Modelling and control by Anthony Rossiter

# SECOND ORDER SYSTEM RESPONSES

This is the first chapter in the theme on modelling and simulation of linear models. Use the left hand toolbar to access the other chapters and themes.

 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 chapter 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.

## Second order responses 1 - over damped systems

An introduction to the step response of a 2nd order system with two real poles. Derives an analytic solution assuming zero initial conditions and illustrates on numerical examples. Solutions tackled in the time domain.

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

#### Quick test question

Over damped 2nd order systems
A. Can often be approximated by a 1st order system.
B. May give signifcant overshoot during transients.
C. Are best handled with the normalised form for 2nd order systems.
D. None of the above.

## Second order responses 2 - very over damped systems

Adds insight to the 1st video on step responses of a 2nd order system with two real poles. This video considers the repercussions of the poles being widely spaced and illustrates with numerical examples. Solutions tackled in the time domain.

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

## Second order responses 3 - over damped systems with Laplace

Repeats the solutions of the first two videos, that is step responses of 2nd order systems with real poles, but this time using Laplace transform techniques rather than the time domain. Illustrates with numerical examples.

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

## Second order responses 4 - under damped systems

An introduction to the step response of a 2nd order system with complex poles and zero initial conditions. Derives solutions from first principles in the time domain and illustrates on numerical examples.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.
• A trial interactive EJSS file showing behaviour of a mass-damper and dependence on spring stiffness. Provided by authors at http://www.um.es/fem/EjsWiki.

## Second order responses 5 - under damped systems with Laplace

An introduction to the step response of a 2nd order system with complex poles and zero initial conditions using Laplace techniques. Includes illustrations on numerical examples.

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

## Second order responses 6 - standard model format

Define the standard form for 2nd order models using damping ratio and natural frequency. Includes numerical examples showing how to compute the key characteristics.

#### Quick test question

A standard format gives insight because:
A. The natural frequency also tells you the speed of oscillation.
B. The damping gives an indication of how quickly the oscillation stops.
C. The damping gives a good insight into the settling time .
D. None of the above.

## Second order responses 7 - steady-state dependence

Using the standard form for 2nd order models, what impact do the parameters and in particular the damping ratio have on the steady-state gain? Assume stable models only.

#### Quick test question

For a 2nd order system:
A. The steady-state solution depends on all the coefficients.
B. The steady-state solution depends on the coefficient of the first derivative.
C. The steady-state solution depends on the coefficients of the maximum derviative.
D. None of the above.

## Second order responses 8 - speed of response

Beginning from the standard form for 2nd order models using damping ratio and natural frequency, how does the speed of covergence depend upon the damping ratio? Includes numerical examples showing how to compute the speed of response and thus illustrate the key insights.

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

## Second order responses 9 - oscillation and overshoot

Beginning from the standard form for 2nd order models using damping ratio and natural frequency, considers how the frequency of oscillation and overshoot depend upon the damping ratio (assumed less than one). Uses zero initial conditions.

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

## Second order responses 10 - sketching under damped step responses

Beginning from the standard form for 2nd order models using damping ratio and natural frequency, shows how some simple and quick analytic computations can be used to produce an accurate sketch of the step response (for zero initial conditions).

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

## econd order responses 11 - tutorial on over damped step responses

Tutorial questions on the computation of step responses, for zero initial condition, for over damped 2nd order systems. Uses time domain and Laplace methods. Includes a few examples on the use of approximation for very over damped systems. Includes worked solutions.

• Talk through video is on youtube.

## Second order responses 12 - tutorial on under damped step responses

Tutorial on the computation of responses for under damped 2nd order systems, using Laplace and time domain methods. Includes worked solutions. .

• Talk through video is on youtube.

## Second order responses 13 - tutorial on normal forms

Questions on the standard form for 2nd order models and thus definitions of damping ratio and natural frequency. Uses ODEs and Laplace transforms. Includes worked solutions.

• Talk through video is on youtube.

## Second order responses 14 - tutorial on sketching under damped step responses

Beginning from the standard form for 2nd order models using damping ratio and natural frequency, gives questions on simple and quick analytic computations can be used to produce an accurate sketch of the step response (for zero initial conditions).Includes worked solutions.

• Talk through video is on youtube.