Pages

Friday, August 26, 2011

Control Systems guidelines for Business

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.


The control model uses the transformation process in the context of a technology system
Slide1.jpg
The inputs consist of primary (raw materials) and secondary (energy, manufacturing plant, workforce). There are also primary and secondary output representing the main product and bi-products (worn out staff, spent resources etc) {Source1}



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
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

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
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

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:
xF' = AFxF + BFu
yF = CFxF + DFu
System 2:
xG' = AGxG + BGyF
yG = CGxG + DGyF
And we can write substitute these equations together form the complete response of system H, that has input u, and output yG:


[Series state equation]
\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


[Series output equation]
\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
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

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
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

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.
Transformation Equation Block Diagram Equivalent Block Diagram
1 Cascaded Blocks Y=\left(P_1 P_2 \right) X Cascaded Blocks.svg Cascaded Blocks Equivalent.svg
2 Combining Blocks in Parallel Y=P_1 X \pm P_2 X Parallel Blocks.svg Parallel Blocks Equivalent 1.svg
3 Removing a Block from a Forward Loop Y=P_1 X \pm P_2 X Parallel Blocks Equivalent 2.svg
4 Eliminating a Feedback Loop Y=P_1 \left( X \mp P_2 Y \right) Feedback Loop.svg Feedback Loop Equivalent 1.svg
5 Removing a Block from a Feedback Loop Y=P_1 \left( X \mp P_2 Y \right) Feedback Loop Equivalent 2.svg
6 Rearranging Summing Junctions Z=W \pm X \pm Y Rearranging Summing Junctions 1.svg Rearranging Summing Junctions 2.svg
Rearranging Summing Junctions 3.svg
7 Moving a Summing Juction in front of a Block Z = P X \pm Y Moving Summing Junction in front of Block 
1.svg Moving Summing Junction in front of Block 
2.svg
8 Moving a Summing Juction beyond a Block Z = P \left( X \pm Y \right) Moving Summing Junction beyond Block 1.svg Moving Summing Junction beyond Block 2.svg
9 Moving a Takeoff Point in front of a Block Y= PX\, Moving Takeoff Point in front of Block 1.svg Moving Takeoff Point in front of Block 2.svg
10 Moving a Takeoff Point beyond a Block Y=PX\, Moving Takeoff Point beyond Block 1.svg Moving Takeoff Point beyond Block 2.svg
11 Moving a Takeoff Point in front of a Summing Junction Z=W \pm X Moving Takeoff Point ahead of a Summing 
Junction 1.svg Moving Takeoff Point ahead of a Summing 
Junction 2.svg
12 Moving a Takeoff Point beyond a Summing Junction Z=X \pm Y Moving Takeoff Point beyond a Summing Junction 
1.svg Moving Takeoff Point beyond a Summing Junction 
2.svg
{Source2}

{Source1: http://www.mbanotes.org/node/9 } 
{Source2: http://en.wikibooks.org/wiki/Control_Systems/Block_Diagrams }  

2 comments:

  1. Howdy! This article couldn’t be written much better!
    Reading through this article reminds me of my previous roommate!
    He always kept talking about this. I'll forward this information to him. Pretty sure he's going to have a very good read. Thank you for sharing! payday loans baton rouge
    my web page - payday loans bad credit

    ReplyDelete
  2. Spot on with this write-up, I honestly feel this website needs far more attention.

    I’ll probably be returning to read more, thanks for the info!
    speaking of
    my site - special info

    ReplyDelete