decentralized Fault Detection and Diagnosis (DFDD) algorithms are very complex. In this paper, we propose a new method for DFDD in large linear stochastic dynamical systems using an unknown input observer estimator scheme. We show that the Kalman filter—Genaralized Likelihood Ratio (GLR) test combination with our decentralized observer—estimator is similar to the approach of Willsky and Jones in centralized situations.

This paper is devoted to linear stochastic differential systems with fractional noise (or fractional Brownian motion) input. On the basis of a convenient Markovian description of such noises, elaborated from a diffusive representation of fractional integrators previously introduced in a deterministic context, the fractional differential system is equivalently transformed into a standard (but infinite-dimensional) one, with white-noise input. Finite dimensional approximations may easily be obtained from classical discretization schemes. With this equivalent representation, the correlation function of processes described by linear fractional stochastic differential systems may be expressed from the solution of standard differential systems, which generalizes, in some way, the well-known differential Lyapunov equation, which appears when computing the covariance matrix associated with standard linear stochastic systems.

Formal synthesis approaches over stochastic systems have received significant attention in the past few years, in view of their ability to provide provably correct controllers for complex logical specifications in an automated fashion. Examples of complex specifications include properties expressed as formulae in linear temporal logic (LTL) or as automata on infinite strings. A general methodology to synthesize controllers for such properties resorts to symbolic models of the given stochastic systems. Symbolic models are finite abstractions of the given concrete systems with the property that a controller designed on the abstraction can be refined (or implemented) into a controller on the original system. Although the recent development of techniques for the construction of symbolic models has been quite encouraging, the general goal of formal synthesis over stochastic control systems is by no means solved. A fundamental issue with the existing techniques is the known curse of dimensionality, which is due to the need to discretize state and input sets. Such discretization generally results in an exponential complexity over the number of state and input variables in the concrete system. In this work we propose a novel abstraction technique for incrementally stable stochastic control systems, which does not require state-space discretization but only input set discretization, and that can be potentially more efficient (and thus scalable) than existing approaches. We elucidate the effectiveness of the proposed approach by synthesizing a schedule for the coordination of two traffic lights under some safety and fairness requirements for a road traffic model. Further we argue that this 5-dimensional linear stochastic control system cannot be studied with existing approaches based on state-space discretization due to the very large number of generated discrete states.

In this paper, an optimal filter for a stochastic linear system with previous stage noise correlation is designed. Based on this result, together with the decomposition techniques of the stochastic singular linear system, the design of an optimal filter for a stochastic singular linear system is given.

Bayesian optimal control for a non-autonomous stochastic discrete time system

A fast state-space self-tuner is developed for suboptimal control of linear stochastic multivariable systems. The suboptimal self-tuner is determined by utilising both the standard recursive-extended-least-squares parameter estimation algorithm and the recently developed matrix sign algorithm, which gives a fast solution of the steady-state discrete Riccati equation. The developed suboptimal state-space self-tuner can be applied to a class of stable/unstable and minimum/non-minimum phase linear stochastic multivariable systems, in which the pair (A, C) is block observable and the pair (A, B) is stablisable. Also, the pair (A, Q?) with Q?Q?T = Q is detectable where A, B and C are system, input and output matrices, respectively, and Q is a weighting matrix in a quadratic performance index.

This Paper deals with the recursive identification problem of stochastic linear dynamical systems , Important Algorithms are explained in the identification system domains that are time-varying, and using a recursive Least Square method with a famous approach to estimate the model parameter, that a forgetting factor, and a Kalman filter approach with different values for a best linear dynamic models, that are identified from the two type of stochastic linear dynamic systems: that the equation error models which contain ARX models and ARMAX models. Output error models which consist of OE and Box-Jenkins models, that are reached by using the suggested instrument in Off-Line Identification, where the exact Linear models reached their parameter stable with Time, Moreover, the Statistic terms are verified from a point of random errors and insignificant cross-correlation between inputs and outputs residual.

The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. Results are derived on subsystem truncation of such systems and it is shown that this truncation preserves the physical realizability property of linear quantum stochastic systems. It is also shown that the property of complete passivity of linear quantum stochastic systems is preserved under subsystem truncation. A necessary and sufficient condition for the existence of a balanced realization of a linear quantum stochastic system under sympletic transformations is derived. Such a condition turns out to be very restrictive and will not be satisfied by generic linear quantum stochastic systems, thus necessary and sufficient conditions for relaxed notions of simultaneous diagonalization of the controllability and observability Gramians of linear quantum stochastic systems under symplectic transformations are also obtained. The notion of a quasi-balanced realization is introduced and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. Moreover, an explicit bound for the subsystem truncation error on a quasi-balanceable linear quantum stochastic system is provided. The results are applied in an example of model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities.

In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered.

We investigate the separation principle for a class of mean-field stochastic optimal control problems with partial information, where the linear stochastic dynamic system is observed through a linear noisy channel. In our model, the time inconsistency caused by the mean-field terms makes the dynamic programming principle (DPP) inapplicable, which immediately leads to difficulties in deriving the separation principle. Utilizing the optimal filtering equations in several dimensions, we can transform the stochastic optimal control problem with partial information into one with complete details. Subsequently, the Hamilton-Jacobi-Bellman (HJB) equation that can solve the optimal control is obtained with the support of the dynamic programming principle. The separation principle for solving the optimal control of the original problem is derived.