Stability Of Linear Stochastic Systems Research Articles

The generalized matrix equation for systematic evaluation of oral Russian self-study

Abstract Generalized Lyapunov matrix equations appear in the fields of controllability analysis and model reduction of bilinear systems, stability analysis and optimal stabilization of linear stochastic systems, etc. The author studies the numerical solution of the generalized Lyapunov matrix equation combined with the evaluation of the Russian spoken language self-study system. Taking the empirical analysis as the research object, we selected 120 oral test samples from 12 colleges and universities in the 2020 national four-level unified test of Russian majors, compared with the teaching syllabus and the examination syllabus, the data were analyzed from the perspectives of speech speed, vocabulary, grammar and vocabulary errors, and a series of important conclusions were drawn, according to the specific problems exposed by the students in the oral test, the current situation of oral Russian teaching is discussed from the aspects of oral teaching materials, oral teachers and oral teaching methods.

High-Order Mean and Moment Exponential Stability Analysis Using Elimination and Duplication Matrices

This study presents a necessary and sufficient condition for high-order moment stability of linear stochastic systems. Such a high-order stability notion is efficient as a metric of both stability and robustness to system randomness. The condition is established using two key matrices called the generalized elimination and duplication matrices, which efficiently represent high-order moment vectors. The high-order vector can be treated as the state variable of another linear deterministic system that involves the generalized matrices. The necessary and sufficient condition for the high-order moment stability is associated with the eigenvalues of the deterministic system. In addition, the high-order moment stability is equivalent to high-order mean stability if the order is even.

Accurate stabilization for linear stochastic systems based on region pole assignment and its applications

On the premise of ensuring the stability of the closed-loop linear stochastic system, this paper uses region stability to further explore the precise control of its dynamic performance, which is an essential consideration in practical control. First, based on the relationship between the poles (eigenvalues) of the system state matrix and system performance, the sufficient conditions are presented for the distribution of system eigenvalues in the specific convex regions, i.e., a novel criterion for the region stability of the linear stochastic systems. Second, the region stabilization conditions are given by LMIs, which can guarantee that the system eigenvalues distribute in the connected region. Third, a new algorithm is designed to solve the stabilization condition of the eigenvalues of the state matrix located in the unconnected region. Accordingly, some dynamic performance indexes of linear stochastic systems can be controlled more accurately. Finally, two cases are provided to show the precise control for the performance indexes of the new stabilization method. • A new criterion of region stability for linear stochastic systems is presented, which can describe the system performance more accurately. • According to the region stability, a more accurate controller design method is provided, which can not only guarantee the stability of the systems, but also control the dynamic performance of the system accurately. • The convergent speed and damping response of the systems can be controlled effectively by using the region stabilization method.

Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

Interval stability and interval stabilization of linear stochastic systems with time‐varying delay

AbstractThis article first investigates the criterion of interval stability for linear stochastic systems with time‐varying delay via equivalent systems and Lyapunov–Krasovskii (L‐K) functionals. Different from existing stability conditions, the criterion of interval stability can be used to make a more accurate judgment of the stability for linear time‐delay systems. In other words, the new criterion can judge not only the stability of the system but also the speed of its convergence. Meanwhile, based on the criterion of the interval stability, the condition of interval stabilization is derived, which cannot only ensure the stability of the linear time‐delay systems, but also adjust the convergence rate of the states to the ideal level. A numerical instance is presented to illustrate the advantages of the resulting interval stabilization criterion.

Pth moment asymptotic interval stability and stabilization of linear stochastic systems via generalized [formula omitted]-representation

The problem of pth moment asymptotic interval stability and stabilization for linear stochastic systems is investigated in this paper. Firstly, we define the generalized H-representation which is very helpful to transform stochastic systems into deterministic systems. Then, the problems of pth moment interval stability, the decay rate of state response and the estimation of pth moment Lyapunov exponent are proposed with the help of the model-transformation method. Furthermore, the criteria for pth moment interval stabilization of the linear stochastic systems are addressed. Some necessary numerical examples are also provided to show the effectiveness of the addressed method.

Stability Analysis for Linear Systems with Time-Varying and Time-Invariant Stochastic Parameters

This paper presents a method to guarantee stability of linear stochastic systems. The systems include both time-varying and time-invariant unknown stochastic parameters simultaneously. For analyzing the stability, such a system is represented by an expanded system that contains only the time-invariant stochastic parameter. This expansion excludes the time-varying parameter from the system, which simplifies the stability analysis. Existing methods on robust stability theory can be thus employed to ensure stability of the expanded system. Guaranteeing stability of the expanded system is a necessary and/or sufficient condition for that of the original system. Consequently, the stability of the original system is evaluated by using linear matrix inequalities.

