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ECMM141 - Multivariable State-Space Control (2019)

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MODULE TITLEMultivariable State-Space Control CREDIT VALUE15
MODULE CODEECMM141 MODULE CONVENER Christopher Edwards (Coordinator)
Number of Students Taking Module (anticipated) 0
DESCRIPTION - summary of the module content

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 ECM2105

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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:


The AHEP document can be viewed in full on the Engineering Council’s website, at


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.

SYLLABUS PLAN - summary of the structure and academic content of the module
  • state-space modelling
  • transfer functions -> state-space and 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
  • the Lyapunov equation
  • LQR optimal controller design;
  • Lur’e systems (Absolute stability and the Small Gain Theorem)
  • Controller robustness
  • Case studies
Scheduled Learning & Teaching Activities 27.00 Guided Independent Study 123.00 Placement / Study Abroad 0.00
Category Hours of study time Description
Scheduled learning & teaching activities 22 Lectures
Scheduled learning & teaching activities 5 Example Classes
Guided independent study 123 Private study, assessment and lecture preparation


FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable      


Coursework 20 Written Exams 80 Practical Exams 0
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Examination - Closed book 80 2 hours - January Exam All Provided on request
Coursework - individual assignement 20 12 hours All Written feedback and model solutions


DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All above Written Examination All August Ref/Def Period



If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by
examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals,
candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original
coursework mark.

INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading:




Web based and Electronic Resources:


Other Resources:


Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set K J Astrom and R M Murray Feedback Systems: An Introduction for Scientists and Engineers 1st Princeton University Press 2008 978-0691135762 [Library]
ORIGIN DATE Tuesday 10 July 2018 LAST REVISION DATE Tuesday 10 July 2018
KEY WORDS SEARCH Control engineering; system dynamics, state-space, multivariable control.