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.

• Parallel systems: If the system is composed of two LTI systems with $h_1(t)$ and $h_2(t)$ connected in parallel, its impulse response is
  
  $$h(t)=h_1(t)+h_2(t)$$
  
  
  or in s-domain
  
  $$H(s)=H_1(s)+H_2(s)$$
  
  

• Serial or cascade system: If the system is composed of two LTI systems with $h_1(t)$ and $h_2(t)$ connected in series, its impulse response is
  
  $$h(t)=h_1(t)*h_2(t)=h_2(t)*h_1(t)$$
  
  
  or in s-domain
  
  $$H(s)=H_1(s)H_2(s)=H_2(s)H_1(s)$$
  
  

• Feedback system: If the system is composed of an LTI system with $h_1(t)$ in a forward path and another LTI system $h_2(t)$ in a feedback path, its output $y(t)$ can be implicitly found in time domain
  
  $$y(t)=h_1(t)*e(t)=h_1(t)*[x(t)+h_2(t)*y(t)]$$
  
  
  or in s-domain
  
  $$Y(s)=H_1(s)E(s)=H_1(s)[X(s)+H_2(s)Y(s)]$$
  
  
  While it is difficult to solve the equation in time domain to find an explicit expression for $h(t)$ so that $y(t)=h(t)*x(t)$, it is easy to solve the algebraic equation in s-domain to find $Y(s)$
  
  $$Y(s)[1-H_1(s)H_2(s)]=H_1(s) X(s)$$
  
  
  and the transfer function can be obtained
  
  $$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1-H_1(s)H_2(s)}$$
  
  
  The feedback could be either positive or negative. For the latter, there will be a negative sign in front of $h_s(t)$ and $H_2(s)$ of the feedback path so that $e(t)=x(t)-h_2(t)*y(t)$ and
  
  $$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$
  
  

Example 1: A first order LTI system

$$\frac{d}{dt}y(t)+3y(t)=x(t)$$

has a transfer function

$$H(s)=\frac{1}{s+3}=\frac{1/s}{1+3/s}$$

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

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

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 the parallel representations as shown above, while the first one is the direct representation which can be derived as follows. First consider a general $H(s)=Y(s)/X(s)=1/(s^2+as+b)$, i.e.,

$$s^2Y(s)+asY(s)+bY(s)=X(s)$$

or

$$s^2Y(s)=X(s)-asY(s)-bY(s)$$

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 $Y(s)$.

Example 3: A second order system with transfer function

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

This system can be represented as a cascade of two systems

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

and

$$Y(s)=H_2(s)Z(s)=(cs^2+ds+e)Z(s)$$

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 represented as shown below. Obviously the block diagram of this example can be generalized to represent any system with a rational transfer function

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