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Dynamic Systems Modelling and Optimal Control
Applications in Management Science
Victoria Miroshnik,Dipak Basu
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eBook - ePub
Dynamic Systems Modelling and Optimal Control
Applications in Management Science
Victoria Miroshnik,Dipak Basu
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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.
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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
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, tf denote the initial and final lime.
Symmetric weighting matrices M,Q and R, are defined as En × En, En × En and Em × Em 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 ...