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0066_Frame_C23 Page 48 Friday, January 18, 2002 5:34 PM

with the same frequency ω and a phase shift φ [6, Chapter 8]. The output amplitude is the input amplitude
scaled by M, the magnitude gain. The magnitude gain M is found by taking the magnitude of G(s)
evaluated at s = jω, i.e.,

M = G s() s=jω (23.129)

Usually, the magnitude gain M is expressed in units of decibels (dB), i.e., M [dB] = 20log M. The phase
shift φ is the angle of G(s) evaluated at s = jω, i.e.,

φ = ∠ Gs() s=jω (23.130)

with units of degrees. The plot of the magnitude gain M and the phase shift φ versus the frequency ω
9
gives a graphical representation of the frequency-response (Bode plots) of a system. These plots can be
obtained experimentally by measuring the magnitude gain and phase shift between the input and output
response of a system over a range of frequencies. Additionally, a system’s transfer-function can be obtained
from an experimental frequency-response data by using curve-fitting software. In section “Experimental
Modeling Using Frequency Response,” we present this approach to determine a model for a system using
experimental frequency-response data.

Transfer-Function to State-Space
In section “State-Space to Transfer-Function,” a transfer-function model was obtained for a system in
state-space form. In the following, an approach for realizing a state-space model from a transfer-function
G(s) is presented. For a realizable transfer function G(s) of a SISO system of the form

b 0 s + b 1 s n−1 + … +
n
Gs() = ---------------------------------------------------- (23.131)
b n
s + a 1 s n−1 + … + a n
n
the controllable canonical state-space form written in terms of the coefficients of G(s) is

…
– a 1 – a 2 – a n−1 – a n 1
…
1 0 0 0 0
x ˙ t() = 0 1 … 0 0 xt() + 0 ut() (23.132)
M M O M M M
…
0 0 1 0 0

yt() = ( [ b 1 – a 1 b 0 ) ( b 2 – a 2 b 0 ) … ( b n – a n b 0 )]xt() + []ut() (23.133)
b 0
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where the number of states n is equal to the highest power of the denominator of G(s). The smallest
possible dimension for realizing a system, referred to as the minimum realization, is an important factor
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to consider in analysis and design. Models of minimum order require less computational power in
simulation and implementation compared to higher order models. For information about other equiv-
alent canonical state-space forms, refer to [7, Chapter 4, sections 3 and 4].

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The MATLAB command bode plots the magnitude gain and phase shift for a linear system.
10
The MATLAB command tf2ss generates the controllable canonical form realization of a transfer function
G(s). Other useful commands include ss2tf, zp2tf, and tf2zp.
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For a detailed discussion of minimal realizations for multi-input multi-output systems, see [7, Chapter 7].
©2002 CRC Press LLC