Optimal control for
linear stochastic system with additional and multiplicative perturbations
1. The preamble
Achievement
of high technical and economic parameters of controlled technological processes
needs to ensure optimal modes of all branches of the plant. At the same time we
must provide economical but effective process control.
The leading branch
of the sugar factory, which determines its performance and power, is the department
of extraction, which involves cutting of raw material for needed quality chip
and "getting sugar" from chip by prepared water, during which we
obtain diffusion sweet juice, which then goes to recycling. Productivity of the
plant depends of the number of processed beet. Therefore it is important to
ensure the quality and continuity of the diffusion department. To provide all
this, engineers should design automation systems, which can maintain optimal
values of process parameters, regardless of the conditions of
flow processes. However, the object is operating under a large number of
permanent obstacles that are often unpredictable, impair the flow of processes,
and we want to eliminate perturbations in real time. Of course, all plants have
a system of automation, but regulators are based on the standard P-, PI
controllers, which are simple to configure and easily implemented using the
microprocessor controllers, but aren’t able to provide high quality control
system.
Therefore
it is necessary to find robust controllers, that aren’t sensitive to external
(additional) and internal (multiplicative) perturbations and are able to save
quality of control system in the changing conditions of flowing processes.
2. The statement of the problem
A control object,
which operates under the action of external and internal random perturbations,
is described by the next system of stochastic differential equations:
(1)
Where is
- dimensional state
vector,
is
- dimensional
control vector,
,
,
,
– set matrix
and
dimensions,
– known vector of dimension
,
– scalar random Wiener
processes, which describe external (additional) and internal (multiplicative)
perturbations acting on control object. There are some expectations for these
perturbations:
, Covariance
,
, where
is Dirac
-function, functions
make a symmetric positive defined
matrix
. The
initial state of the system
is Gaussian random
-dimension vector, such as
,
, where
– given
-dimension vector of means,
is known symmetric positive defined covariance matrix of
dimension.
The resolve of the system of stochastic
differential equations was founded in Ito meaning.
We will review the next criterion of
control:
(2)
Where are given symmetric positive
defined weight matrix respective dimensions.
The problem is to
find the optimal control which minimizes the functional (2) to the solutions of
(1).
3. The results
It is possible to show that
the optimal control can be find as a linear feedback from system state
.
Where the feedback matrix and the additional vector
are discovered from the next
equations
.
Here unknown
matrix meets the next
differential Riccati equations.
And vector is a solution of ordinary differential equations of the form:
The minimum value of the criterion in this case is discovered from the
expression:
Where can be found from:
4. The conclusion
The optimal control
of stochastic systems with additional and multiplicative perturbations was
found as a linear matrix controller from the output of the system, and its
parameters can be calculated prior to the control.