Category:Classical Control

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Classical Control
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1 Classical Controls

Open-loop systems lack feedback and therefore cannot compensate for errors. Prior to the advent of sensors and control theory, people provided the feedback of control for these systems. As the complexity and speed of machines grew people were no longer capable of providing adequate feedback.

Today we have a large variety of sensors for feedback allowing for fully automated control. With automated control comes the ability to make adjustments to a given system at rate well above any human's capacity (i.e. a 1 kHz update is reasonable for a computer and impossible for a person). Higher update rates allow a properly designed system to make corrections sooner. These corrections will be smaller since errors have had less time to accumulate.

Classical control techniques can be broken up into Frequency Domain techniques and Time Domain techniques. Frequency Domain techniques are a suite of analysis and design tools that include Root Locus, Pole Placement, Bode Plots, and Nyquist. State-Space techniques are the only part of Classical control that is done in the Time Domain.

Single-Input Single-Output (SISO) systems are the primary focus of Classical controls. Multiple-Input Multiple-Output (MIMO) systems can be controlled through Classical control techniques but to do so usually requires particular conditions be true. For example, an SISO controller can be used on an MIMO system if each input and output can be decoupled. In this case the system can be treated as a bunch of SISO systems and each controller can be designed wihtout consideration of the others.

1.1 Qualitative Example: Air Conditioning

In order to keep a room at a cool temperature on a hot day, the room needs air conditioning - a system to blow cool or cold air into the room. In order to keep the room temperature within some comfortable band a thermostat is used to measure the temperature and turn the cold air on or off.

This is a controller that simply turns the air conditioner on when the room is too hot or turns the air conditioner off when the room is too cold.

1.2 Qualitative Example: An Automobile's Cruise Control

Many systems today would not be possible without closed-loop control. For example, an automobile's cruise control. The driver pushes a button when the automobile achieves the desired speed. If this were open-loop the engine's throttle would remain stuck in that position. However, when going up a hill the automobile's speed would decrease. When going down a hill the speed would increase. Sometimes these speed changes would be neglible; sometimes those speed changes become dangerous.

For safety, the automobile must maintain the driver's specified speed within a small envelope. This requires feedback from the speedometer. But feedback from the speedometer alone isn't enough. The driver doesn't want the engine's throttle at 100% when the car's speed drops by 1%. The desired performance requires the throttle to be change smoothly. Smooth changes require throttle changes that proportional to the difference (error) between the desired speed and the automobile's current speed.

Achieving smooth adjustments to the automobile's speed while maintaining that speed within designed boundaries requires a controller. The designed controller can be thought of as a filter that shapes commands to the system plant in order to achieve teh desired system behavior.

2 The Big Picture



Goal
Design the pre-filter (W) and controller (K) on the basis of a nominal model LaTeX:  P_0 for the plant LaTeX:  P such that the feedback system exhibits the following properties:

  1. Stability: if the system is perturbed then the system will return to equilibrium
  2. Small tracking error
  • Good low frequency command following
  • Good low frequency disturbance attenuation
  • Good high frequency noise attenuation

Note
The stated goals must be achieved in the presence of the following sources of uncertainty

  • LaTeX: P_0 \ne P
  • LaTeX: H is not known exactly
  • LaTeX: d_i and LaTeX: d_o are not known exactly
  • LaTeX: n is not known exactly

3 Category Table of Contents


  1. Feedback
  2. Stability
    1. Characteristic Polynomial
    2. Routh's Algorithm
    3. Root Locus
    4. Bounded-Input Bounded-Output
    5. Frequency Response
      1. Nyquist
      2. Nichols
    6. Stability Margins
      1. Phase Margin
      2. Gain Margin
    7. Stability Robustness
      1. Robustness
      2. Metrics - How stable is the loop?
  3. Controller Design
    1. PD
    2. PI
    3. PID
      1. Ziegler-Nichols Tuning
      2. Reaction Curve Tuning
    4. Lead
    5. Lag
    6. Lead-Lag Compensators
    7. Polynomial Approach
    8. Root Locus
    9. Pole Assignment
    10. Controllers aren't Magic - The Real World and Limitations
      1. Bode Integral
      2. Sensors
        1. Noise
        2. Saturation
        3. Dynamics and Frequency Response
      3. Actuators
        1. Saturation
        2. Disturbances
        3. Model Error
        4. Time Delays
      4. Anti-Wind-up
        1. What is Wind-up?
        2. Anti-Wind-up Schemes