Basics of control system

 

Part A: Control Systems

1. Introduction to Control Systems

• Definition: A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops.

• Types:

• Open-loop system: Output is not fed back (e.g., electric toaster).

• Closed-loop system: Output is fed back to input for correction (e.g., automatic temperature control).

2. Block Diagrams & Reduction Techniques

• Block Diagram: Graphical representation of a system showing functional relationships.

• Reduction Techniques:

  • Series connection: Multiply transfer functions.

  • Parallel connection: Add transfer functions.

  • Feedback connection:

$$T(s)=\frac{G(s)}{1+G(s)H(s)}$$

• Significance: Simplifies complex systems into a single transfer function for analysis.

3. Signal Flow Graphs

• Definition: Alternative to block diagrams using nodes and branches.

• Mason’s Gain Formula:

$$T=\frac{\sum \text{(Gain of forward paths)}\cdot {\mathrm{\Delta }}_k}{\mathrm{\Delta }}$$

where $\mathrm{\Delta }$ = determinant of graph, ${\mathrm{<br>}}_{\mathrm{<br>}}$ = cofactor for path.

4. Time-Domain Analysis

• Standard Test Signals: Step, Ramp, Impulse.

• Performance Measures:

• Rise time

• Settling time

• Overshoot

• Steady-state error

5. Frequency-Domain Analysis

• Bode Plot: Logarithmic plot of magnitude and phase vs frequency.

• Nyquist Plot: Stability analysis using contour mapping.

• Gain Margin & Phase Margin: Indicators of system stability.

6. Stability Analysis

• Routh-Hurwitz Criterion: Determines stability without solving roots.

• Root Locus: Graphical method showing how poles move with varying gain

7. Controllers

• Types:

  • P (Proportional): Improves speed but leaves steady-state error.

  • PI (Proportional-Integral): Eliminates steady-state error.

  • PD (Proportional-Derivative): Improves transient response.

  • PID: Combines all three for optimal control.