# Chapter seven

State space methods

## Section two: STATE SPACE BEHAVIOURS

This section focuses on the behaviours associated to state space models.

### 1. Introduction

This section introduces the concept of the state transition matrix using Laplace transforms to derive this. This matrix shows how the future behaviour of a state is linked to the initial condition, assuming there is no system input.

The required algebra is numerically intensive in general so once the principles are established, students are encouraged to use computer software tools such as MATLAB.

A talk through video is on YouTube. Read the notes (PDF, 350KB).

### 2. Eigenvalues

This resource emphasises the link between exponential behaviours and the eigenvalues of the A matrix. It gives an alternative definition of the state transition matrix based on an eigenvalue/vector decomposition (considers distinct eigenvalues only). It is useful for insight, but not a paper and pen exercise.

A talk through video is on YouTube. Read the notes (PDF, 352KB).

### 3. Step response

This resource shows how the step response can be derived either using Laplace transforms or a convolution integral. It also links the step response to the state transition matrix.

A talk through video is on YouTube. Read the notes (PDF, 371KB).

### 4. Eigenmodes

This resource introduces the concept of the phase plane, so the behaviour of the state over time and how each component of this is linked to the eigenvalue/vector decomposition.

It shows how the decay along each eigenvector direction is linked explicitly to the corresponding eigenvalue. It does not discuss non-simple Jordan forms.

MATLAB files are provided for the figures in the numerical examples. (Phaseplane.m, phaseplane2.m, phaseplane3.m, phaseplane4.m)

A talk through video is on YouTube. Read the notes (PDF, 462KB).

### 5. Oscillatory modes

Where a system has complex poles or equivalently eigenvalues, then the response oscillates. In the phase plane this is observed as a spiralling around the origin.

This resource shows how the eigenvalue/vector decomposition exposes this spiralling in an explicit manner. Concepts are demonstrated on examples with two states to make illustration easier.

Viewers are recommended to use computer tools for analysis due to the tedium of pen and paper exercises. MATLAB files are provided for the figures in the numerical examples (phaseplane5.m, phaseplane6.m).

A talk through video is on YouTube. Read the notes (PDF, 422KB).

### 6. Cayley-Hamilton Theorem

Read the notes (PDF, 283KB).

### 7. Discrete systems

This covers the same content as the first five videos in this section, but now for discrete systems. It uses analogies with the continuous time observations to show that very similar results and insights apply.

Consequently, the overview is presented briefly and viewers are referred back to the first five videos for a slower derivation of core principles.

MATLAB files are provided for the figures in the numerical examples (phaseplane7.m).

A talk through video is on YouTube. Read the notes (PDF, 440KB).