Semester 1 Flashcards
What are the two main categories that systems can be broadly classified into?
Linear and non-linear
What are the main 3 ways a system can be described?
1) Differential equations
2) Input-output relationship
3) Transfer functions
What letter is used to represent the state of a system?
x
What letter is used to represent the input of a system?
u, the inputs are the signals coming from the actuators
What letter is used to represent the output/measurements of a system?
y, the outputs/measurements are the signals coming from the sensors.
What letter is used to describe the order of a system?
n
What is another name for linear systems?
time invariant systems
What type of system can be described by transfer function?
Linear, time-invariant systems
How is a linear system described by transfer function?
H(s) = Y(s)/U(s) = N(s)/D(s)
What is U(s) in a linear systems transfer function?
U(s) = L [u(t)]
What is Y(s) in a linear systems transfer function?
Y(s) = L [y(t)]
How can you tell how many state variables a system has?
Look at the order of the system (the number of x’s)
How can you tell how many outputs (sensors) a system has?
Look at the number of y’s
How can you tell the number of inputs there are in a system?
Look at the number of u’s
What is the main difference between a linear system and a non-linear system?
A linear system can be described by transfer function whereas a non linear system can’t.
How do you analyse the stability of a system?
Use the characteristic equation P(s) = det(sI-A) to compute the eigenvalues of a.
How can you tell from the eigenvalues whether a system is stable or not?
If the system is stable the eigenvalues will lie in the left-half complex plane (be negative)
What is the transfer function of a system, involving the term (sI-A)?
H(s) = C 1/(sI-A) B
Why is it necessary to analyse the state controllability of a system before designing the controller for the system?
You cannot design a state feedback for a system that isn’t state controllable.
Why is it necessary to analyse the output controllability of a system before designing the systems controller?
If it is not output controllable then an output feedback cannot be designed for the controller.
How do you check whether a system is state controllable?
Check whether the rank of the matrix [B, AB] is equal to 2.
How do you check whether a system is output controllable?
Check whether the rank of the matrix [CB, CAB] is equal to 1.
What are the advantages and disadvantages of a state feedback controller?
It is not directly implementable since it doesn’t measure x2, however there is a greater degree of freedom when designing the controller.
What are the advantages and disadvantages of the output feedback controller?
It is readily implementable however there is a lesser degree of freedom in designing the controller than a state feedback controller.
What does proportional gain do in an analog PID controller?
Gives rise to a steady state error.
What does integral gain do in an analog PID controller?
Eliminates the steady state error caused by the proportional gain, but it slows down the response time.
What does derivative gain do in an analog PID controller?
Increases the rapidity of the response but might give rise to overshoots.
How can a discrete PID be implemented in practice?
With an AD converter on the front and DA converter on the back.
What are the advantages and disadvantages of describing a system using a transfer function?
The stability can easily be analysed by computing the poles of the system, but since the initial conditions are assumed to be zero it id only suitable for linear, time-invariant systems.
What is the drawback of describing a system by input-output relationship?
It’s generally described using an nth order differential equation, making it difficult to analyse.
What is the main advantage of describing a system by state-space approach?
It is a general approach and can be employed for both linear and non linear systems that can decompose an nth order differential into n first order differential equations.
What step-by-step procedure would you follow when designing a controller in an industrial environment?
1 - Model the system
2- Identify the parameters
3- Analyse properties of the system (stability, controllability etc)
4- Start designing the controller
Why is a proportional controller not recommended in practice?
It gives rise to steady state error, causing the output to not reach its target.
What is the transfer function of a system in terms of its input and output?
H(s) = Output (S) / input (S)
How do you check that the rank of a 2x2 matrix is 2?
it has a rank of 2 if it has a non-zero determinant
When u(t) = Kx(t) + v(t), what is the controller design given by?
u=k1x1+k2x2+v
When u(t) = Kx(t) + v(t), what is the closed loop system given by?
dx/dt = (A+BK)+Bv
