Constrained regulation problem for continuous-time stochastic systems under state and control constraints

2021 ◽  
pp. 107754632110280
Author(s):  
Xindong Si ◽  
Hongli Yang
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Invariant Set ◽  
Point Of View ◽  
Linear Feedback ◽  
Stochastic Nonlinear Systems ◽  
Computation Method ◽  
Computational Point ◽  
Linear Feedback Controller ◽  
Optimization Methodology

Constrained regulation problem (CRP) for continuous-time stochastic systems is investigated in this article. New existence conditions of linear feedback control law for continuous-time stochastic systems under constraints are proposed. The computation method for solving constrained regulation problem of stochastic systems considered in this article is also presented. Continuous-time stochastic linear systems and stochastic nonlinear systems are focused on, respectively. First, the condition of polyhedral invariance for stochastic systems is established by using the theory of positive invariant set and the principle of comparison. Second, the asymptotic stability conditions in the sense of expectation for two types of stochastic systems are established. Finally, finding the linear feedback controller model and corresponding algorithm of constrained regulation problem for two types of stochastic systems are also proposed by using the obtained condition. The presented model of the stochastic constrained regulation problem in this article is formulated as a linear programming problem, which can be easily implemented from a computational point of view. Our approach establishes a connection between the stochastic constrained regulation problem and positively invariant set theory, as well as provides the possibility of using optimization methodology to find the solution of stochastic constrained regulation problem, which differs from other methods. Numerical examples illustrate the proposed method.

SET STABILIZATION OF A MODIFIED CHUA'S CIRCUIT

2005 ◽  
Vol 15 (02) ◽  
pp. 567-604 ◽  
Author(s):  
SHIHUA LI ◽  
YU-PING TIAN
Keyword(s):  
Equilibrium Point ◽  
Lyapunov Stability ◽  
Closed Loop ◽  
Invariant Set ◽  
Linear Feedback ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Linear Feedback Controller ◽  
Simulation Results ◽  
First Time

In this paper, we develop a simple linear feedback controller, which employs only one of the states of the system, to stabilize the modified Chua's circuit to an invariant set which consists of its nontrivial equilibria. Moreover, we show for the first time that the closed loop modified Chua's circuit satisfies set stability which can be considered as a generalization of common Lyapunov stability of an equilibrium point. Simulation results are presented to verify our method.

Optimal Linear Feedback Control for a Class of Nonlinear Nonquadratic Non-Gaussian Problems

1991 ◽  
Vol 113 (4) ◽  
pp. 568-574 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Control System ◽  
Stochastic Systems ◽  
Feedback Controller ◽  
Performance Criteria ◽  
Linear Feedback ◽  
Feedback Gain ◽  
Linear Feedback Controller ◽  
Gaussian Method ◽  
Optimal Linear ◽  
Non Gaussian

An optimal linear feedback controller designed for a class of nonlinear stochastic systems with nonquadratic performance criteria by a non-Gaussian approach is presented. The non-Gaussian method is developed through expressing the unknown stationary output density function as a weighted sum of the Gaussian densities with undetermined parameters. With the aid of a Gaussian-sum density, the optimal feedback gain for a control system with complete state information is derived. By assuming that the separation principle is valid for the class of stochastic systems, a nonlinear precomputed-gain filter is then implemented. The method is illustrated by a Duffing-type control system and the performance of a linear feedback controller designed through both quadratic and nonquadratic performance indices is compared.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

1993 ◽  
Vol 25 (3) ◽  
pp. 518-548 ◽  
Author(s):  
Sean P. Meyn ◽  
R. L. Tweedie
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Linear Feedback ◽  
Test Function ◽  
Markovian Processes ◽  
Harris Recurrence ◽  
Lyapunov Inequalities ◽  
Continuous Time Processes ◽  
Risk Processes ◽  
Stability Concepts

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator.Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula.We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case.We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

1993 ◽  
Vol 25 (03) ◽  
pp. 518-548 ◽  
Author(s):  
Sean P. Meyn ◽  
R. L. Tweedie
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Linear Feedback ◽  
Test Function ◽  
Markovian Processes ◽  
Harris Recurrence ◽  
Lyapunov Inequalities ◽  
Continuous Time Processes ◽  
Risk Processes ◽  
Stability Concepts

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

Preserving partial balance in continuous-time Markov chains

1987 ◽  
Vol 19 (2) ◽  
pp. 431-453 ◽  
Author(s):  
P. K. Pollett
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Equilibrium Distribution ◽  
Point Of View ◽  
Equilibrium Analysis ◽  
Practical Point ◽  
Continuous Time Markov Chains ◽  
Coupling Procedure ◽  
The Individual ◽  
Number Of Authors

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved.In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.

Preserving partial balance in continuous-time Markov chains

1987 ◽  
Vol 19 (02) ◽  
pp. 431-453 ◽  
Author(s):  
P. K. Pollett
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Equilibrium Distribution ◽  
Point Of View ◽  
Equilibrium Analysis ◽  
Practical Point ◽  
Continuous Time Markov Chains ◽  
Coupling Procedure ◽  
The Individual ◽  
Number Of Authors

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved. In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

2020 ◽  
Vol 7 (2) ◽  
pp. 34-41
Author(s):  
VLADIMIR NIKONOV ◽  
◽  
ANTON ZOBOV ◽  
Keyword(s):  
Mathematical Expectation ◽  
Point Of View ◽  
Small Range ◽  
Computational Point ◽  
Wide Range ◽  
Bijective Function ◽  
The Difference ◽  
Element Base ◽  
Selection Of

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xindong Si ◽  
Hongli Yang
Keyword(s):  
Feedback Control ◽  
Fractional Order ◽  
State Feedback Control ◽  
Invariant Sets ◽  
Linear Feedback ◽  
Control Law ◽  
Optimization Approach ◽  
Fractional Order Systems ◽  
Linear Feedback Control ◽  
Computational Point

AbstractThis paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

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