Lecture times: weeks 11–15
Monday 11am Management School LT 11
Monday 2pm Fylde Lecture 2
Tuesday 10 Management School LT 11
Tuesday 1pm Fylde Lecture 3
Workshop times: weeks 11–15
Tuesday 2pm SAT A001a Old Engineering (James Groves)
Tuesday 3pm SAT A001a Old Engineering (Gordon Blower)
Tuesday 4pm SAT A001a Old Engineering (Harjit Hullait)
Tuesday 5pm SAT A001a Old Engineering (Em Poulter)
Assessment: Examination 70 percent, project 20 percent and weekly coursework 10 percent. The examination has section A worth 40 percent, with all questions compulsory; and section B, worth 30 percent, with a choice of one questions from two. The coursework is based upon four assignments, due in weeks 12,13,14,15. Project topics are included with the exercises.
Coursework submission time: Monday 4pm, weeks 12,13,14,15.
Office hours: Monday 9:00 and 12noon, B8 Fylde
Lecturer: Gordon Blower, email: g.blower@lancaster.ac.uk
Tutors: Dr James Groves, Harjit Hullat, Em Poulter
Aims
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics courses and provide opportunities for further study. Linear systems is engineering mathematics. In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input. Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute. The course then describes feedback control for linear systems. The main result describes the linear controllers that stabilize a (A,B,C,D) system. The module covers graphical methods such as Nyquist and Bode plots.
Bibliography: Chi-Tsong Chen, Linear System Theory and Design, Fourth Edition, Oxford University Press, 2012 Learning outcomes
Subject Specific On successful completion of this module, students will be able to:
Derive linear differential equations from block diagrams;
Express, manipulate and solve linear differential equations in matrix form;
Calculate the transfer function of a (A,B,C,D) system, by hand and by appropriate computing software;
Identify bounded-input bounded-output systems in terms of poles of the transfer function;
Compute the stabilizing controllers of a single input single output linear system.
General outcomes
On successful completion of this module, students will be able to:
Interpret correctly the concepts of stability, resonance, positive and negative feedback for a (A,B,C,D) linear system;
Interpret the conclusions of mathematical results about linear systems in simple applications.
Modern applications
Electrical powers systems can have many inputs (generators) and may outputs (consumers).
Heart rate monitors such as pacemakers are used to control irregular heart beats.
In electronic music, feedback loops are frequently used.
Timeline for control theory
1680: Papin described a self-regulating pressure cooker.
1772: Determinants considered by Laplace and Vandermonde.
1775: Watt’s steam engines with governors.
1785: Laplace transform discovered.
1858: Cayley–Hamilton theorem for matrices.
1866: Transatlantic telegraph cable installed.
1868: Maxwell’s paper on governors.
1880: Heaviside invented the coaxial cable, and started ‘operational calculus’.
1895: Routh and Hurwitz characterized stable polynomials.
1901: Marconi’s first transatlantic radio transmission.
1907: Thermionic triode valve invented.
1915: First telephone call from New York to San Francisco.
1920: Oil tanker J.A. Moffet sails with a nautical autopilot.
1927: Black invented the negative feedback amplifier.
1932: Nyquist’s paper on generation theory.
1936: Bode invented the self-adjusting amplifier, and produced his plots.
1947: Transistor invented.
1957: Sputnik 1 satellite.
1962: Telephone via Telstar satellite.