LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Transient Impulse Response Bound
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The System[edit | edit source]
For the single-input multi-output discrete-time LTI system with state-space realization,
where , and and it is assumed that is invertible. It is also considered that be the unit impulse response of the system.
The Data[edit | edit source]
, and
The LMI[edit | edit source]
If the Euclidean norm of the impulse response satisfies. and if there exist and ,where P > 0, such that
Proof[edit | edit source]
- The proof follows same procedure as for transient output bound for Discrete time autonomous LTI sysyems,but taking as the initial condition, so that the unit impulse response matching the free response .
- This yields the result.
- Using the non-strict Schur complement, the matrix inequality is equivalent to the inequality .Substituting this and into .gives the desired result.
Conclusion[edit | edit source]
This LMI can be used to analyze the Discrete-Time Transient Impulse Response Bound for the given system
Implementation[edit | edit source]
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.