DYNAMICS and CONTROL
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1 DYNAMICS and CONTROL MODULE I (App) Model of Sytem & Signal Math review Preented by Pedro Alberto Profeor of Sytem Engineering and Control - UPV
2 Y(z) 2
3 G() 3
4 Model ABSTRACTION!!! 4
5 Dealing with Signal: Sinuoidal y( t) Y 0en( wt j) Parameter Magnitude: Yo Frequency: w Phae: j Operation: Sum Linear combination Delay Derivative Integral 5
6 Dealing with Signal yt () Laplace Tranform: t Y( ) e y( t) dt 0 Laplace tranform propertie Unicity y( t) Y( ) L Linearity L ay ( t) by ( t) ay ( ) by ( ) 2 2 Integral t 0 L y( ) d Y () Delay Derivative y( t ) e Y( ) dy() t L dt L Y ( ) y(0) 6
7 Laplace Tranform of typical ignal f() t F () DYNAMICS & CONTROL Unitary Impule Impulounitario Ecalónunitario Unitary Step Rampaunitaria Unitary Ramp 2 Delay e e at e a at en t 2 2 a a at co t 2 2 a T y( t T) e Y() 7
8 Laplace Tranform of typical ignal f() t F () DYNAMICS & CONTROL Unitary Impule Impulounitario Ecalónunitario Unitary Step Rampaunitaria Unitary Ramp 2 Delay e e at e a at en t 2 2 a a at co t 2 2 a T y( t T) e Y() 7
9 Laplace Tranform of typical ignal f() t F () Unitary Impule Impulounitario DYNAMICS & CONTROL e Ecalónunitario Unitary Step Rampaunitaria Unitary Ramp 2 Delay e e at e a at en t 2 2 a a at co t 2 2 a T y( t T) e Y() 7
10 Laplace Tranform of typical ignal f() t F () Unitary Impule Impulounitario DYNAMICS & CONTROL e Ecalónunitario Unitary Step Rampaunitaria Unitary Ramp 2 Delay e e at e a at en t 2 2 a a at co t 2 2 a T y( t T) e Y() 7
11 Invere Laplace Tranform L at y() t e a DYNAMICS & CONTROL Y () ( )( 2) 2 y( t) 2e t 2e 2t 8
12 Invere Laplace Tranform L at y() t e a DYNAMICS & CONTROL Y () ( )( 2) 2 y( t) 2e t 2e 2t 8
13 Invere Laplace Tranform L at y() t e a DYNAMICS & CONTROL Y () ( )( 2) 2 y( t) 2e t 2e 2t 8
14 Magnitude Dicrete-time Signal y k DYNAMICS & CONTROL y( z) 3z 3z 3 z... 2 Amplitud tiempo Time 9
15 y k Amplitud Magnitude Dicrete-time Signal DYNAMICS & CONTROL tiempo Time 9
16 Correpondence between CT and DT dy t dt y y () k k T
17 Correpondence between CT and DT dy t dt Y. ( ) () k k L y T Z y z z. Y ( z ) T T z T But, the delay: T z e z e T
18 Sytem to be modeled By the attached ignal Continuou / Dicrete Logic/Binary Determinitic / etocatic Approximated / concrete Monovariable / multivariable By the operator Linear / nonlinear Etatic / Dynamic Time variant / Invariant Concentrated / Ditributed 2
19 Model a an operator V C Voltage balance Laplace tranform DYNAMICS & CONTROL dvc () t dt RC V ( t) V ( t) Vc( ) V ( ) Vc( ) RC Vc ( ) V ( ) RC c Set of differential equation dx() t ax( t) a2 y( t) bu ( t) dt dx( t) dy( t) 2 a2x( t) a22 y( t) b2u( t) dt dt Set of algebraic equation x( ) a x( ) a y( ) b u( ) 2 2 x( ) y( ) a x( ) a y( ) b u( )
20 u(k) SYSTEM y(k) Compact Sytem repreentation: y(k) 0.95* y(k) u(k) By mean of an operator between tranformed ignal y( z) u( z) z 0.95 Tranfer Function : G( z) z
21 Block Diagram Serie compoition: U(z) G (z) G 2 (z) Y(z) G 2 G () z X(z) Y() z () z X() z X() z U() z U(z) G( z) G ( z) G ( z) G(z) 2 Y(z) 5
22 Block Diagram Paralell compoition: Y( z) Y ( z) Y ( z) 2 G () z Y() z U() z Y() z + G () z 2 Y () z 2 G () z G 2 Y () z U() z Y2 () z () z U() z G( z) G ( z) G ( z) U(z) 2 G(z) Y(z) 6
23 Block Diagram U() z Ez () + - M() z Loop arrangement: Gz () H() z Y() z Y ( z) G( z) E( z) M ( z) H ( z) Y ( z) E( z) U ( z) M ( z) Y() z Gz () G ( z ) H ( z ) 7
24 Simulation tool MATLAB 0 G () ( 2)( 5) Programming >> =zpk(''); >> G=(+0)/(+2)/(+5) Exploiting >> tep(g) Zero/pole/gain: (+0) (+2) (+5) 8
25 Simulation tool MATLAB 0 G () ( 2)( 5) Programming >> =zpk(''); >> G=(+0)/(+2)/(+5) Exploiting >> tep(g) Zero/pole/gain: (+0) (+2) (+5) 8
26 Simulation tool MATLAB 0 G () ( 2)( 5) Programming >> =zpk(''); >> G=(+0)/(+2)/(+5) Exploiting >> tep(g) Zero/pole/gain: (+0) (+2) (+5) 8
27 Simulation tool MATLAB 0 G () ( 2)( 5) Programming >> =zpk(''); >> G=(+0)/(+2)/(+5) Exploiting >> tep(g) Zero/pole/gain: (+0) (+2) (+5) 8
28 Simulation tool MATLAB 0 G () ( 2)( 5) Programming >> =zpk(''); >> G=(+0)/(+2)/(+5) Exploiting >> tep(g) Zero/pole/gain: (+0) (+2) (+5) 8
29 What have we een today? A bit of Math around modeling Parameterizing the ignal information Laplace and Z- Tranformation Typical tranformed ignal Model of ytem a operator Sytem connection and tructure Modeling and imulation tool 9
30 What i next? Module: Example of ytem and ignal Model of ytem and ignal Controlled ytem: propertie Dynamic and tatic behavior Senitivity and Robutne Control ytem deign Control benefit DYNAMICS & CONTROL Topic to tudy 20
31 Thank you! The ource of ome of thee figure are: Slide 2- Author: Andy Dingley Public Domain. Slide Author: Emocope. GNU Free Documentation Licene Slide 3- By Dr. Mirko Junge (Own work) [CC-BY-3.0 ( via Wikimedia Common Slide 4. Catalonia_Terraa_mNATEC_MaquinaDeVapor_ReguladorDeWatt.jpg. Author Friviere GNU 2
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