AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and...
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Transcript of AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and...
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
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 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
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