Section 2

Question 1

What is a transfer function?

Answer 1

A mathematical function that models the systems output for every possible input.

Question 2

What is meant by discrete time?

Answer 2

In discrete time variables are viewed as having values at distinct seperate points in time. They are unchanged thoughout each non-zero region of time.

Question 3

What is a model?

Answer 3

A scientific hypothesis about the behaviour of the system under study.

Question 4

Why do we model systems?

Answer 4

• Scientific understanding
• Forecasting
• Simulation
• Management and operational planning
• Optimisation
• Control system design

Question 5

There are many times of model, how do we know which is the right one?

Answer 5

There is no one right model. The most appropriate model for a given case will depend on the project objectives.

Question 6

What is the main assumption of Linear Systems Theory?

Answer 6

Linear Systems Theory assumes a cause and effect relationship between the input and output variables.

Question 7

What are the three types of linear model?

Answer 7

1. Mechanistic model
2. Black box model (data based models)
3. Grey box model

Can you define them?

Question 8

Write the first order, discrete time difference equation
Bonus: show the steps to find the transfer function equivalent

Answer 8

y(k)=-ay(k-1)+bu(k-1)

Question 9

What does the backwards shift operator ‘z’ represent?

Answer 9

A single sample time delay.

Question 10

What is the difference between a transfer function and difference equation?

Answer 10

A transfer function gives the output as a function of the control input.
The transfer function gives the output in as a function of previous outputs and inputs.

Question 11

Write the generalized differential equation.
Bonus: show the steps to find the generalized transfer function

Answer 11

y(k)=-a(1)y(k-1)-a(2)y(k-2)-…-a(n)y(k-n)+b(0)u(k)+b(1)u(k-1)+…+b(m)u(k-m)

Question 12

How can you tell the order of a model from the difference or transfer function?

Answer 12

The model order depends on the output terms. For example the generalized form is nth order.

Question 13

How can you tell the time delay of a model from the difference equation or transfer function?

Answer 13

The order of the lowest input term is equal to the number of samples time delay.
Eg: if the equation has b(0)u(k) term there is no delay
If the first output term is b(3)u(k-3) then there is a delay of 3 samples

Question 14

What are the units of the Stead State Gain?

Answer 14

The Steady State Gain has no units, it is the ratio of the output to input at steady state.
The Steady State Value however will have units depending on the system in question.

Question 15

Define Steady State

Answer 15

When the output has settled towards a constant value as time tends towards infinity.

Question 16

How can you find the Stead State Gain?

Answer 16

In the transfer function, replace all values of z with 1.
Because: if there is no change between output values then y(k)-y(k-1)=0
Therefore: y(k)*(1-z(-1))=0
Therefore 1-z(-1) = 0
Therefore z(-1)=1

Question 17

Define Time Constant

Answer 17

For First Order systems the Time Constant is the time taken to reach 63% of the Steady State value.
T=-deltat/loge(-a1)

Question 18

Define Stable System

Answer 18

If a system that is initially in equilibrium is disturbed, it is said to be stable if it eventually returns to an equilibrium condition.

Question 19

Describe the steps to find the poles of a model and whether the system is stable

Answer 19

1. Find the characteristic equation by taking the characteristic polynomial (Transfer Function denomination) and setting it eqaul to 0.
2. Convert to the Z domain by multiplying through by Z. Algebraic rules apply.
3. Solve for Z. If the magnitude of each pole is less than 1 the system is stable. If a pole = 1 the system is marginally stable and if any pole is greater than 1 the system is unstable.

Question 20

What does the presence of complex poles indicate?

Answer 20

That a model might exhibit oscillatory behavior.

Question 21

What does the lack of complex poles indicate?

Answer 21

That a model will be exponential (stable or unstable).

Question 22

What is a non-minimum phase system?

Answer 22

If the zeros (roots of the transfer function numerator) are greater than 1 the system is non-minimum phase.
Non-minimum phase systems sometime respond strangely to inputs. For example by initially responding in the opposite direction (negative) before correcting after a few samples.

Question 23

How can you simplify two Transfer Functions in series in Block Diagram Analysis?

Answer 23

Series Transfer functions can be multiplied together.

Question 24

Prove the Negetive Feedback Rule for Block Diagram Analysis

Answer 24

Should result in :
X=U(G1/1+G1*G2)