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Y=H(s)X
where
$H(s)=\frac{\sum _{m=0}^{M}{b}_{m}{s}^{m}}{\sum _{n=0}^{N}{a}_{n}{s}^{n}}$
b/ System function definition
$H(s)=\frac{\sum _{m=0}^{M}{b}_{m}{s}^{m}}{\sum _{n=0}^{N}{a}_{n}{s}^{n}}$
Example — reconstruction of differential equation from H(s)
Suppose
$H(s)=\frac{Y}{X}=\frac{s}{s+1}$
what is the differential equation that relates y(t) to x(t)? Cross-multiply the equation and multiply both sides by ${e}^{\text{st}}$ to obtain
$(s+1){\text{Ye}}^{\text{st}}={\text{sXe}}^{\text{st}}$
which yields
${\text{sYe}}^{\text{st}}+{\text{Ye}}^{\text{st}}={\text{sXe}}^{\text{st}}$
from which we can obtain the differential equation
$\frac{\text{dy}(t)}{\text{dt}}+y(t)=\frac{\text{dx}(t)}{\text{dt}}$
2/ Poles and zeros
H(s) can be expressed in factored form as follows
$H(s)=K\frac{\prod _{m=1}^{M}(s-{z}_{m})}{\prod _{n-1}^{N}(s-{p}_{n})}$
where $K=\frac{{b}_{M}}{{a}_{M}}$
a/ Poles are the natural frequencies
Note that poles of H(s) are the natural frequencies of the system. Recall that natural frequencies are given by the roots of the characteristic polynomial
$(\sum _{n=0}^{N}{a}_{n}{\lambda}^{n})=0$
and the poles are the roots of denominator polynomial of H(s)
$(\sum _{n=0}^{N}{a}_{n}{s}^{n})=0$
Both originate from the left-hand side of the differential equation
$\sum _{n=0}^{N}{a}_{n}\frac{{d}^{n}y(t)}{{\text{dt}}^{n}}$
b/ Pole-zero diagram
H(s) characterizes the differential equation and H(s) is characterized by N + M + 1 numbers: N poles, M zeros, and the constant K. Except for the multiplication factor K, H(s) is characterized by a pole-zero diagram which is a plot of the locations of poles and zeros in the complex-s plane. The ordinate is $\text{jI}\{s\}=\mathrm{j\varpi}$ and the abscissa is $R\{s\}=\sigma $ where
$s=\sigma +\mathrm{j\varpi}$
Example — system function of a network
The differential equation relating v(t) to i(t) is
$\frac{\text{di}(t)}{\text{dt}}=C(\frac{{d}^{2}v(t)}{{\text{dt}}^{2}}+\frac{1}{\text{RC}}\frac{\text{dv}(t)}{\text{dt}}+\frac{v(t)}{\text{LC}})$
The particular solution is obtained from
$\text{sI}=C({s}^{2}+\frac{1}{\text{RC}}s+\frac{1}{\text{LC}})V$
With v(t) as the output and i(t) as the input, the system function of the RLC network is
$H(s)=\frac{V}{I}=\frac{1}{C}(\frac{s}{{s}^{2}+\frac{1}{\text{RC}}s+\frac{1}{\text{LC}}})$
Two-minute miniquiz problem
Problem 3-1
Given the system function
$H(s)=\frac{s+2}{(s+3)(s+4)}$
Solution
$(s+3)(s+4){\text{Ye}}^{\text{st}}=(s+2){\text{Xe}}^{\text{st}}$
$({s}^{2}+\mathrm{7s}+\text{12}){\text{Ye}}^{\text{st}}=(s+2){\text{Xe}}^{\text{st}}$
so that
$\frac{{d}^{2}y(t)}{{\text{dt}}^{2}}+7\frac{\text{dy}(t)}{\text{dt}}+\text{12}y(t)=\frac{\text{dx}(t)}{\text{dt}}+\mathrm{2x}(t)$
VIII. TOTAL SOLUTION
The general solution is
$y(t)=\sum _{n=1}^{N}{A}_{n}{e}^{{\lambda}_{n}t}+\text{XH}(s){e}^{\text{st}}\text{for}t>0$
and
y(t)=0 for t<0
Hence, provided there are no singularity functions (e.g., impulses) at t = 0, the general solution can be written compactly as follows
$y(t)=(\sum _{n=1}^{N}{A}_{n}{e}^{{\lambda}_{n}t}+\text{XH}(s){e}^{\text{st}})u(t)$
As we shall see later, no singularity functions occur in the response provided the order of the numerator polynomial of H(s) does not exceed that of the denominator.
1/ Initial conditions
To completely determine the total solution we need to determine the N coefficients
$\{{A}_{1},{A}_{2},\text{.}\text{.}\text{.},{A}_{N}\}$
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