Necessary conditions of optimality for vector-valued impulsive control problems

A vector-valued impulsive control problem is considered whose dynamics, defined by a differential inclusion, are such that the vector fields associated with the singular term do not satisfy the so-called Frobenius condition. A concept of robust solution based on a new reparametrization procedure is adopted in order to derive necessary conditions of optimality. These conditions are obtained by taking a limit of those for an appropriate sequence of auxiliary “standard” optimal control problems approximating the original one. An example to illustrate the nature of the new optimality conditions is provided.     

Measurement-based optimization of batch processes: Meeting terminal constraints on-line via trajectory following

The context of this paper is the use of process measurements to optimize batch processes in the presence of uncertainty. The optimal solution consists of (i) keeping certain path and terminal constraints active and (ii) driving the sensitivities to zero. In particular, the problem of meeting the active terminal constraints in each run is considered here, which is important when these constraints have a larger bearing on the cost than the sensitivities. A two-time-scale methodology is proposed, whereby the task of meeting the active terminal constraints is addressed on-line using trajectory following, while pushing the sensitivities to zero is implemented on a run-to-run basis. The proposed methodology is illustrated via the simulation of a batch distillation system.     

Dynamic optimization in the presence of uncertainty: From off-line nominal solution to measurement-based implementation

The problem of optimizing a dynamic system in the presence of uncertainty is typically tackled using measurements. The method most widely documented in the literature is based on repetitive optimization of an updated process model. Recently, a computationally less expensive alternative that is based on the real-time adaptation of a solution model using measurements has been proposed. The solution model, which relates elements of the input profiles to the set of active constraints and to appropriate sensitivities, has typically been derived manually from physical insight and intuition. This paper presents a systematic and automated approach to generate a solution model based on recent results in numerical optimization of dynamic systems. This concept provides the first step toward an entirely automated procedure for constrained dynamic optimization of uncertain large-scale processes.     

Necessary and sufficient conditions for a bounded solution to the optimality equation in average reward Markov decision chains

We consider average reward Markov decision processes with discrete time parameter and denumerable state space. We are concerned with the following problem: Find necessary and sufficient conditions so that, for arbitrary bounded reward function, the corresponding average reward optimality equation has a bounded solution. This problem is solved for a class of systems including the case in which, under the action of any stationary policy, the state space is an irreducible positive recurrent class.     

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

NCO tracking and self-optimizing control in the context of real-time optimization

This paper reviews the role of self-optimizing control (SOC) and necessary conditions of optimality tracking (NCO tracking) as presented by [1]. We show that self-optimizing control is not an alternative to NCO tracking for steady state optimization, but is to be seen as complementary. In self-optimizing control, offline calculations are used to determine controlled variables (CVs), which by use of a lower layer feedback controller indirectly keep the process close to the optimum when a disturbance enters the process. Preferably, the setpoints are kept constant, but they may be adjusted by some optimization layer. Good CVs reduce the need for frequent setpoint changes. When selecting self-optimizing CVs, a set of disturbances must be assumed, as unexpected disturbances are not rejected in SOC. On the other hand, the presented NCO tracking procedure adapts the inputs at given sample times without a model or any assumptions on the set of disturbances. However, disturbances with high frequencies or those which do not lead to a steady state are not rejected. By using NCO tracking in the optimization layer and SOC in the lower control layer, we demonstrate that the methods complement each other, with SOC giving fast optimal correction for expected disturbances, while other disturbances are compensated by the model free NCO tracking procedure on a slower time scale.     

Necessary first- and second-order optimality conditions in problems of control described by a system of Volterra difference equations

The terminal problem of optimal control described by a system of Volterra difference equations is considered. Constructively verifiable necessary first- and second-order optimality conditions are obtained under the assumption that the domain of control is open.     

Source localization and calibration using TDOA and FDOA measurements in the presence of sensor location uncertainty

Source localization accuracy is very sensitive to sensor location error.This paper performs analysis and develops a solution for locating a moving source using time difference of arrival(TDOA)and frequency difference of arrival(FDOA)measurements with the use of a calibration emitter.Using a Gaussian random signal model,we first derive the Cram′er-Rao lower bound(CRLB)for source location estimate in this scenario.Then we analyze the differential calibration technique which is commonly used in Global Positioning System.It is indicated that the differential calibration cannot attain the CRLB accuracy in most cases.A closed-form solution is then proposed which takes a calibration emitter into account to reduce sensor location error.It is shown analytically that under some mild approximations,our approach is able to reach the CRLB accuracy.Numerical simulations are included to corroborate the theoretical developments.     

Necessary and sufficient optimality conditions of a design decision on the structure of a local computer network

A problem of nonlinear programming with Boolean variables is considered, for which the possibility of extension of the optimality conditions of solving a special problem of nonlinear programming (with continuous variables) is shown by statistical estimation of possible design decisions. This contributed to the solution of the initial problem by simpler methods. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 133–137, September–October, 1999.     

On necessary conditions for the existence of finite-dimensional filters in discrete time

Under some regularity assumptions we show that, if a finite-dimensional filter in discrete time exists, then the observation, prediction, and filtering distributions are all of exponential class. Our results, motivated by analogous results in the field of statistics, hold for arbitrary (finite) dimensions of the state and observation spaces as well as of the filter itself.