Let $C$ be a given (known) matrix and let $\theta$ be a given (known) positive real. We would like to determine the matrices $X$ and $Y$ and diagonal matrix $P$ solving the following inequality

\begin{equation} (XC)^{\text{T}}e{}^{(-YC)^{\text{T}}\theta}Pe{}^{(-YC)\theta}(XC)-P\prec0, \end{equation}

which is equivalent to \begin{equation} P-(XC)^{\text{T}}e{}^{(-YC)^{\text{T}}\theta}Pe{}^{(-YC)\theta}(XC)\succ0. \end{equation} or \begin{equation} P-(XC)^{\text{T}}e{}^{(-YC)^{\text{T}}\theta}PP^{-1}Pe{}^{(-YC)\theta}(XC)\succ0. \end{equation} Using Schur complement, it is equivalent to \begin{equation} \begin{bmatrix}P & (XC)^{\text{T}}e{}^{(-YC)^{\text{T}}\theta}\\ e^{(-YC)\theta}(XC) & P^{-1} \end{bmatrix}\succ0, \end{equation} or \begin{equation} \begin{bmatrix}P & (XC)^{\text{T}}e{}^{(-YC)^{\text{T}}\theta}P\\ Pe^{(-YC)\theta}(XC) & P \end{bmatrix}\succ0. \end{equation}

Does anyone know how to transform these inequalities into linear matrix inequalities (LMIs) please? Or does anyone know how to solve these inequalities in order to find the matrices $X$ and $Y$ please? Thanks.

  • $\begingroup$ Please provide background and motivation. Does this come from the stability of discrete-time linear systems? $\endgroup$ – Rodrigo de Azevedo May 14 '17 at 11:56
  • $\begingroup$ Thanks Rodrigo. Yes, it comes from the stability of discrete-time linear systems. I am actually dealing with linear impulsive systems. $\endgroup$ – G. Trav May 15 '17 at 19:35

Choosing $\mathrm X = \mathrm I$ and $\mathrm Y = \mathrm O$, we obtain the following Lyapunov linear matrix inequality (LMI) in $\mathrm P$

$$\mathrm C^{\top} \mathrm P \, \mathrm C - \mathrm P \prec \mathrm O$$

We can choose a positive definite matrix $\mathrm Q$ and solve the following Lyapunov equation in $\mathrm P$

$$\mathrm C^{\top} \mathrm P \, \mathrm C - \mathrm P + \mathrm Q = \mathrm O$$

Vectorizing, we obtain a system of linear equations

$$\left( \mathrm I - (\mathrm C \otimes \mathrm C)^{\top} \right) \mbox{vec} (\mathrm P) = \mbox{vec} (\mathrm Q)$$

  • $\begingroup$ Thank you very much for your answer Rodrigo. The case $X = I$ and $Y = 0$ is a trivial case. However, we are more interested in how to solve this inequality in general cases. $\endgroup$ – G. Trav May 15 '17 at 13:10

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