Question

1. b)the system has transfer function

g(s)=\frac{y(s)}{u(s)}=\frac{2s^{2}+6s + 5}{s^{3}+4s^{2}+5s + 2}

obtain state - space model for above system using parallel decomposition

Explanation:

Step1: Factor the denominator

First, factor the denominator $s^{3}+4s^{2}+5s + 2$. By trial - and - error, we find that $s=-1$ is a root. Using polynomial long - division or synthetic division, we get $s^{3}+4s^{2}+5s + 2=(s + 1)(s^{2}+3s + 2)=(s + 1)^{2}(s+2)$.

Step2: Perform partial - fraction decomposition

We write $G(s)=\frac{2s^{2}+6s + 5}{(s + 1)^{2}(s + 2)}=\frac{A}{s+1}+\frac{B}{(s + 1)^{2}}+\frac{C}{s+2}$.

$$2s^{2}+6s + 5=A(s + 1)(s + 2)+B(s + 2)+C(s + 1)^{2}$$

Let $s=-1$, then $2-6 + 5=B$, so $B = 1$.
Let $s=-2$, then $8-12 + 5=C$, so $C = 1$.
Comparing the coefficients of $s^{2}$: $2=A + C$, since $C = 1$, then $A=1$.
So $G(s)=\frac{1}{s+1}+\frac{1}{(s + 1)^{2}}+\frac{1}{s+2}$.

Step3: Define state variables

Let $X_1(s)=\frac{1}{s + 1}U(s)$, $X_2(s)=\frac{1}{(s + 1)^{2}}U(s)$, $X_3(s)=\frac{1}{s+2}U(s)$. Then $(s + 1)X_1(s)=U(s)$, $(s + 1)^{2}X_2(s)=U(s)$, $(s + 2)X_3(s)=U(s)$.
In the time - domain, $\dot{x}_1=-x_1+u$, $\dot{x}_2=-2x_2 - x_1+u$, $\dot{x}_3=-2x_3+u$.
And $Y(s)=X_1(s)+X_2(s)+X_3(s)$, so in the time - domain $y=x_1+x_2+x_3$.
The state - space model is:

$$
LATEXBLOCK0
=
LATEXBLOCK1

LATEXBLOCK2
+
LATEXBLOCK3
u$$

$$y=
LATEXBLOCK4

LATEXBLOCK5
$$

Answer:

$$
LATEXBLOCK0
$$