• Comprehend · Control Engineering

    The Role of Feedback in Error Generation

    Moving from a verbal description of control systems towards the mathematical and visual representation of reference, measurement, feedback and error.

    Contents

    Key Learning Outcomes

    By the end of this section, you should be able to:

    01

    Define the reference, measured output, feedback and error within a control system.

    02

    Calculate the error between a reference and measured output using e = r − y.

    03

    Interpret a basic feedback control block diagram.

    04

    Explain how feedback enables the measured output to be compared with the desired reference.

    The Role of Feedback in Error Generation

    Building on the previous section, the focus now moves from a verbal description of control systems towards their mathematical and visual representation using block diagrams.

    At a fundamental level, feedback control requires knowledge of both what we want the system to achieve and what the system is actually doing.

    The desired value is referred to as the reference (or setpoint) and is denoted by r. For example, in the indoor temperature-control system introduced previously, the reference temperature may be set to:

    r = 25°C

    The actual behaviour of the process is determined using a measurement device. The resulting measured value is referred to as the measured output and is denoted by y.

    Through feedback, the measured output is returned and compared with the reference. The difference between the desired reference and the measured output is known as the error, denoted by e:

    e = r − y (1)

    Conceptually, the error represents the difference between what is desired and what is actually occurring.

    At this stage, the emphasis is on understanding how feedback is used to generate this error. How the error is subsequently used to determine an appropriate control action will be introduced later.

    Block Diagram Form

    The components and signals within a control system are commonly represented using block diagrams. A block diagram provides an abstract visual representation of a system, allowing us to focus on how information flows and how signals interact without initially considering all of the underlying physical detail.

    The key elements introduced here are:

    B

    Blocks

    Represent components, processes or operations within the system.

    →

    Arrows

    Indicate the direction of signal or information flow.

    Σ

    Summing junctions

    Combine signals through addition or subtraction.

    In Figure 2, the measurement device is represented by a block, while the reference, measured output and error are represented as signals. The arrows show the direction in which these signals flow.

    The summing junction compares the reference with the measured output and generates the error, as given by Equation (1):

    e = r − y

    The positive (+) and negative (−) signs at the summing junction indicate that the measured output is subtracted from the reference.

    For this comparison to be meaningful, the reference and measured output must represent the same physical quantity and use compatible units or scaling. For example, a temperature reference expressed in degrees Celsius must be compared with a measured output representing the same temperature quantity.

    Figure 3 applies the same structure to the indoor temperature-control example, demonstrating how the general representation in Figure 2 can be related to a physical engineering system.

    Block diagram form for error generation A reference r enters the positive side of a summing junction. The measured output y is returned through feedback to the negative side. The resulting signal is the error e. A measurement device sits within the process environment. reference, r + − error, e Measurement device Process environment measured output, y feedback
    Figure 2: Block Diagram Form for Error Generation.
    Error generation for an indoor temperature-control system A 25 degree Celsius reference is compared with a measured output of 20 degrees Celsius from a temperature sensor. Negative feedback produces an error of 5 degrees Celsius. reference, r 25°C + − error, e 5°C e = r − y e = 25 − 20 = 5 Temperature sensor Building (inside) measured output, y 20°C feedback
    Figure 3: Block Diagram Form for Error Generation for the Temperature Control System.

    Exercises

    1

    Signal compatibility

    A temperature sensor produces an electrical output of 2.0 V when the measured room temperature is 20°C. The temperature reference is:

    r = 25°C

    A student proposes calculating the error as:

    e = 25 − 2 = 23
    1. Explain why this calculation is not meaningful.
    2. What must happen before the reference and measured output can be compared?
    3. What physical quantity should r and y represent at the summing junction?
    4. Why is this important when constructing a block diagram?
    2

    Construct and interpret a feedback block diagram

    An autonomous vehicle is required to travel at a reference speed of 15 m/s. A wheel-speed sensor measures the actual vehicle speed as 12.5 m/s.

    Using the block-diagram conventions introduced in this section:

    1. Draw a block diagram showing:
      • the reference r;
      • a summing junction;
      • the error e;
      • the process;
      • the measured output y;
      • the measurement device; and
      • the feedback path.
    2. Clearly indicate the positive and negative inputs to the summing junction.
    3. Calculate the resulting error.
    4. Explain, using the block diagram, how information returns from the process to the summing junction.
    5. Identify which elements are signals and which are system components.
    3

    Apply the framework to an engineering system

    Choose one of the following:

    • autonomous vehicle;
    • unmanned aerial vehicle;
    • industrial robotic arm;
    • surgical robot; or
    • autonomous delivery vehicle.

    For your chosen application:

    1. Identify a suitable controlled variable.
    2. Define an appropriate reference r.
    3. Identify how the corresponding measured output y could be obtained.
    4. State the physical units of r, y and e.
    5. Propose numerical values for r and y, and calculate e.
    6. Sketch the corresponding feedback/error-generation block diagram.
    7. Explain what a positive, negative and zero error would mean physically for your chosen system.

    Concluding Remarks

    Feedback provides a control system with information about what the process is actually doing.

    By feeding the measured output y back and comparing it with the desired reference r, an error can be generated:

    e = r − y

    The error provides a mathematical representation of the difference between desired behaviour and measured behaviour.

    We have also introduced block diagrams as a graphical method for representing control-system components and signal flow. Blocks represent components or operations, arrows indicate the direction of signal flow, and summing junctions allow signals to be combined or compared.

    At this stage, the key idea is therefore:

    reference→ measurement→ feedback→ comparison→ error

    The next stage is to consider how this error can be used to determine an appropriate control action and influence the behaviour of the process.

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