Optimal Stabilization Problem for the Quasilinear System with Controllable Parameters

The main concern of this paper is the problem of optimal stabilization of a quasilinear stochastic system with controllable parameters. Systems of this type are described by linear stochastic differential equations with multiplicative noises whose matrices, in general case, are nonlinear functions of control. The performance criterion is a modification of the classic quadratic performance cost. The goal is to minimize the criterion on the set of admissible control processes. This formulation of the problem is interesting because it allows us to study a wide range of optimization problems of linear systems with multiplicative perturbations, including: optimization of design parameters of the system, the problem of optimal stabilization under constraints on the gain matrix of the linear regulator in the form of inequalities, the problem of optimal stabilization of linear stochastic systems under information constraints. The main result of this paper is the necessary conditions for the optimal vector in the problem of stabilization of a quasilinear stochastic system with controllable parameters.The numerical gradient-type procedure for synthesis of the optimal stabilizing vector is also proposed. In addition, using obtained results we construct the algorithm for synthesis of a suboptimal time-dependent control. The result of the proposed algorithm is piecewise constant control, which gives the value of the criterion is guaranteed not worse than for the optimal stabilizing vector. This algorithm is relatively simple and one may use it for calculations in real time. The obtained results are applied to the problem of optimal stabilization under information constraints, in which the necessary optimality conditions are also obtained and the gradient-type procedure for the synthesis of the optimal control is proposed. The use of the obtained results is demonstrated by a model example.

Dynamic event-triggered control for linear stochastic systems with sporadic measurements and communication delays

This paper deals with the stability of linear stochastic systems in the presence of sporadic measurements and communication delays. Motivated by considerations about the efficient use of the available resources, two so-called dynamic discrete-time event-triggered control (DTETC) strategies tailored to the system are designed, including some existing periodic event-triggered control strategies as particular cases. An impulsive switched system approach and a switched time-delay system approach are developed for modeling and stability analysis of the resulting closed-loop systems, respectively. In addition, it is shown that under the same decay rate, the designed dynamic DTETC may further reduce the conservatism than their static counterparts. Interestingly, the proposed impulsive switched system approach can capture the stabilizing effect of the communication delays. A numerical example is provided to illustrate the theoretical results.

On Event-Triggered Control of Nonlinear Stochastic Systems

This technical note discusses the stability analysis and design procedure of event-triggered control (ETC) for nonlinear stochastic systems with state-dependent noise. Periodic event-generators and continuous event-generators are considered in both static and dynamic cases. All the event-generators we proposed have a guaranteed positive minimum interevent time for every sample path solution of the systems. It is shown that under the same decay rate, the dynamic event-generators may further reduce the conservatism than their static counterparts. In addition, Riccati-like conditions for global mean square exponential stability of linear stochastic systems under dynamic ETC are established. The theoretical results are illustrated by means of a numerical example.

Stabilization of linear stochastic systems with a discount: Modeling and estimation of the long-term effects from the application of optimal control strategies

The paper is devoted to the problem of stabilizing a linear stochastic control system. The quadratic cost functional measures the total loss caused by deviation from the fixed (target) levels and control trajectories, as well as a decision-maker’s time preferences expressed in the discount function. The long-term impacts of the use of decision-making, optimal on average, over an infinite-time horizon are taken as estimates of the deviation of the optimal trajectory from its target in the mean square sense and with the probability of 1.

Robust stability and stabilization of linear stochastic systems with Markovian switching and uncertain transition rates

This paper is concerned with the robust stabilization problem for a class of linear uncertain stochastic systems with Markovian switching. The uncertain stochastic system with Markovian switching under consideration involves parameter uncertainties both in the system matrices and in the mode transition rates matrix. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback controllers. A numerical example is given to illustrate the effectiveness of our results.

Spectral characterisation for stability and stabilisation of linear stochastic systems with Markovian switching and its applications

This study is concerned with the stability analysis and stabilisation of linear stochastic systems with Markovian switching by using the spectral technique. The notion of spectrum for Markovian jump linear stochastic systems (MJLSSs) is introduced and the relationship between the spectrum and the asymptotical mean-square stability is revealed. As applications, a necessary and sufficient condition for regional stability of MJLSSs and a result on generalised Lyapunov equations are derived. Examples are presented to illustrate the results established.

