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.