Multivariable State-Space Control - 2024 entry

Control theory is concerned with forcing the measured outputs of a system to follow a desired reference command, through the manipulation of certain input variables to the system. Ideally this tracking should be accomplished in the face of uncertain knowledge of the system and external disturbances. A powerful concept in this field is the notion of feedback – whereby the measured outputs of the system are compared in real-time with the reference signal, and the errors are processed to compute updates of the manipulated system inputs. Control systems are often a `hidden technology’ and exist all around us and are often a key aspect of many of the devices and products that that we rely upon. For example, control systems are a vital `component’ in hard disk drives, aircraft, communications devices, robots, chemical plants, space exploration, motors and drives, and land-vehicles. This module will build on ideas from ECM2105 which considered these ideas when posed in the framework of single-input single-output systems. Real engineering systems are often intrinsically multi-variable in nature, and a change to one input simultaneously affects many outputs e.g., aircraft. Whilst it is possible to try to decouple multi-input multi-output systems into several single-input single-output loops, a more elegant approach is to retain the multi-variable nature of the problem from the outset, and to consider a so-called state-space approach.

Pre-requisite ECM3018

The aim of this course is to introduce the concept of a state-space system, and how such a representation can be used for the systematic development of control laws for multivariable systems. The course will consider how to create state-space models from other representations (such as transfer functions and higher order differential equations), and the properties of state-space systems will be analysed. The key notion of controllability will be described and different paradigms will be introduced to provide systematic ways of designing feedback controls laws – including so-called observer based strategies.

This is a constituent module of one or more degree programmes which are accredited by a professional engineering institution under licence from the Engineering Council. The learning outcomes for this module have been mapped to the output standards required for an accredited programme, as listed in the current version of the Engineering Council’s ‘Accreditation of Higher Education Programmes’ document (AHEP-V3).

This module contributes to learning outcomes: SM2m, SM3m, SM4m, SM5m, EA2m, EA3m, EA5m, D3m, EP8m

A full list of the referenced outcomes is provided online: http://intranet.exeter.ac.uk/emps/subjects/engineering/accreditation/

The AHEP document can be viewed in full on the Engineering Council’s website, at http://www.engc.org.uk/

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge: SM2m, SM3m, SM4m, SM5m, D3m, EP8m

1. create state-space models of multivariable physical systems - including electrical and mechanical systems;

2. understand how a state-space system relates to a transfer function representation and be able to appreciate the importance of minimal realisations;

3. be familiar with the solution to a linear state-space representation, its dependence on initial conditions, and the concept of a state-transition matrix;

4. be able to perform modal decomposition of linear systems and relate the eigenvalues to poles of a transfer function;

5. be able to determine whether a given state-space system is stable;

6. determine whether a given system can be controlled and observed ;

7. design a feedback controller to achieve a desired response in the time domain for a given (single input) system;

8. understand the concept of an observer for a state-space system and determine the conditions (observability) when this can be achieved;

9. design observers to estimate the states of the dynamic system;

10. appreciate the duality of controller and observer design;

11. be familiar with the separation principle and its ramifications;

12. understand the principles of LQR optimal controller design;

13. implement state space feedback controllers and observers in Matlab;

14. see how state-space representations can be extended to include (static) nonlinearities in the feedback loop (Lur’e systems).

 

Discipline Specific Skills and Knowledge: EA2m, EA3m, EA5m

15. translate a physical problem into an appropriate mathematical system;

16. interpret solutions of these equations in physical terms.

 

Personal and Key Transferable/ Employment Skills and Knowledge: SM5m, EA5m, D3m

17. demonstrate enhanced ability to formulate and analyse real physical problems using a variety of tools of applied mathematics

18. show enhanced modelling, problem-solving and computing skills;

19. display knowledge of tools that are widely used in scientific research and modelling.

state-space modelling

transfer functions -> state-space

state-space -> transfer functions minimal realisations;

explicit solutions to the linear state-space equations

the state-transition matrix;

modal decomposition of linear systems

poles, eigenvalues and eigenvectors

bounded input stability

controllability

feedback controller design

observers for a state-space system

observability

observer design

the duality of controller and observer design;

the separation principles

co-prime factorization;

the Lyapunov equation

LQR optimal controller design;

Small Gain Theorem;

controller robustness (H-infinity);

Case studies

Reassessment will be by a single written exam only worth 100% of the module. For deferred candidates, the mark will be uncapped. For referred candidates, the mark will be capped at 50%.

Basic reading:

ELE: http://vle.exeter.ac.uk

Web based and Electronic Resources:

Other Resources:

Reading list for this module: