Sufficient conditions for hyperstability of a class of nonlinear systems

The concept of hyperstability is applied to obtain a new stability criterion for a class of nonlinear systems. The physical system considered is the chemical reactor which gives rise to a very interesting aspect of stability study.     

On the hyperstability of a class of hybrid systems

Sufficient conditions of the hyperstability of a class of single-input single-output hybrid systems, composed of interconnected continuous and discrete subsystems, are given. Hyperstability is guaranteed by that of a discrete extended system which is built by incorporating samples of the continuous substrate. Asymptotic hyperstability is guaranteed by that of one discrete system, together with that of a continuous one. Both systems are related to an augmented state representation of the linear part of the overall system for all the nonlinear output-feedback time-varying actuators that satisfy generalized Popov-type inequalities.     

Lyapunov conditions certifying stability and recurrence for a class of stochastic hybrid systems

Lyapunov-based conditions for stability and recurrence are presented for a class of stochastic hybrid systems where solutions are not necessarily unique, either due to nontrivial overlap of the flow and jump sets, a set-valued jump map, or a set-valued flow map. Randomness enters exclusively through the jump map, yet the framework covers systems with spontaneous transitions. Regularity conditions are given that guarantee the existence of random solutions and robustness of the Lyapunov conditions.     

A generalization of the hyperstability conditions for model reference adaptive systems

A generalization of the hyperstability criterion given in the literature is discussed. The new hyperstability conditions were obtained by utilizing the properties of the positive definite kernels. The results presented here are directly applicable for the synthesis of hyperstable adaptation algorithms for various applications.     

A result on the hyperstability of a class of hybrid dynamic systems

This paper presents a hyperstability theorem for a class of hybrid dynamic systems composed of coupled differential and difference equations subject to (possibly) time-varying nonlinearities satisfying a Popov-type inequality. The nonlinear controller generates the plant input at all times from its sampled values by defining an extended discrete system. The hyperstability results are obtained from this discrete system of special type whose state consists of the sampled continuous substate and the digital substate of the given hybrid system. Some corollaries and related physical interpretations are also given     

Sampling average approximation method for a class of stochastic Nash equilibrium problems

We consider a class of stochastic Nash equilibrium problems (SNEP). Under some mild conditions, we reformulate the SNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce a semismooth Newton method to solve the SAA problems. The comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas–oil markets.     

Output-feedback control of a class of stochastic nonlinear systems with power growth conditions

This paper investigates the problem of output-feedback stabilization for a class of stochastic nonlinear systems in which the nonlinear terms depend on unmeasurable states besides measurable input. We extend linear growth conditions to power growth conditions and reduce the control effort. By using backstepping technique, choosing a high-gain parameter, an output-feedback controller is designed to ensure the closed-loop system globally asymptotically stable in probability. The efficiency of the output-feedback controller is demonstrated by a simulation example.     

Adaptive stochastic control for a class of linear systems

The problem considered in this paper deals with the control of linear discrete-time stochastic systems with unknown (possibly time-varying and random) gain parameters. The philosophy of control is based on the use of an open-loop feedback optimal (OLFO) control using a quadratic index of performance. It is shown that the OLFO system consists of 1) an identifier that estimates the system state variables and gain parameters and 2) a controller described by an "adaptive" gain and correction term. Several qualitative properties and asymptotic properties of the OLFO adaptive system are discussed. Simulation results dealing with the control of stable and unstable third-order plants are presented. The key quantitative result is the precise variation of the control system adaptive gains as a function of the future expected uncertainty of the parameters; thus, in this problem the ordinary "separation theorem" does not hold.     

Identification of a class of non-linear systems with gaussian non-white inputs

An identification algorithm for open-loop non-linear systems composed of linear dynamic and static non-linear elements is developed for the case of gaussian non-white inputs. It is shown that correlation analysis provides estimates of the parametrized linear and second-order kernels from which the parameters of the linear subsystems can be obtained.     

Averaged dynamics and pole assignment for a class of stochastic systems

For a linear system with Markov jump parameters, the moment Liapunov exponents are computed explicitly in terms of averaged dynamics. Besides giving a necessary and sufficient condition for moment stability, the computation suggests that control design for these systems can be aimed at assigning desired closed-loop eigenvalues. This is shown to be equivalent to a decentralized control problem to which the homotopy method applies and the control gains are obtained by continuation from the solution of the decoupled (no jump) problem.     

