Automatic Control Systems Toolbox v.3.0
Contents
1. All input delays will be resetted lt SYNTAX gt sersys series sys1 sys2 OUT 1 IN2 where sersys is the name of the resultant LTI system obtained by series interconnection of sys1 and sys2 The outputs of sys1 specified by OUT1 are connected to the inputs of sys2 specified by IN2 see fig 8 lt EXEMPLE gt We have two systems sys1 with three inputs and four outputs and sys2 with five inputs and three outputs We want to connect sys1 to sys2 in series by connecting outputs 3 and 2 of sys1 with inputs 1 and 4 of sys2 from the Command Window we must enter sersys series sys1 sys2 3 2 1 4 sys1 and sys2 must be preventively stored in sys1 and sys2 var All input delays will be resetted Command parallel lt PURPOSE gt Parallel Connection of two LTI SISO or MIMO systems All input delays will be resetted lt SYNTAX gt parsys series sys1 sys2 IN1 IN2 0UT1 0UT2 where parsys is the name of the resultant LTI system obtained by parallel interconnection of sys1 and sys2 The inputs specified by IN1 and IN2 are connected and the outputs specified by OUT1 and OUT2 are summed The resulting system parsys maps u1 u u2 to y1 y y2 The vectors IN1 and IN2 contain indexes into the input vectors of sys1 and sys2 respectively and define the input channels v12 and v21 in the diagram Similarly the vectors OUT1 and OUT2 contain indexes into the outputs of these two systems2. and define the output channels z12 and z21 in the diagram see fig 9 lt EXEMPLE gt We have two systems sys1 with three inputs and four outputs and sys2 with one input and one output We want to connect sys1 and sys2 in parallel by connecting input 1 of sys1 with the input 1 of sys2 and by summing output 3 of sys1 and the output of sys2 from the Command Window we must enter parsys parallel sys1 sys2 1 1 3 1 or parsys parallel sys1 sys2 1 1 3 1 sys1 and sys2 must be preventively stored in sys1 and sys2 var All input delays will be resetted pag 11 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 12 20 Command feedback lt PURPOSE gt Feedback Connection of two LTI SISO or MIMO systems All input delays will be resetted lt SYNTAX gt fosys feedback sys1 sys2 INPUT OUTPUT where fbsys is the name of the resultant LTI system obtained by feedback interconnection of sys1 name or expression and sys2 INPUT is the vector that contains indices of the input vector of sys1 and specifies which inputs are involved in the feedback loop OUTPUT specifies which outputs of sys1 are used for feedback INPUT and sys2 outputs must have matching dimensions OUTPUT and sys2 inputs must have matching dimensions INPUT contains the indices of v2 INPUT i it s connected with yO i OUTPUT contains the indices of y2 OUTPUTI i it s connected
3. lt F4 gt lt 1 gt load a saved PIC file press lt F4 gt lt 2 gt Seale sek impulsetg 153 o g feedback gl gsintl3 1 13 gt gt stepta 152 AEST RAD AUTO FUNC 21 30 fig 26 G s modelling and commands A EAS ACST RAD AU TO RAD AUTO fig 27 impulse response of G s fig 28 step response of G s FUNC ACST FUNC pag 17 20 automatic control systems toolbox v 3 0 Command simul lt PURPOSE gt Plot the time response of continuous or discrete linear systems to arbitrary L transformable inputs lt SYNTAX gt simul SYS INPUT StopTime where ACST SYS it s the name of system under analysis user s manual 2000 2001 92BROTHERS pag 18 20 INPUT it s the vector column of inputs INPUT must have a row for each imput of SYS number of colums of SYS INPUT elements must be L transormable StopTime it s the stop time in seconds for the simulation lt EXEMPLE gt 1 3 3 1 fig 29 The inputs of global system G s are control u t 1 the control input see fig 31 noise 0 2 u t 10 the noise see fig 31 The commands to define inputs and run the simulation are illustrated in fig 31 gt gt utt 1 4control gt gt 0 2 uCt 10 4noise gt gt simul g control noise 20 RAD AUTO FUNC 25430 fig 31 After time response it plotted fig 32 you can return to command window press lt
