# Chapter seven

State space methods

## Section three: STATE SPACE OBSERVABILITY AND CONTROLLABILITY

The focus of this section 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.

Therefore, a good understanding of these properties is important before moving to control design.

### 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. This is why they are summarised and illustrated in this resource.

A talk through video is on YouTube. Read the notes (PDF, 474KB).

### 2. Controllability

This resource looks at the concept of controllability. 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 talk through video is on YouTube. Read the notes (PDF, 670KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 514KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 535KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 463KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 647KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 596KB).

### 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 talk through video is on YouTube. Read the notes (PDF, 446KB).