Modelling and control by Anthony Rossiter

# STATE SPACE BEHAVIOURS

This is the 2nd section in the chapter on state space models, behaviours and control. The focus here is the behaviours associated to state space models. 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 behaviours 1 - introduction

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

## State space behaviours 2 - eigenvalues

Emphasises the link between exponential behaviours and the eigenvalues of the A matrix. Gives an alternative definition of the state transition matrix based on an eigenvalue/vector decomposition (considers distinct eigenvalues only). Useful for insight but not a paper and pen exercise.

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

## State space behaviours 3 - step response.

Shows how the step response can be derived either using Laplace transforms or a convolution integral. Also links the step response to the state transition matrix.

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

## State space behaviours 4 - eigenmodes

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. Shows how the decay along each eigenvector direction is linked explicitly to the corresponding eigenvalue. Does not discuss non-simple Jordan forms.MATLAB files are provided on main resource website for the figures in the numerical examples (phaseplane.m, phaseplane2.m, phaseplane3.m, phaseplane4.m)

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

## State space behaviours 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 2 states to make illustration easier and again viewers are recommended to use computer tools for analysis due to the tedium of pen and paper exercises. MATLAB files are provided on main resource website for the figures in the numerical examples (phaseplane5.m, phaseplane6.m).

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

## State space behaviours 6 - Cayley-Hamilton Theorem.

Statement only . No video.

• A summary of key facts is given in the pdf file.

## State space behaviours 7 - discrete systems

Covers the same content as videos 1-5 but now for discrete systems. Uses analogies with the continuous time observations so show that very similar results and insights apply. Consequently the overview is presented briefly and viewers are referred back to videos 1-5 for a slower derivation of core principles. MATLAB files are provided on main resource website for the figures in the numerical examples (phaseplane7.m).

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