Mathematics and MATLAB
Objectives and Syllabus
Establishes a mathematical basis for the more theoretical control modules which follow. Topics covered are - matrices; determinants, notation, linear algebra: ordinary differential equations; Laplace transforms, transfer functions, the s-plane, poles & zeroes: numerical methods; numerical integration, iteration, convergence: data typing & structured text: statistics; mean, variance etc., probability, regression, correlation coefficients.
Practical work is based upon the use of the Matlab and Simulink packages available on the School network. For assignments, students are expected to make their own arrangements for access to these packages at home and/or at work.
|Code:||CME 8360 (formerly ACS 660)|
|Time Allocation:||Lectures||40 hours|
|Private Study||70 hours|
|Pre-requisites:||First degree of equivalent in an appropriate discipline|
|Assessment:||By report on assignment
By 1 x 2 hour examination
To establish a mathematical basis for understanding the more theoretical aspects of control and automation and to introduce the Matlab and Simulink packages for use in subsequent modules.
- To review and/or revise the use of matrices and vectors as a basis for studying linear control theory.
- To become familiar with the use of Laplace transforms for solving differential equations in the context of control systems.
- To provide a grounding in statistical methods as a basis for handling data driven relationships in automation.
- To introduce and/or revise numerical methods of integration.
- To provide an insight of the potential for using mathematics for the analysis of dynamics and design of control systems.
- To introduce the functionality of the Matlab and Simulink packages as a basis for problem solving and for modelling and simulating control systems.
This is a stand-alone module and has no pre-requisites as such.
This module is of one week's full-time intensive study consisting of a variety of lectures, informal tutorials for problem solving and structured computer-based laboratory work. It is followed by an assignment to be carried out in the student’s own time.
The time allocation for practical work provides for PC based activity using the Matlab and Simulink packages. Students are taught how to use these packages and familiarised with their functionality. The packages are used for simple control system design, modelling and optimisation exercises. These are structured to reinforce the syllabus content.
The assignment typically consists of the analysis of the dynamics of a simple control system using Matlab. Students are expected to make their own arrangements for access to Matlab/Simulink for the assignment.
- Dabney J B & Harman T L, Mastering Simulink, Prentice Hall, 2004.
- Dutton K, Thompson S & Barraclough B, The Art of Control Engineering, Addison Wesley, 1997.
- Hanselman D & Littlefield B, Mastering Matlab, Prentice Hall, 2005.
- Jeffrey A, Advanced Engineering Mathematics, Harcourt Academic Press, 2002.
- Love J, Process Automation Handbook, Springer, 2007
- Stroud K A, Further Engineering Mathematics, 4th Edition, MacMillan (Palgrave), 1996.
- Wilkie, J, Johnson M and Katebi R, Control Engineering: An Introductory Course, McMillan (Palgrave), 2002.
Revision: Power series. Maclaurin’s series: sin, cos and exp functions. Binomial expansion. Taylor’s series. First order approximation and application to function of two or more variables. Complex numbers. Cartesian, polar and exponential forms. Basic sin, cos and exp identities. Addition, multiplication and division of complex numbers. Modulus and argument of product of complex numbers.
Matrices: Scalar, vector and matrix quantities. Determinants. Matrix notation. Special matrices, eg transpose. Multiplication and inversion. Matrix algebra. Matrix solutions to linear equations. Matrix properties: eigenvalues, canonical transformations, singular values. Representation of multivariable processes.
Ordinary differential equations: Formulation of ODEs. Terminology. Inputs & outputs. First order systems. Response to simple forcing functions. Definition of the Laplace transform. Transforms of functions. Inverse transforms. Properties of transforms. Solution of ODEs using Laplace transforms. Transfer functions for lead, lag, integrator, delay and controller. Concept of transfer functions and block diagrams. Principle of superposition. Introduction to Control System Toolbox.
Numerical methods: Concept of iteration. Numerical stability and convergence. Accuracy and speed considerations. Convergence methods, eg interval halving and Newton-Raphson. Numerical integration of ODEs. Explicit and implicit methods, eg Euler and Runge-Kutta. Choice of step length and run time. Initial and boundary conditions.
Statistics: Graphical methods, eg cumulative frequency diagrams. Summary statistics, eg mean, variance, standard deviation, etc and estimation thereof. Multivariate statistics: covariance and covariance matrix. Correlation coefficients. Probability distributions, eg normal, bi-nomial and Poisson. Significance tests, eg t-test. Auto & cross correlation functions. Simple linear regression models. Least squares method of finding regression coefficients.