EC-7.5 Poles, Zeros and Stability

Work through these

• The transfer function, and where it comes from

  Taking the Laplace transform of the impulse response gives a ratio of two polynomials for most systems of interest. That ratio holds the same information as the impulse response in a form you can factor.

  NPTEL: Principles of Signals and Systems · Course

• Poles and zeros, and the diagram they live on

  The roots of the denominator are the poles and the roots of the numerator are the zeros, and both are marked on the complex plane. Everything the system does can be argued from where those marks sit.

  NPTEL: Signals and Systems (IIT Kanpur) · Course

• The region of convergence, and why the same algebra can describe two systems

  A transfer function without its region of convergence is ambiguous, because the same expression can belong to a causal unstable system or to a non-causal stable one. Stating the region is part of stating the answer.

  MIT RES.6-007: Signals and Systems · Course

• Pole positions and stability in continuous time

  A causal continuous-time system is stable when every pole lies strictly to the left of the imaginary axis. A pole on the axis or to its right gives a response that never settles, which is the whole of the stability question in one sentence.

  NPTEL: Signals and Systems (IISER Bhopal) · Course

• Reading the shape of the response off the pole locations

  The distance of a pole from the imaginary axis sets how fast its contribution decays, and its distance along the axis sets how fast it oscillates. A pair close to the axis is the ringing you can see on an oscilloscope.

• The steady state, and what happens long after the input begins

  The response splits into a part that dies away and a part that persists, and only a stable system has the second part determined by the input alone. This split is what the time response work in control systems formalises.