Slovak University of Technology
Faculty of Material Science and Technology in Trnava
AUTOMATIC
CONTROL THEORY II
Optimal control

Formulation of optimal control problems
The 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
Optimal control

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
 the
 to
performance index is given by
system dynamics obey
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
Optimal control

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
Optimal control

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