Page 739 - The Mechatronics Handbook

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Relationship between State Equations and Transfer-Functions
State-Space to Transfer-Function
The input-to-output relationship of a dynamic system in the frequency-domain is represented by a
transfer-function, which can be obtained by taking the Laplace transform of (23.107) and (23.108) with
zero initial conditions as follows [8, Chapter 3, section 5]:
sX s() = AX s() + BU s(), (23.123)
Ys() = CX s() + DU s(), (23.124)

where s is the Laplace variable. Solving (23.123) for X(s) and substituting into (23.124), the ratio of the
output Y(s) to input U(s) for a single-input single-output system (SISO) can be found as
Ys()
(
Gs() = ----------- = CsI A) B + D
1
–
–
Us()
Ns()
= ----------- (23.125)
Ds()
where I is an n × n identity matrix. In Eq. (23.125), N(s) and D(s) are referred to as the numerator and
denominator polynomial of G(s), respectively. 7
Analogous to the state-space equation, the boundedness of the output response y(t) to a bounded
input u(t) is characterized by the roots of the denominator polynomial D(s), i.e., the values of s for which
D(s) = 0. In particular, the output y(t) will be bounded for any bounded input, i.e., system is stable if
8
the real parts of all the roots of D(s) are less than zero (strictly negative). Alternatively, a convenient
method to determine stability without having to find the roots of D(s) explicitly is the Routh–Hurwitz
stability criterion [6, Chapter 6].
Example
With the state-space description of the mass-spring-damper system defined in Eqs. (23.112) and (23.113),
the transfer-function realization using Eq. (23.125) becomes

– 1
Ys()
Gs() = ----------- = 10 s 10 0 1 0 + 0 []
Us() –
01 ( – k/m) – ( c/m) b/m
b/m
= ------------------------------------------ (23.126)
s + ( c/m)s + k/m
2
The input to the system is the applied voltage V z (t) and the output is the displacement of the mass z(t).
Frequency-Response Using Transfer-Functions
Consider a linear single-input single-output (SISO) stable system with transfer-function description G(s).
When the system G(s) is excited by a sinusoidal input of the form

ut() = Psin ( wt) (23.127)

with amplitude P and frequency ω, the output response (after the transients decay) will also be a sinusoid
of the form

yt() = MPsin ( wt + f) (23.128)

7
The MATLAB command ss2tf can be used to convert a state-space realization to a transfer-function.
8
The MATLAB command roots (den) can be used to find the roots of den, where den is the coefficients of D(s).

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