Part II

Question 1

How can you check the stability of a system x. = Ax + Bu?

Answer 1

1

Question 2

What does it mean for a system to be controllable?

Answer 2

Given any initial state and a desired state, if you are able to drive the system from the initial stage to the desired state, the system is completely controllable.

Question 3

How can you check the controllability of a system?

Answer 3

3

Question 4

What does it mean for a system to be observable?

Answer 4

From the measurement outputs of the system, you can derive the initial state of the system

Question 5

How can you check if a system is observable?

Answer 5

4

Question 6

How can you transform a system into a controllable form?

Answer 6

5

Question 7

How can you transform a system into different state variables?

Answer 7

2

Question 8

How can you transform a system into an observable form?

Answer 8

6

Question 9

What does it mean for a system to be stabilizable?

Answer 9

In controllable canonical form, if the uncontrollable part is stable (in LHP), the system can be stabilised.

Question 10

What does it mean for a system to be detectable?

Answer 10

In observable canonical form, if the unobservable part is stable (in LHP), the system can be detected.

Question 11

What are the assumptions made in state feedback controller design?

Answer 11

• Assume we can measure all the states
• Assume system is completely controllable

Question 12

How can you design a state feedback controller for a SISO system?

Answer 12

7

Question 13

How can you design a state feedback controller for a MIMO system?

Answer 13

8

Question 14

How can you design a full order observer?

Answer 14

9

Question 15

How can you design a reduced order observer?

Answer 15

10

Question 16

When designing an LQR controller, if you want the states to converge faster, what should you do?

Answer 16

Put Q higher and R lower. This is however at the cost of controller effort (amplitude)

Question 17

What are two of the good properties of LQR control?

Answer 17

Infinite gain margin
Phase margin >= 60 deg

Question 18

Illustrate gain and phase margin

Answer 18

11

Question 19

How to design an LQR controller?

Answer 19

12

Question 20

What is the closed loop equation of an LQR controlled system?

Answer 20

|sI - (A-BF)| = 0
where F is the controller gain
When solving, expand to matrix form, simplify for s = …

Question 21

Why do we use a Kalman Filter / LQG controller?

Answer 21

In the presence of noise in the measurements or uncertainties in the model, we can estimate the states using a Kalman Filter.

Question 22

How can you design a Kalman Filter / LQG controller?

Answer 22

13

Question 23

What are the problems with the LQG controller?

Answer 23

There are no guaranteed gain and phase margins as with the LQR controller.
Even small perturbations in input can cause the system to go unstable for large values of q and noise.

Question 24

What are the basic steps of applying Loop Transfer Recovery?

Answer 24

14
Choose an appropriate Kalman filter gain Ke such that Llqg(s) approximates Llqr(s) over a certain range of frequencies. Higher frequencies does not work as much.

Question 25

How can you design a H infinity controller?How would you do a filter?

Answer 25

15

Question 26

What are the steps / required information for rejecting sinusoidal disturbances?

Answer 26

We need to know the frequency of the disturbance.
Magnitude and phase of disturbance is governed by initial conditions
Combine a disturbance model with the plant and design an observer
Estimate the disturbance d and design a controller u = - Kxhat - Gdhat to cancel out the disturbance