Suvarna Garge (Editor)

Negative imaginary systems

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

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
(Text) CC BY-SA


Similar Topics