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    Step response specifications

    Step response specifications

    For a unit step reference (often ), common time-domain specifications are:

    • Rise time : time to go from 10% to 90% of the steady-state value .
    • Peak time : time at which reaches its maximum.
    • Percent overshoot %OS: .
    • Settling time (5%): time after which stays within of .
    • Steady-state error : .

    These specs are what we use to describe (and design for) a “good” transient response.

    Simple SISO reference tracking (constant reference, full observability)

    Consider the LTI system

    and suppose we want the output to track a constant reference , i.e., . A common controller is

    where:

    • choose so is asymptotically stable;
    • choose to make the steady-state output match the reference.

    If is constant and is stable, then at steady state

    so

    To get , pick

    assuming the scalar .

    This is the simpler SISO/full-observability tracking setup. For a more general treatment, see Reference tracking.