Dynamic Systems Modelling and Optimal Control

Applications in Management Science

Victoria Miroshnik, Dipak Basu

• Publisher: Palgrave Macmillan
• Year: 2016
• Print ISBN: 9781137508935
• eBook ISBN: 9781137508959
• Topic: Business
• Subtopic: Business Mathematics
• Index: Business

Dynamic Systems Modelling and Optimal Control explores the applications of oil field development, energy system modelling, resource modelling, time varying control of dynamic system of national economy, and investment planning.

1

Optimal Control Problem

Solution of a continuous-time optimal control problem

Pontryagin (1962) and his associates developed the maximum principle for solving continuous-time control problems. Basically, the maximum (or minimum) principle provides a set of local necessary conditions for optimality. According to this method, variables analogous to the Lagrange multipliers should be introduced. These variables, usually denoted by p, are often called the co-state or adjoint-system variables. A scalar-value function H, which generally is a function of x,p,u (state, co-state, control vector) and t, named Hamiltonian function of the problem, is also considered. An economic model can be presented as:

or

where

and x ϵ Eⁿ, u ϵ E^m, are the state and control vectors. Matrices A and B are defined on Eⁿ × Eⁿ and Eⁿ × E^m respectively. The system described is stable if:

The trajectory of a non-stable system exhibits explosive oscillations. The general solution of the system presented in (1.1) has the form:

where x(t₀) is fixed.

We can rewrite it as (Pontryagin, 1962):

where matrix Φ(t) and the vector r(t) can be computed given t and u(t). Φ(t) is known as the state transition equation or fundamental matrix of solutions.

In an optimal control problem, nominal state and control trajectories, denoted by ẍ(t) and ü(t), are specified, and the performance function to be minimized is:

where t₀, t_f denote the initial and final lime.

Symmetric weighting matrices M,Q and R, are defined as Eⁿ × Eⁿ, Eⁿ × Eⁿ and E^m × E^m respectively. Matrix R is assumed to be positive definite whilst the other two may be positive semi-definite.

The control problem is as follows

Minimize the cost function (1.4), with the constraints:

where, x(t₀) and t are fixed.

To solve ...

Table of contents

• Cover
• HalfTitle
• Title
• Copyright
• Dedication
• Contents
• List of Figures
• List of Tables
• Preface
• 1  Optimal Control Problem
• 2  Optimal Planning with Exhaustible Resource
• 3  Management of Oil Field Development
• 4  Energy System Modelling
• 5  Control System Modelling for Investment Planning
• 6  National Economic Management with Stochastic Optimal Control: Simulations over History
• 7  Anticipations and Time-Varying Modelling in Adaptive Control System
• 8  Dynamics of Information and Resource Management
• Conclusion
• References
• Index