4. commands use this manual for full help on commands operator can be used to assign right values to left values ex var expression assign expression to van ggraph to run a simple graphic editor to edit graphical results preventively saved control toolbox commands bode to perform the bode analysis of a SISO or MIMO system feedback to connect two SISO and or MIMO systems by feedback connection impulse to simulate the time impulse response of SISO or MIMO systems margin to evaluate gain and phase margins of SISO or MIMO systems mmimo to crate MIMO systems using SISO systems nichols to plot the Nichols chart of a SISO or MIMO system nyquist to plot the Nyquist diagram of a SISO or MIMO system parallel to connect two SISO and or MIMO systems by parallel connection rlocus to plot the root locus of a SISO system tf to specify the transfer function of a system by numerator and denominator coefficients series to connect two SISO and or MIMO systems by series connection simul simulate the time response of SISO or MIMO system to arbitrary inputs step simulate the time response of SISO or MIMO systems to step input ss to specify the transfer function of a system by state space model of a system zpk to specify the transfer function of a system by zeros and poles of a transfer function zpmap to map zeros and poles of a SISO system Some of this commands can be runned from the HOME window of the calculator but it s
5. menu and select Send Select desktop acst acst 92g file from the dialog window and click on Add Click on Retain Folder to send the files back to their original folder Click on OK to transmit the files 3 control that a new fold named acst was created and it contain all listed files acst PRGM bode PRGM errorms PRGM feedback FUNC ggraph PRGM ilaplace ilapsub laplace lapsub impulse PRGM intro PIC invlap FUNC kernel PRGM laplace FUNC margin mmimo nichols PRGM nyquist PRGM parallel FUNC rlocus PRGM series FUNC simul PRGM step PRGM zpmap PRGM 4 ACST v2 2 it s now installed ilaplace ilapsub laplace and lapsub are included as a library and are developed by Lars Frederiksen pag 6 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 7 20 The Command Window The Command Window CW allow ACST s users to input their commands and see the result The gt gt symbol it s the prompt and it s displayed when ACST it s ready to get commands To input a command simply type it and press lt ENTER gt to run it There are two kinds of commands TI BASIC commands operator excluded and user defined commands function and or programs ACST commands There are two kinds of ACST commands see next paragraphs to more explains basic commands quit to quit ACST commands clear to refresh the command window help to get quick help on
6. recommended to run the commands from the ACST Command Window CW that perform a control on syntax or convert to the rigth sintax and suggest you the right syntax or return the error number Running the commands from the CW moreover allow ACST Kernel control changes on the mode settings of your calc and the storage of temporay files into your calculator memory that can occur after a runtime error pag 7 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 8 20 LTI Models LTI systems are rappresented as a matrix of transfer functions If the system has M inputs and N outputs the matrix has N rows and M coluns and SYSTEMI i j it s the transfer function from the input j to the output i To define SISO system you can specify the coefficients of numerator and denominator of the transfer function the poles and the zeros of the transfer function the state space model note that you can t specify the transfer function as a function of s as in the older versions This limit is due to a new way to stori LTI systems models to have best performaces A special SISO system it s the Linear Gain or Gain To specify a gain you must specify the gain value The corresponding commands are sys tf NUM DEN sys zpk ZEROS POLES k sys ss A B C 1 D sys gain GainValue To Define MIMO M inputs and N outputs system you can define MxN SISO systems and co
