LMIs in Control/pages/NI-Lemma

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LMIs in Control/pages/NI-Lemma Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

The System[edit]

Consider a square continuous time Linear Time invariant system, with the state space realization

The Data[edit]

nxn, nxm , mxn , m

The LMI: LMI for Negative Imaginary Lemma[edit]

According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions

  • There exists a n,where , such that,

  • There exists a n,where , such that,


The system is strictly negative if det() 0 and either of the above LMIs are feasible with resulting or


A link to the Matlab code for a simple implementation of this problem in the Github repository:


Related LMIs[edit]

Positive Real Lemma

External Links[edit]

Return to Main Page:[edit]