New approach in analysis and synthesis

problems for distributed and nonstationary systems

V.A. Afanas'ev, A.A. Baloev, A.S. Meshchanov

Kazan state teУhnical university of A.N.Tupolev's name (KAI)

e-mail: mas41@mi.ru

 

It is known that our country has relatively low level of automatization in pipelines operated for hydrocarbon raw materials (gas, crude oil and products of its refining). Therefore there is significant pulsation in pressure, temperature and other parameters into the pipelines, which strongly influences in consumption of gas, oil and oil products by consumers. At a great length of the pipelines (hundreds kilometers) the oscillation process in pressure can be very long with impermissible overshoots which inevitable causes a damage of pipelines that is an accident (local technogenetic disaster) with losses in the fuel energy and ecology. A decision about this problem is to equip the pipelines of hydrocarbon raw materials and refining products with automatic devices which, taking in consideration readings of a number of sensors (pressure, temperature and other parameters) placed in certain points of the pipelines, will provide a given level of pressure (fulfillment of restrictions) and other parameters along its whole length by using even this incomplete data about the pipeline's conditions. These automatic devices will be able to remote the pulsations in a minimum time at a permissible overshoot in pressure into the pipeline.

In the paper we discuss the new approaches to problems how to analyze the stability and synthesize optimal control systems with distributed parameters at complete and incomplete data about their conditions. To further develop methods and approaches to research of the distributed systems and also as an independent fundamental and science-capacious problem we suggest to develop the methods and approaches to an enough simple analyze of stability and synthesis of effective control actions for complicated many-dimensional systems with linear and nonlinear objects taking into consideration constant effect of uncertain external and parametric disturbances and restraints. In particular on analysis of stability and synthesis of linear nonstationary systems with uncertain and unstable parametric disturbances we offer to resolve a problem how to find new effective criterions of the exponential stability for a trivial solution and exponential decay in a preassigned times of the Euclidean norm for a vector of deviations from the trivial solution. The solution of this problem for a nominal system with a certain matrix of coefficients varying during transient processes (not constant as it is assumed for well-known interval matrices) is based on usage of Vazhevsky's inequality, Kharitonov's theorem and some well-known criterions (of Routh-Hurwits, Silvester, Lienar and Chipar). Other new methods suggested for examining the stability and synthesizing linear nonstationary controlling systems are based on usage of such Lyapunov's transformation that generates the second summand in the right side (in matrix at the state vector) in addition to a usual product of a reciprocal matrix of Lyapunov's transform by its derivative as a matrix of Jordan form. When there are uncertain disturbances (external and parametric) acting constantly on a system we suggest a method of multistep terminal control that has a property of adaptability to the mentioned disturbances of different nature and is to construct control actions anew in the nominal system in the beginning of each sufficiently small steps dividing the whole time interval of the terminal control and to use during the control construction data about conditions of the real system generated by some sensors at the beginning of each step.