7. with vO i see fig 10 fig 10 lt EXEMPLE gt We want to make a system as shown in the next picture fig 11 ul ue us from the Command Window we must enter fosys feedback sys1 sys2 3 2 3 4 sys1 and sys2 must be preventively stored in sys1 and sys2 var All input delays will be resetted pag 12 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 13 20 Analysis in Frequency Domain Analysis of LTI MIMO systems can be performed by Bode Plots plots the magnitude and phase of the frequency response of LTI systems Full semilog grid is also displayed Bode plots analysis support SISO or MIMO systems and systems with time delays Nichols Plot plots the frequency response of an LTI MIMO system and plots it in the Nichols coordinates It s also displayed the 3dB curve of the Nichols Chart Nichols analysis support SISO or MIMO systems and systems with time delays Nyquist Plot plots the Nyquist frequency response of LTI MIMO systems It s also displayed the 1 0 point and the unitary circle Nyquist analysis support SISO or MIMO systems and systems with time delays Gain and Phase margins calulates the gain margin and the phase margin and the crossover frequencyes of a system Gain and Phase margins support SISO or MIMO systems but not support systems with time delays Command bode lt PURPOSE gt Bode plot of a LTI SISO or MIMO system lt SYNTA
8. 01 92BROTHERS pag 14 20 Command nichols lt PURPOSE gt Nichols plot of a LTI SISO or MIMO system lt SYNTAX gt nichols SYS wmin wmax or nichols SYS where SYS it s a the name or the expression of a LTI SISO or MIMO system wmin wmax it s a two elements list that specifies the frequency range of the plot If it s used the syntax bode SYS wmin and wmax will be setted by the program lt EXEMPLE gt E Pi E NER BE We want the Nichols Plot of G s SISO system zeros 10 10 poles 0 1 5 gain 1 We want that the frequency range will be auto selected After the Nichols plot it s plotted fig 17 by the toolbar at the top you can return Command Window press lt F1 gt change the zoom press lt F2 gt select plot MAG Phase type press lt F3 gt show windows dims press lt F4 gt lt 1 gt get a point info press lt F4 gt lt 2 gt save the plot as a PIC file press lt F5 gt lt 1 gt load a saved PIC file press lt F5 gt lt 2 gt ss 1J 3 5 delay 0 ACST DE PAR G APPROX fig 17 Nichols plot of G s pag 14 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 15 20 Command nyquist lt PURPOSE gt Nyquist plot of a LTI SISO or MIMO system lt SYNTAX gt nyquist SYS wmin wmax or nyquist SYS where SYS it s a the name or the expression of a LTI SISO or MIMO system wmin wmax it s
9. Automatic Control Systems Toolbox v 3 0 user s manual automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 2 20 INDEX Note OIC SMSC toi E 3 Note for TI 92 and Tl 89 WSers casei a a ss a ee 3 Used Symbols and Abbreviations crearan ad ane 4 What s Automatic Control Systems TO0lDOX oooooccccccccccccccooccccnnnccnnnnnnnannccnnnnnnnn nro ccnnnnnnnnnnnn nn nnccnnnnns 5 What its new inversion g Dioni has BO Ar eee 5 Installing Automatic Control Systems Toolbox ACST V 3 0 ooooncccnnnccnnccccnnnncncnnnnnannncnnnnnnnnnnnencnnnnn 6 The GCommand WINdOW 2 ecimiento 7 ETLModelS 0 8 CAIMAN e a o e et ae ni ati tae ales 8 Command A ol se escheat ec etack A ces cl basteNcnen T 9 GOMIMANG ESS LE A AA ias 9 GOMMANG Gal aine ea aa lalalala e 9 Command MMim6 ii A ei a Wea a lena dices een Onell aa 9 Gommand S tc Slay seis se a laa odiar apa aero 10 Command S rie cairo io te ai le oie nadie 11 Command parallel coiiniiosi aa 11 Command Tecdhack loo 12 Analysis in Frequency DOMAiN ccocccnnnnoncccncnnnonanonnnnnnnnnonnnnnnnnnnnnnnnnrnnnnnnnnnnnnnnnnrrnnnnennrnnn nr nnnnnnnncnn 13 Command bode cion AA 13 Command NCNOlS sariaren aira dadaa orada ara aa aaa adoura aa a aaa naa dap aeda eaaa 14 GOMIMANG EM ANAS UN EA EEEE EEE E A EE EE ET 15 Command MA Nuria aida 15 Poles and Zeros Analysis ic cocinas 16 GOMIMANG HOCUS E EE T eh ved Miata Tied eee eee haserteAnevectaeceetien eels ent eleven h
10. F1 gt set zoom factor press lt F2 gt select the output MIMO systems press lt F3 gt show plot dimensions press lt F4 gt lt 1 gt trace the plot press lt F4 gt lt 2 gt save the plot as a PIC file press lt F5 gt lt 1 gt noes a ENEN PIC lle press lt F5 gt lt 2 gt pun eos ae RAD AU fig 31 response of G s We want to study the response of G s described in fig 29 for 20 sec G s can be formed by feedback connection of G1 s and unitary gain system fig 30 G1 s it s a system with two inputs control input and noise input and one output in the feedback connection only input 1 is envolved For G1 s the tranfer function from input 1 and the output is g1 1 1 1 s s 1 the tranfer function from input 2 and the ouput is g1 1 2 1 fig 30 system name number of inputs number of outputs insert system from input 1 to output 1 tfetil1 01 1 073 insert system from input 2 ai om 10 0 fig 30 G s see fig 29 model pag 18 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 19 20 Elaborating Graphical Results Command ggraph Graphical results obtained by bode nyquist nichols rlocus zomap impulse step simul stored in PIC variables see prev sections can be elaborated by a simple graphical application not so powerful but a simple tool that allows to elaborate plots under ACST environme
