Modelling, dynamics and control by Anthony Rossiter

Section on responses of 1st order systems

This section is part of the chapter 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 1st order systems [ T dx/dt +x =Cu ] with an emphasis on constant inputs.
1. How do I characterise the behaviours and thus form analogies between different systems?
2. Time constant forms and their use in modelling and analysis.
Students should note that the time constant form is particular to 1st order systems, although some principles are still useful for higher order systems.

1. Free Response: Behaviour dependence on initial condition and with no input. Illustrates core role of parameter, the time constant T, in characterisation.
i. PDF notes on responses with no input.
ii. Video talk through on youtube.

2. Step response: Considers the behaviour for a step input, with zero initial conditions. Again demonstrates the core role of the parameter, the time constant T, as well as the role of the gain parameter C.
i. PDF notes on step response.
ii. Video talk through on youtube.

3. Step input and non-zero initial conditions: Builds on the insight of the first two notes and using superposition again demonstrates the significance of parameters T,C, in characterising behaviour.
i. PDF notes on complete first order response.
ii. Video talk through on youtube.

4. Using Laplace transforms: Illustrates how Laplace transforms can be used to derive the solutions given in the first 3 notes. This is a preparation for higher order responses where Laplace is more useful.
i. Using Laplace to find 1st order responses.
ii. Video talk through 1.
iii. Video talk through 2.

5. Sketching 1st order responses: Given the characterisation in temrs 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.
ii. PDF notes on sketching 1st order responses.
ii. Video talk on youtube.

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 and indeed this can be used to guide parameter design.
i. Parameter dependence of responses pdf file.
ii. Video talk on youtube.

7. Modelling from a 1st order step response: Given the characterisation in terms of parameters T,C, x(0), it is possible to form links between behaviour and parameters and, assuming some knowledge of the underlying model structure, determine the model parameters from the response.
i. Modelling from responses pdf file .
ii. Video talk through.

8. Using MATLAB: Gives a quick overview of using dsolve.m in MATLAB code to solve 1st order models and plot the behaviour. Coding requirements are typically only 2-3 lines.
i. Use of MATLAB dsolve.m for 1st order systems.
ii. Video talk through on dsolve.m .
iii. Use of MATLAB with Laplace transforms.


Use the main MATLAB page to see all resources. Topic relevant resources are referenced directly as used in the sections.
To use these files in MATLAB, save the files into your own folder and then open, for example from the command window write: "open filename" .
More precise instructions for function files are provided where required.