0.1 Lecture 2: introduction to systems  (Page 5/8)

Y=H(s)X

where

$H\left(s\right)=\frac{\sum _{m=0}^{M}{b}_{m}{s}^{m}}{\sum _{n=0}^{N}{a}_{n}{s}^{n}}$

b/ System function definition

$H\left(s\right)=\frac{\sum _{m=0}^{M}{b}_{m}{s}^{m}}{\sum _{n=0}^{N}{a}_{n}{s}^{n}}$

• is called the system function
• is a rational function in s
• is a skeleton of the differential equation
• characterizes the relation between x(t) and y(t)

Example — reconstruction of differential equation from H(s)

Suppose

$H\left(s\right)=\frac{Y}{X}=\frac{s}{s+1}$

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

$\left(s+1\right){\text{Ye}}^{\text{st}}={\text{sXe}}^{\text{st}}$

which yields

${\text{sYe}}^{\text{st}}+{\text{Ye}}^{\text{st}}={\text{sXe}}^{\text{st}}$

from which we can obtain the differential equation

$\frac{\text{dy}\left(t\right)}{\text{dt}}+y\left(t\right)=\frac{\text{dx}\left(t\right)}{\text{dt}}$

2/ Poles and zeros

H(s) can be expressed in factored form as follows

$H\left(s\right)=K\frac{\prod _{m=1}^{M}\left(s-{z}_{m}\right)}{\prod _{n-1}^{N}\left(s-{p}_{n}\right)}$

where $K=\frac{{b}_{M}}{{a}_{M}}$

• $\left\{{z}_{1},{z}_{2},\text{.}\text{.}\text{.},{z}_{M}\right\}$ are the roots of the numerator polynomial and are called zeros of H(s) because these are the values of s for which H(s) = 0.
• $\left\{{p}_{1},{p}_{2},\text{.}\text{.}\text{.},{p}_{M}\right\}$ are the roots of the denominator polynomial and are called poles of H(s) because these are the values of s for which $H\left(s\right)=\infty$ .

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

$\left(\sum _{n=0}^{N}{a}_{n}{\lambda }^{n}\right)=0$

and the poles are the roots of denominator polynomial of H(s)

$\left(\sum _{n=0}^{N}{a}_{n}{s}^{n}\right)=0$

Both originate from the left-hand side of the differential equation

$\sum _{n=0}^{N}{a}_{n}\frac{{d}^{n}y\left(t\right)}{{\text{dt}}^{n}}$

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 $\text{jI}\left\{s\right\}=\mathrm{j\varpi }$ and the abscissa is $R\left\{s\right\}=\sigma$ where

$s=\sigma +\mathrm{j\varpi }$

Example — system function of a network

The differential equation relating v(t) to i(t) is

$\frac{\text{di}\left(t\right)}{\text{dt}}=C\left(\frac{{d}^{2}v\left(t\right)}{{\text{dt}}^{2}}+\frac{1}{\text{RC}}\frac{\text{dv}\left(t\right)}{\text{dt}}+\frac{v\left(t\right)}{\text{LC}}\right)$

The particular solution is obtained from

$\text{sI}=C\left({s}^{2}+\frac{1}{\text{RC}}s+\frac{1}{\text{LC}}\right)V$

With v(t) as the output and i(t) as the input, the system function of the RLC network is

$H\left(s\right)=\frac{V}{I}=\frac{1}{C}\left(\frac{s}{{s}^{2}+\frac{1}{\text{RC}}s+\frac{1}{\text{LC}}}\right)$

Two-minute miniquiz problem

Problem 3-1

Given the system function

$H\left(s\right)=\frac{s+2}{\left(s+3\right)\left(s+4\right)}$

• Determine the natural frequencies of the system.
• Determine a differential equation that relates x(t) and y(t).

Solution

• The natural frequencies of the system are the poles of the system function and are −3 and −4.
• The differential equation can be obtained by cross multiplying and multiplying by ${e}^{\text{st}}$ to obtain

$\left(s+3\right)\left(s+4\right){\text{Ye}}^{\text{st}}=\left(s+2\right){\text{Xe}}^{\text{st}}$

$\left({s}^{2}+7s+\text{12}\right){\text{Ye}}^{\text{st}}=\left(s+2\right){\text{Xe}}^{\text{st}}$

so that

$\frac{{d}^{2}y\left(t\right)}{{\text{dt}}^{2}}+7\frac{\text{dy}\left(t\right)}{\text{dt}}+\text{12}y\left(t\right)=\frac{\text{dx}\left(t\right)}{\text{dt}}+2x\left(t\right)$

VIII. TOTAL SOLUTION

The general solution is

$y\left(t\right)=\sum _{n=1}^{N}{A}_{n}{e}^{{\lambda }_{n}t}+\text{XH}\left(s\right){e}^{\text{st}}\text{for}t>0$

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

$y\left(t\right)=\left(\sum _{n=1}^{N}{A}_{n}{e}^{{\lambda }_{n}t}+\text{XH}\left(s\right){e}^{\text{st}}\right)u\left(t\right)$

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

$\left\{{A}_{1},{A}_{2},\text{.}\text{.}\text{.},{A}_{N}\right\}$

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