Matrix Minimum Principle Research Articles

Solvability for indefinite mean‐field stochastic linear quadratic optimal control with random jumps and its applications

SummaryIn this article, we study the open‐loop and closed‐loop solvability for indefinite mean‐field stochastic linear quadratic (MF‐SLQ) optimal control problem and its application in finance, where the controlled stochastic system is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes. The intrinsic property of stochastic systems results in the inequivalence of those two solvabilities, which is different from deterministic case. Based on a well‐posedness result of the problem, it is shown that the uniform convexity of cost functional is sufficient for the open‐loop solvability of the problem. By a matrix minimum principle, the necessity condition of regular solvability for a decoupled Riccati equation is established, meanwhile, the closed‐loop solvability is turned out to be equivalent to the regular solvability of Riccati equations with some constraints on the solutions of disturbances equations, moreover, the optimal closed‐loop strategy is characterized by regular solutions of Riccati equations and adapted solutions of disturbances equations. And then a mean‐variance model is considered to solve optimal portfolio selection strategy problem of the insurance company with liability. Our study fills a gap in the field of solvability for mean‐field stochastic optimal control with random jumps and provides a different way for solving mean‐variance portfolio selection model.

Indefinite LQ optimal control with equality constraint for discrete-time uncertain systems

Based on uncertainty theory, this paper studies a kind of discrete-time uncertain linear quadratic (LQ) optimal control with equality constraint for the terminal state, allowing the state and control weighting matrices in the cost function to be indefinite. First, we transform the uncertain LQ optimal control problem into an equivalent deterministic optimal control problem. Then, a necessary condition for the existence of optimal linear state feedback control is presented by means of matrix minimum principle. Moreover, the well-posedness of the uncertain LQ problem is proved by applying the technique of completing squares. Finally, an example is provided to demonstrate the effectiveness of our theoretical results.

Stochastic optimal control under randomly varying distributed delays

An output feedback control law is presented, hereafter called the linear quadratic coupled delay compensator (LQCDC), for application to processes that are subjected to randomly varying distributed delays. An example is future generation aircraft, which are equipped with computer networks to serve as the communications link for the vehicle management system. The LQCDC is synthesized via dynamic programming in the stochastic setting. A pair of discrete-time modified matrix Riccati equations and a pair of modified matrix Lyapunov equations are constructed by using lagrangian multipliers and the matrix minimum principle. The performance cost is formulated as the conditional expectation of a quadratic functional adjoined with an equality constraint involving the dynamics of the conditional covariance of the closed-loop system state. Results of simulation experiments are presented to demonstrate the efficacy of the LQCDC for control of longitudinal motion of an advanced aircraft.

Determination of the minimal number and optimal sensor location in a nuclear system with fixed incore detectors

The determination of the minimal number of sensors and the optimal sensor location in a nuclear system with fixed incore detectors, which is represented by a linear stochastic distributed parameter system, was studied in this work. The partial differential equation representing nuclear reactor dynamics was approximated to the finite dimensional ordinary differential equation by the modal expansion. A scalar measure of the covariance matrix error in the optimal filter was minimized with respect to the sensor locations. The necessary conditions for optimal sensor location were derived using the matrix minimum principle, thus making the calculations computationally more attractive. The locations of sensors were guessed initially through sensitivity analysis to reach solutions of the optimal location quickly. A method to determine the minimum number of sensors is suggested based on the observability and admissible error bound. Several numerical simulations are performed to determine the minimal number and optimal sensor location for a one-dimensional slab reactor and a two-dimensional ABB Combustion Engineering type reactor with fixed incore detectors. Through the simulations the possibility of practical implementation and the rapid convergence of the algorithm are verified.

Modal Testing and Suboptimal Vibration Control of Flexible Rotor Bearing System by Using a Magnetic Bearing

A suboptimal output feedback controller is designed based on a reduced order model and applied to a flexible rotor bearing system in order to control the unstable or lightly damped vibrations occurring during operation. The reduced order model is a truncated modal equation of the distributed parameter system obtained through the singular perturbation. The instability problem arising from the spillover effects caused by the uncontrolled high frequency modes is prevented through the constrained optimization by incorporating the spillover term into the performance index. The application of the matrix minimum principle yields a set of matrix equation similar to the unconstrained output feedback controller. The efficiency of the proposed method is demonstrated experimentally with a flexible rotor by using a magnetic bearing.

