The Oscillatory Stable Regime of Nonlinear Systems, with two time constants

Transcription

1 6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec The Oscillator Stable Regime of Nonlinear Sstems with two time constants NUŢU VSILE * ROTRIU DRIN ** CÎRMCI MRIUS-VLERIU *** * Department of viation Integrated Sstems ** Department of Mechanics *** Department of mmnitions and Explosives Militarechnical cadem George Coşbc vene 8-83 Bcharest ROMNI bstract: - The object of this paper is the nonlinear sstems that inclde a rela with a hsteresis loop characteristic propert The linear parts of sstems that are taken into consideration are characterized b an order two transfer fnction with two time constants The paper shows a method to evalate the period of the limit stabile ccles which are proper for these nonlinear sstems e Words: - nonlinear sstems oscillator stable regime rela hsteresis loop phase-plane method calcls algorithm Introdction The sstems that inclde linear and nonlinear parts in assembl are nonlinear sstems These sstems are realized in order to maintain for an parameters a specified vales or a vales in a specified band The figre presents a tpical strctre for these nonlinear sstems r H N H (s) m H P (s) (s) H T ig Nonlinear sstem destined to control of one parameter The previos schema presents a strctre that ses the feedback principle The linear components of sstem are: the fixed part of the controlled process the actators and the loop s transdcer described form the transfer fnctions H P (s) H (s) and H T (s) Usall the transdcer can be considerate that a pre proportional element The error signal is the difference between the inpt r and the image T (via transdcers) of otpt Based on the error signal a nonlinear device described b a nonlinear fnction H N generates the command The command is applied to actators The actator otpt m is the inpt for the fixed part of the controlled process The nonlinear component inclded in the sstems can indce a pnctal instabilit Globall the response of theses sstems to constant inpts bring to an oscillator stable state The characteristics of sch state in man sitations can be analticall obtained and in other sitations onl b simlations Sometimes the presented nonlinear sstems can have a pnctall stationar state This paper proposes an algorithm to calcls the period of oscillator stable state for the nonlinear sstems that inclde a hsteresis loop rela and the linear components is described that a two order sstem withot integrator elements The case of linear parts described b a two order transfer fnction with on or two integrators are presented in [] The std model In the following we consider a nonlinear sstem which has identicall strctre that figre dditionall is considered that the nonlinear component is a hsteresis loop rela with ± the level of otpt The hsteresis band is centered in and is the bandwidth The transdcer is pre proportional and T is

2 6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec its gain The actators and the plant of process are described b order one transfer fnction: k H ( ) P s s T () k H ( ) s s T and the time constants achieved the condition: < T < T () Using the previos hpotheses the strctre presented in the figre can be presented sch as the figre r ( st ) ( st ) T In the previos relation the sign of identicall with the sign of initial vale of st 3 The oscillator stable regime std In the following we ll consider that the loop gain is so great that achieved the condition: < (6) or the sstem linear parts we ll se an eqivalent model which is a parallel connection showed in the figre 4 ± ( st ) ( st ) is ig The nonlinear std model In order to simplif the next considerations in the following we ll consider that the sstem otpt coincides with the transdcer otpt The shape of and evoltions are identicall bt the scale differ b the transdcers gain If the sstem otpt is considered the transdcer otpt the strctre can be presented more compact like in the figre 3 r - ( st ) ( st ) ig 3 The compacted std model The linear parts of previos schema is an order two sstem with two negatives poles and the gain of loop is: T (3) In case a constant vale of inpt the sstem otpt tends to one static work points if the following condition is achieved: > (4) In this case the hsteresis bandwidth is so large or the gain is so small The stationar vale of sstem otpt will be: st ± (5) ± s T s T ig 4 The eqivalent parallel model The gains of the two order one elements of parallel connection are positives and satisfied the relations: T T T (7) T T T Observation: if the linear parts has a zero (this case will not treated in this paper): s (8) s T s ( ) ( ) T then it can be represented like a parallel connection of two elements with the gains: T T T (9) T T T or the two elements of the eqivalent connection we ll associate the state variables

