Feedback Control

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What is control engineering?

e.g. Automated highway

Objective:

Assumptions:

Solution 1: Open loop

Solution 2: Closed Loop

e.g. Web server

(See 1.4.1 in the notes)

Control objectives

  1. Don't let the queue length get too large (buffer may overflow and requests will be lost)
  2. Don't let queue length get to zero (it would be a waste of resources to have the server idle)

Consequently, we want to keep the queue length at some known value \(r(t)\).

The difficulty is that the service rate is not known, as it depends on many things (e.g. number of clients). Because of this, we will model it as a disturbance \(d(t)\).

We must decide on the request rate \(u(t)\) based on \(r(t)\) and \(y(t)\) (queue length).

Definition

Control engineering attempts to change the behaviour of a system (called a "plant") in a useful way despite the presence of external influences ("disturbances") and despite model uncertainty.

We change the behaviour of the plant by connecting it to another system ("controller"). Feedback is the most powerful interconncetion strategy.

Control Cycle

  1. Sense the operation of a system
  2. Compare against a desired behaviour
  3. Compute a correctibe action informed bu a model of the sustem's respnse to external stimuli
  4. Actuate the sustem to affect the desired change

The sense, compute, actuate cycle loops.

Design Cycle

  1. Study the system to be controlled, decide on sensors and actuators. Sensors change what information is at your disposal, and actuators represent choices you can make in response to the inputs.
  2. Model the resulting system
  3. Simplify model if necessary
  4. Analyze the resulting system
  5. Determine specifications: stability, good steady-state behaviour, robustness, good transient performance
  6. Decide on type of controller
  7. Design the controller
  8. Simulate (usually using Matlab)
  9. Return to step 1 if neecessary
  10. Implement controller

e.g. follower is \(u(t) = -K_p (r(kT)-y(kT)), \quad kT \le t \lt (k+1)T\)

Analog to digital:

Digital to analog: