5. NON-LINER BLOCKS Non-linear standard blocks

Transcription

1 5. NON-LINER BLOCKS In previou chapter continuou tem or tem where to the change of the input a change of the output correponded, which in the whole range of the ignal value could be expreed b one equation, were conidered. Such conception i ome-how inaccurate, but in mot cae, b drive, for intance, the ariing error remain le than the accurac cla of the meauring device. In addition, the error i reduced b the fact that the operation occur on the linear part of the characteritic mainl. Neverthele, ituation arie, where the tem ha to be decribed a non-linear block, or, the equation decribing the change of the output depending on the change of the input varie depending on how large i the input ignal, and ometime on that, how large ha been the input ignal at previou time moment (b the hterei) [6] 5.1. Non-linear tandard block a) b) c) d) e) 1 Figure 5.1. Non-linear tandard block a) linear, b) aturating, c) with inenibilit zone d) with hterei, e) two-point characteritic A aturating block all amplifing block hould be decribed in fact, therefore that b real amplifier the growth of the output i limited b the uppl energ. For example, uing a thritor bridge, the output of the converter could be varied in he range V, at the ame time, b the D-regulator the intantaneou value of the voltage could exceed one kilovolt limit. Mathematicall, the equation characteriing thi block are preented a one tem: = 1, kui 1 =K, kui 1 2 = 2, kui 2. (5.1)

2 The inenibilit (dead) zone exit b all device, but relativel larget inenibilit zone exit b meauring device and tranfer. For intance, b a magnet-electric meaurement device the meaured current ha to overcame the reitance force of the pring and bearing firt. B the tranfer, the inenibilit i characteried b the backlah between two gear or the elaticit of the belt of a variable peed drive. The mathematical repreentation i = K, kui 1 =0, kui 1 2 = K, kui 2. (5.2) The hterei could be oberved b the rela with magnetic core, where, for intance, the reidual magnetim of the core hold contact cloed, whereb the correponding for ignal ha diappeared alread. B operating of the ame rela the magnetic flux ha to cro the air gap, therefore for the operation higher input ignal i required than for reetting : Expreed a a tem of equation it i a follow: = 1, kui 1 = 1, kui 1 2, enne = 1 = 2, kui 1 2, enne = 2 = 2, kui 2. (5.3) Two-point characteritic decribe the rela in general b on value of input ignal a jump in the output occur (i.e. b cloing of rela contact) and b further growth of the input, the output doe not change an more. However, the output ignal will change tepwie b decreaing of the input ignal (b opening of the contact). It mathematical repreentation = 1, kui 0 = 2, kui 0. (5.4) In fact, non-linear block are combined from tandard block uuall. So could oberve the hterei b the bi-metal rela, which i decribed well b the two-point characteritic. B the real meauring device once the meaurement range come to an end and that could be decribed b becoming aturated. Proceed from that all nonlinear part of automatic control tem hould be conidered carefull, to decribe them with proper tem of equation and onl if falling of the device into non-linear operation i excluded, then decribe the device a linear one

3 5.2. Non-linear regulator For regulator of continuou action the following relation i valid: u t = f e t, which i repreented in the figure 5.2. If the declination change continuoul in the range 0...e 1, then alo the control action change in thi range continuoul. If e 1 =0, then the control Jooni 5.2. Pideva üteemi karakteritik action ha two poible value onl u=0 or u=u max or, one ha to do with a two-point regulator. Such characteritic do have rela, bi-metal witche etc. And a a continuou regulator o the non-linear regulator react to the deviation. B given characteritic on the poitive deviation the regulator would witch on, and b negative deviation, it would witch off. [6] Although the non-linear regulator are technicall imple, the have one diadvantage vibrating control action. Such ignal i caued b continuou witching of the regulator, whereb the witching period equal to the vibration period and the longer i the witching period the bigger i the ocillation amplitude. So a the mechanicall witching device have limited number of witching, thu, earlier uch regulator were ued for impler tak for the control of room temperature, water level etc. Modern witching regulator have grater witching number, therefore in a welladjuted tem.the amplitude of ariing ocillation remain le than the dead zone of the meaurement device. Thi fact enable the application of uch regulator for more complicated tak [6].

