1 Introduction. r + _

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1 A method and an algorithm for obtaining the Stable Oscillator Regimes Parameters of the Nonlinear Sstems, with two time constants and Rela with Dela and Hsteresis NUŢU VASILE, MOLDOVEANU CRISTIAN-EMIL, ŞOMOIAG PAMFIL Department of Mechanical Engineering, Militar Technical Academ No.8-83, George Cosbc, Sector 5, Bcharest, ROMANIA Abstract: - The object of this paper is the oscillator stable regime of a particlar nonlinear sstem. This particlar nonlinear sstem incldes a rela, and its linear part is characterized b a transfer fnction with two time constants. In the beginning, the paper shows a method that can be sed in the calcls of the parameters for the limit stable ccle, which is appropriated for these nonlinear sstems. After that, the method is particlarized for two different commtation lows of the rela nonlinearit: rela with hsteresis and rela with dela time. The different commtation laws indced, for the same linear parts, different shapes and parameters for the oscillator stable regimes proper to assembl nonlinear sstem. These differences are presented in a case std. e-words: - nonlinear sstems, oscillator regime, rela, dela time, hsteresis, calcls method, algorithm. Introdction The sstems that inclde linear and nonlinear parts, in assembl, are nonlinear sstems. The figre shows a tpical strctre for these nonlinear sstems. r ε H N (s) H L Fig.. A tpical nonlinear sstem The previos strctre ses the feedbac principle, with a nitar reaction. The controlled process (or plant) is described b a linear transfer fnction, H L (s), which also incldes the linear parts of the actator. The reglator fnctionalit and the nonlinear actator characteristic are described b a nonlinear fnction, H N. The error signal, ε, is the difference between the sstem inpt (or reference), r, and sstem otpt,. Based on the error signal, the nonlinear parts generates the command, that are the inpt of the plant. Following, we will consider that the ind of the nonlinearit is a real rela, with dela time and hsteresis. This is the model of certain actators, freqentl inclded in man indstrial and civilian applications, even in advanced eqipments: liqid alimentation parts, air or gas conditioning devices. The main fnction of theses nonlinear sstems is to maintain the vales of an sstem parameters, in specified ranges []. Usall, these sstems wor with a constant inpt vales. According to the sstem inpt level, the response of those sstems bring to a pnctall stationar state or to an oscillator stable regime. The oscillator stable regime is exclsivel indced b the sstem nonlinearit. The characteristics of oscillator stable regime, if it appears, can be analticall expressed [], [3], [4], [8], and in other sitations can be obtained onl b simlations. The rela model is proper for the actators that wor in maximal regime. An ideal rela model of the sstem nonlinearit can indce a ver fast oscillator regime, which increases the actators stress and wearing. Nevertheless, in man atomatic sstems, is not preferred the ideal rela wor for the nonlinear part. The rela model can be deliberatel depreciated, sing delas or hsteresis characteristics, in order to increase the sstems oscillator regime periods. In this wa, the commtations freqenc of the actators can be redced, the actators wearing decreases and the ISSN: Isse 8, Volme 4, Agst 9

