Chapter two

This chapter is on the theme of linear models. For example:
A d³x/dt³ + B d²x/dt² + C dx/dt + D x = K u
where x(t) is the state, u(t) the input and A, B, C, D, K are model parameters.

This section focuses on understanding the behaviour of first order systems (T dx/dt + x = Cu) with an emphasis on constant inputs. It looks at the following principles:

• How do I characterise the behaviours and thus form analogies between different systems?

• Time constant forms and their use in modelling and analysis.

Students should note that the time constant form is particular to first order systems, although some principles are still useful for higher order systems.

1. Free response

This section looks at behaviour dependence on initial condition and with no input. It illustrates the core role of parameter, the time constant T, in characterisation.

A talk through video is on YouTube. View the notes (PDF, 254KB).

2. Step response

This section considers the behaviour for a step input, with zero initial conditions. It demonstrates the core role of the parameter, the time constant T, as well as the role of the gain parameter C.

A talk through video is on YouTube. View the notes (PDF, 302KB).

3. Step input and non-zero initial conditions

This section builds on the insight of the first two notes. Using superposition again, it demonstrates the significance of parameters T, C, in characterising behaviour.

A talk through video is on YouTube. View the notes (PDF, 608KB).

4. Using Laplace transforms

This section illustrates how Laplace transforms can be used to derive the solutions given in the first three notes. This is a preparation for higher order responses where Laplace is more useful.

Two talk through videos are on YouTube (video one, video two). View the notes (PDF, 297KB).

5. Sketching first order responses

Given the characterisation in terms of parameters T,C, x(0), a sketch of the response can be developed quickly and with minimal numerical computations. This re-emphasises the value of simple characterisation.

A talk through video is on YouTube. View the notes (PDF, 279KB).

6. Parameter dependence and modelling

Given the characterisation of responses in terms of parameters T, C, one can find strong links between model parameters and the behaviour. This can be used to guide parameter design.

A talk through video is on YouTube. View the notes (PDF, 205KB).

7. Modelling from a first order step response

Given the characterisation in terms of parameters T, C, x(0), it is possible to form links between behaviour and parameters. Assuming some knowledge of the underlying model structure, you can determine the model parameters from the response.

A talk through video is on YouTube. View the notes (PDF, 289KB).

8. Using MATLAB

This section gives a quick overview of using dsolve.m in MATLAB code to solve first order models and plot the behaviour. The coding requirements are typically only two to three lines.

A talk through video is on YouTube. View the notes on the use of MATLAB dsolve.m for first order systems (PDF, 497KB). View the notes on the use of MATLAB with Laplace transforms (PDF, 808KB).