Modelling and control by Anthony Rossiter

# STATE SPACE MODEL DEFINITIONS

This is the first section in the chapter on state space models, behaviours and control. The focus here is an introduction to state space models and thus basic definitions, origins and equivalences.

 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 1 - origins of the model

Introduces the concept of taking first principles models for systems and converting them into state space form. Explains the key assumption in a state space model is that one can write an equation for all the key dynamics in terms of their 1st order derivatives. Gives simple examples from 1st order engineering systems (mass-damper, resistor-capacitor, tank system).

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

## State space 2 - basic modelling

Extends the concept of taking first principles models for systems and converting them into state space form. Uses some 2nd and 3rd order examples (mass-spring-damper, RLC circuit, dc servo and pendulum) to demonstrate the process of constructing a state-space equivalent. Introduces the concept that state space descriptions for a given system are not unique as they depend on the selection and ordering of states.

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

## State space 3 - equivalent models for a given ODE.

In some cases a system model is supplied solely as an ODE rather than separate 1st principles equations. This resource shows how an equivalent state space model can be derived from an ODE. It is assumed that the results are given in canonical forms but again emphasis is made on the state space matrices not being unique as they depend on the selection and ordering of states.

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

## State space 4 - defining the system output

State space models have numerous states but the user may only be interested in a subset of these. The selected states are denoted as outputs; outputs are only those states you want to measure. This resource shows how the extraction of outputs from a state space model leads to another matrix definition or set of equations which are therefore part ofÂ  the state space model.

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

## State space 5 - equivalent transfer functions

It is useful to understand the relationship between state space models and transfer function models. This resource shows how one can form an equivalent transfer function model from a state space model. Several numerical examples are given but it is emphasised that the process is numerically intensive and thus in general should be performed on a computer and not by hand. Tools like MATLAB are demonstrated. It is also noted that the system poles correspond to the eigenvalues of the A matrix.

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

## State space 6 - equivalent models for a given transfer function.

This resource shows how one can form a state space model from a transfer function. The process is analogous to that used for ODEs but with the extra subtlety of allowing more complex numerators than a constant. The resource gives the controllable canonical form only as this can be constructed by inspection from the transfer function parameters.

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

## State space 7 - diagonal canonical form

Canonical forms can be useful for giving insight into behaviours and also for feedbacl design. A common canonical form is the diagonal one. This resource shows how such a form can be derived from the partial fraction expansion of a transfer function, or an eigenvalue/vector decomposition of the state transformation matrix. There us a brief discussion of the differences that arise with repeated roots. Minor typo around 11.14min where two residues are back to front.

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

## State space 8 - similarity transformations

State space models are not unique in that one can get models with equivalent input/output behaviour but very different state definitions. This resources introduces the concept of equivalence and shows how one can from state space models with a desirable structure. Readers are reminded that changing the model structure and changing the states may mean there is no longer a physical interpretation for the model states. Also looks at transformation possible using an eigenvalue/vector decomposition which exposes the underlying behaviours as being linked to the eigenvalues.

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

## State space 9 - use of MATLAB and numerical examples.

This resource shows how MATLAB can be used for much of the number crunching associated to state space analysis and manipulation as this is rather tedious on pen andpaper. Many useful built in functions are providded by MATLAB. Demonstrates the basic state space object available in MATLAB and how this object can be used in a number of other built in functions for typical analysis such as poles, responses and transformations.

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

## State space 10 - models form a difference equation.

Introduces state space models for systems described by difference equations. Conversions from z-transform transfer function to state space and vice versa. Uses analogies with continuous time conversions.

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

## State space 11 - tutorial and worked examples.

Goes through in real time the solution of questions on creation of state space models from ODEs and transfer functions. Also conversions back to transfer function from a state space model and use of a similarity transform.