11. Nyquist diagram MIMO systems Gain and Phase margins MIMO systems Root locus SISO systems Poles Zeros map SISO systems systems simulation by step response MIMO systems impulse response MIMO systems generic multiple input response MIMO systems save and modify graphical outputs What it s new in version 3 0 new LTI modelling that allow to set the input delay separately by the LTI transfer function to speed up the analysis of systems now you can use auto frequency ranging for bode nichols and nyquist analysis more flexible systems simulation specific command to create MIMO systems using SISO systems bugs fixed faster pag 5 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 6 20 Installing Automatic Control Systems Toolbox ACST v 3 0 To install ACST into your calculator you need software TI GRAPH LINK software http www ti com calc docs link htm acst zip htto Awww 92brothers net or http web tiscalinet it 92brothers that must contain acst 92g for TI 92 readme txt license txt acst manual pdf this manual hardware Windows Macintosh Gray TI GRAPH LINK cable Windows Only Black TI GRAPH LINK cable To install ACST into your calculator must 1 unzip the acst zip file into a folder ex desktop acst 2 send files from your PC to a TI 92 run the Tl Graph link program Open the Link
12. X gt bode SYS wmin wmax or bode SYS where SYS it s a the name or the expression of a LTI SISO or MIMO system wmin wmax it s a two elements list that specifies the frequency range of the plot If it s used the syntax bode SYS wmin and wmax will be setted by the program lt EXEMPLE gt E We want the Bode Plot of G s SISO system zeros 10 10 poles 0 1 1 10 5i 5i gain 1 in the frequency eke 10 101 6 1 1 10 54 54 1 15 range of 0 1 100 rad sec After we run the command fig 12 a pop up menu will ask what to plot MAG plot PHASE plot or MAG and Phase plot in split ea screen mode In fig 13 14 15 are illustrated the MAG plot the s sti s 101 32 25 Phase plot and MAG and Phase plot in split screen mode delay 0 a 3 respectively bodeca 8 1 10077 After the selected plot it s plotted by the toolbar at acy CA Fc the top you can return Command Window press lt F1 gt O anil change the zoom press lt F2 gt att ARTE select plot MAG Phase type press lt F3 gt Ho show windows dims press lt F4 gt lt 1 gt get a point info press lt F4 gt lt 2 gt save the plot as a PIC file press lt F5 gt lt 1 gt load a saved PIC file press lt F5 gt lt 2 gt Aaa A TNU A SNS CAI CUL io TA CU g e O LS DO O A O O O un Co 7 GiH FUNC fig 13 14 15 Bode plots of G s pag 13 20 automatic control systems toolbox v 3 0 user s manual 2000 20
13. a two elements list that specifies the frequency range of the plot If it s used the syntax nyquist SYS wmin and wmax will be setted by the program lt EXEMPLE gt al la aime Gos We want the Nyquist Plot of G s SISO system NUM 2 DEN 10s 1 with time delay of 2sec in the frequency range of 0 001 5 rad sec fig 18 After the Nyquist plot it s plotted fig 19 by the toolbar at the top you can return Command Window press lt F1 gt gtfrel2 010 115 change the zoom press lt F2 gt geta point info press lt F4 gt 2 ReGraph the plot press lt F5 gt I s 1 save the plot as a PIC file press lt F6 gt lt 1 gt gorane ig load a saved PIC file press lt F6 gt lt 2 gt nuquistig 10 004 575 ALST RAD AUTO fig 18 ex running command nyquist EsTrizcon ser 1 o rracelecrapnfrite a a A ACST RAD APPROX PAR fig 19 Nyquist plot of G s Command margin lt PURPOSE gt Evaluate Gain Margin and Phase Margin return also crossover frequencyes lt SYNTAX gt margin SYS where SYS it s a the name or the expression of a LTI SISO or MIMO system lt EXEMPLE gt We want the Gain Margin Gm the Phase Margin Pm and the crossover frequencyes Wcg and Wcp of the system G s ZEROS 10 POLES 1 2 3 Gain 1 See fig 20 for syntax and result 213 s reo delay 0 margin Ma 15 00 Wcg 5 10 gt Mp 95 31 Wop 1 01 gt fig 20 exemple of command
