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 second order systems [ B d²x/dt² + C dx/dt + D x = K u ] with an emphasis on constant inputs. It looks at:

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

• Normalised forms involving damping and natural frequency and their use in modelling and analysis.

• Divergent responses (unstable systems) are not discussed in this section.

Students should note that the normalised form is particular to under damped second order systems, although some principles are still useful for higher order systems.

1. Over damped systems

Over damped second order systems have two stable real poles. It is called this because a mass-spring-damper system with a large damping leads to this scenario.

Over damped systems video and notes (PDF, 167KB).

Very over damped systems video and notes (PDF, 261KB).

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.

A talk through video is on YouTube. View the notes (PDF, 228KB).

3. Laplace transforms

This section uses Laplace methods to solve both over and under damped systems, without normalised forms.

Laplace and over damped solution video and notes (PDF, 296KB).

Laplace and under damped solution video and notes (PDF, 287KB).

4. Normalised forms

This section gives the definition of a normalised form for a second order system in terms of damping ratio and natural frequency. It shows how solutions are characterised in terms of the normalised parameters.

Normalised forms video and notes (PDF, 213KB).

Oscillation, overshoot and decay rates video and notes (PDF, 578KB).

5. Analysis and sketching with normalised forms

Normalised forms give useful insight and generalisations that facilitate quick sketching and characterisation of behaviour.

Steady-state behaviour video and notes (PDF, 258KB).

Convergence rate dependence on damping video and notes (PDF, 300KB).

Sketching video and notes (PDF, 255KB).

6. Using MATLAB

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

Use of MATLAB for higher order systems notes (PDF, 265KB).

Using Laplace transforms and MATLAB notes (PDF, 315KB).