Optimal scanning measurement problem for a stochastic distributed-parameter system

This paper deals with the problem of determining the optimal measurement scheduling for a stochastic distributed-parameter system (DPS) based on spatially continuous and discrete-scanning observations. These two types of measurement are realized by the optimal motion of spatially-movable sensors and the optimal selection of measurement data from spatially-fixed sensors, respectively. For the continuous scanning case, the existence of optimal solutions for the problem is proved and the N-modal approximation problem is established. For the discrete case, however, it is impossible to derive the existence conditions. Therefore, it is shown that there exist optimal relaxed solutions by introducing the relaxed control theory. A practical method for constructing an approximate solution for the relaxed problem is proposed. Necessary conditions for approximate-optimality are represented in the form of a matrix minimum principle, and a feasible algorithm is developed to determine the approximate solution. A numerical example is considered and the present two types of optimal measurement trajectory are compared.

Suboptimal sensors location in the state estimation problem for stochastic non-linear distributed parameter systems

The present paper deals with the minimal number sensor choice and their optimal location for the estimation in non-linear stochastic distributed parameter systems described by parabolic and hyperbolic partial differential equations. The necessary condition for the optimal sensor location by using the matrix minimum principle was obtained. In turn, the computational algorithm of the sensors location was determined on the basis of the necessary condition, applying the optimal control theory. The computational efficiency of this algorithm is defined by the suboptimal filtering algorithm which does not require solving of the matrix Riccati equation for the filter error covariance. Finally, one example is given to demonstrate the effectiveness of the present approach.

Optimal scanning measurement problem for a stochastic distributed-parameter system

This paper deals with the problem of determining the optimal measurement scheduling for a stochastic distributed-parameter system (DPS) based on spatially continuous and discrete-scanning observations. These two types of measurement are realized by the optimal motion of spatially-movable sensors and the optimal selection of measurement data from spatially-fixed sensors, respectively. For the continuous scanning case, the existence of optimal solutions for the problem is proved and the N-modal approximation problem is established. For the discrete case, however, it is impossible to derive the existence conditions. Therefore, it is shown that there exist optimal relaxed solutions by introducing the relaxed control theory. A practical method for constructing an approximate solution for the relaxed problem is proposed. Necessary conditions for approximate-optimality are represented in the form of a matrix minimum principle, and a feasible algorithm is developed to determine the approximate solution. A numeric...

Low-sensitivity observer-compensator design for two-dimensional digital systems

The problem considered in the paper refers to the general area of developing design and computational algorithms for analysing low-sensitivity time-invariant multivariable two-dimensional systems using output feedback. It is seen that the matrix-minimum principle can be used to study two-dimensional digital systems with incomplete state information for the implementation of a low-sensitivity feedback law. The results show that an observer can be used in addition to the output vector to estimate the state of the system, which in turn is used for implementing the feedback law.

Optimal measurement problem for a stochastic distributed parameter system with movable sensors

Abstract This paper deals with the problem of determining optimal measurements for a stochastic distributed parameter system (DPS). This problem can be formulated as an optimal control problem for a system described by the Riccati equation, whose solution gives, generally a measurement strategy of the scanning type. However, the specified strategy is not always practicable, and therefore a more realistic formulation is required. The optimal measurement problem is reformulated as that of minimizing not only the trace of the estimation error covariance, but also the total kinetic energy of the measurement sensors by taking the sensor velocity as the control variable. The existence of an optimal solution for the reformulated problem is proved, and the necessary conditions for optimality are derived in the form of the matrix minimum principle. Based on successive approximation of the Riccati equation, a computationally advantageous approach is developed to obtain the optimal measurement of the scanning type w...

Matrix minimum principle for distributed parameter control systems

Abstract It is shown that the matrix minimum principle can be used to study both deterministic and stochastic control systems described by partial differential equation models. It in shown that the matrix formulations of the control problem reduce to a state canonical equation which is identical with the Riccati equation of the conventional state vector representation. The technique employs the powerful optimization techniques of the calculus of variations and dynamic programming, and leads in a natural way to the separation principle for the case of stochastic optimal control problems. The results of this study show once more the close relationship between the tools used for finite-dimensional control problems and their distributed-parameter (infinite-dimensional) counterparts.

Optimization of Systems with Control Delay through Output Feedback

The infinite time optimal control of linear time-invariant (LTI) system with delays in control action only is discussed with the regulator constraint that the control vector is a linear time-invariant function of the output only. The quadratic performance criteria which is a quadratic functional in x(t), u(t) and u(t—T) is minimized using matrix minimum principle and feedback gain matrix F* is found; where T is the delay.The problem leads to two types of optimization: (i) optimization in te(T,∞), which is handled by a parameter optimization technique; (ii) a finite time optimization within the delay interval te(0, T), simultaneously minimizing to zero the response due to initial states of the delay which enters as a disturbance; a tracking problem.