14. aedlee 16 Command Zi Clea a deel vane ae te ea ne 16 Analysis in Time Domain Simulation 6 sic coronan dis 17 Commands impulse and SP ivi das 17 GommandssiMU stc tiraron diia oral OT 18 Elaborating Graphical Result A Pa Ral eA ne eae ee a 19 COmMMand Garaph fries EEE ias 19 Multiple Plots Using PIC Files 204 ccatleie edie didi dd a 19 Register 92 BROT HE RS programs dae 20 About Os 2 BROT AER Sato 20 pag 2 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 3 20 Note on License Automatic Control System Toolbox Copyright C 2000 2001 92BROTHERS G Luca Troiani This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program if not write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Full GNU GENERAL PUBLIC LICENSE Version 2 June 1991 license txt file Note for Tl 92 and TI 89 Users Automatic Control Systems Toolbox v 3 0 was tested on a TI 89 Hard
15. bout92BROTHERS For more info on 92BROTHERS programs http www 92broters net For other 92BROTHERS programs http programs 92brothers net or support on 92BROTHERS programs http support 92brothers net Autors 92 BROTHERS www http www 92brothers net e mail 92brothers infinito it pag 20 20
16. c file by 4500 k pressing lt F4 gt lt 1 gt When the plot with k 7 248 it s plotted recall z 361 al the stored zoom by pressing lt F2 gt lt 6 gt and recall xpic by delay 101 pressing lt F4 gt lt 2 gt and save the new pic see fig 34 in xpic file eee by pressing lt F4 gt lt 1 gt E ip Fir Y Fer Far Fur istepia 0 0507 EXIT ZOOM PLOT INFO FIEL ssid Po ip step 0 059 iPP 7 2484k gt steptg 0 057 ACET RAD AUTO FUNC_ 0430 fig 33 fig 34 step rE DOE for ke 181 2 14 5 7 248 In fig 35 it s showed the last version of xpic fig 34 modified oa o tools A A 3 1 0 amp 2 2 De 2 3 0e 2 4 0e 2 ACST RAD AUTO fig 35 graphical laboration of fig 34 pag 19 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 20 20 Register 92 BROTHERS programs It s recommended to register your programs ABSOLUTELY FREE to have information about new versions bugs etc by e mail The only data required to register are e Name or Nickname e e mail address to send you informative e mail e mail address will not given to third parts e program and version you want to register e calculator model e where the program was downloaded Note e mail sentby 9 2 BR OT HE RS don t have promotional messages To register send the following form at http register 92brothers net or http web tiscalinet it 92brothers register htm A
17. calculates the unit impulse response of a linear system The impulse response is the response to a Dirac input d t Inpulse response support SISO or MIMO systems and systems with time delays step calculates the unit step response of a linear system Step response support SISO or MIMO systems and systems with time delays simul simulates the time response of continuous or discrete linear systems to arbitrary L transformable inputs Time response support SISO or MIMO systems and systems with time delays Commands impulse and step lt PURPOSE gt Plot the unit impulse response and unit step response of a linear system SISO or MIMO time delays supported lt SYNTAX gt impulse SYS StopTime step SYS StopTime where SYS it s the system name or expression and StopTime is the maximum time in seconds of the simulation plot it s from 0 to StopTime sec lt EXEMPLE gt We want to study the impulse and step response of G s described in fig 25 for 15 sec G s can be formed by feedback connection of G1 s 1 s s 1 and unitary gain system fig 26 Then we can run impulse command and step command separately fig26 fig 25 G s Stet After plots are plotted figg 27 28 we can delay 9 return to command window press lt F1 gt set zoom factor press lt F2 gt show plot dimensions press lt F3 gt lt 1 gt trace the plot press lt F3 gt lt 2 gt save the plot as a PIC file press
