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Negative imaginary systems

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Negative imaginary (NI) systems theory was introduced by Lanzon and Petersen in. A generalization of the theory was presented in In the single-input single-output (SISO) case, such systems are defined by considering the properties of the imaginary part of the frequency response G(jω) and requiring the condition j ( G ( j ω ) − G ( j ω ) ∗ ) > 0 for all ω in (0, ∞). Roughly speaking, SISO NI systems are stable systems having a phase lag between 0 and -π for all ω> 0.

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Negative Imaginary Definition

A square transfer function matrix G ( s ) is NI if the following conditions are satisfied:

  1. G ( s ) has no pole in R e [ s ] > 0 .
  2. For all ω ≥ 0 such that j ω is not a pole of G ( s ) and j ( G ( j ω ) − G ( j ω ) ∗ ) ≥ 0 .
  3. If s = j ω 0 , ω 0 > 0 is a pole of G ( s ) , then it is a simple pole and furthermore, the residual matrix K = lim s ⟶ j ω 0 ( s − j ω 0 ) j G ( s ) is Hermitian and positive semidefinite.
  4. If s = 0 is a pole of G ( s ) , then lim s ⟶ 0 s k G ( s ) = 0 for all k ≥ 3 and lim s ⟶ 0 s 2 G ( s ) is Hermitian and positive semidefinite.

Negative Imaginary Lemma

Let [ A B C D ] be a minimal realization of the transfer function matrix G ( s ) . Then, G ( s ) is NI if and only if D = D T and there exists a matrix

P = P T ≥ 0 ,   W ∈ R m × m , and  L ∈ R m × n such that the following LMI is satisfied:

[ P A + A T P P B − A T C T B T P − C A − ( C B + B T C T ) ] = [ − L T L − L T W − W T L − W T W ] ≤ 0.

References

Negative imaginary systems Wikipedia
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