Stability is a fundamental concept in control systems. It ensures that a system behaves predictably and does not produce unbounded outputs in response to bounded inputs. Let’s break down stability analysis into simple, manageable steps.
What is Stability?
A control system is stable if its output remains finite and bounded for a finite input over time. In simpler terms:
- Stable: Output settles to a steady value or oscillates within a fixed range.
- Unstable: Output grows without limit (diverges).
Key Methods to Analyze Stability
1. Pole-Zero Analysis (S-Domain Approach)
- Poles are the roots of the denominator of the transfer function.
- The location of poles determines system stability:
- Stable: Poles are in the left-half of the s-plane (negative real part).
- Unstable: Poles are in the right-half of the s-plane (positive real part).
- Marginally Stable: Poles are on the imaginary axis (purely imaginary roots).
Steps:
- Find the transfer function
\(E=mc^2\) - Solve D(s)=0 to find the poles.
- Analyze the location of the poles in the s-plane.
2. Routh-Hurwitz Criterion
This method checks stability without explicitly finding pole locations.
Steps:
- Write the characteristic equation
- Construct the Routh array:
- The first row contains coefficients an .
- The second row contains
- Use a recursive formula to fill the rest of the array: determinant of the blockleading coefficient of the previous row