Stability of linear stochastic systems via Lyapunov exponents and applications to power systems

This paper studies linear systems under sustained random perturbations. The stochastic perturbation model is given by a bounded Markov diffusion process, as it appears, e.g., in the description of load or generation uncertainties of power systems. For such systems, the Lyapunov exponents describe necessary and sufficient conditions for almost sure asymptotic stability. The paper presents several numerical methods for the computation of Lyapunov exponents and applies this methodology to linear oscillators in dimension 2 and 3, and to a one machine – infinite bus electric power system.

Interval Stability and Stabilization of Linear Stochastic Systems

This technical note applies the spectrum technique to deal with the interval stability and stabilization of linear stochastic time-invariant systems. First, the notion of (-beta,-alpha) -stability of stochastic systems is defined and some relationships among the (-beta,-alpha) -stability, the decay rate of system state response, and the second-order moment Lyapunov exponent are revealed. We then address the (-beta,-alpha)-stabilization problem and give a sufficient condition via a linear matrix inequality approach.

Almost Sure Stability of Stochastic Linear Systems with Ergodic Parameters

This paper deals with the notion of almost sure stability for linear stochastic systems whose dynamic matrix depends on an ergodic process. It is shown that such systems are exponentially almost surely stable if and only if their transition matrix is averagely contractive over a finite, yet unknown, time interval. In order to test this condition, an efficient computational procedure based on a Monte Carlo strategy is proposed. Moreover, an H∞ condition ensuring exponential almost sure stability is proven. The condition requires that the mean square value of the ergodic process is less than a constant involving the H∞-norm of an appropriate transfer function. The applicability of the proposed methods is illustrated by means of a numerical example.

Discussion on: “An Algorithm for Solving a Perturbed Algebraic Riccati Equation”

ATXþ XA XSXþQþ FðXÞ 1⁄4 0, with A,S,Q 2 R n being real matrices and F(X) representing a monotonically increasing operator in the positive semi-definite sense, U V ) FðUÞ FðVÞ. Riccati equations of this type appear in several control theory areas such as optimal control of linear systems with state dependent noise [3], mean square exponential stability of linear stochastic systems [4], minimax control of switched systems under sampling [5], and optimal control of jump parameter linear stochastic systems [6]. In [5,6], the linearly coupled systems of algebraic Riccati equations (CARE) were derived. It was shown in [1] that the CARE can be treated as a special case of the PARE. The algorithm of [1] for solving the above algebraic equation is based on ARE iterations of the form

On the relationship between the sample path and moment lyapunov exponents for jump linear systems

In this paper, we study the relationship between the sample and moment Lyapunov exponents for jump linear systems. Using a large deviation theorem, a modified version of Arnold's formula for connecting sample path and moment Lyapunov exponents for continuous-time linear stochastic systems is extended to discrete-time jump linear systems. Sample path stability properties of linear stochastic systems are determined by the top Lyapunov exponent and relating sample and moment Lyapunov exponents may be useful for developing computationally efficient methods for determining the almost-sure (sample path) stability of linear stochastic systems.

Almost sure stability of linear stochastic differential equations with jumps

Under the nondegenerate condition as in the diffusion case, see [14, 21, 6], the linear stochastic jump-diffusion process projected on the unit sphere is a strong Feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump-diffusions and the corresponding diffusions due to [22]. The unique deterministic Lyapunov exponent can be represented by the Furstenberg-Khas'minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign. The critical case of zero Lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated. Several examples are treated for illustration.

Almost-Sure Stochastic Stability of Viscoelastic Plates in Supersonic Flow

The dynamic stability of a viscoelastic plate in a supersonic gas flow and subjected to a stochastically fluctuating axial thrust is performed within the concept of the Lyapunov exponent. The constitutive relation is modeled in an integral form by using the Boltzmann superposition principle. The piston theory as a quasi-first-order approximation is used to represent the aerodynamic loading on the plate. The stochastic averaging method is used and the Khasminskii's technique [Khasminskii, R. A., Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic System, Theory of Probability and Its Application, Vol. 12, No. 1, 1967, pp. 144-147 (English translation)] is employed to obtain the stability boundaries. The influence of the various plate and flow parameters and the random loading spectral densities on the stability are investigated.