DYNAMICS and CONTROL
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1 DYNAMICS and CONTROL MODULE 1I (III) Models of Systems & Signals Formalism Presented by Pedro Albertos Professor of Systems Engineering and Control - UPV 1
2 Modules: Examples of systems and signals Models of systems and signals Representations Analogies Formalism Controlled systems: properties Control systems design Control benefits Topics to study 2
3 Compact representation of CT Signals y( t) Y 0sen( wt ) Or by means of transformed representations. Laplace Transform: y( t) Y( s) L ws y( s) Y s w y 1 ( t at ) Y e 0 ; Y ( s ) Y 0 s a 3
4 Magnitude DYNAMICS & CONTROL Amplitud Discrete-time Signals y k Compact representation of DT Signals y( k) 3(1 0.5 Or using transformed representations y( 1.5z It is a sequence 1 k ) 2.25z tiempo Time y z 1 (1 z Delay operator 1 1.5z 1 )(1 0.5z ( 1 ) 4
5 Output: y(k) DYNAMICS & CONTROL Computer sequence Input Const. ALGORITHM y(0) = 25; for k=1 to k=100 y(k+1)=0.95*y(k)+10; Represent y(k) MODEL: y(k+1) = 0.95*y(k) + u(k); u(k) = 10; y(0) = 25 5
6 u(k) SYSTEM y(k) Compact System representation: y(k1) 0.95* y(k) u(k) zy( 0.95*y( u( Or by means of an operator between transformed signals 1 y( u( z 0.95 T.F. : G( z T.F. : Transfer Function z 1 Delay operator
7 Tank system Water balance Outlet flow q ( t) q ( t) q ( t) i r o q ( ) ( ) o t F h t Retained flow q r Adh() t dt dh( t) 1 dt A q ( ) ( ) o t F h t 7
8 Electric Circuit V C V( t) R. I( t) V ( t) Q( t) I( ) d DYNAMICS & CONTROL Voltage balance Capacitor Accumulated charge t c Q( t) V c ( t) C dvc I() t C dt dvc ( ) ( ) () t V t Vc t RC dt dvc () t 1 dt RC V ( t) V ( t) c
9 Model as an operator Voltage balance DYNAMICS & CONTROL Capacitor dvc () t 1 dt RC V ( t) V ( t) c V C Unicity Laplace transform properties (see Appendix) y( t) Y( s) L 1 svc( s) V ( s) Vc( s) RC Linearity Derivative 1 Vc ( s) V ( s) 1 RCs L ay ( t) by ( t) ay ( s) by ( s) dy() t dt Gs () L sy() s 1 1 RCs
10 Experimental Modeling u(k) SYSTEM y(k) u k u( z 1 10 Gz () y yz ( ) 1 u( z 0.95 ( 1 (1 z 10 )( z 0.95) 10
11 Block Diagrams Series composition: U( G 1 ( G 2 ( Y( U( G 2 ( G 1 ( Y( Paralell composition: U( G 1 ( G 2 ( + Y( + U( G 1 (+G 2 ( Y( Loop arrangement: U( + G( H( Y( G( Y( U( 1G(H( 11
12 Kind of systems By the attached signals Continuous / Discrete Logic/Binary Deterministic / stochastic Approximated/concrete Monovariable / multivariable By the operator Linear / nonlinear Static / Dynamic Time variant / Invariant Concentrated / Distributed 12
13 Kind of Models Black box Input / Output relationship Experimental modeling Use of analogies Null initial conditions (in equilibrium) White box Internal and external relationships Behavioral description: first principles Stated as equations Initial conditions are considered 13
14 Representations Signals: TRANSFORMED y(t) Y ( b( a( Systems: OPERATOR G( b( a( Transfer Function 14
15 What have we seen today? Formalism in modeling systems and signals Parameterizing the signals information Expressing in equivalent representations Transformations Models of systems as operators Models of systems as set of equations Systems connection and structure 15
16 A historical curiosity The steem generator 16
17 A historical curiosity The steem generator 16
18 A historical curiosity The steem generator 16
19 A historical curiosity The steem generator and the Watt s regulator ( ) How it works? On Governors J. Clerk Maxwell (1868) 17
20 Let s explore the control systems, their structure, their goals, and the benefits. 18
21 What is next? Modules: Examples of systems and signals Models of systems and signals Controlled systems: properties Dynamic and static behavior Sensitivity and Robustness Control systems design Control benefits DYNAMICS & CONTROL Topics to study 19
22 Thank you! The sources of some of these figures are: Slide Author: Andy Dingley Public Domain. Slide Author: Emoscopes. GNU Free Documentation License Slide By Dr. Mirko Junge (Own work) [CC-BY-3.0 ( via Wikimedia Commons Slide Catalonia_Terrassa_mNATEC_MaquinaDeVapor_ReguladorDeWatt.jpg. Author Friviere GNU 20
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