LMIs in Control/Click here to continue/LMIs in system and stability Theory/Transient Impulse Response Bound for Non-Autonomous LTI systems
Transient Impulse Response Bound
The System edit
For a single-input multi-output continuous-time LTI system with state-space realization
where , and .
The Data edit
, and .
Also it is assumed that Z(t)=C B be the unit impulsive response of the system.
The LMI edit
If the Euclidean norm of the impulse response satisfies. and if there exist and ,where P > 0, such that
Proof edit
- The proof follows same procedure as the proof for transient output Bound for Autonomous LTI systems, but in this case taking as the initial condition that yields the result .
- Using the non-strict Schur complement, the matrix inequality in is equivalent to . Substituting this and into gives the desired result.
Discrete Time Transient Impulse Response edit
The System edit
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
, and
The LMI edit
If the Euclidean norm of the impulse response satisfies. and if there exist and ,where P > 0, such that
Proof edit
- 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
The above LMIs can be used to analyze the Transient Impulse Response Bound and analyze the Discrete-Time Transient Impulse Response Bound for the given system.
Implementation edit
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.