3 6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec and Using the previos notations the sstem otpt will be: () or a constant rela otpt is possible to obtain analtical expressions for and : t / T ( ( ) e () t / T ( ( ) e where and are the initial vales of and If the rela otpt is constant is obvios those indifferentl on the initial vales and tend to vales: ( t ) () ( t ) In the plane of and variables which is the phase s plane the state trajector of the eqivalent sstem tend to the point ( ) which is an accmlation point Using the relation () we can obtain an analtical eqation of state trajector in the phase-plane: T / T ( ( (3) The relation (3) is an eqation of a power crve Indifferentl of the initial conditions or of the vale of ratio a T / T > the state trajector will tend to the accmlation point If the parameters of the initial sstem components are known is possible to obtain a crve famil who inclde the state trajector The aspect of these crve famil is present in the next figre In case of nll vale of reference the commtation lines in the phase-plane are described trogh the eqations: ( ( (4) When the rela otpt are negative the famil of crves that inclde the trajector has same graphical shape bt the accmlation point will be other ( ) The two crves families and the commtation lines are presented in the next figre > > ig 6 The two trajectories crves families and the commtation lines In the previos figre we can observe: - if the accmlation points are inclded in the band delimited b the two commtation lines according to the initial state the sstem otpt tend to one or other of theses points In this case the condition (4) is achieved; - if the accmlation points aren t inclded in the band delimited b the two commtation lines the condition (4) isn t achieved The accmlation points are inclded in the domains that correspond to a rela otpt with an opposite sign If we consider the tpical shape of trajectories is evident that the sstem otpt tends to a limit stable ccle ig 5 The state trajector famil in the phase-plane st st ig 7 The limit stable ccle to a nll reference

4 6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec The limit stable ccle is smmetric relative to the coordinate sstem origin The extreme points of the ccle are positioned to the two commtation lines and the real otpt of sstem will be inclded in an interval: ( [ / T / T ] (5) If st is the maximm vale of state variable this vale satisfied the relation: st st st st T / T E (6) The period of limit stable ccle T can be obtained sing the relation: T T st ln st (7) In case of one no nll constant reference we mst cont that the commtation lines are placed in other position and its eqations will be: r (8) Becase the trajectories crve families and the accmlation points aren t modified the limit stable ccle are displaced in the variable and plane like in the next figre r st r st The vales and will satisfied the relations: T / T st r st () st r st and: T / T st r st () st r st The solving of last two eqations sstem permits to obtain the extremes point of limit stable ccle Using the obtained vales and st st st we can calclate the period sch as: T T st ln () st This relations exactl is difficlt to applied The previos considerations are tre if the accmlation points are positioned in the phase s plane relative to the commtation lines like in figre 8 If the reference vales r is too high the state trajector tends asmptoticall to an accmlation point The limit level of reference modle is: r (3) lim T If the reference absolte vale exceeds the limit vale r lim the sstem not tends to an oscillator stable regime st r ig 8 The limit stable ccle in case a no nll reference The otpt of sstem will oscillate in a band centered to the reference vale: ( [ r / T r / T ] (9) The limit stable ccle will be delimited in the phase s plane b two extreme points positioned to the commtations lines: ( ) st st r ( ) st st r 4 Example We ll std the case of a sstem characterized b the next transfer fnctions for the linear parts: 5 H ( s) 3 s H ( s) 5 s T We consider for the rela the next vales:

5 6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec ccording the presented relations will be obtained: 5 > and the limit of the reference absolte vales is: r lim 99 If the reference satisfied the condition: r lim < r < r lim the vales of sstem otpt will be in the band: r r [ r r ] T T The gains of two linear components of eqivalent connections will be: The eqation (6) has the next soltion: st and sing (7) we obtain the oscillations period for a nll reference: T 5 ln 385( s) rom simlation it was obtained the vales of oscillations period for sstem otpt sing non nll vales for references These vales are presented in the next table r r r lim T T T T 6 6 r r rlim T T T T The reslts obtained b simlation are practicall invariable The variations of the limit stable ccle periods relative to the reslt of calcls relation (7) are insignificant 5 Conclsions or the nonlinear sstems that can be reshaped in an eqivalent strctre similarl of those which presented in the figre the paper offers the possibilit to calclate the elements of limit stable ccle The relation (7) can be sed to evalate the period of the limit stable ccle not onl in the nll reference case becase the errors indced in case of non nll reference are redced References [] Belea C: tomatică neliniară Editra Tehnică Bcreşti 983; [] Belea C: Teoria sistemelor Sisteme neliniare Editra Didactică şi Pedagogică Bcreşti 984; [3] Ch Y: Digital Simlation of Continos Sstem Mc Graw-Hill Book New York 97; [4] Dmitrache I; Dmitri S; Mih I; Mntean ; Mscă Gh; Calcev C: tomatizări indstriale Editra Didactică şi Pedagogică Bcreşti 993; [5] Ionesc Vlad: Teoria sistemelor Editra Didactică şi Pedagogică Bcreşti 985; [6] Mihoc D; Ceapâr M; Iliesc S St; Borangi T: Teoria şi elementele sistemelor de reglare atomată Editra Didactică şi Pedagogică Bcreşti 98; [7] Stăncilesc l: naliza şi simlarea sistemelor neliniare Probleme neliniare în electrotehnică electronică şi atomatică Editra cademiei RSR Bcreşti 974; [8] Nţ Vasile: Elemente de analiza sistemelor atomate liniare Editra Univers Ştiinţific Bcreşti 7