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Transfer functionFrom frequency- to time domain and vise versus

Important components in frequency domain

Spacemaster Preparation CourseModeling in Frequency Domain

Lei Ma

Department of Computer Science VII: Robotics and TelematicsUniversity of Würzburg

09.10.2007

Lei Ma Pre-Course: Frequency domian



Outline

1 Transfer functionDefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems

2 From frequency- to time domain and vise versus

3 Important components in frequency domainThe proportional components




References

Ogata,K. Modern control engineering. Prentice Hall.

Brogan,W.L. Modern control theory. Prentice Hall.

Lunze,J. Regelungstechnik 1,2. Springer.

Jeffrey, A. Mathematics for Engineers and Scientists.Chapman & Hall/CRC 2004

Stroud, K.A., Booth, D.J. Engineering Mathematics.Industrial Press 2001

Dettman, J.W. Introduction to Linear Algebra andDifferential Equations. Dover Publications 1986.

Golub, G.H., Van Loan, C.F. Matrix Computations (JohnsHopkins Studies in Mathematical Sciences). The JohnsHopkins University Press 1996




Why frequency domain?

DEs in the time domain become algebraic equations in thefrequency domain.

Frequency-domain analysis provides a deep understandingof the system behaviors at different frequencies.

An almost complete set of analysis/synthesis established.

Decomposition of signals into sinus signals;

Separate calculation of system response to individualsinus signals;

Calculation of output by superposition.




DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems

Outline








Obtaining transfer functions

Obtaining transfer functions from differential equations,

Obtaining transfer functions from state-space models





Outline








G1(s) G2(s)

U Y1 Y

= G1(s)G2(s)

U Y

Cascaded system

G1(s)

G2(s)

U

Y1

Y2

Y

= G1(s)+G2(s)

U Y

Parallel system

-G1(s)

G1(s)

U U1 Y

Y2

=G1(s)

G1(s)G2(s)1+

U Y

Feedback system

Figure: Transfer functions of some basic circuits




Obtaining time-domain response based on transferfunctions

Time Domain

Frequency Domain

Laplace

transform

Inverse

Laplace

transform

ODE/SS

Transfer

Function G(s)

u(t)

U(s) Y(s)=G(s)U(s)

y(t)




Time-domain behaviors revisited

Step response: h(t) =∫ t

0 c′Φ(t − τ)bdτ + d

Impulse response: g(t) = c′Φ(t − τ)b + dδ(t)

Therefore the follows are true: h(t) =∫ t

0 g(τ)dτ , g(t) = dhdt

and the input-output behavior:

y(t) =∫ t

0 c′Φ(t − τ)bu(τ)dτ + du(t)

=∫ t

0 g(t − τ)u(τ)dτ

= g(t) ⋆ h(t).




Step- and impulse responses in frequency domain




State-transient matrix and transfer function



Important components in frequency domainThe proportional components

Outline







The PT1-Component




The PT2-Component

The transfer function and the response

The damping ratio ζ and the undamped natural frequencyωn

Underdamped case ζ < 1

Critically damped case ζ = 1

Overdamped case ζ > 1




A bit more about poles


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