EEEN40010 Control Theory

Academic Year 2022/2023

Systems do not in general naturally behave in a manner which accords with the user’s wishes. Systems must in general be extended by the addition of a controller in order to force them to behave in an acceptable fashion. The controller may be a human (as in the case of the driver of a car for example), but the controller may also be a human-designed engineering system in its own right. In the latter case the controller is called an automatic controller. This module addresses the need for, the value of and the design of automatic controllers for some of the most common classes of engineering systems. Automatic controllers appear in more or less every engineering environment, from automotive/aerospace to biomedical equipment and including almost everything in between.

Outline: Mathematical background and problem statement. Continuous-time systems: models, block diagrams, open-loop and closed-loop control. Feedback: Performance, unit step response, effect of pole locations, dominant poles and model order reduction. PID control and effect on dominant pole location. Stability: Pole location and stability. Type: steady-state error, system type. Root locus: control system design using root locus. Bode Plots: frequency response, Bode plots and system identification, control system design using frequency response. Nyquist plot: Nyquist stability criterion, stability margins, relationship to dominant pole locations, lead/lag controller design. State-space: State space models, stability, controllability and observability, linear full-state feedback and pole-placement. Luenberger observer: Observer design using pole placement. Digital control: Zero order hold, equivalent discrete-time systems, digital PID and linear state feedback control design.

Learning Outcomes Rationale:

LO1: A critical component of control theory, and indeed of engineering in general, is the translation of the engineering problem (good system behaviour) into a purely mathematical problem (root locations of polynomials). An appropriate sense of what specifications are reasonable needs to be developed. Also the student needs to develop an ability to recognise when a purported solution is purely academic and when it is realistic, i.e. when it is forgiving of the idealisations and approximations made in the process of acquiring the model. The requirements of safety and, in particular, of the need to fail safe are paramount. The ability to identify the presence of non-minimum phase zeros and to appreciate the deleterious effect of such zeros on system performance is of great importance, both for control and for system modelling in general. The terminology of this field is widespread in engineering practice as are several of its key ideas, most notably negative feedback.

LO2: PID controllers remain dominant in the field. The proper design of such controllers, and of controllers in general, can improve safety, decrease wear, raise productivity and reduce energy consumption (although of course the latter phrase, although ubiquitous, is in error. We do not consume energy, since energy must be conserved, actually we raise entropy). Human experts always produce PID designs which comfortably outperform self-tuning controllers. Linear state-feedback can, in principal, achieve even more significant gains in performance. The benefit, both to economy and environment, cannot be overstated. The use in design of the package MATLAB is standard, both in academia and in industry.

LO3: The system property of stability is, in many cases, virtually indistinguishable from that of safety. The property of observability can be almost equally significant. An unobservable and marginally stable, or even unstable, state comprises for the designer a nightmare scenario, where the system suddenly and almost inexplicably switches from good behaviour to appalling and potentially dangerous behaviour. The knowledge that such hostages to fortune can exist is vital. It is extremely important when modelling to make full sure that all marginally stable, unobservable and/or uncontrollable states have been identified, just as it is vital to identify all resonances. That many practicing engineers fail to do so in no way undermines the importance of this task. Equally important is a proper understanding of how to interpret the standard stability criteria and an ability to determine when they comprise actual proof of stability and when they do not. The highest ethical responsibility is to design safe systems. It is unforgivable, although very common, to employ a stability test and to deduce stability when that stability test does not apply and when the system is in fact unstable.

LO4: It hardly needs to be stated that digital controllers are becoming more widespread. Accordingly the design of such controllers is a valuable skill. Offering increased versatility these controllers come with the usual slew of attendant benefits both to profit and to the environment. The use of MATLAB appears to make sense, since it is effective and nearly ubiquitous.

LO5: Towards the design of controllers we consider the Bode method of presenting the frequency response data and of identifying the system using this data. It is fairly obvious that system identification transcends all branches of engineering and therefore comprises an absolutely fundamental engineering skill. Much of the terminology of this field pervades engineering.

Module Plan:

Section 0: Introduction

Week 1:
Lectures: 3 lectures on mathematical background and historical introduction to core problems of control system design.
Laboratories: offering 1 of Introduction to MATLAB, general commands.

Section 1: Dominant-pole placement via root locus

Week 2:
Lectures: 3 lectures on negative feedback and time-domain specifications of desirable closed-loop performance.
Laboratories: offering 2 of Introduction to MATLAB, general commands.

Week 3:
Lectures: 3 lectures on desirable locations of dominant poles and definition of root locus.
Laboratories: offering 1 of Introduction to MATLAB, module-specific commands.

Week 4:
Lectures: 3 lectures on dominant pole-placement via root locus control design procedure.
Laboratories: offering 2 of Introduction to MATLAB, module-specific commands.

Section 2: Basic Loop Shaping

Week 5:
Lectures: 3 lectures on Bode plots, Nyquist plot and Nyquist criterion.
Minor Project (first session): offering 1: modelling and linearisation, root locus

Week 6:
Lectures: 3 lectures on stability margins and lead/lag controllers.
Minor Project (first session): offering 2: modelling and linearization, root locus.

Week 7:
Lectures: 3 lectures on control system design using frequency-domain methods (i.e. loop shaping).
Minor Project (second session): offering 1: root locus and frequency domain.

Section 3: State-Space Methods

Week 8:
Lectures: 3 lectures on Linear State feedback.
Minor Project (second session): offering 2: root locus and frequency domain.

Section 4: Digital Control

Week 9:
Lectures: 3 lectures on mathematical background to analysis and design of discrete-time systems.
Laboratories: offering 1 of Linear State Feedback.

Week 10:
Lectures: 3 lectures on dominant pole placement via root locus design for discrete-time systems.
Laboratories: offering 2 of Linear State Feedback.

Week 11:
Lectures: 3 lectures on linear state feedback for discrete-time systems.
Laboratories: offering 1 of Digital Control.

Week 12:
Lectures: 2 lectures on Minor Project support and review.
Laboratories: offering 2 of Digital Control.