On the zero dynamics of one class of distributed discrete systems

The problem of optimal control of a linear dynamical system under set-membership uncertainty is studied: it is required to steer the system to the terminal set with a guarantee and to maximize the guaranteed value of the quality criterion. The sets of the initial and current a preposteriori distributions of the states of the dynamical system are introduced; they are used to determine a positional solution of the problem of optimal a preposteriori³ observation with the help of inaccurate measurements of input and output signals of the observation object by two measuring devices. The obtained solution is used for determining a positional solution of the optimal control problem under uncertainty. Depending on the amount of the information used, optimal closable and closed output feedbacks are determined. The method of quasiimplementation of optimal feedbacks by means of optimal estimators and a regulator producing real-time control actions is described. The results are illustrated by examples.     

Solution of a problem in the optimal estimation of the states of linear dynamical systems

A problem in the optimal estimation of a linear dynamical system is solved under the condition that the effects on the system of noise and observation error belong to a prescribed parallelepiped region.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 110–114, 1989.     

Optimal estimation of a state based on deterministic and stochastic discrete measurements

A problem of constructing algorithms of optimal estimation of states of a discrete system accounting for deterministic and stochastic measurements is considered. A low order estimation algorithm is proposed in the form of a two-stage procedure for each instant of time obtained by direct application of the Kaiman optimal filtering algorithm.Translated from Dinamicheskie Sistemy, No. 4, pp. 85–89, 1985.     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.     

New algorithm for optimal parameter estimation with linear constraints

This paper presents a new algorithm for optimal parameter estimation problems with linear constraints. The algorithm developed is based on least absolute-value approximations. The problem is solved first using a least-error-square technique, where we add to the cost function the equality constraints via Lagrange multipliers, to obtain a good estimate for the residuals of the measurements, having gained this information, we choose a number of measurements with the smallest residuals. This number equals the number of parameters to be estimated minus the number of constraints. Using these measurements together with the constraints, we obtain a number of observations equal to the number of parameters to be estimated. By using this technique, we show that there is no need to either iterate or use linear programming to obtain the estimation.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant A4146.     

Reconstruction of boundary controls in parabolic systems

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system’s motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.     

Optimal estimation of parameters and states in stochastic time-varying systems with time delay

In this study estimation of parameters and states in stochastic linear and nonlinear delay differential systems with time-varying coefficients and constant delay is explored. The approach consists of first employing a continuous time approximation to approximate the stochastic delay differential equation with a set of stochastic ordinary differential equations. Then the problem of parameter estimation in the resulting stochastic differential system is represented as an optimal filtering problem using a state augmentation technique. By adapting the extended Kalman–Bucy filter to the resulting system, the unknown parameters of the time-delayed system are estimated from noise-corrupted, possibly incomplete measurements of the states.     

Optimal observation and control in linear systems

The paper consists of two parts. In Part I, we consider the optimal observation problem for a nondeterministic linear system by examining results of a deep processing of an output signal of a dynamical measurement device (sensor). The problem studied is an auxiliary problem for studying the optimal control problem for dynamical systems under set-uncertainty conditions in Part II. The methods for the construction of a posteriori, program, and positional solutions are described. The results are illustrated by examining an example of an optimal observation and control problem of a fourth-order mechanical system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 23, Optimal Control, 2005.     

An approximation method for stochastic control problems with partial observation of the state — a method for constructing ∈-optimal controls

We consider a continuous-time stochastic control problem with partial observations. Given some assumptions, we reduce the problem in successive approximation steps to a discrete-time, complete-observation, stochastic control problem with a finite number of possible states and controls. For the latter problem an optimal control can always be explicitly computed. Convergence of the approximations is shown, which in turn implies that an optimal control for the last-stage approximating problem is ∈-optimal for the original problem.     

A robust optimization approach to experimental design for model discrimination of dynamical systems

A high-ranking goal of interdisciplinary modeling approaches in science and engineering are quantitative prediction of system dynamics and model based optimization. Quantitative modeling has to be closely related to experimental investigations if the model is supposed to be used for mechanistic analysis and model predictions. Typically, before an appropriate model of an experimental system is found different hypothetical models might be reasonable and consistent with previous knowledge and available data. The parameters of the models up to an estimated confidence region are generally not known a priori. Therefore one has to incorporate possible parameter configurations of different models into a model discrimination algorithm which leads to the need for robustification. In this article we present a numerical algorithm which calculates a design of experiments allowing optimal discrimination of different hypothetic candidate models of a given dynamical system for the most inappropriate (worst case) parameter configurations within a parameter range. The design comprises initial values, system perturbations and the optimal placement of measurement time points, the number of measurements as well as the time points are subject to design. The statistical discrimination criterion is worked out rigorously for these settings, a derivation from the Kullback-Leibler divergence as optimization objective is presented for the case of discontinuous Heaviside-functions modeling the measurement decision which are replaced by continuous approximations during the optimization procedure. The resulting problem can be classified as a semi-infinite optimization problem which we solve in an outer approximations approach stabilized by a suggested homotopy strategy whose efficiency is demonstrated. We present the theoretical framework, algorithmic realization and numerical results.     