2 lifetime increases. This paper proposes a calcls model for the parameters of the oscillator stable regime proper to the nonlinear sstems that inclde non-ideal rela nonlinearit. The rela model sed incldes a dela time and a hsteresis characteristic. Separatel, the case of rela with hsteresis characteristic and the case of a rela with dela time was treated in [4] and [8]. Using the previosl assmptions, the sstem can be presented as in the figre 3. r The nonlinear fnction ε * ε sτ ε e - ( st ) ( st ) The std model Frther down, we will consider a nonlinear sstem with the strctre showed in the figre. For the sstem nonlinearit, described b a nonlinear fnction H N, we will consider a rela smmetrical characteristic with nitar gain: ( t), () ε ct and the real gains of actators are inclding in the gain,, of the linear controlled process model. The pattern of the linear controlled process (or plant) has two real time constants, withot integrator elements, and it is modeled b a linear strictl stable transfer fnction, H L (s) : where: H L ( s), () ( s T ) ( s T ) < T < T. (3) The nonlinear fnction H N can be presented in an analticall form, in order to emphasize the effect of the rela dela time τ : ( ε( t) ) H ( ε( t τ) H N ). (4) N Fig. 3. The initial std model In order to simplif the following considerations, we will se an eqivalent strctre of this sstem. This strctre is presented in the figre 4. In the figre (4) we can remar that, the central parts of the eqivalent strctre are a simplified nonlinear sstem. This simplified nonlinear sstem has a nitar linear part with nitar gain, and for the eqivalent rela the hsteresis bandwidth is ε. r the central parts of the eqivalent sstem ε ε sτ ε e ( st )( st ) - Fig. 4. The eqivalent model The following considerations will refer to a nonlinear sstem lie the central parts of the eqivalent sstem, presented in figre 4. In the relation (4) H N ( ε) is the rela with hsteresis characteristic. This characteristic, presented in figre, will be considered smmetrical and its hsteresis bandwidth will be considered ε (in order to simplif the next assmptions). r ε ε - ε * sτ e Fig. 5. The std model ( st ) ( st ) ε ε - ε * Fig.. The nonlinear characteristic H ( ε) N The std sstem is presented in the figre 5, and for this sstem, the transfer fnction H L (s) is: H L ( s). ( ) ( s T ) ( s T ) ISSN: Isse 8, Volme 4, Agst 9

3 The differences between the initial sstem and the std model consist in the scales of its inpts and otpts and the hsteresis rela bandwidth. The dnamics of internal parameters for the two sstems will have identicall shapes at different scales. 3 The conditions for apparition of the oscillator stable regime All the following considerations will be based on the assmption that the sstem inpt are constant. Onl in this case we can spea abot a stationar regime. Starting to the strctral representation of the std model, we can emphasize an important aspect: becase the linear part of std sstem is strictl stable, when the error signal is constant, the sstem otpt tends to a limit vale: ( t) sign( ε) ε ct st. (5) If the sstem inpt level is high enogh or low enogh, for an initial condition, the error signal will not change its sign. In all cases, the sstem otpts are limited, and we can conclde that the assembl nonlinear sstem is globall stable, in great. In an initial conditions, for redced vales of references [4], [8], the error signal can change its sign, the sstem evoltion tends to an oscillator stable regime, and its otpts tend alternatel toward vales or -. In these cases, we can conclde that the sstem is pnctall instable, in small. We are interested to evalate the parameters of the stable oscillator regime, becase it emphasizes the effectiveness of this sort of atomatic sstem. The principal parameter of the oscillator stable regime is the oscillations period. The apparition of the oscillator stable regime is possible if the reference modle achieves the next condition: r ( t) < r lim ε. (6) The previos condition is applicable and for the sstems presented in figre 3 and it do not depend on the dela time. 4 The oscillator stable regime std method Frther, we will se a particlarization for the phases-plane method (the phases plane will be noted PF). Will consider the linear sstem part as a parallel connection of two order-one elements, as it is presented in figre 6. ( st ) ( st ) s T s T Fig. 6. The eqivalent parallel connection The gains of the two order one elements, noted and, are the following: T T ( T T ), ( T T ). (7) The gains are positive and flfill the condition. This is the reason to express the gain trogh. The state variables associated to each element of connection are noted and. If the plant inpt has a constant vale, the evoltions of the state variables can be calclated [6], [7], depending on the initial conditions, (, ) and the time, t: ( t) ( t) ( ) ( ) e t / T e t / T. (8) In the case, the evoltions of the state variables are described b eqations: ( t) ( t) ( ) ( ) e t / T e t / T. (8 ) Based on (8), we will obtain, in PF, the eqations of trajectories, which are spra-nitar power crves ( T / T > ) : ( t) ( t) T ( t) T ( t) / T / T,,. (9) ISSN: Isse 8, Volme 4, Agst 9

