Systeme-Eingenschaften im Zeit und Frequenzbereich M d Q Tavares , TB425 dqtm@zhaw.ch

Applied Mathematics (SiSy) SiSy Overview Signals step / impulse / rect / sinc continuous / discret periodic / aperiodic deterministic / random representation in time / frequency domains power and energy in freq domain Systems linearisation feedback and stability discretisation Transforms FR ; FT ; DFT/FFT : Laplace : Z-Transformation : DCT : Applications Biomedical / Operational Research/ Sensorics & Messtechnik Automation / Location & Telecommunication Systems Data Compression & Cryptography / Image Processing / Audio Synthesis & Analysis Statistical Signal Processing LTI DGl; BSB; ZVD; h(t); g(t); G(ω); G(s) u(t) U(ω) u[n] U(z) y(t) Y(ω) y[n] Y(z) LTD DzGl; BSB; ZVD; g[n]; G(ω); G(z) ω t f S-Plane Control (RT) Telecomm (NTM) SigProc (DSV, ASV) Z-Plane n Applied Mathematics (SiSy) x y Mathematics

Inhalt Relationship LTI views & System Properties System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

LTI Systeme : Modellierung & Darstellung Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Block schaltbild Zustands variable Genauigkeit? Messverfahren Kontinuierliche Systeme Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation Physikalisches Modell nicht genug

State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order Free-response, with u(t) =0 to satisfy differential equation, x(t) must have exponential form λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A Solving for λ (non-trivial solution) nth-order polynom characteristic equation

State Space (ZVD) Free-response & Eingenvalues Solving for v (eigenvectors) satisfy this equation with 1 normalised component Given superposition (linearity) principle, if n distinct eigenvalues Apply initial conditions to solve For the αi factors (*) Compare to known free-response of normalised system 2.order with parameters : k, d and ω0 ; or σ and ωe . (*) For larger n, symbolic solution gets complicated, but there are computationally efficient numerical techniques.

State Space (ZVD) Forced response: response to a step input Complete solution (sum) With homogeneous solution And input or stimuli Simplifying for t > 0 The particular solution has form similar to input, and for t>0 derivative equals zero, so: Therefore complete solution Obs.: use initial conditions to solve for αi

State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order System response as homogeneous plus particular solution with λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A αi = constants determined by n initial conditions

Inhalt Relationship LTI views & System Properties System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

State Space (ZVD) ZVD with Laplace Operator & Transfer Function G(s) representation of LTI system nth-order & Laplace Transform (with initial conditions equal zero) Manipulating to isolate the transfer function : G(s) = Y(s) / U(s) G(s) Übung 2 : Aufgabe 3

State Space (ZVD) ZVD with Laplace Operator & Transfer Function (Übertragungsfunktion) Examples: Generic 2nd order system (from BSB exercise with direct form I & II) - Sallen-Key Butterworth TPF

ZVD & System-Antwort u(t) y(t) b2 1/s -a1 b1 -a0 b0 Example 1: Generic 2nd order system direct form II 1/s -a1 -a0 b1 b2 b0 u(t) y(t)

ZVD & System-Antwort Example 1: Generic 2nd order system Calculating the eigenvalues: and the respective eigenvectors: Gives the complete system response:

ZVD & Übertragungsfunktion Example 1: Generic 2nd order system Übung 2 - Auf 3 Similar but easier with b2 =0

ZVD & Übertragungsfunktion Example 2: Sallen-Key Butterworth TPF For Butterworth pattern:

Inhalt Relationship LTI views & System Properties System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

LTI Systeme : Modellierung & Darstellung Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Block schaltbild Zustands variable Genauigkeit? Messverfahren Kontinuierliche Systeme Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation Physikalisches Modell nicht genug

Relationship among LTI views Skript: Kapitel 6 S. 73

Relationship among LTI views Known relationships (visualisation : Matlab ltiview) Differential Equation Frequence Response G(jω) Differential Equation Transfer Function G(s) Frequency Response Impulse Response g(t) Impulse Response System Output y(t) Transfer Function System Output Y(s) Step Response h(t) Impulse Response g(t) LTI u(t) U(s) y(t) Y(s) Fourier Transformation + isolate {Y(jω) / U(jω)} s=jω Laplace Transformation Inv. Fourier Transformation Convolution with input fct Multiplication with input fct Derivative in time domain

Inhalt Relationship LTI views & System Properties System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

Laplace Transformation : PN-Map Relationship: Transfer Function & Frequency Response (Übertragungsfunktion & Frequenzgang) S-Plane (S-Ebene) Im{s} X Re{s} X Sweep ω from 0 to +∞ , and check for minimum & maximum points

Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} x x o Re{s} Im{s} Re{s} Im{s} x x o o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} Tiefpass Hochpass x x o Re{s} Im{s} Re{s} Im{s} x x Bandpass o Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} Tiefpass Hochpass x x o Re{s} Im{s} Re{s} Im{s} x x Bandpass o Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

Inhalt Relationship LTI views & System Properties System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

Pol-Nullstellen-Darstellung Beispiel 1

Pol-Nullstellen-Darstellung Beispiel 2

Pol-Nullstellen-Darstellung Beispiel 2

Pol-Nullstellen-Darstellung Beispiel 3

Pol-Nullstellen-Darstellung Beispiel 3

Pol-Nullstellen-Darstellung Erkenntnisse

Pol-Nullstellen-Darstellung Erkenntnisse

Pol-Nullstellen-Darstellung Erkenntnisse

Pol-Nullstellen-Darstellung Erkenntnisse

Pol-Nullstellen-Darstellung Erkenntnisse

Pol-Nullstellen-Darstellung Beispiel 4

Pol-Nullstellen-Darstellung Beispiel 4

Pol-Nullstellen-Darstellung Beispiel 4

Pol-Nullstellen-Darstellung Beispiel 4

Pol-Nullstellen-Darstellung Beispiel 4