# On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems

## Abstract

**:**

## 1. Introduction

## 2. A Multirate Sampling System with Two Sampling Rates

**Assumption**

**1.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**i**) ${r}_{11}={r}_{{A}_{11}}<1$, ${r}_{22}={r}_{{A}_{22}}<1-{r}_{11}^{q}$,

**ii**) $max\left({\Vert {A}_{12}\Vert}_{2},{\Vert {A}_{21}\Vert}_{2}\right)\in \left({\delta}_{1},{\delta}_{2}\right)$, where:

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

## 3. A Multirate Sampling System with $\mathit{p}$ Sampling Rates

**Assumption**

**2.**

**a**) The multirate sampling system is subject to $p+1$ distinct sampling periods $\left\{{T}_{1},{T}_{2},\dots ,{T}_{p}=T\right\}$ such that ${T}_{i+1}/{T}_{i}={q}_{i}$ where ${q}_{i}\left(\ge 2\right)\in {\mathit{Z}}_{+}$, i.e., each one is an integer multiple of all its preceding ones (then, ${T}_{i}={\dot{T}}_{j-1}$; $\forall j\in \overline{i}$);$\forall i\in \overline{p}$.

**b**) The sampling period ${T}_{i}$ runs a set of ${n}_{i}$ variables grouped in a vector ${x}_{i}\in {\mathit{R}}^{{n}_{i}}$; $\forall i\in \overline{p}$ so that the combined system (3) of state vector ${x}_{k}={\left({x}_{1k}^{T},{x}_{2k}^{T},\dots ,{x}_{p,k}^{T}\right)}^{T}$; $\forall k\in {\mathit{Z}}_{0+}$ is described by a square real matrix:

**c**)${M}_{k}={G}_{1}=\left[\begin{array}{c}{A}_{11}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{A}_{12}\hspace{1em}\cdots \hspace{1em}{A}_{1p}\\ 0\hspace{1em}BlockDiag\left({I}_{{n}_{2}},\dots ,{I}_{{n}_{p}}\right)\end{array}\right]$; $\forall k\notin {\dot{q}}_{1}$,

**Proposition**

**2.**

**Theorem**

**2.**

**i**) It is globally stable if ${\prod}_{j=2}^{p}}{\Vert {G}_{j}\Vert}_{2}^{{q}_{j-1}}\le 1$, and it is globally asymptotically stable if${\prod}_{j=2}^{p}}{\Vert {G}_{j}\Vert}_{2}^{{q}_{j-1}}<1$.

**ii**) It is globally stable if ${\sum}_{j=2}^{p}{q}_{j-1}ln{\lambda}_{max}^{1/2}}\left({G}_{j}^{T}{G}_{j}\right)\le 0$, and it is globally asymptotically stable if${\sum}_{j=2}^{p}{q}_{j-1}ln{\lambda}_{max}^{1/2}}\left({G}_{j}^{T}{G}_{j}\right)<0$.

**Proof.**

**Theorem**

**3.**

**i**) It is globally asymptotically stable if $W={a}^{-1}{\left(I+Z\right)}^{-1/2}\left({\displaystyle {\prod}_{j=2}^{p}\left[{G}_{j}^{{q}_{j-1}}\right]}\right){\left(I-Z\right)}^{-1/2}$ is a (non-necessarily strict) contraction for some real $a\in \left(0,1\right)$ and some symmetric strict contraction $Z$. A sufficient condition for the global asymptotic stability is that ${W}^{2}$ is a strict contraction.

**ii**) It is globally stable if $W={\left(I+Z\right)}^{-1/2}\left({\displaystyle {\prod}_{j=2}^{p}\left[{G}_{j}^{{q}_{j-1}}\right]}\right){\left(I-Z\right)}^{-1/2}$ is a (non-necessarily strict) contraction for some real $a\in \left(0,1\right)$ and some symmetric strict contraction$Z$.

**Proof.**

**Remark**

**2.**

**Theorem**

**4.**

## 4. Multirate Input Sampling with Combined Pole-Placement and Closed-Loop Stability Analysis via Linear State-Feedback

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proposition**

**7.**

**Step 1**: If condition (37) of Proposition 7 is fulfilled for some $1\le {q}_{1}\le {q}_{1}-1$ and some prefixed targeted matrix of stable spectrum ${\overline{A}}_{c{0}_{11}}^{\ast}$ for ${\overline{A}}_{c{0}_{11}}^{}$, then solve (38), equivalent to (36), to calculate some compatible controller gain ${K}_{{q}_{1}}$ for given controller gains ${K}_{i}$ for $i\ne {q}_{1}\left(\ne 0\right)$.

**Step 2**: If $\left(D,F\right)$ is controllable, then calculate the controller gain ${G}_{0}$, according to Proposition 6, to prefix ${\overline{A}}_{c{0}_{22}}^{}$ to some prefixed value ${\overline{A}}_{c{0}_{22}}^{\ast}$ with a stable spectrum.

**Step 3**: If the condition (33) of Proposition 5 holds for some prefixed ${\tilde{\overline{A}}}_{c{0}_{21}}^{\ast}$ value for ${\tilde{\overline{A}}}_{c{0}_{21}}$, then synthesize the controller gain ${K}_{0}$ so that the first equation of (32) holds.

**Step 4**: Assume that the condition (34) of Proposition 5 is modified to re-allocate the left-hand-side term in ${G}_{0}$ (already calculated in Step 2) to its right part and the resulting solvability rank condition holds for some prefixed ${\tilde{\overline{A}}}_{c{0}_{12}}^{\ast}$ value for ${\tilde{\overline{A}}}_{c{0}_{12}}$. Then, synthesize the controller gains ${G}_{i}$ for $i\in \overline{q-1}$ so that the second modified equation of (32), associated with the above modification of the condition (34) holds.

**Assertion**

**1.**

**Proof.**

**Assertion**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

De la Sen, M.
On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems. *Mathematics* **2018**, *6*, 78.
https://doi.org/10.3390/math6050078

**AMA Style**

De la Sen M.
On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems. *Mathematics*. 2018; 6(5):78.
https://doi.org/10.3390/math6050078

**Chicago/Turabian Style**

De la Sen, M.
2018. "On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems" *Mathematics* 6, no. 5: 78.
https://doi.org/10.3390/math6050078