# LMIs in Control/Matrix and LMI Properties and Tools/Negative Imaginary System DC Constraint

**Introduction**

[edit | edit source]These 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.

**The System**

[edit | edit source]Consider a square continuous time Linear Time invariant system, with the state space realization

**The Data**

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**The LMI**

[edit | edit source]Consider an NI transfer matrix and an NI transfer matrix . The condition λ̅ is satisfied if and only if

- ,

- ,

**Conclusion**

[edit | edit source]The above equation holds true if and only if .

**Implementation**

[edit | edit source]This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.