# EEEN40010 Control Theory

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.

Module 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, 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 (specifically root locations of polynomials in the case of control theory). 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. 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 packages MATLAB or Python 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 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 resonances. 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.

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 or Python appears to make sense, since they are 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.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this subject the student will be able to:

1. Describe the desirable closed-loop dominant pole locations for given time and/or frequency domain specifications.

2. Design PID, lead/lag, pole-zero and/or linear state-feedback/output feedback controllers to meet given control specifications.

3. Determine the stability, controllability and/or observability of a given system.

4. Design a digital controller to meet given design specifications.

5. Partially identify a system from its frequency response data.

Indicative Module Content:

Module Plan:

Lec 1-2: Laplace transform method for solving linear, constant-coefficient ODEs. Transfer function.
Lec 3-4: Feedback, basic rules for control
Lec 5-6: Dominant pole theory
Lec 7-9: Dominant pole placement via root locus, PID controller
Lec 10-12: Bode plot, stability margins
Lec 13-17: System type, lead/lag controllers, basic loop-shaping
Lec 18-20: Kalman state-space description, linear state feedback, Luenberger observer
Lec 21: Controllability and Observability
Lec 22-24: Equivalent discrete-time plant, dominant pole theory for discrete-time systems
Lec 25: Dominant pole placement via root locus for discrete-time systems
Lec 26: Linear state feedback and observer for discrete-time systems
Lec 27: Basic loop shaping for discrete-time systems

Week 1: Practical Computer Laboratory: introduction to MATLAB/Python
Week 2: Practical Computer Laboratory: specialised MATLAB/Python for Control Theory
Week 5: Practical Computer Laboratory: dominant pole placement via root locus
Week 8: Practical Computer Laboratory: basic system identification, basic loop-shaping. linear state-feedback with observer
Week 11: Practical Computer Laboratory: digital control

Student Effort Hours:
Student Effort Type Hours
Lectures

28

Laboratories

10

Specified Learning Activities

17

Autonomous Student Learning

56

Total

111

Approaches to Teaching and Learning:
Project-based and problem-based continuous assessment. Emphasis on practical applications and utilisation o appropriate software tools.
Requirements, Exclusions and Recommendations
Learning Requirements:

Transform theory to level of Signals and Systems (EEEN30110) or equivalent.
Matrix Theory to advanced level 2 or better.
Differential Equations to advanced level 1 or better.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Report(s): Practical/Computer Laboratory takes place in week 5, report due in week 6. Week 5, Week 6 Graded No

33

No
Report(s): Practical/Computer Laboratory takes place in week 8, report due in week 9. Week 8, Week 9 Graded No

33

No
Report(s): Practical/Computer Laboratory takes place in week 11, report due in week 12. Week 11, Week 12 Graded No

33

No

Carry forward of passed components
No

Resit In Terminal Exam
Spring No