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
For a single-input multi-output continuous-time LTI system with state-space realization



\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y&=Cx\\ \end{align}$$

where $$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times 1}$$ and $$C \in \mathbb{R}^{p \times 1}$$.

The Data
$$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times 1}$$ and $$C \in \mathbb{R}^{p \times 1}$$.

Also it is assumed that Z(t)=C$e^{\text{At}}$ B be the unit impulsive response of the system.

The LMI
If the Euclidean norm of the impulse response satisfies. $$\lVert z(T) \rVert \leq \gamma, \forall T \in \mathbb{R}_{\geq 0}$$ and if there exist $$P \in \mathbb{S}^p$$ and $$\gamma \in \mathbb{R}_{> 0}$$,where P > 0, such that


 * $$\begin{bmatrix} P & PB \\ * & \gamma \end{bmatrix} \geq 0, $$


 * $$\begin{bmatrix} P & C^T \\ * & -\gamma 1 \end{bmatrix} \geq 0 . $$


 * $$PA+AP \leq 0 $$

Proof

 * The proof follows same procedure as the proof for transient output Bound for Autonomous LTI systems, but in this case taking $$ X_{0}=B $$as the initial condition that yields the result $$X^{T}(T)PX(T)\leq B^{T}PB$$.


 * Using the non-strict Schur complement, the matrix inequality in $$\begin{bmatrix} P & C^T \\ * & -\gamma 1 \end{bmatrix} \geq 0 . $$ is equivalent to $$B^{T}PB\leq \gamma $$. Substituting this and $$\begin{bmatrix} P  & C^T \\ * & -\gamma 1 \end{bmatrix} \geq 0 $$ into $$X^{T}(T)PX(T)\leq B^{T}PB$$ gives the desired result.

The System
For the single-input multi-output discrete-time LTI system with state-space realization,



\begin{align} \ x_{k+1}&=A_{d}x_{k}+B_{d}u_{k}\\ y_{k}&=C_{d}x_{k}\\ \end{align}$$

where $$A \in \mathbb{R}^{n \times n}$$, $$B_{d} \in \mathbb{R}^{n \times 1}$$ and $$C_{d} \in \mathbb{R}^{p \times n}$$ and it is assumed that $$A_{d}$$ is invertible. It is also considered that $$ Z_{k}C_{d}A^{k-1}_{d}B_{d}$$ be the unit impulse response of the system.

The Data
$$A \in \mathbb{R}^{n \times n}$$, $$B_{d} \in \mathbb{R}^{n \times 1}$$ and $$C_{d} \in \mathbb{R}^{p \times n}$$

The LMI
If the Euclidean norm of the impulse response satisfies. $$\lVert z_{k} \rVert_{2} \leq \gamma, \forall k \in \mathbb{Z}_{\geq 0}$$ and if there exist $$P \in \mathbb{S}^p$$ and $$\gamma \in \mathbb{R}_{> 0}$$,where P > 0, such that


 * $$\begin{bmatrix} P & PA_{d}^{-1}B_{d} \\ * & \gamma \end{bmatrix} \geq 0, $$


 * $$\begin{bmatrix} P & C^T_{d} \\ * & \gamma 1 \end{bmatrix} \geq 0, $$


 * $$A^T_{d}PA_{d}-P \leq 0. $$

Proof

 * The proof follows same procedure as for transient output bound for Discrete time autonomous LTI sysyems,but taking $$X_{0}=A_{d}^{-1}B_{d} $$as the initial condition, so that the unit impulse response matching the free response $$Z_{k}=C_{d}A_{d}^{k}X_{0}$$.


 * This yields the result$$X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$$.


 * Using the non-strict Schur complement, the matrix inequality $$\begin{bmatrix} P & PA_{d}^-1B_{d} \\ * & \gamma \end{bmatrix} \geq 0, $$ is equivalent to the inequality $$ B_{d}^TA_{d}^-TPA_{d}^-1B_{d}\leq \gamma$$.Substituting this and  $$\begin{bmatrix} P  & C^T_{d} \\ * & \gamma 1 \end{bmatrix} \geq 0 , $$ into $$X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$$.gives the desired result.

Conclusion
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
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.