Control Systems Basics
It is good to revisit the basics of control engineering.
BEWARE: this post cointains Mathematics, not for the fein of heart!
Control loop description
A general schematic of a control loop is depicted in the following figure (for simplicity, we deal with a Single Input Single Output - SISO system):
The notations are the following:
r: reference signal, the signal our controller wants to track
y: output of the system
e: tracking error, e= r-y
cs: controller signal, driving the actuator
as: actuator signal, the signal acting on the plant, aka our system
d: disturbances, in real life no sensor measurement is perfect, to model this often a disturbance signal is introduced to account for imperfection
o: output, an abstracted physical output of the system which is measurable by our sensors
y: output of the sensor, the electrical (in most cases) signal representation of the measured attribute
These level of details matter really a lot when we talk about real life system, for the sake of simplicity of demonstration we result to the following:
The plant basically will comprise all the physical systems together, so real plant + sensor(s) + actuator(s)
Laplace transformation
The math, unless there is a need for it later, at this point, is kept to a minimum. But there are some unavoidable details I need to mention. The formal definition of Laplace transformation is the following:
This will prove useful for practical applications in modeling, for brevity, I mention a couple of common shortcuts from the toolbox.
Model descriptors
In the first place, we restrict ourselves to linear models. Linear models, as we are often used to, can be described in their time-dependent form (assuming that all output and their derived states are initially 0):
This can be easily converted using the Laplace transformation into a so called transfer function:
Or our last option, a very frequently used state-space form:
From which with some linear algebra the transfer function, can be easily determined:
Interesting to note, that the denominator of the transfer function equals the canonical equation of the A matrix. Therefore, the eigenvalues of A will become the poles of the transfer function. Poles and zeros are important to determine stability and performance criteria to our system, as we will see it later. The numerator and the denominator of a linear function can be expressed as polynomials with real (in a mathematical sense, not as in we did not imagine them :) ) coefficient. Then the zeros and the poles are where the numerator and denominator would yield to zero respectively. It can be proved that for linear systems they are real numbers or complex conjugated pairs: a ± bj.
Real physical systems can often be represented as second-order systems, a common representation of these systems as follows:
Time domain response, with its important elements:
final value : 1
max overshoot: highest peak compare to the settled value: 1.25
settling time: time for the system to stay in between the a specified(usually 5%) error band of the final value: 4.8 s
rise time: time in between 10% and 90% of the final value: 0.43 s
delay: usually the time it takes for the signal to reach 50%: 1.2 s
Bode plot:
Phase margin: The stability margin is the ansolute difference between the pahse when the amplitude curve crosses 0 dB (unit gain) and -180 degrees.
Gain margin: The amplitude difference between 0dB and the current value when the phase is -180 degrees.
In our system this is 68.9 degrees while the amplitude gain is infitie, due to the fact that the phase curve never reaches -180 degrees.
Pole zero plot: poles need to be on the negative half imaginary plane for a system to remain stable.
Modelling basics
As it was outlined in my first post, the first step of the journey is to model our system. Often this step involves research, consulting with domain experts, and obtaining information from relevant books and academic papers. There is not one ubiquitous way for modeling. Often determined by the available information, because having the mathematical model of the system is not enough we need to have precise data regarding the parameters of our models.
Generally we can distinguish three separate types of models.
1. White box models
White box models: White box models are often described as models as transparent consisting of easily interpretable parts. This is what I have described so far.
2. Black box models
When we don’t know much about the inner workings and intricacies of the system. It gives some output for some inputs. An example could be a propeller curve for ships. Often that is just a plot where we know what output torque(force) is presented on the blade for given pitch and angular velocity (often represented in rpm-s).
3. Grey box models
As you might imagine as the grey color is in between black and white, the apporach is similar. Having some insight and some empirical knowledge into the observed system.
DIY
Given a m mass attached to a wall via a spring-damper system(k,b) and we can excercise Force on the mass. Assuming that our x coordinate is 0 at the middle of the mass. We are interested in the following:
State-space description of the model: A,B,C,D
Transfer function: W(s)
Analytical calculation of the time constant, damping coefficient, undamped eigen(natural) frequency
Plot step response
Determine, phase and gain margin
Find the zeroes and poles of this system
What is the maximum of the overshoot, and in what time
Settling time
BONUS: How does the cover picture of the post relates to it?
Parameters are, m=2kg, k=0.5N/m, b = 0.25Ns/m
The system can simply be depicted as:
We will cover the answer and analyse this system in the next edition
Stay Courious!