18. m g The to puto 1 transfer function by input j to output i will be asked gij Peeri de ca g 11 it s the TF from input 1 to output 1 to output 1 g12 it s the TF from input 2 to output 1 al E AEE EE g21 it s the TF from input 1 to output 2 to output 2 EEN C a g22 it s the TF from input 2 to output 2 insert system from input 2 to output 2 We have to make two step 22 1 using tf zpk or ss commands we define g11 g22 HET FAD AUTO UT 2 weruncommand mmimo see fig 4 5 fig 4 5 exemple of command mmimo pag 9 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 10 20 Command setdelay lt PURPOSE gt Specify input delay s of SISO or MIMO systems lt SYNTAX gt CTI S sten 2 s 10 setdaly SYS DELAY Li E where delay 10 01 SYS it s the name of an existent SISO or MIMO system gt gt DELAY is a row vector of inputs delays DELAYS must have an element for each input of SYS If SYS is a SISO system Bap AUTO FUNC 20730 DELAY can be a number Example is illustrated in fig 6 7 gt gt setdelay sys 2 5 LTI System 1 2 s 10 s 1 s 2 s s 5 delay 2 5 gt gt ACST RAD AUTO FUNC 20430 fig 6 7 exemple of command setdelay pag 10 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 11 20 Command series lt PURPOSE gt Series Connection of two LTI SISO or MIMO systems
19. margin pag 15 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 16 20 Poles and Zeros Analysis To analyze the poles and zeros characteristics of a SISO system are available two commands rlocus zpmap Command rlocus rlocus computes the Root Locus of a SISO open loop system The root locus gives the closed loop pole trajectories as a function of the feedback gain assuming negative feedback pzmap plots the pole zero map of a SISO system rlocus and zomap support only SISO systems without time delays Poles are rappresented by X zeros are rappresented by Ll the evolution of poles it s rappresented Command rlocus lt PURPOSE gt Plot the root locus of a SISO system without time delays lt SYNTAX gt s 3 ts 1 rlocus SYS GainMin GainMax Pee ar ene or ee A ier sees rlocus SYS SpecificGains where SYS is the name or expression of the system under analysis GainMin GainMax it s the gain interval of study a dialog asking how many point plot in the gain interval will appear SpecificGains it s a row vector of specific gains see the exemple in fig 21 lt EXEMPLE gt We want the root locus of G s SISO system zeros 1 3 poles 1 2 2 i 2 i gain 1 at the specific gains 0 05 0 1 0 5 1 5 15 50 100 200 300 gt gt OO a eros 1 5 15 50 100 200 30 fig 21 After the Root Locus is plotted fig 22 you can AT DES AFETE FINE return to c
20. mbine single systems in a NxM matrix using commamd mmimo SISO and MIMO systems support delays to specify systems delays there is a specific command setdelay SISO and MIMO systems can be interconnected by series interconnection of two systems SISO and or MIMO parallel interconnection of two systems SISO and or MIMO feedback interconnection of two systems SISO and or MIMO The corresponding commands are sys series SYS1 SYS2 OUTPUT 1 INPUT2 sys parallel SYS1 SYS2 INPUT1 INPUT2 OUTPUT1 OUTPUT2 sys feedback SYS1 SYS2 INPUT 1 OUTPUT1 The next pages will show the use of commands tf zpk ss gain mmimo setdelay series parallel feedback Command tf lt PURPOSE gt o Specify transfer function of SISO system by numerator zus tfC 2 51 1 1 0 and denominator coefficients sys lt SYNTAX gt LTI System sys tf NUM DEN Eear where delay 0 NUM and DEN are the row vectors of numerator and gt gt denominator coefficients ordered in descending powers of s ACST RAD AUTO FUNC 11 30 fig 1 exemple of command tf pag 8 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 9 20 Command zpk Specify transfer function of SISO system by zeros poles AA and gain sys lt SYNTAX gt LTI System 2 5e0 T PRERASTE Ee 1 where delay 0 ZEROS is the row vector of zeros set if n
21. nt see fig 32 To call the graphical tool from in the CW run the ggraph command type ggraph and press lt ENTER gt Re ggraph tool you can load a saved PIC file lt F1 gt lt 1 gt save the image in a PIC file lt F1 gt lt 2 gt insert horizontal line lt F2 gt lt 1 gt insert vertical line lt F2 gt lt 2 gt Aia insert point to point line lt F2 gt lt 3 gt A 2 8 07 sec insert text lt F2 gt lt 4 gt undo last lt F2 gt lt 5 gt undo all lt F2 gt lt 6 gt save and exit lt F3 gt lt 1 gt exit without saving lt F3 gt lt 2 gt fig 32 graphical elaboration of fig 28 Multiple Plots Using PIC Files Graphical commands that return plots ex bode can save the result in a PIC file and can save current Zoom Factor This functionality can used to have multple plots Exemple G s it s paramteric with parameter k 4500 k G s 5 s 361 2 5 4500 k and we want to plot the step response with k 181 2 k 14 5 k 7 248 for 0 05 sec see fig 33 When the plot with k 181 2 it s plotted save the plot in a PIC file named ex xpic by pressing lt F4 gt lt 1 gt and store the zoom factor by pressing lt F2 gt lt 5 gt When the plot with k 14 5 it s s tP 14500 k1 1 361 2 45004k 1 plotted recall the stored zoom by pressing lt F2 gt lt 6 gt and recall s E y LTI System xpic by pressing lt F4 gt lt 2 gt and save the new pic in xpi