4 In the case, the trajectories converge to an accmlation point, (, ), placed in the first qadrant and noted PAP. The figre 7 presents this point and this trajector crves famil. ε r cn Δt τ εn cp εp Δt τ εp cp PCP PHP PAP PCN ε n ε r cn PHN Fig. 9 The main points of the stable limit ccle Fig. 7 The PAP add the famil of trajector crves in case of If, all trajectories tend to another accmlation point, (, ), placed in the third qadrant and noted PAN. The points PAP and PAN and the two famil of trajector crves are smmetricall positions and shapes in the PF plane. If the conditions (6) are achieved, the sstem state trajector tends to a stable limit ccle, CLS, this sitation are presented in the figre 8. PAN CLS Fig. 8 The stable limit ccle PAP Starting from an arbitrar initial state of the sstem (, ), after some steps, the state sstem trajector bring the CLS. In order to find the parameters of the oscillator stable regime, will se the next figre. The main points of the CLS, presented in the previos figre are: - the commtations points, PCP and PCN. In these points, the nonlinear part of sstem effectivel commtes. The coordinates of this point are: ( cp, cp ), for the PCP and ( cn, cn ) for the PCN - the hsteresis commtations points, PHP and PHN. In these points, the conditions for nonlinear part commtation are achieved, excepting the dela. The coordinates of this point are: ( εp, εp ), for the PCP and ( ε n, εn ) for the PCN. Using the hsteresis characteristic, for the PHP and PHN points we can obtain two relations between its coordinates: εp εn εp εn r ε r ε. () Using the state trajectories eqations, we can obtain other relations: and: cp εp cn εn cp cn cn cp cp εp cn εn cp cn cn cp T / T T / T T / T T / T,. () () ISSN: Isse 8, Volme 4, Agst 9

5 Based on the eqations (8) and the dela time, it reslts another other relations: cp cn ( ) εp ( ) εn e e τ / T τ / T. (3) B adding of relations (), (), () and (3), we obtain a nonlinear eqation sstem, with eight eqations: εp εp r ε εn εn r ε cp cp εp εp cn cn εn εn cp cp cn cn cn cn cp cp τ / T cp ( εp ) e τ / T cn ( εn ) e T / T T / T T / T T / T. (4) The solving of the nonlinear eqation sstem (4) permit to obtain the elements of the CLS. The main parameters are the coordinates of PCP and PCN, ( cp, cp ) and ( cn, cn ), and the times between the two commtations. We will se the notations: - t n, the time of transition between PCP and PHN - t p, the time of transition between PCN and PHP. The previos two times can be calclate sing the relations: t t p n T T ln ln cn cp cp cn τ τ.. (5) Becase the times of transitions between the points PHP and PCP and between the points PHN and PCN are τ, the CLS period, T, will be: T t n t τ (6) p or: T T cp ln cn. (7) cp cn 5 The calcls of otpt sstem characteristics The eqations of sstem (4), completed with (5) and (6) permit to find ot the main parameters of the sstem oscillator stable regime. Additionall to all that were presented, we need to determine the sstem otpt characteristics: the mean vale, the extreme vales and the otpt bandwidth. The otpts mean vale can be calclated sing an integral relation: T T ( ( t) ( t ) ) dt. (8) Using the CLS arcs eqations, after same calcls and transformations we obtain a short reslt: t p tn. (9) T For the calcls of the extreme otpt vales, we will begin with same observations: - the slope of a tangent of the CLS arc which converge to the PAP from the point PCN, can be express sch as: d d T T and after some transformation: d d () T. () / ( ) cn ( ) / T T T cn In the previos relations (, ) are the coordinate of the tangenc point. Becase the vale is positive, it is obviosl that the tangent slope continosl decreases this conclsion will be also obtained for the other CLS arc - the extremes sstem otpt vales are characterized b a extremes vales of the difference - for the points where CLS cross the line between the points PAP and PAN, the coordinates satisfied the condition: ISSN: Isse 8, Volme 4, Agst 9

