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# MSc Advanced Control and Systems Engineering / Course details

Year of entry: 2020

## Course unit details:State-Space and Multivariable Control

Unit code EEEN60109 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 1 Department of Electrical & Electronic Engineering No

### Overview

Brief Description of the Unit:

• Introduction to state variables, order and state equations
• State space modelling of dynamical systems - applications and examples
• Controllable canonical form and observer canonical form
• State space realisation of transfer functions
• State transition matrix, matrix exponential and time response of state space models
• Laplace transform of state equations and matrix transfer functions for MIMO systems
• Use of eigenvalues and eigenvectors to diagonalise a state space model
• Modal form, modal behaviour and system dynamics
• Jacobi linearisation on nonlinear systems
• Controllability and observability and links to modal analysis
• Minimal representations
• State feedback control design
• Observer design
• Output feedback design and separation principle
• Discrete-time state-space counterparts to continuous-time
• State space and transfer function matrix representations of multivariable linear systems
• Realisation of a MIMO transfer function matrix: minimal representation
• Multivariable poles and zeros; directionality
• Diagonal controllers and decoupling
• Internal model control structure design for both single variable and multivariable control
• Inner-outer factorisations
• Design example

### Aims

The course unit aims to:·

• Give students a sound understanding of control systems analysis and synthesis using state space techniques
• Introduce multivariable control systems
• Give students an introduction to internal model control design.

### Learning outcomes

Students will be able to:

Knowledge and Understanding:

• Describe dynamical systems in mathematical terms using state space representations;
• Define terms such as eigenvalue, eigenvector, system state matrix, system observability and controllability
• Design controllers using state space pole assignment and/or observers
• Understand design limitations due to uncontrollable and unobservable states
• Demonstrate knowledge of multivariable control and understand differences between SISO systems and MIMO systems.
• Understand internal model control (IMC)

Intellectual Skills

• Assess system stability, time response, controllability and observability from state space models
• Derive linearisation of nonlinear systems
• Derive expressions for transfer function from state space representation and state space representation from differential equations and transfer functions
• Derive canonical forms of the system from the state space representation and vice versa applying basis changes and transformations
• Derive minimal realizations of a MIMO transfer function.
• Derive factorizations for IMC

Practical skills

• Analyse continuous linear system's time response, stability, controllability and observability
• Design controller and observer in the state space representation
• Design IMC controller

Transferable skills and personal qualities

• Development of a critical attitude in the assessment of analytical results
• Encouragement of physical interpretation where possible
• Development of capability in problem solving using mathematical models
• Understand the relation between computer-aided-design and practical implementation.

### Teaching and learning methods

 Number of Hours Allocated to: Lectures Tutorials Practical Work/ Laboratory Private Study Total 30 3 6 111 150

### Knowledge and understanding

• Describe dynamical systems in mathematical terms using state space representations;
• Define terms such as eigenvalue, eigenvector, system state matrix, system observability and controllability
• Design controllers using state space pole assignment and/or observers
• Understand design limitations due to uncontrollable and unobservable states
• Demonstrate knowledge of multivariable control and understand differences between SISO systems and MIMO systems.
• Understand internal model control (IMC)

### Intellectual skills

• Assess system stability, time response, controllability and observability from state space models
• Derive linearisation of nonlinear systems
• Derive expressions for transfer function from state space representation and state space representation from differential equations and transfer functions
• Derive canonical forms of the system from the state space representation and vice versa applying basis changes and transformations
• Derive minimal realizations of a MIMO transfer function.
• Derive factorizations for IMC

### Practical skills

• Analyse continuous linear system's time response, stability, controllability and observability
• Design controller and observer in the state space representation
• Design IMC controller

### Transferable skills and personal qualities

• Development of a critical attitude in the assessment of analytical results
• Encouragement of physical interpretation where possible
• Development of capability in problem solving using mathematical models
• Understand the relation between computer-aided-design and practical implementation.

### Assessment methods

Method Weight
Other 80%
Written exam 20%

Written Examination:

Length of examination: 3 hours

Calculators are permitted, providing they cannot store text.

The written examination forms 80% of the total unit assessment

Coursework - Laboratories

The number of laboratories attended is 2

• Lab 1: 3 hours - 10% Short report
• Lab 2: 3 hours -  10% Short report

### Study hours

Scheduled activity hours
Lectures 30
Practical classes & workshops 6
Tutorials 3
Independent study hours
Independent study 111

### Teaching staff

Staff member Role
Alexander Lanzon Unit coordinator
Long Zhang Unit coordinator