Negative imaginary systems


Negative imaginary systems theory was introduced by Lanzon and Petersen in. A generalization of the theory was presented in
In the single-input single-output case, such systems are defined by considering the properties of the imaginary part of the frequency response G and require the system to have no poles in the right half plane and > 0 for all ω in. This means that a system is Negative imaginary if it is both stable and a nyquist plot will have a phase lag between for all ω > 0.

Negative Imaginary Definition

Source:
A square transfer function matrix is NI if the following conditions are satisfied:
  1. has no pole in.
  2. For all such that is not a pole of and.
  3. If is a pole of, then it is a simple pole and furthermore, the residual matrix is Hermitian and positive semidefinite.
  4. If is a pole of, then for all and is Hermitian and positive semidefinite.
These conditions can be summarized as:
  1. The system is stable.
  2. For all positive frequencies, the nyquist diagram of the system response is between .

Negative Imaginary Lemma

Source:
Let be a minimal realization of the transfer function matrix . Then, is NI if and only if and there exists a matrix
such that the following LMI is satisfied:
This result comes from positive real theory after converting the negative imaginary system to a positive real system for analysis.