6 T T, () and the slope of the tangent of the CLS are nitar: T d. (3) T d Using the previos observations, we can conclde that the extremes otpt vales correspond to the points where CLS cross the line between the points PAP and PAN. The figre presents theses crossing points. The relations (5), (6), (9), (4), (5), (6) and (7) will complete the sstem eqations (4). In this wa, we obtain a complete eqations sstem, which can be sed for the calcls of the CLS parameters and of the oscillator stable regime parameters. The complete eqations sstem is too complex and nonlinear. The solving of the complete eqation sstem is possible, sing an iterative algorithm. This algorithm reprodces the sstem evoltion toward the CLS. Δ max min min ( ) min n PAN p PAP max ( ) max 6 The calcls algorithm In order to find all the parameters of CLS and of the oscillator stable regime, the next steps will be passed: a. choose an arbitrar vale εp b. calclate the vale εp, based on (): εp εp r ε Fig. The points of the CLS where the sstem otpt has the extreme vales We will se the notation p and n for the vales of the variable which corresponds to max and min, the sstem otpt extreme vales. Using the last conclsion and the CLS parameters, we will obtain the following relations for the calcls of the vales p and n : ( ) cp n ( ) cp ( ) cp p. ( ) cp (4) The maximm vale of the sstem otpt will be: max p /, (5) and the minimm of the sstem otpt: min n /, (6) Finall, the otpt bandwidth can be calclated b sing the relation: Δ ( p n ) /. (7) c. calclate the vales cp and cp sing the state variable evoltion: cp cp ( εp ) ( ) εp e τ / T e τ / T d. calclate the vale t n, lie soltion of eqation: cp exp cp ( t T ) exp( t T ) n e. calclate the vales cn and cn : n r ε ( τ t ) T ( ) cn cp e n ( τ t ) T ( ) n cn cp e, f. calclate the vale t p, lie soltion of eqation: exp cn cn ( t T ) exp( t T ) p g. calclate the vales cp and cp : p, r ε cp cp exp ( τ t p ) T ) cp cp, exp ( τ t p ) T ) ISSN: Isse 8, Volme 4, Agst 9

7 h. if the differences between the sccessive vales of cp and cp are not acceptable go to step d i. calclate the CLS period: T t n t p τ, j. calclate the otpt parameters:. STOP. ( t p tn ) T Δ ( p n ) /, max p / min n /. The presented algorithm strongl converges (the steps d-h) to the soltion the next example emphasizes this fact. Table. The reslts in case r r. parameters Iteration Iteration Iteration 3 εp εp cp cp t n τ cn cn t p τ cp cp Δ cp Δ cp T.5886 Example. The calcls of CLS parameters b sing the above presented algorithm, in case of a nonlinear sstem parameterized b the next vales: T 5s, T 3s, ε.5, τ.s. For the reference, we will consider the vales: r, r., r 3.5, and the reslts are presented in the next tables. Table. The reslts in case r r. parameters Iteration Iteration Iteration 3 εp εp cp cp t n τ cn cn t p τ cp cp Δ cp Δ cp T.58 Table 3. The reslts in case r r 3. parameters Iteration Iteration Iteration 3 εp.5 εp.8 cp cp t n τ cn cn t p τ cp cp Δ cp Δ cp T A case std In this example we will consider a nonlinear sstem, with the strctre presented in figre. R The Nonlinear part τ U The Linear part 5 H( s) (5s )(3s ) Y Fig. Example of a nonlinear sstem The parameters of the linear part are: T 5s, T 3s and the gain L 5. The stationar otpt vale of the nonlinear part is U. ISSN: Isse 8, Volme 4, Agst 9

8 We will se the presented algorithm to determinate the characteristics of its oscillator stable regime, for an vales of the dela τ and of the inpt hsteresis bandwidth, ΔH. In order to appl the presented calcls method, we will se an eqivalent sstem, as in figre. R r τ H( s) (5s )(3s ) Fig. The eqivalent shape for the exemplified nonlinear sstem The open loop gain for the initial sstem is: L U. st The central part of the eqivalent sstem flfills the conditions of the std model. The interval of hsteresis bandwidth vales for the eqivalent sstem is: ΔH ε. For the central part of the eqivalent sstem, the reference r and the otpt represent normed vales: r R /, Y /. If the parameters of the otpt of the eqivalent sstem central part are, Δ, min and max, the parameters of the initial sstem otpt Y will be: Y Y, ΔY Δ,, Y. min min max max The oscillator regime appears if the next condition is achieved: r < ε. For the initial sstem, the previos condition becomes: ΔH R <. Y 5/.5, 3/.5. Based on the presented algorithm, in an possible cases ( τ,ε ), for differed reference vales it was obtained the parameters of the CLS and of the oscillator stable regime. The next tables and figres present the reslts τ[s] ε. 5 ε. 35 ε. ε.. 5 ε.75 ε. T[s] 5 Fig. 3 The oscillator stable regime period depending on rela dela time for a nll inpt ε..9 τ τ τ..4.3 τ... 5 τ. τ.5 T[s] 5 Fig. 4 The oscillator stable regime period depending on rela inpt bandwidth for a nll inpt Table 4. The period of the oscillator stable regime for a nll inpt ε τ[s] The gains of the two elements, which decompose the linear parts of the eqivalent sstem, are: ISSN: Isse 8, Volme 4, Agst 9

