Transient Impulse Response Bound
For a single-input multi-output continuous-time LTI system with state-space realization
![{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y&=Cx\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a6d53e0bc98fba247acc1ca7213e8c26bad16a)
where
,
and
.
,
and
.
Also it is assumed that Z(t)=C
B be the unit impulsive response of the system.
If the Euclidean norm of the impulse response satisfies.
and if there exist
and
,where P > 0, such that
![{\displaystyle {\begin{bmatrix}P&PB\\*&\gamma \end{bmatrix}}\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a08d303bbb1de688bd836ba204d0d0aa374b5d7)
![{\displaystyle {\begin{bmatrix}P&C^{T}\\*&-\gamma 1\end{bmatrix}}\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54976a9abaed8fd12a61c593500490f59df9f620)
![{\displaystyle PA+AP\leq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/984a599676d7b99381296fd21a12b2c2295f79bb)
- 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.
For the single-input multi-output discrete-time LTI system with state-space realization,
![{\displaystyle {\begin{aligned}\ x_{k+1}&=A_{d}x_{k}+B_{d}u_{k}\\y_{k}&=C_{d}x_{k}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b3aa7107bd132219f69ef46c34e518ce394b1a)
where
,
and
and it is assumed that
is invertible. It is also considered that
be the unit impulse response of the system.
,
and
If the Euclidean norm of the impulse response satisfies.
and if there exist
and
,where P > 0, such that
![{\displaystyle {\begin{bmatrix}P&PA_{d}^{-1}B_{d}\\*&\gamma \end{bmatrix}}\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd39180018c5837728fb0d46a9477ed9673099b)
![{\displaystyle {\begin{bmatrix}P&C_{d}^{T}\\*&\gamma 1\end{bmatrix}}\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc6c77763073ee97669f87b496dc2154f0dda31)
![{\displaystyle A_{d}^{T}PA_{d}-P\leq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fde8dc8587788968f637ef75178aa2807fbd17e)
- 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.
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.
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.