Modelling and control by Anthony Rossiter

# STATE SPACE OBSERVABILITY AND CONTROLLABILITY

This is the 3rd section in the chapter on state space models, behaviours and control. The focus here is on state space observability and controllability. If a system has poor observability or poor controllability, it may be difficult to ensure the desired behaviours and hence a good understanding of these properties is important before moving to control design.

 It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar.

This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## State space analysis 1 - concepts of stability

It is important to define the term stability before moving on to deeper analysis. Stability can be defined in various ways and hence these are summarised and illustrated here.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 2 - controllability

This resource looks at the concept of controllability which means, can a given position x(T) in the state plane be achieved from an arbitrary start point x(0)? Modal forms via eigenvector/eigenvalue decompositions are deployed to demonstrate key concepts and insights.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 3 - controllability matrix

This resources introduces a core test for controllability, that is the ability to take a state to a specified point in a specified time. Numerical examples are given and these are also supported by illustrations of how to use MATLAB tools. The concept of non-minimal forms is introduced.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 4 - controllability for discrete systems

This resource extends the concepts of controllability to discrete state space systems. It is demonstrated that the same tests and insights apply as used for continuous time systems.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 5 - controllability worked examples

This resource shows how the controllable canonical form and modal canonical forms are guaranteed controllable. Some discussion follows on minimal realisations.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 6 - observability.

Observability links to the potential for inferring a state correctly from a set of output measurements. Modal forms via eigenvector/eigenvalue decompositions are deployed to demonstrate key concepts and insights. It is shown that this concept is analogous to controllability. Numerical examples and MATLAB are used to demonstrate the results.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 7 – observability continued

This resource defines the so called observability matrix which is an easier test for observability. Numerical and MATLAB examples are given to demonstrate the usage.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space analysis 8 – detectability and stabilisability

This resource give a little more insight into the consequences and causes of losing observability and/or controllability. The important concepts of detectability and stabilisability are defined and illustrated with some numerical examples.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.