# Control Systems/Block Diagrams

When designing or analyzing a system, often it is useful to model the
system graphically. **Block Diagrams** are a useful and simple method
for analyzing a system graphically. A \"block\" looks on paper exactly
what it means:

## Systems in Series

When two or more systems are in series, they can be combined into a
single representative system, with a transfer function that is the
product of the individual systems.

![](Time_Series_Block.svg "Time_Series_Block.svg")

If we have two systems, *f(t)* and *g(t)*, we can put them in series
with one another so that the output of system *f(t)* is the input to
system *g(t)*. Now, we can analyze them depending on whether we are
using our classical or modern methods.

If we define the output of the first system as *h(t)*, we can define
*h(t)* as:

$$h(t) = x(t) * f(t)$$

Now, we can define the system output *y(t)* in terms of *h(t)* as:

$$y(t) = h(t) * g(t)$$

We can expand *h(t)*:

$$y(t) = [x(t) * f(t)] * g(t)$$

But, since convolution is associative, we can re-write this as:

$$y(t) = x(t) * [f(t) * g(t)]$$

Our system can be simplified therefore as such:

![](Time_Convolution_Block.svg "Time_Convolution_Block.svg")

### Series Transfer Functions

If two or more systems are in series with one another, the total
transfer function of the series is the product of all the individual
system transfer functions.

![](S-Domain_Series_Block.svg "S-Domain_Series_Block.svg")

In the time-domain we know that:

$$y(t) = x(t) * [f(t) * g(t)]$$

But, in the frequency domain we know that convolution becomes
multiplication, so we can re-write this as:

$$Y(s) = X(s)[F(s)G(s)]$$

We can represent our system in the frequency domain as:

![](S-Domain_Multiplication_Block.svg "S-Domain_Multiplication_Block.svg")

### Series State Space

If we have two systems in series (say system F and system G), where the
output of F is the input to system G, we can write out the state-space
equations for each individual system.

System 1:

$$x_F' = A_Fx_F + B_Fu$$

$$y_F = C_Fx_F + D_Fu$$

System 2:

$$x_G' = A_Gx_G + B_Gy_F$$

$$y_G = C_Gx_G + D_Gy_F$$

And we can write substitute these equations together form the complete
response of system H, that has input u, and output y~G~:

$$\begin{bmatrix}x_G' \\ x_F'\end{bmatrix} 
     = \begin{bmatrix}A_G & B_GC_F \\ 0 & A_F\end{bmatrix} 
       \begin{bmatrix}x_G \\ x_F\end{bmatrix} +
       \begin{bmatrix}B_GD_F \\ B_F\end{bmatrix}u$$


$$\begin{bmatrix}y_G \\ y_F\end{bmatrix} 
     = \begin{bmatrix}C_G & D_GC_F \\ 0 & C_F\end{bmatrix} 
       \begin{bmatrix}x_G \\ x_F\end{bmatrix} +
       \begin{bmatrix}D_GD_F \\ D_F\end{bmatrix}u$$

## Systems in Parallel

![](S-Domain_Parallel_Block.svg "S-Domain_Parallel_Block.svg")

Blocks may not be placed in parallel without the use of an adder. Blocks
connected by an adder as shown above have a total transfer function of:

$$Y(s) = X(s) [F(s) + G(s)]$$

Since the Laplace transform is linear, we can easily transfer this to
the time domain by converting the multiplication to convolution:

$$y(t) = x(t) * [f(t) + g(t)]$$

![](S-Domain_Addition_Block.svg "S-Domain_Addition_Block.svg")

## State Space Model

The state-space equations, with non-zero A, B, C, and D matrices
conceptually model the following system:

![](Typical_State_Space_Model_(General).svg "Typical_State_Space_Model_(General).svg")

In this image, the strange-looking block in the center is either an
integrator or an ideal delay, and can be represented in the transfer
domain as:

$$\frac{1}{s}$$ or $\frac{1}{z}$

Depending on the time characteristics of the system. If we only consider
continuous-time systems, we can replace the funny block in the center
with an integrator:

![](Typical_State_Space_Model_(CT).svg "Typical_State_Space_Model_(CT).svg")

### In the Laplace Domain

The state space model of the above system, if *A*, *B*, *C*, and *D* are
transfer functions *A(s)*, *B(s)*, *C(s)* and *D(s)* of the individual
subsystems, and if *U(s)* and *Y(s)* represent a single input and
output, can be written as follows:

$$\frac{Y(s)}{U(s)} = B(s)\left(\frac{1}{s - A(s)}\right)C(s) + D(s)$$

We will explain how we got this result, and how we deal with feedforward
and feedback loop structures in the next chapter.

## Adders and Multipliers

Some systems may have dedicated summation or multiplication devices,
that automatically add or multiply the transfer functions of multiple
systems together

## Simplifying Block Diagrams

Block diagrams can be systematically simplified. Note that this table is
from Schaum\'s Outline: Feedback and Controls Systems by DiStefano et al

  Transformation                                                                                      Equation                                                Block Diagram                        Equivalent Block Diagram
  --------------------------------------------------------------------------------------------------- ------------------------------------------------------- ------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------
  1                                                                                                   Cascaded Blocks                                         $Y=\left(P_1 P_2 \right) X$          image:Cascaded Blocks.svg
  2                                                                                                   Combining Blocks in Parallel                            $Y=P_1 X \pm P_2 X$                  image:Parallel Blocks.svg
  3                                                                                                   Removing a Block from a Forward Loop                    $Y=P_1 X \pm P_2 X$                  image:Parallel Blocks Equivalent 2.svg
  4                                                                                                   Eliminating a Feedback Loop                             $Y=P_1 \left( X \mp P_2 Y \right)$   image:Feedback Loop.svg
  5                                                                                                   Removing a Block from a Feedback Loop                   $Y=P_1 \left( X \mp P_2 Y \right)$   image:Feedback Loop Equivalent 2.svg
  6                                                                                                   Rearranging Summing Junctions                           $Z=W \pm X \pm Y$                    image:Rearranging Summing Junctions 1.svg
  image:Rearranging Summing Junctions 3.svg                                                                                                
  7                                                                                                   Moving a Summing Junction in front of a Block           $Z = P X \pm Y$                      image:Moving Summing Junction in front of Block 1.svg
  8                                                                                                   Moving a Summing Junction beyond a Block                $Z = P \left( X \pm Y \right)$       ![](Moving_Summing_Junction_beyond_Block_1.svg "Moving_Summing_Junction_beyond_Block_1.svg")
  9                                                                                                   Moving a Takeoff Point in front of a Block              $Y= PX\,$                            image:Moving Takeoff Point in front of Block 1.svg
  10                                                                                                  Moving a Takeoff Point beyond a Block                   $Y=PX\,$                             image:Moving Takeoff Point beyond Block 1.svg
  11                                                                                                  Moving a Takeoff Point in front of a Summing Junction   $Z=W \pm X$                          image:Moving Takeoff Point ahead of a Summing Junction 1.svg
  12                                                                                                  Moving a Takeoff Point beyond a Summing Junction        $Z=X \pm Y$                          file:Moving Takeoff Point beyond a Summing Junction 1.svg

## External links

-   SISO Block Diagram with transfer functions on
    ControlTheoryPro.com

