The coursework deals with modelling, control and sensitivity/uncertainty analysis for the         longitudinal axis of a B747- 100/200 aircraft. A typical longitudinal axis states and inputs of an aircraft are given inFigure 1.

Positive

deflection

Stabiliser

Elevator

Positive

rotation

Figure 1: aircraft longitudinal axis (states and inputs)

A simplified model, known as a linear parameter varying (LPV) model, can be used to       represent the actual B747- 100/200 aircraft. The states of the model are () ∈ ℝ5×1  given by:

() = [ ℎ] (1)

which represents pitch rate (rad/s), speed (m/s), angle of attack (rad), pitch angle (rad) and altitude (m). The inputs are () ∈ ℝ3×1  given by:

() = [ ] (2)

which represent elevator (rad), stabilizer (rad) and thrust (N). The underlying equation of the LPV model is given by the following equation:

̇ () = ()() + ()() (3)

where () ∈ ℝ5×5  and () ∈ ℝ5×3  defined as:

() = 0  + ∑

=1

() = 0  + ∑

=1

The seven LPV varying parameters are given by:

=  [1 2 3 4 5 6 7] = [ 2

which depends on the states angle of attack and speed .

2 3 4] (6)

It is required that the flight path angle (rad) (defined as = − ), and the velocity is to be controlled, with the tracking performance specification are given inTable 1below:

Table 1: Controlled states and its performance specification

The typical range for the inputs is given below:

Table 2: Inputs and its typical range of operation

The signals and used in LPV parameters in(6), have uncertainty ranges specified below:

Table 3: The LPV signal source uncertainties

Preparation

To complete this coursework, you will need to download the zip file called                               "B747_LPVplant.zip" file from the Coursework section on the ELE page for this module.   To use this file within MATLAB, you need to save it to a local directory and select this             directory as your current folder in MATLAB or any folder that you prefer. The zip file contains two files. The "B747_LPVplant.mat" file contains all the matrices for () and ()               associated with the LPV model (see equations(3)-(6)above) while                                           "B747_LPVplant_init.m" will load the mat file. You should use the                                     "B747_LPVplant_init.m" file to initialise all the parameters in the model and other            programs/codes associated with this coursework.

Tasks

Using MATLAB and SIMULINK:

1)  Create the SIMULINK model to represent the longitudinal axis of the B74-100/200       using equations(1)-(6)given above. To test whether your model is implemented          correctly in Simulink, you can perform a simple test by perturbing the pitch rate states by 1 deg/s at the beginning of the simulation and check the behaviour of all the states in the model.

Notes:

• Some info on simple aircraft modelling can be foundhere.

• States perturbation can be made by setting nonzero initial conditions in the states integrator.

• Be careful about units while modelling. Note that some of the units for the inputs and states of the model(1)-(6)are in rad or rad/s, while the specifications given inTable 1-Table 3are in deg or deg/s.

2)  Perform a sensitivity/uncertainty analysis of the open-loop systems using the one-

factor-at-a-time method and plot the tornado graph. Identify the parameters (input/uncertainties) that affect the system's states the most.

Notes:

• Note that, for sensitivity/uncertainty analysis, the nominal values of all the states () and inputs () are zero.

• The sensitivity/uncertainty analysis must be performed for all inputs () as well as the uncertainty in the LPV parameters.

• Step inputs (with magnitude set as a variable) can be used as a type ofinput signals ().

3)   Implement two PID controllers to track flight path angle γ (using the elevator) and       velocity V (using thrust). Tune the PID controllers to ensure the tracking performance inTable 2is satisfied, with minimal coupling effect.

Notes:

• Use square inputs as the command signals. For example, for the command signals for flight path angle , first, perform +3 deg for 10 s and followed by - 3deg for the following 10 s to return the command signal back to zero.

• To ensure decoupled performance, the command signals for flight path angle

and velocity need to be set at a different time.

• The SIMULINK standard PID block to tune the PID controllers can be used to aid the tuning process.

4)   Implement the "Latin Hypercube sampling" method (or at least the "brute-force"        method) and perform the sensitivity/uncertainty analysis for the closed-loop system. Identify the parameters that affect the performance of the two PID controllers (the    tracking of flight path angle and speed ).

Notes:

• For sensitivity/uncertainty analysis, the nominal values of all the states () and inputs () are zero.

• The sensitivity/uncertainty analysis must be performed for all command signals as well as the uncertainty in the LPV parameters.

• To analyse the performance of the PID controllers, you can analyse the error signals () between the commanded signals and the actual measurements.

• To create a random number, use Matlab command “rand”

5)  Discuss the possibility of using the genetic algorithm to help tune the controller that will be able to handle all variations in the command signals and uncertainties. How  this can be achieved using MATLAB and/or SIMULINK.

Notes:

• You can use the knowledge provided in Workshop 7 or 8 or lecture notes on genetic algorithms to help you with this question.

Instructions

• This coursework submission has two parts:

a) A report in PDF

b) Simulation files that implement the model * . It should contain:

i. One m file and

ii. one Simulink (slx) file

• ZIP all three files (PDF, m and slx) before submitting to eBart**

• File naming convention***:

o eBartCandidateNumber.pdf

o eBartCandidateNumber_init.m

o eBartCandidateNumber.slx

REMARK:

* The submitted simulation files should run task 4 (task 1-3 can be commented). ** Make sure that files submitted are the correct file type. Incorrect file type may fail to open and thus preventing it from being marked.

*** If you are not sure what your eBart candidate is, check the eBart submission page.

The ZIP file for this coursework must be submitted to e-BART (https://bart.exeter.ac.uk -       School: CEMPS, location: Harrison) by 12:00 noon on the date indicated on the front page of this document.

In the report, you should address all questions/tasks posed above, which should be brief and to the point. Include all the modifications made to the original code and functions (if     any). Furthermore, you should include the MATLAB commands or SIMULINK diagrams you used, and the text or image output produced. Finally, you should include the reasoning and  motivation behind the choices made.

Matlab Version requirement:

If you are using Matlab later than R2021a, please save your Simulink file to R2021a version – In Simulink, Menu>File>Export Model to>previous version>select R2021a models.

Report (PDF) formatting requirement:

•    The report should not exceed ten A4 pages (or approximately 4000 words - when excluding plots/figures/codes).

•    Only Arial or Calibri Light font styles may be used and text should be justified left.

•    You should use a minimum font size of 11.

•    Margins must be at least 2cm in all directions.

•    The document size should be standard A4 size, single column.

•    All pages must be numbered.

•    The report needs to include an introduction and conclusion sections (both need to be as brief as possible).