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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
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
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.
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
Obtaining transfer functions
Obtaining transfer functions from differential equations,
Obtaining transfer functions from state-space models
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
DefinitionObtaining transfer functionsPoles and zerosTransfer functions of control systems
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
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)
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
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).
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
Step- and impulse responses in frequency domain
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domain
State-transient matrix and transfer function
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domainThe proportional components
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domainThe proportional components
The PT1-Component
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domainThe proportional components
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
Lei Ma Pre-Course: Frequency domian
Transfer functionFrom frequency- to time domain and vise versus
Important components in frequency domainThe proportional components
Lei Ma Pre-Course: Frequency domian