Adaptive tracking control for a class of stochastic switched systems

The problem of adaptive tracking is considered for a class of stochastic switched systems, in this paper. As preliminaries, the criterion of global asymptotical practical stability in probability is first presented by the aid of common Lyapunov function method. Based on the Lyapunov stability criterion, adaptive backstepping controllers are designed to guarantee that the closed-loop system has a unique global solution, which is globally asymptotically practically stable in probability, and the tracking error in the fourth moment converges to an arbitrarily small neighbourhood of zero. Simulation examples are given to demonstrate the efficiency of the proposed schemes.     

Optimal nonlinear estimation for a class of discrete-time stochastic systems

Recursive estimation for nonlinear discrete-time stochastic systems with additive white Gaussian observation noise is investigated. It is proved that for certain classes of systems, described either by finite Volterra series expansions or by state-linear realizations under certain algebraic conditions, the optimal conditional mean estimator is recursive and of fixed finite dimension. An example is presented to illustrate the structure of the estimators.     

Infinite time optimal control for a class of stochastic systems

Optimization results developed by the author (Brandeberry and Wu 1970) for a class of stochastic regulator problems will be extended from the finite time interval to the infinite time interval. Conditions are given for the existence and stability of the infinite constant feedback of the system state; necessary conditions will also be obtained for the inverse problem (i.e. conditions for a linear constant control law to be optimal for some cost functional).     

A minimum principle for a class of discrete-time stochastic systems

A minimum principle is obtained for discrete-time stochastic systems described by the stochastic difference equationx_{k+1} = A_{k}x_{k} + phi_{k}(u_{k})+w_{k}where{w_{k}, k = 0, ... ,N - }is la sequence of independent random vector variables. The control action ukis constrained to belong to a compact set Uk, and the setphi_{k}(U_{k}), k = 0,..., N - 1is convex. The system is open-loop.     

Near optimal control for a class of stochastic hybrid systems

To address a computationally intractable optimal control problem for a class of stochastic hybrid systems, this paper proposes a near optimal state feedback control scheme, which is constructed by using a statistical prediction method based on approximate numerical solution that samples over the entire state space. A numerical example illustrates the potential of the approach.     

Adaptive observer-based control for a class of nonlinear stochastic systems

In this paper, the adaptive state estimation and state-feedback stabilization problems for a class of nonlinear stochastic systems with unknown constant parameters are studied. The sequential design methods are proposed to construct the adaptive controllers. Adaptive state and parameter estimators are designed by using a stochastic Lyapunov method and the separation theory of the design for the state-feedback gain and observer gain, which guarantees that the closed-loop system is asymptotically stable in the mean-square sense. Sufficient conditions for the existence of parameters estimator are given in terms of linear matrix inequalities. Finally, the numerical examples are provided to illustrate the feasibility of the proposed theoretical results.     

Hyperstability of linear time-varying discrete systems

The hyperstability of a class of linear time-varying discrete systems is studied. Sufficient conditions for hyperstability and for stability in the sense of Lyapunov are obtained.     

Frequency-domain criteria for stability of a class of nonlinear stochastic systems

The problem of finding frequency-domain conditions that are sufficient to ensure asymptotic stability with probability one (ASWP 1) of a Lure-type system with white-noise input disturbance is considered. It is shown that, if the noise is linearly related to the state of the system, a relatively simple frequency-domain inequality that guarantees ASWP 1 of the system exists. The stochastic version of the second method of Lyapunov, along with a Meyer-Kalman-Yakubovich(MKY)-type lemma, is used to derive such a condition, assuming first that only sector information of the nonlinearity is available. A modification of this lemma is subsequently used to derive an improved stability condition, assuming that both sector and slope information of the nonlinearity are available. Finally, the case of a differential system with white-noise parameter perturbation is considered and an illustrative example is presented.     

A control scheme for a class of discrete nonlinear stochastic systems

A control scheme is presented for effective stabilization of discrete-time, nonlinear stochastic systems where the nonlinearity involves a zero-mean, independent random sequence. The control is of a constant feedback type and makes use of finite time solutions of a Riccati-like matrix difference equation. Stability results are both in terms of mean square and almost sure stochastic stability. Moreover, a matrix inequality is given to check the existence of weighting matrices which would result in a stable regulator problem.