Modelling and control by Anthony Rossiter

# OFFSET IN SIMPLE FEEDBACK LOOPS

This is a section in the chapter on introduction to the importance and impact of feedback. 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 section 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.

## Offset 1 - steady-state for signals

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The first video looks at definitions of steady-state values and the final value theorem (or FVT).

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

#### Quick test question

The following can be used to find the steady-state value of a signal.
A. The final value theorem can always be used.
B. The final value theorem can sometimes be used.
C. Just let s=0 in the Laplace transform of the signal.
D. None of the above.

## Offset 2 - steady-state gain in the closed-loop

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The second video looks at the definition of steady-state system gain and how this can be used to determine steady-state values and offset for a closed-loop.

• A summary of key facts and derivations is given in the pdf file.
• Quick tutorial sheet on basics with outline answers.
• A talk through video is on youtube.

#### Quick test question

The following can always be used to find the steady-state gain of a transfer function.
A. The final value theorem is used on the transfer function.
B. The final value theorem is used on the Laplace transform of the system output.
C. Put s=0 into the transfer function.
D. None of the above.

## Offset 3 - steady-state offset in the closed-loop (introduction)

Assume a system is placed within a feedback loop. These videos focus on whether the output from that loop with track the desired output asymptotically. The third video looks at closed-loop offsets.

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

## Offset 4 - steady-state offset in the closed-loop analysis

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The fourth video continues the focus on steady-state offset and discusses the role of an integrator.

## Offset 5 - steady-state offset with input disturbances

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The fifth video looks at the impact of input disturbances on steady-state values and offsets.

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

## Offset 6 - steady-state offset with output disturbances

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The sixth video looks at the impact of output disturbances on steady-state values and offsets.

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

#### Quick test question

To find the impact of output disturbances.
A. Use superposition.
B. First find the overall output, do inverse Laplace and analyse the result.
C. Check whether the system has an integrator.
D. None of the above.

## Offset 7 - steady-state offset with a sensor

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop with track the desired output asymptotically. The seventh video looks at the impact of an output sensor on steady-state values and offsets.

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

#### Quick test question

A sensor will affect the steady-state offset if:
A. The dynamics are significant.
B. The steady-state gain is not unity.
C. Always.
D. None of the above.

## Offset 8 - steady-state offset to a ramp

Assume a system is placed within a feedback loop.  These videos focus on whether the output from that loop will track the desired output asymptotically. The eighth video looks at offsets for ramp targets.

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

## Offset 9 - tutorial on steady-state signals and system gain

A tutorial sheet to reinforce videos in this series. This tutorial gives questions on how to find asymptotic values of signals which are the output of a system with a know input. Also introduces questions on system steady-state gain. Provides worked solutions for each question.

## Offset 10 - tutorial on steady-state offset to a step

A tutorial sheet to reinforce videos in this series. This tutorial gives questions on how to find asymptotic values of system outputs from a feedback loop, where the loop input is a constant. Provides worked solutions for each question.

## Offset 11 - tutorial on steady-state offset to input and output disturbances

A tutorial sheet to reinforce videos in this series. This tutorial gives questions on how to find asymptotic impact on the system output of a disturbance signal entering the loop either on the input or output. Provides worked solutions for each question.

## Offset 12 - tutorial on steady-state offset to a ramp

A tutorial sheet to reinforce videos in this series. This tutorial gives questions on how to find asymptotic errors between loop outputs and inputs when the loop input is a ramp. Provides worked solutions for each question.

## Offset 13 - Matlab GUIs: satellite tracking with a radar and lag compensator design

This GUI is focussed on the process of tracking a satellite, or other flying object, with a radar. The GUI assumes given radar dynamics and that the tracking problem is equivalent to following a ramp target. Lag compensation is used to increase the low frequency gain. Students can test the efficacy of different uplifts in low frequency gain.

• A short video introduction to the GUI is available here.
• A summary of the GUI and the underlying context is given in the pdf file.
• The GUI (a MATLAB p-file and fig file) are available here (both needed).  