22. ommand window press lt F1 gt fig 22 root locus for G s set zoom factor press lt F2 gt see fig 23 for ZoomBox exemple show plot dimensions press lt F3 gt ReGraph the root locus press lt F4 gt save the plot as a PIC file press lt F5 gt lt 1 gt load a saved PIC file press lt F5 gt lt 2 gt To decide the values of the row vector of specific gains it s suggested to try little interval and find the most significant gains ZoomBox is very helpful to set the most significant area of the ACST DEG APPROX FUNC plot fig 23 zoom of fig 22 Command zpmap lt PURPOSE gt Plot the poles and zeros map of a SISO system without time delays zpmap correspond to rlocus command with SpecificGains 0 lt SYNTAX gt zpmap SYS where SYS is the name or expression of the system under analysis lt EXEMPLE gt We want the root locus of G s the same system used for the exemple of rlocus command see fig 23 24 for the resultant plot gezek 1 3 151 2 204 2441 1 J LTI System s 311S 1 EXtrizoom enor tnrolkecrashlette x a ifs 2 s24 4 s45 DEG AUTO ACST DEG APPROX FUNC fig 23 exemple of command zpmap fig 24 zpmap for G s pag 16 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 17 20 Analysis in Time Domain Simulation Simulation of LTI MIMO systems can be performed by inpulse
23. one gt POLES is the row vector of poles a LTI system must have I at least one pole ACST RAD AUTO FUNC 11 30 fig 2 exemple of command zpk kis the gain Command ss lt PURPOSE gt Brik sic peren if Dimis dente Specify transfer function of SISO and MIMO system by a 11 10511 12 5 01 0 state space model sys lt SYNTAX gt LTI System sys ss A B C IDI ews where A B C1 D are the matrix of the delay 101 system described by gt gt X A X B U YI I CI XI 00 1U a Setting D 0 is interpreted as the zero matrix of adequate ngs exemple ol commang Ss dimensions Command gain lt PURPOSE gt Specify SISO gain system lt SYNTAX gt g gain GainValue where gis the name of the gain system GainValue it s the value of the gain Command mmimo lt PURPOSE gt Hi malg pi Generate a MIMO system model using pre existent SISO gatem systems All input delays will be resetted 1 2 s 10 lt SYNTAX gt eri lera het 1 1 mimo G Sri 36 10 it will appear a dialog window fig 4 that will ask 110 01 the name of new MIMO system the number of inputs the number of outputs then you must insert the names of pre existent SISO systems from input 1 2 to output 1 2 fig 5 As an exemple we want to create a MIMO systems with two insert sustem from input 1 inputs and two output We will ask this MIMO syste
24. ware version 2 00 Advanced Mathematics software version 2 05 Screenshots are captured on a TI 92 in TI 89 the graphical results and screen messages can look different case of the smaller screen pag 3 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 4 20 Used Symbols and Abbreviations e matrix iE element of a matrix placed in row i and column j array row vector 111 element i of the array row vector lt gt key of TI 92 s keyboard SYSTEM matrix associated to a system see LTI Models paragraph LTI linear time invariant system SISO Single Input Single Output system MIMO Multiple Input Multiple Output system ACST Automatic Control System Toolbox program CW Command Window pag 4 20 automatic control systems toolbox v 3 0 user s manual 2000 2001 92BROTHERS pag 5 20 What s Automatic Control Systems Toolbox Automatic Control Systems Toolbox ACST it s a powerful toolbox to design analyze and simulate control systems ACST has powerful tools for systems design by specifying the transfer function specifying poles and zeros of the transfer function specifying the state space SS model specifying MIMO systems using SISO systems connecting two MIMO Multiple Input Multiple Output systems by series connection parallel connection feedback connection system analysis by Bode plots MIMO systems Nichols chart MIMO systems
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