# Chapter three

Introduction to feedback

## Section two: Basic Analysis of Feedback Loops

This chapter is on the theme of linear feedback control, for example with G(s) representing a system, M(s) a compensator and d an input disturbance signal:

This section focuses on an introduction to analysis of feedback loops, for example:

How does the closed-loop behave and how might I modify the behaviour?

How is behaviour quantified and what tools will help with this?

Students can use the behaviour section to remind themselves of core concepts. Also, block diagram developments with video support are covered more carefully in the introduction to block diagrams section.

### 1. What is a transfer function?

This overlaps with developments in the modelling chapter and is repeated for convenience. Students need to be confident with block diagram representations linking signals and systems.

What is a transfer function (PDF, 493 KB)

Systems in series with block diagrams (PDF, 472 KB)

Presented more slowly in block diagrams section.

### 2. Feedback loop expressions

What is the block diagram representation of a feedback loop and hence how do I derive closed-loop transfer function expressions to represent the behaviour?

Introduction to closed-loop transfer functions (PDF, 723 KB)

Numerical examples (PDF, 557 KB)

Make computation quicker and easier (PDF, 702 KB)

Presented more slowly in block diagrams section.

### 3. Steady-state gain

A core property of a closed-loop is the steady-state output, for a given input; this is inferred from the steady-state gain. The concept of gain was also discussed in the modelling chapter.

Steady-state gain and tracking errors (PDF, 509 KB)

### 4. Steady-state offset

A core property of a feedback loop is how well it tracks targets asymptotically; this is known as steady-state offset which ideally should be zero.

Steady-state offset video and notes (PDF, 506 KB).

Impact of an integrator on offset video and notes (PDF, 528 KB).

### 5. Closed-loop poles

A second core property of a feedback loop is the poles as these indicate the dynamical behaviour. These notes show how poles depend upon compensator gain.

First order examples (PDF, 493 KB)

Second order examples (PDF, 592 KB)

High order examples (PDF, 587 KB) need more advanced analysis and/or software tools.

### 6. Proportional compensator design

In general the user wants to specify a closed-loop with given properties such as meeting offset and convergence time requirements. These notes show how simple proportional gain choices can be made for low order models.

Achieving desired gain characteristics (PDF, 733 KB)

Achieving desired closed-loop pole (PDF, 704 KB)

Achieving desired closed-loop time constant (PDF, 694 KB)

### 7. Advanced offset scenarios

Some notes considering how to define and compute offset for more realistic scenarios. Content likely to be beyond requirements of an introductory course.

Offset with input disturbances video and notes (PDF, 528 KB).

Offset with output disturbances video and notes (PDF, 465 KB).

Impact of a sensor on offset video and notes (PDF, 511 KB).

Offset to ramps video and notes (PDF, 463 KB).

### 8. Case study

A case study on a radar dish tracking problem - involves tracking a ramp and a MATLAB GUI for illustrations.

Case study - radar dish tracking (PDF, 594 KB)

Satellite tracking with a radar video

GUI files are available here (both needed):

MATLAB p-file (19 KB)

fig file (11 KB)

### 9. MATLAB tools

Summarises some basic MATLAB commands that can be used to help in the analysis of simple feedback systems.

MATLAB commands for feedback (PDF, 522 KB)