Optimal observation of multistage systems

The problem of optimal observation of a nonlinear step system is considered. The task is to obtain guaranteed estimates of the initial state from given incomplete and inaccurate measurements of the current states. The cases of fixed and unfixed transition times in a step system are examined. Algorithms for solving a posteriori and positional observation problems are proposed. The results are illustrated by examples.     

阶段结构远海-近海渔业模型的稳定性及最优收获

§ 1　IntroductionAlong with the development of science and technology,the capacity and output offishing have increased.For example,in China,traditional and backward manual tools areused and the inshore coastlines are limited for fishing,but now gradual improvements inthe technical efficiency of fishing gear and vessels have radically changed the fishing sce-nario.With the advent of sophisticated fishing instruments,the fishing scope has been ex-panded from inshore area to offshore area.As a …     

Markov chain approximations to filtering equations for reflecting diffusion processes

Herein, we consider direct Markov chain approximations to the Duncan–Mortensen–Zakai equations for nonlinear filtering problems on regular, bounded domains. For clarity of presentation, we restrict our attention to reflecting diffusion signals with symmetrizable generators. Our Markov chains are constructed by employing a wide band observation noise approximation, dividing the signal state space into cells, and utilizing an empirical measure process estimation. The upshot of our approximation is an efficient, effective algorithm for implementing such filtering problems. We prove that our approximations converge to the desired conditional distribution of the signal given the observation. Moreover, we use simulations to compare computational efficiency of this new method to the previously developed branching particle filter and interacting particle filter methods. This Markov chain method is demonstrated to outperform the two-particle filter methods on our simulated test problem, which is motivated by the fish farming industry.     

A simple procedure for determining order quantities under a fill rate constraint and normally distributed lead-time demand

One of the most common practical inventory control problems is considered. A single-echelon inventory system is controlled by a continuous review (R, Q) policy. The lead-time demand is normally distributed. We wish to minimize holding and ordering costs under a fill rate constraint. Although, it is not especially complicated to derive the optimal solution, it is much more common in practice to use a simple approximate two-step procedure where the order quantity is determined from a deterministic model in the first step. We provide an alternative, equally simple technique, which is based on the observation that the considered problem for each considered fill rate has a single parameter only. The optimal solution for a grid of parameter values is stored in a file. When solving the problem for an item we use interpolation, or for parameter values outside the grid special approximations. The approximation errors turn out to be negligible. As an alternative to the interpolation we also provide polynomial approximations.     

Optimal Control of a Class of Phase Change Processes with Terminal State Observation

A class of optimal control problems for nonlinear evolutionary processes governed by two-phase Stefan problems is analyzed. The processes with terminal state observation are considered in the case of one space dimension. Approximate optimal solutions (controls, as well as the corresponding states and adjoint states), referring to the problems with time-averaged state observation are shown to converge to the appropriate solutions for the original problem.     

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.     

Properties of the Optimal Ellipsoids Approximating the Reachable Sets of Uncertain Systems

The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.     

Synthesis of the optimal control of dynamical structures

The problem of synthesizing an optimal control by choosing the structure, in non-linear dynamical systems a with random structure, is formulated. One of the possible approaches to solving this problem is considered: it uses a method from the theory of the optimal control of systems with distributed parameters and enables one to construct the density vector of the distributions of the process under consideration for all states in such a way as to guarantee an optimum of the selected probability functional. An example is given to illustrate the practical possibilities of the approach.     

Dynamic Nonlinear Modelization of Operational Supply Chain Systems

Supply Chain Management (SCM) is an important activity in all producing facilities and in many organizations to enable vendors, manufacturers and suppliers to interact gainfully and plan optimally their flow of goods and services. To realize this, a dynamic modelling approach for characterizing supply chain activities is opportune, so as to plan efficiently the set of activities over a distributed network in a formal and scientific way. The dynamical system will result so complex that it is not generally possible to specify the functional forms and the parameters of interest, relating outputs to inputs, states and stochastic terms by experiential specification methods. Thus the algorithm that will presented is Data Driven, determining simultaneously the functional forms, the parameters and the optimal control policy from the data available for the supply chain. The aim of this paper is to present this methodology, by considering dynamical aspects of the system, the presence of nonlinear relationships and unbiased estimation procedures to quantify these relations, leading to a nonlinear and stochastic dynamical system representation of the SCM problem. Moreover, the convergence of the algorithm will be proved and the satisfaction of the required statistical conditions demonstrated. Thus SCM problems may be formulated as formal scientific procedures, with well defined algorithms and a precise calculation sequence to determine the best alternative to enact. A “Certainty equivalent principle” will be indicated to ensure that the effects of the inevitable uncertainties will not lead to indeterminate results, allowing the formulation of demonstrably asymptotically optimal management plans.     

Stability of Discrete Approximations and Necessary Optimality Conditions for Delay-Differential Inclusions

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler–Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data.