Estimation for discrete non-linear time-delayed systems and measurements with correlated and coloured noise processes

Thia paper presents new filtering and smoothing algorithms for discrete non-linear time-delayed systems with multi-channel delayed measurement signals embedded in both the coloured noise sequence and the correlated message and measurement noise processes. The derivation assumes that the smoothing estimator introduces new data in a linear additive fashion and makes use of the matrix minimum principle to minimise the error variance cost functional. The numerical example simulated indicates that the two-stage estimation algorithm developed in this paper does not have the narrow bandwidth problem, this is due to the dynamics of its two-stage operation.

State estimation for nonlinear distributed-parameter systems involving multiple delays

This correspondence presents new estimation algorithms for nonlinear distributed-parameter systems involving multiple time-delays. The results are first derived in discrete-time form through the use of matrix minimum principle and are then formally converted to continuous-time form using Kalman's limiting procedure.

On symmetric matrices and the matrix minimum principle

The gradient matrix has come into wide use during the past decade. When gradients are taken with respect to symmetric matrices in the context of optimization problems, special care must be taken to insure the validity of the results. This note considers the necessary conditions of the matrix minimum principle [1] when the system matrix is constrained to be symmetric.

New filtering and smoothing algorithms for discrete non-linear delayed system with coloured noise

This paper presents a new dynamic filtering and smoothing algorithm for discrete non-linear time-delayed systems with coloured noise. The derivation makes use of the matrix minimum principle, to minimize the error variance taken to be the estimation criterion. The algorithms obtained can be approximated to arbitrary order under the assumption that the conditional probability density function of the state variables are Gaussian.

Synthesis of a high-speed tracked vehicle suspension system--Part II: Definition and solution of the control problem

This two-part paper is concerned with the problem of synthesizing a high-speed vehicle suspension system. In part I [13] a new suspension structure was developed which features three independent parts. All three parts are calculated for a vehicle which travels along a rigid guideway with arbitrary vertical and lateral curves and grades. In this part in developing the system control the use of instantaneous state feedback and the presence of inaccessible states demand the use of Itô-calculus and the Athans matrix minimum principle in order to solve the problems of vibration control synthesis. It is concluded that the application of the principles and methods of advanced dynamics, deterministic and stochastic optimal control theory, and applied mathematics to the problem of synthesizing a high-speed vehicle suspension is successful and results in the identification of a new suspension structure which is proposed as a starting point in the development and design of a high-speed vehicle suspension system. Augmented by a thorough system simulation and hardward tests for verification of all assumptions made and by implementation of the sensor-and actuator-dynamics, the described synthesis procedure will lead to an effective suspension system.

New estimation algorithms for discrete non-linear systems and observations with multiple time delays

This paper derives new estimation algorithms for discrete non-linear systems and observations with multiple time delays. The results directly yield the fixed-interval, fixed-lag, fixed-point smoothing and the filtering algorithms. The derivation makes use of the matrix minimum principle to minimize the error variance, taken to be the estimation criterion. For systems with polynomial or product type non-linearities, the algorithms are physically realizable, under the assumption that the conditional probability density functions of the estimator errors are Gaussian. Examples illustrating the use of the algorithms are presented.

Exact and approximate state estimation for nonlinear dynamic systems

This paper presents a modified approach to solve state estimation problems of nonlinear dynamic systems involving noise free, uncorrelated and correlated state and measurement noise processes. The basic approach makes use of the matrix minimum principle together with the Kolmogorov and Kushner's equations to minimize the error-variance, taken to be the estimation criterion. The filtering equations obtained for nonlinear systems with white noise process are exact, but for non-white noise processes the results obtained are approximate. For systems with polynomial or product types non-linearities, the proposed algorithms can be evaluated without the need of approximation under the assumption that the estimator errors are Gaussian. Such an assumption is significantly different from the most commonly used assumption that the state is Gaussian. Simulation results obtained from the proposed filtering algorithms are compared to various other approximate nonlinear filters. The results indicate the superiority of the proposed filter over those of other filters investigated.

Optimal location of process measurements

The problem of optimally locating a given number of Bensors for observing a general linear distributed parameter system is considered. Measurements at the sensors are assumed to be available continuously in time, and the design criterion is minimization of a scalar measure of the covariance of the estimate error in the optimal linear filter. Necessary conditions for optimality are derived based on the formulation of a distributed parameter matrix minimum principle. A computational algorithm is developed for determining the optimum set of measurement locations. The algorithm is applied to the problem of optimally locating temperature sensors in a solid undergoing transient heat conduction.