A Class of Learning/Estimation Algorithms Using Nominal Values: Asymptotic Analysis and Applications

A class of estimation/learning algorithms using stochastic approximation in conjunction with two kernel functions is developed. This algorithm is recursive in form and uses known nominal values and other observed quantities. Its convergence analysis is carried out; the rate of convergence is also evaluated. Applications to a nonlinear chemical engineering system are examined through simulation study. The estimates obtained will be useful in process operation and control, and in on-line monitoring and fault detection.     

Robust Stability and Stabilizability of Markov Jump Linear Uncertain Systems with Mode-Dependent Time Delays

This paper studies the class of uncertain linear systems with time delay and Markov jump disturbance, in which the time delay is assumed to be dependent on the system mode. An LMI-based condition for this class of systems to be robustly stable is established. Sufficient conditions for the robust stabilizability under a state feedback controller are developed, and an LMI-based method to design the state feedback is proposed. Numerical examples are worked out to show the usefulness of the theoretical results.     

Random-Direction Optimization Algorithms with Applications to Threshold Controls

This work develops a class of stochastic optimization algorithms. It aims to provide numerical procedures for solving threshold-type optimal control problems. The main motivation stems from applications involving optimal or suboptimal hedging policies, for example, production planning of manufacturing systems including random demand and stochastic machine capacity. The proposed algorithm is a constrained stochastic approximation procedure that uses random-direction finite-difference gradient estimates. Under fairly general conditions, the convergence of the algorithm is established and the rate of convergence is also derived. A numerical example is reported to demonstrate the performance of the algorithm.     

Necessary and Sufficient Condition for Robust Stability and Stabilizability of Continuous-Time Linear Systems with Markovian Jumps

In this paper, we investigate the quadratic stability and quadratic stabilizability of the class of continuous-time linear systems with Markovian jumps and norm-bound uncertainties in the parameters. Under some appropriate assumptions, a necessary and sufficient condition is established for mean-square quadratic stability and mean-square quadratic stabilizability of this class of systems. The quadratic guaranteed cost control problem is also addressed via a LMI optimization problem.     

An example of a quantum exponential process

In [3],R. L. Hudson andK. R. Parthasarathy showed that the Fock space based on the Heisenberg—Weyl algebra hosts Brownian motion and Poisson processes. In this paper we construct a quantum exponential process acting on the Fock space based on the finite-difference algebra ofP. J. Feinsilver ([2]).     

Renormalization and central extensions

The stochastic limit of quantum theory [1] motivated a new approach to the renormalization program. Subsequent investigations brought to light unexpected connections with conformal field theory and some subtle relationships between renormalization and central extensions. In the present paper we review the path that has lead to these connections at the light of some recent results.     

Robust Stabilization by Dynamic Combined State and Output Feedback Compensator for Nonlinear Systems with Jumps

This paper deals with the robustness of the class of nonlinear systems with Markovian jumping parameters and unknown but bounded uncertainties. Under the assumption that the Markovian jump process (disturbance) is irreducible and under complete access to the system state and its mode, we establish robust stability results in two cases: (i) under matching conditions; and (ii) under bounded uncertainties.Research of this author was supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0036444     

Nonfragile Robust Controller for Linear Markovian Jumping Parameter Systems with Multiplicative Brownian Disturbance

This paper deals with the uncertain class of continuous-time linear systems with Markovian jumping parameters and multiplicative Brownian disturbance. A design method for a nonfragile robust controller for this class of systems is proposed when the uncertainties are of the norm-bounded type. An LMI based sufficient condition is developed. The methodology used is based mainly on the Lyapunov approach. A numerical example is presented to show the usefulness of the proposed results.     

Controlled Markov Chains with Weak and Strong Interactions: Asymptotic Optimality and Applications to Manufacturing

This paper deals with the asymptotic optimality of a stochastic dynamic system driven by a singularly perturbed Markov chain with finite state space. The states of the Markov chain belong to several groups such that transitions among the states within each group occur much more frequently than transitions among the states in different groups. Aggregating the states of the Markov chain leads to a limit control problem, which is obtained by replacing the states in each group by the corresponding average distribution. The limit control problem is simpler to solve as compared with the original one. A nearly-optimal solution for the original problem is constructed by using the optimal solution to the limit problem. To demonstrate, the suggested approach of asymptotic optimal control is applied to examples of manufacturing systems of production planning.