Control of Linear Systems

 This assignment should be written as an academic report, with all working shown and explanations given for each step taken. Marks will be awarded for correct understanding of the subject and mathematical calculations from each step, but not for the final answers, if it is not obvious how the answers are obtained. All sources of information should be correctly referenced.

1. This question relates to the system represented in the following block diagram, where 𝑁 is a number representing the last two digits of your student registration number.

2. Given a system having the following state-space model:

(a) Draw its simulation diagram, containing no vectors, matrices, or transfer functions other

(b) By reducing the simulation diagram to a transfer function using block diagram reduction techniques (i.e. not by using the rank tests), determine and explain any values of the gain a11 for which the system would be:

(i) unstable.

(ii) uncontrollable.

(iii) unobservable.

(c) Confirm the results of parts (b) (ii) and (iii) using rank tests.

3. A system has a transfer function

(a) Obtain a state-space model of the system

(b) There are several ways of showing that this system is unstable. Your task is to stabilise it

using state variable feedback (SVF) design, assuming that you can measure all the state variables. Here are some matters to consider when designing the SVF controller. Your responses to these will depend on your state space model and, even if you have generated the same model as a colleague, you still have an infinite number of choices for the closed- loop pole locations, for example, so different solutions are quite likely.

• Is it necessary to perform a controllability test? If so, do it, if not, why not?
• Is it necessary to perform a stabilisability test? If so, do it, if not, why not?
• Is it necessary to perform an observability test? If so, do it, if not, why not?
• Where are you going to place your closed-loop poles, and why

(c) Produce a simulation diagram of the closed-loop system determined in (b), containing no vectors, matrices, or transfer functions other than simple integrators and gain blocks, showing how your controller interfaces with the plant.

(d) Work out whether your controlled system will exhibit any steady-state error following a step input (I have in mind an analytical solution, but doing it by simulation is OK so long as you provide all relevant details). Describe (but do not design) how a tracking controller could be designed to overcome such behaviour.

4. 

A lateral beam guidance system has an inner loop as shown in Figure Q4, where the transfer function for the coordinated aircraft is

23

𝐺(𝑠) ---------- = 

𝑠 +N

and 𝑁 is a number representing the last two digits of your student registration number. Consider the PI controller

(a) Design a control system to meet both of the following specifications:

• Settling time (with a 2% criterion) to a unit step input of less than 1 
• second; Steady-state tracking error for a unit ramp input of less than 0.1.

(b) Verify the design by a Simulink simulation.

Note: Although you will probably consult colleagues, notes, books and staff during the execution of this assignment, the work you hand in MUST BE DONE BY YOURSELF. The University takes cheating very seriously and the appropriate investigation procedures will be invoked if necessary!