4 Two-point regulator and PT 1 -T h control object In the figure 5.3. a flow-though water heating tem i preented, that i controlled b a contact thermometer. B witching the device on, the rela operate and the heater get the uppl, in reult of what the temperature of water will rie. Proceed from that, the mercur column in the thermometer rie, which, reaching the upper (adjutable) contact hort the coil of the rela, upon which it contact open and the heater remain without uppl. With water temperature lowering the thermometer contact will witch off and the heater will witched on he uppl again. [6]. L N Küttekeha Kontakttermomeeter Figure 5.3. Water heater If to handle thi tem from the point of automation, then one ma notice, that the temperature of the heater, water and mercur column are growing aperiodicall and the time lag in the tem i introduced b cell wall of the thermometer of the heater. The hterei of the rela could be omitted here, o a it i negligible relative to other time contant. Or, umming up, one ha to do with PT n -T h -tem, that could be approximated to a PT 1 -T h -tem. The parameter of thi tem could be obtained b method imilar to that, which wa conidered b CHR nthei method, and the tem could be repreented with the following tructure diagram:

5 K P, T 1, T h e u _ Figure 5.4. Equivalent circuit of a water heater One more implification that will be made in thi application i the levelling of time contant of heating and cooling. what i the fact in practice. For proving that, one might tud the urface of the bod, through of which the heat i given to the bod and to compare it with the urface through which the heat i given up from the bod or it i cooled. The ditortion could be introduced b the motion of the heat or motion caued b heat in the bod a well. If the tem i et into operation ( 0 =0 ), it output, after lag period, et to grow aperiodicall. Without a regulator, up to = K P. Uing regulator and feedback it i poible to reduce the output of the tem (reducing power, for intance), reducing tem time contant virtuall at the ame time. T 1 loomulik max 2 0 min 0.5*u T h t 1 T h t 2 Figure 5.5. Tranient characteritic of proce quantitie of the water heater T h

6 For the determination of the of the witching duration of the regulator it could be expreed t t =T h t 2 (5.5) and imilarl expre the paue duration t p =T h t 1. (5.6) The determined duration together give the witching period, which at the ame time i the ocillation period of the output. T lp =t t t. (5.7) The upper limit of the output i expreed a max = 1 e T h T 1, (5.8) where = K P u. (5.9) The lower limit of the output correpondingl min = e T h T 1. (5.10) Deleting from the equation (5.8) the equation (5.10) the range of ocillation could be determined 2 0 = max min = 1 e T h T 1. (5.11) It follow from (5.11), that the ocillation range i proportional to the tem time lag and reciprocal to the tem time contant. It hould be mentioned that the ocillation range doe not depend on the et value, but on the tem time contant onl. In the figure 5.5. a ituation i repreented, where the et value = ½. In the cae the output ocillate around the mean, determined b the et value. If to increae or decreae the et value, then the ocillation mean doe not coincide with the value determined b the et value. Thi difference i called the mean value hift and it i calculated from the relation:

7 k = max min 2, (5.12) kn = k = T h 2 1 e T 1. (5.13) Figure 5.6. Graphical example of the mean value hift The equation of the ocillation period i given in (5.7). Previoul the time t 1 and t 2 hould be determined, which are calculate b equation t 1 =T 1 ln[ 1 T 1] h e T, (5.14) t 2 =T 1 ln e T h T 1. (5.15) Placing the previou equation into equation (5.7), the ocillation period will be expreed a

8 T lp =2 T h T 1 ln[ 1 1 e T h T 1 e T h T 1 ]. (5.16) In figure 5.7.the dependence of the witching period and witching duration on the et value i repreented for the cae if T h T 1 = On can ee, that the witching period i minimal, if =0.5, In the ame point the witching duration equal to the witching paue. Regulation range of thi tem b the input i , wherea outide of thi range the ocillation period will increae harpl, alo the mean value hift. In cae if the ocillation frequenc approximate to zero, the ocillation period approximate to zero alo. Switching frequenc, being reciprocal to the ocillation period, approximate to infinit, which et evere requirement to the rela-baed regulator due to the limited witching number of regulator. So, in the nthei proce the compromie i found between witching frequenc and witching range. a) b) Figure 5.7. Dependence of the tem parameter on the et value a) witching period, b) witching duration Two-point regulator with hterei and PT 1 control object Mot real device of rela action do have hterei, let it be introduced or b magnetim or b friction, etc. Thereb, the hterei ha not to be mmetrical in

9 relation to the zero point, however, the law hold, that the operation value of the rela i greater than it reet value the [6]. In the figure 5.8. a regulator with hterei i repreented controlling an aperiodical proce. Similar to the implification made in the previou diviion it could be aumed, that the time contant of the proce in the growth of the output equal to the time contant of the decline of the output. For the implification, the rela with mmetrical hterei alo i elected. e u K P, T 1 _ Figure 5.8. Two-point regulator with hterei and aperiodic proce Switching thi tem into operation i output tart to grow exponentiall till the value h, after that the rela reet and the output diminihe until it operate again. Thi proce proceed periodicall, however, the difference from the above conidered he mean value of ocillation coincide with the et value. Figure 5.9.Tranient characteritic of the tem with a regulator having hterei. t t t p The witching duration of the rela could be calculated b the equation T lp

