Modelling, dynamics and control by Anthony Rossiter

Section on responses of 2nd 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 2nd order systems [ B d2x/dt2 + C dx/dt + D x = K u ] with an emphasis on constant inputs.
1. How do I characterise the behaviours and thus form analogies between different systems?
2. Normalised forms involving damping and natural frequency and their use in modelling and analysis.
3. Divergent responses (unstable systems) are not discussed in this section.
Students should note that the normalised form is particular to under-damped 2nd order systems, although some principles are still useful for higher order systems.

1. Over damped systems: Over damped 2nd order systems have two stable real poles. So called because a mass-spring-damper system with a large damping leads to this scenario.
i. Over damped systems.
ii. Very over damped systems.
iii. A talk through video on youtube
iv. Video on very over-damped.

2. Under damped systems: Under damped essentially means the system will oscillate to some extent. This is an introductory note giving a basic solution method, but without exploiting normalised forms.
i. Under damped response solution.
ii. Talk through video on youtube.

3. Laplace transforms: Using Laplace methods to solve both over and under damped systems, without normalised forms.
i. Laplace and over damped solution.
ii. Laplace and under damped solution.
iii. Video on overdamped with Laplace.
iv. Video on underdamped with Laplace.

4. Normalised forms: Gives definition of a normalised form for a 2nd order system in terms of damping ratio and natural frequency. Shows how solutions are characterised in terms of the normalised parameters.
i. Normalised form definition.
ii. Overshoot and decay rate.
iii. Video on normalised forms.
iv. Video on oscillation and overshoot.

5. Analysis and sketching with normalised forms: Normalised forms give useful insight and generalisations that facilitate quick sketching and characterisation of behaviour.
ii. Convergence rate dependence on damping.
iii. Sketching.