9 Table 5. The otpt bandwidth, Δ[%], of the oscillator stable regime for a nll inpt ε τ[s] T[s] τ.75, ε. 5 τ.75, ε τ, ε. 5 5 Fig. 5 The oscillator stable regime period depending on the inpt τ.75, ε. 5 τ.75, ε [%] τ, ε. 5 5 Δ Δ r[%] r[%] Fig. 6 The otpt and the otpt bandwidth depending on the inpt 8 Conclsions For the tpe of nonlinear sstems taen into consideration, the paper offers certain instrments sefl for evalating the performances of the limit stable ccle which characterize its woring. Compared with the se of the nmerical simlation methods, the se of the calcls relations is more advantageos, being able to offer a more comprehensive view on the limits of these atomatic strctres. Regarding to the reslts, some conclsions have to be nderlined: - the period of the oscillator regime growing with the dela and with the hsteresis rela bandwidth - the most redced periods of the oscillator regime correspond to a nll references - the mean otpt of vale deviates from reference this deviation can be considered linear and it can be eventall compensated sing a gain correction in the reaction loop - for the dela vales comparable with the major time constant of the sstem and for high vales of hsteresis rela bandwidth, the sstem s performances deteriorate considerabl - the otpt bandwidths has a ver low variation with the inpt increase, bt it increase with the dela and with the hsteresis rela bandwidth. 9 References: [] Mihoc, D., Ceapâr, M., Iliesc, S. St., Borangi, T., The Theor and the Elements of the atomatic control sstems, Didactical and Pedagogical Pblishing Hose, Bcharest, 98 (rom) [] Belea, C., Nonlinear Atomatics, Technical Pblishing Hose, Bcharest, 983 (rom) [3] Belea, C., Sstems Theor. Nonlinear Sstems, Didactical and Pedagogical Pblishing Hose, Bcharest, 984 (rom) [4] Nţ Vasile, Rotari Adrian, Cîrmaci Maris- Valeri, The Oscillator Stable Regime of Nonlinear Sstems, with two time constants, 6th WSEAS International Conference on CIRCUITS, SYSTEMS, ELECTRONICS, CONTROL & SIGNAL PROCESSING (CSECS 7), Cairo, Egpt, December 9/3, 7, ISSN: 79-57, ISBN , Part II, Sstems Theor, page 5-9 ISSN: Isse 8, Volme 4, Agst 9

10 [5] Stăncilesc, Florin, The Analze and the Simlation of nonlinear sstems. Nonlinear problems in electronics, electrotechnics and atomatics, Romanian Academ Pblishing Hose, Bcharest, 974 (rom) [6] Ionesc, Vlad, Sstems Theor, Didactical and Pedagogical Pblishing Hose, Bcharest, 985 (rom) [7] Nţ, Vasile, Elements of analsis of linear atomatic sstems, Scientific Univers Pblishing Hose, Bcharest, 7 (rom) [8] Nţ Vasile, Moldovean Cristian, Somoiag Pamfil, Some aspects of the Oscillator Regime of Nonlinear Sstems with Rela and Dela Time, th WSEAS International Conference on AUTOMATIC CONTROL, MODELLING and SIMULATION (ACMOS 9), Istanbl, Tre, Ma 3-Jne, 9, ISSN: 79-57, ISBN , page ISSN: Isse 8, Volme 4, Agst 9