10 t t =T 1 ln h h (5.17) and the paue duration correpondingl t p =T 1 ln h h. (5.18) Adding equation (5.17) and (5.18) the witching period could be calculated b T lp =T 1 [ ln h ln h h]. (5.19) Similar to the tem conidered in the previou diviion a time lag occur in thi tem, however, it doe no depend on the proce parameter but i determined b the hterei of the regulator Comparing equation (5.16) and (5.19) one might notice, that the witching period depend in thi tem alo on the relation, which i graphicall preented in figure 5.10, provided that h =0.1. Given relation correpond in principle to the relation T h T 1 that wa preented in the previou tem Figure Dependence of the witching period on the et value

11 Two-point regulator with local feedback The behaviour of tem, condered in two preiou diviion, wa trongl influenced b the parameter a of the proce a well the parameter of the rela itelf, couing ocillation of large ampliude and long lating in the output.. However, uing local feedback b rela-tpe regulator it becom poible toreduce, t and in ideal cae even get rid full the winging of the operating poin.[3] K r, T r e _ e r u K P, T 1 _ Figure Feedback tem with two-point regulator B witching thi tem into operation the rela-action devie operate and the output of the tem and the value of local feedback tart to grow. The objective of local feedback i to create to the input of the rela a virtual output quantit, that will force it to witch out earlier, thu reducing the ocillation amplitude of the operating point. Conidering the control devie a being independent, one can mention, that the control device act a PD-regulator. In the figure the value of input, feedback, deviation and output ignal in time are preented. Oberving tha lat characteritic, repreenting the mean of the output value, one ma be convinced that it i imilar to the characteritic of a continuou PD-characteritic. For thi regulator the following parameter could be preented: K R =K r u max (5.20) and T R =T r. (5.21)

12 Figure 5.12.Characreritic of a control device with local feedback. To determine thee parameter for the whole tem, the tem i conidered without local feedback, and from it output characteritic the critical ocillation range and the critical period are tated. krit T krit Figure Characteritic of a tem without local feedback

13 The parameter of the regulator could be expreed though the critical parameter a follow: K R =2 krit (2.22) T R = T krit 16. (2.23) Figure Comparion of regulator of continuou and rela action So a on hat to do with a regulator of PD-action, the van permanent operation error of tha tem arie, however, if to compare the regulator of the rela action with the continuou action one, one could notice, that both are operating imilarl, having different tatic error, that b the device of rela action i le due to the ocillation, induced b the hterei. Partl, the difference in contingent to the inaccurac of the model, o a mathematicall the D-block i inducing tatic error and with a rela device one ha to do with non-linearit Three-point regulator with hterei and I- output tranducer A diadvantage of two-point regulator i, that it i impoible with them to control

14 reverible procee, electrical motor, for intance, or tage-wie witching, o a with them it i poible to witch between two dicrete tate onl. Three point regulator are, however, a combination of two two-point regulator, enabling thu to control reverible procee. Such control i jutified b actuator, when one ha to deal with mall capacitie and the motor ha to be often tarted in both direction, for the opening and cloing of motor valve, for intance. Thereb, the actuator could be conidered a a part of the control device, that could be well decribed b an I-block [3] Figure Temperature regulation with three-point regulator Switching thi tem into operation the polarit of the rela winding voltage will be determined, depending on the deired room temperature and the poition of the valve. For he explanation of the operating principle it could be aumed, that preliminar the rela get poitive voltage from the bridge circuit, cloing it contact o, that the actuator motor open the valve. In a proce the uppl voltage of the winding will change (the poition enor of the valve), however, the rela will not reet due to the hterei, except cae where the valve i alread full opened. In thoe cae, the uppl voltage will decreae o much, that the motor uppl interrupt. With the rie of the temperature the value reitance of the reitance-thermometer will change, in reult of what the rela uppl voltage will change until it polarit change and rela witche negative uppl voltage to the motor, in reult of what the motor will cloe the valve. Simultaneoul the value of the poition enor of the valve and the

15 thermometer reitance will change. From a certain moment, the tem tabilie around the point determined with et value and it parameter. [6]. Three-point regulator could in ome cae be repreented with the following tructure diagram. K R, T R e _ e r K I u Figure Three-point regulator with local feedback and integrating output Figure Internal tranient characteritic of a three-point regulator Oberving the lat characteritic, one might notice, that the regulator act a IP-block, however, in large the regulator act like two-point regulator Exercie to the chapter 5 Exercie 14. Decribe the timel progre of ignal of the tem repreented in the figure 5.18, uing attached coordinate tem

16 K = 1-1 u e u Figure A tem with three-point regulator t t e t u t Figure Characteritic for the decription of the ignal progreing