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System Algebra and Block Diagram

Laplace transform converts many time-domain operations such as differentiation, integration, convolution, time shifting into algebraic operations in s-domain. Moreover, the behavior of complex systems composed of a set of interconnected LTI systems can also be easily analyzed in s-domain. We first consider some simple interconnections of LTI systems.

Example 1: A first order LTI system

\begin{displaymath}\frac{d}{dt}y(t)+3y(t)=\dot{y}(t)+3 y(t)=x(t),\;\;\;\;\mbox{or}\;\;\;\;\;
\dot{y}(t)=x(t)-3 y(t) \end{displaymath}

which can be represented in the block diagram shown below:

blockdiagram5.gif

Alternatively, the system can be described in s-domain by its transfer function:

\begin{displaymath}H(s)=\frac{Y(s)}{X(s)}=\frac{1}{s+3}=\frac{1/s}{1+3/s} \end{displaymath}

Comparing this $H(s)$ with the transfer function of the feedback system, we see that a first order system can be represented as a feedback system with $H_1(s)=1/s$ (an integrator implementable by an operational amplifier) in the forward path, and $H_2(s)=3$ (a feedback coefficient) in the negative feedback path.

Example 2: Consider a second order system with transfer function

\begin{displaymath}
H(s)=\frac{1}{s^2+3s+2}=\frac{1}{s+1}\;\frac{1}{s+2}=\frac{1}{s+1}-\frac{1}{s+2}
\end{displaymath}

These three expressions of $H(s)$ correspond to three different block diagram representations of the system. The last two expressions are, respectively, the cascade and parallel forms composed of two sub-systems, and they can be easily implemented as shown below:

blockdiagram7.gif

Alternatively, the first expression, a direct form, can also be used. To do so, we first consider a general $H(s)=Y(s)/X(s)=1/(s^2+as+b)$, i.e.,

\begin{displaymath}s^2Y(s)+asY(s)+bY(s)=X(s),\;\;\;\;\;\mbox{or}\;\;\;\;\;\;
s^2Y(s)=X(s)-asY(s)-bY(s) \end{displaymath}

Given $s^2Y(s)$, we can first obtain $sY(s)$ by an integrator $1/s$, and then obtain the output $Y(s)$ from $sY(s)$ by another integrator $1/s$. We see that this system can be represented as a feedback system with two negative feedback paths of $a=3$ from $sY(s)$ and $b=2$ from $Y(s)$.

blockdiagram4.gif

Example 3: A second order system with transfer function

\begin{displaymath}H(s)=\frac{cs^2+ds+e}{s^2+as+b}=\frac{1}{s^2+as+b}(cs^2+ds+e) \end{displaymath}

This system can be represented as a cascade of two systems

\begin{displaymath}Z(s)=H_1(s)X(s)=\frac{1}{s^2+as+b}X(s) \end{displaymath}

and

\begin{displaymath}Y(s)=H_2(s)Z(s)=(cs^2+ds+e)Z(s) \end{displaymath}

The first system $H_1(s)$ can be implemented by two integrators with proper feedback paths as shown in the previous example, and the second system is a linear combination of $s^2Z(s)$, $sZ(s)$ and $Z(s)$, all of which are available along the forward path of the first system. The over all system can therefore by represented as shown below.

blockdiagram6.gif

Obviously the block diagram of this example can be generalized to represent any system with a rational transfer function:

\begin{displaymath}
H(s)=\frac{\sum_{k=0}^M b_k s^k}{\sum_{k=0}^N a_k s^k}\;\;\;\;(M \le N)
\end{displaymath}

If $M>N$, $H(s)$ can be separated into several terms (by long-division) which can be individually implemented and then combined to generate the overall output $Y(s)$.


next up previous
Next: Unilateral Laplace Transform Up: Laplace_Transform Previous: Higher Order Systems
Ruye Wang 2012-01-28