AUTOMATIC CONTROL THEORY II

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

Optimal control

Formulation of optimal control problemsThe formulation of an optimal control problem requires the

following: a mathematical model of the system to be controlled a specification of the performance index a specification of all boundary conditions on states,

and constraints to be satisfied by states and controls a statement of what variables are free

Optimal control

General case with fixed final time and no terminal or path constraints Problem 1: Find the control vector trajectory

to minimize the performance index

subject to

Optimal control

Problem 1 is known as the Bolza problem If

then the problem is known as the Mayer problem if

it is known as the Lagrange problem

define an augmented performance index

Optimal control

Define the Hamiltonian function H as follows

such that can be written

variation in the performance index

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For a minimum, it is necessary that

This gives the stationarity condition

These necessary optimality conditions, which define a two point boundary value problem, are very useful as they allow to find analytical solutions to special types of optimal control problems, and to define numerical algorithms to search for solutions in general cases.

Optimal control

The linear quadratic regulator The performance index is given by

the system dynamics obey

to find that the optimal control law can be expressed as a linear state feedback

Optimal control

the state feedback gain is given by

the solution to the differential Ricatti equation

it is possible to express the optimal control law as a state feedback but with constant gain

Optimal control

the positive definite solution to the algebraic Ricatti equation

the closed loop system

is asymptotically stable

Optimal control

This is an important result, as the linear quadratic regulator provides a way of stabilizing any linear system that is stabilizable.

An extension of the LQR concept to systems with gaussian additive noise, which is known as the linear quadratic gaussian (LQG) controller, has been widely applied.

Optimal control

Minimum time problems to reach a terminal constraint in minimum time Find and to minimise

subject to

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Problems with path constraints Sometimes it is necessary to restrict state and control

trajectories such that a set of constraints is satisfied within the interval of interest

where

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it may be required that the state satisfies equality constraints at some intermediate point in time

These are known as interior point constraints and can be expressed as follows

where