The control system of the thermal object

THE CONTROL SYSTEM OF THE THERMAL OBJECT

Andrey N. Lavrentiev

Taganrog State University of  Radioengineering

Aleksandrovskay av. 30, Taganrog, 347900, RUSSIA

Tel: 89185702956, E-mail : lan_tan2005@mail.ru

Abstract. During the solving of any control theory problem by human or with the help of computer it’s necessary to consider a control object and its mathematic model.  Sometimes there is no possibility to apply the methods of classical control theory to some control objects because those mathematic models are either nonlinear or have some distributed constants. Today there are many different methods of nonlinear objects control. These methods allow building of quit precise control systems. In case of the distributed control objects there are no well developed methods of such control systems construction because development of distributed control objects’ mathematic models sometimes is a very difficult problem [1].

In this report the control system synthesis of distributed object with method of logarithmic frequency characteristic is described. The main problem consists in special corrector’s constants calculation which can ensure necessary dynamic and operating characteristics of whole system. The control object is the heating furnace with flat electric heater. The controlled quantity is temperature in the given section of the furnace working place. The control action is a thermal current produced with a flat electric heater as well as positions of the regulating authorities which control air intake and gas drawing out. The external action is mass and composition of warmed-up matter changing and flat electric heater voltage changing [2].

1.  CONTROL OBJECT TRANSFER FUNCTION DESIGN

1.1. The heat transfer description.

Theheat transfer process is one of the most difficult physical phenomena because it is the result of three different physical processes such as heat transfer, radiation and convection. In the general case the heat transfer mathematic model is partial derivative differential equation proposed by Fourier [3]

,                         (1.1)

where  is temperature of point with  working place co-ordinate at the moment t,  are heat conduction coefficients in the line of  those describe medium constants and physical properties. In the case of isotropic medium  the equation (1.1) is transformed into

,                                                                   (1.2)    

where  is Laplacian. For solving equation (1.2) it is necessary to determinate boundary condition

 при ,                                                                   (1.3)

 при ,                                                         (1.4)

where  is a height of heating furnace,  is a flat electric heater level.

Let a working place of heating furnace like rectangle and flat electric heater is placed in the  plane. In this case a thermal current spreads along axis z and so for solving equation (1.2) it is necessary to resolve quantity into Fourier series with operator eigenfunction of this equation. In that way the solving is like

,                                                         (1.5)

where  are Fourier coefficients which define throughout solving process,  are basic functions of some orthogonal series such as Fourier trigonometric series[4].

Set (1.5) into (1.2) and pass on to Laplace representations. As a result there is infinite numbers of linear differential equations according to representations unknown functions:     

,                  .                             (1.6)

The solving of each equation looks in the following way:

,                                              (1.7)

.                                                 (1.8)

With boundary condition (1.3), (1.4) every  - constituents of solving (1.2) look in the following way:

, .         (1.9)

1.2. Control object transfer function

Let temperature distribution on the flat electric heater is described by equations

                      (1.10)

where  are heating furnace linear dimensions.

As shown above, input control action is distributed thermal current  produced with a flat electric heater. For input and out of control object model concordance resolve function  into series similar (1.5) i.e. let us assume