# Chapter seven

State space methods

This chapter gives a summary of key methods and concepts around state space models.

The content is primarily targeted at students doing a single course in state space methods. Therefore, the content does not dwell on some fine details which would be covered in a second course or in research applications.

An example is non-simple Jordan forms, and another is finding approximate state space models by linearisation of first principles models.

Once the principles are understood clearly, students are encouraged to use tools like MATLAB for some of the numerical aspects. Manipulation of state space models is not usually a paper and pen exercise.

The focus of these sections is on state space analysis methods. This begins with definitions and origins of state space models alongside a discussion of their equivalences with transfer function models.

This is followed by analysis of the associated system behaviours and links to the state space model parameters. The final sections focuses on control design and concepts such as controllability, observability and control design methods.

A number of MATLAB files are available to support the learning of this topic - see section 6.13 for more details.

## Sections in chapter seven

### Section one: State space model definitions

An introduction to state space models including basic definitions, origins and equivalences.

### Section two: State space behaviours

A focus on the behaviours associated to state space models.

### Section three: State space observability and controllability

A focus 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.

### Section four: State space feedback control and observers

This section first focuses on state feedback design; starting with basic pole-placement approaches and then a brief mention of optimal control. However, state space control laws are often based on state information and this information may not be readily available.

Consequently and equally important topic is observer design, where the role of the observer is to estimate the state information for use in the feedback.