A Primer On Linear Control Systems Analysis

This page is everything I know about Linear Control Systems Analysis. The point of building this page was to help me understand it better. If it helps you at all or if you have any comments, please let me know here. All the information here is gathered from what I learned in APCS 401 at S.A.I.T. and surfing the web, and research at various libraries. A complete bibliography is here.

Introduction:
Automated Control Systems are everywhere you look. They are integrated into the office towers we work in, help us travel safely from place to place, and make manufacturing systems possible. Any relatively complex system you can think of probably includes an automatic control system. Something as common as an elevator is a control system. This type of system is actually a closed-loop (feedback) control system and is the primary focus of this paper.

A closed-loop system includes feedback. The feedback is compared to a reference point and then the system is changed to try to reach that point. The resulting change is then fed back to the reference and compared again, and so on, until the reference point matches the output. See figure 1 below to see how this works.


Figure 1 - the feedback control loop

This is the simplified diagram for ALL feedback systems. Something that is interesting in this field of study is that ALL systems tend to look similar when they are broken down to their math equivalents. This is an interesting thought really, and one that will be explored further later on in this paper.

From figure 1, you can see that there is a reference point (r) a feedback signal (b) and a controlled variable (c). The feedback element (H) sends the signal back to the reference point and the Physical Plant (G) is the actual system we are working with. For instance, take the elevator discussed earlier. The elevator, the cable attached to it, the pulley the cable goes around, motor, and counterweight are all the Physical Plant (G). The feedback system (H) may be position sensors or torque sensors in the motor, or some other way to find out where the elevator is. So someone steps on the elevator and pushes the 3rd floor button. That is the reference point (r) that is fed to the physical plant (G). The controlled variable (c) is the actual position of the elevator. The feedback element (H) sends the actual position back (b) to the reference for comparison. If the reference, or desired position, is not equal to the actual position, then b will be the difference and the error (e) is how much change is required for the system change. Eventually b will become zero when the reference will be the same as the actual position and the system will be at rest again (at the 3rd floor).

This is one of the simplest way to demonstrate a feedback (closed loop) control system, but elevators are a fairly simple example. The job of a Control Systems Designer is to understand these systems and the only way we can do that effectively is by understanding the math and physics behind them.

Here is a little tidbit of information that took me a while to get, but understanding this up front, right at the beginning can help in understanding the 'big picture'.

Everything in the universe, whether it is frogs, or elevators, or space ships, or blades of grass - ALL things share one thing in common.... ENERGY. Now, knowing that , you should be able to write an energy equation for just about anything, and, if you can write an equation for it, you can equate it to other things that are similar. So, if everything contains energy, and you can write an energy equation for anything, then it follows that you should be able to relate everything through their energy equations. Cool, eh ?

Somewhere along the line, we also found out that every system, with the possible exception of thermal energy systems, contains at least 3 energy elements. There is always an energy dissipater and usually at least 2 energy storage elements.

Too much? - sorry. Let's try an example.

Figure 2 - RLC Circuit

You don't have to understand electricity, just have some faith that the equations that follow are true.

In the example above (figure 2) there is a resistor, inductor, and a capacitor in series. This is a very common electronic circuit. To examine the Control System Equations for this, we can consider the way each device deals with energy. A resistor dissipates energy generally in the form of heat and because we are using voltage (V) as an energy source, the equation for the resistor should be in terms of voltage. Ohm's law says the voltage across a resistor equals the resistance multiplied by the current flowing through it, or .
For the capacitor storing energy, we can use Faraday's law and get .
For the inductor storing energy, we can use Lenz's law to get .

To make this look a little simpler, we will replace all the Greek stuff with D for differential and 1/D for integral, so Lenz now looks like this V=LDi and Faraday is V=(1/CD) i.

Now, writing the equation for the circuit is as easy as adding the energy components and factoring out the feedback factor, like this:

Figure 3 - the RLC equation.

That's the easy part. Now you need to select physical variables for each energy STORAGE element based on it's energy equation. The capacitor and the inductor are the storage element is this system and the capacitor's energy equation is , so Vc is chosen as the first variable (x1). The inductor's energy equation is ,
so iL is chosen as the second variable (x2). From basic electronics knowledge we know that iC = IL, and we know that IC=CDVC so we can substitute x1 and x2 to get x2=CDx1. Now the D operator in this equation can act on the x1 and differentiate it. We can show this Dx1 as . So now the equation looks like this: x2=C and we can rearrange it to get =(1/C)x2. That is the first state equation. Here it is again and also the other derivation for x2.

Figure 4 - the state equations

These state equations represent first order differential equations for each variable that will describe the system in terms of math and that is something we can place in a matrix for further calculation or we can use Laplace Transforms on the full system equation to find the needed solution. We will look at the Matrix equation first.

Next -- Matrix Manipulation

Bibliography:

Linear Control System Analysis and Design - Fourth Edition
by John J. D'azzo and Constantine H. Houpis
Copyright 1995, 1988, 1981, 1975 McGraw-Hill Inc.
ISBN# 0-07-016321-9

Basic Technical Mathematics With Calculus - Fourth Edition
by Allyn J. Washington
Copyright 1985, The Benjamin/Cummings Publishing Company Inc.
ISBN# 0-8053-9545-8

Introduction to Electric Circuits - Fifth Edition
by Herbert W. Jackson
Copyright 1981, Prentice-Hall Inc.
ISBN# 0-13-481432-0

Everything I ever heard from Russ Hersberger,
SAIT Instructor, APCS 401
http://learnat.sait.ab.ca/ict/apcs401/instructor.htm
russ.hersberger@sait.ab.ca

Virtually everything I read at http://www.efunda.com
but in particular :
      http://www.efunda.com/math/laplace_transform/index.cfm
      http://www.efunda.com/math/ode/firstorder_ode.cfm

Lecture Notes from University of Calgary Course ELEN441
http://www.enel.ucalgary.ca/People/Westwick/Courses/ENEL441/LectureNotes/index.html
Dr. David T. Westwick, Instructor

University of Newcastle's Department of Electrical and Computer Engineering
Control System Design [http://cpsu.newcastle.edu.au/control/index.html]
in particular, http://cpsu.newcastle.edu.au/control/book_slds_ch03/sld03_008.html

Stanford University Course EE263 Lecture
"Solution via Laplace transform and matrix exponential"
http://www.stanford.edu/class/ee263/expm.pdf