# Stochastic Stabilization for Discrete-Time System with Input Delay and Multiplicative Noise in Control Variable

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Preliminaries

**Remark**

**1.**

**Definition**

**1.**

**Lemma**

**1.**

**.**The equivalent conditions for the mean square stabilization of system $[A,B,C|d]$ are given as follows.

- (a)
- System $[A,B,C|d]$ is stabilizable in the mean square sense, if and only if there admit K and $P>0$ satisfying the following equation$$P=Q+(A+BK)P{(A+BK)}^{\prime}+{\sigma}^{2}{A}^{d}CKP{K}^{\prime}{C}^{\prime}{({A}^{\prime})}^{d},\phantom{\rule{3.33333pt}{0ex}}\forall Q>0.$$
- (b)
- System $[A,B,C|d]$ is stabilizable in the mean square sense, if and only if there exists a constant matrix K such that $|\rho ({\mathcal{F}}_{K}^{[A,B,C|d]})|<1$.
- (c)
- The mean square stabilization of system $[A,B,C|d]$ is equivalent to that of the following delay free system$${z}_{k+1}=A{z}_{k}+B{u}_{k}+{e}_{k}{A}^{d}C{u}_{k}.$$
- (d)
- The mean square stabilizable of system $[A,B,C|d]$ is equivalent to the following DARE$$P={A}^{\prime}PA-{L}^{\prime}{U}^{-1}L+Q,\phantom{\rule{3.33333pt}{0ex}}\forall R>0,\phantom{\rule{3.33333pt}{0ex}}Q>0.$$

## 3. The Necessary Condition of Asymptotic Mean Square Stabilization

**Lemma**

**2.**

- (a)
- For any $M\ge 0$,$${\mathcal{R}}^{[A,B,C|d]}(M)=\underset{K}{min}{\mathcal{G}}_{K}^{[A,B,C|d]}(M).$$
- (b)
- If ${M}_{2}\ge {M}_{1}\ge 0$, then,$${\mathcal{R}}^{[A,B,C|d]}({M}_{2})\ge {\mathcal{R}}^{[A,B,C|d]}({M}_{1}).$$
- (c)
- For system $[A,B,C|d]$, define ${X}_{k+1}={\mathcal{R}}^{[A,B,C|d]}({X}_{k})$ and ${Y}_{k+1}={\mathcal{R}}^{[A,B,C|d]}({Y}_{k})$, then, for any $k\ge 0$,$${X}_{0}\ge {Y}_{0}\ge 0\Rightarrow {X}_{k}\ge {Y}_{k}\ge 0.$$

**Lemma**

**3.**

**Theorem**

**1.**

**Proof.**

## 4. Critical Stabilization

**Definition**

**2.**

**Theorem**

**2.**

- (i)
- System $[A,B,C|d]$ is critical stabilization.
- (ii)
- There exists matrix K, $K\in {S}^{n}$, such that for any $\alpha >1$ and $Q>0$, the following DLE$$P=Q+(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)P{(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)}^{\prime}+{\sigma}^{2}{(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}KP{\left[{(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}K\right]}^{\prime},$$
- (iii)
- There exists matrix K, $K\in {S}^{n}$, such that for any $\alpha >1$, the following inequality$$P-(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)P{(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)}^{\prime}-{\sigma}^{2}{(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}KP{\left[{(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}K\right]}^{\prime}>0,$$
- (iv)
- There exists matrices K, $K\in {S}^{n}$, such that for any $\alpha >1$ and $Q>0$, the following inequality$$P=Q+{(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)}^{\prime}P(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)+{\sigma}^{2}{K}^{\prime}\frac{{C}^{\prime}}{\sqrt{\alpha}}{(\frac{{A}^{\prime}}{\sqrt{\alpha}})}^{d}P{(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}K,$$

**Proof.**

## 5. Essential Destabilization

**Definition**

**3.**

- Matrix $\mathbb{T}({n}^{2},\frac{n(n+1)}{2})$ has column full rank.
- Matrix ${\mathbb{T}}^{\prime}({n}^{2},\frac{n(n+1)}{2})\mathbb{T}({n}^{2},\frac{n(n+1)}{2})$ is invertible.

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**3.**

**Proof.**

- (i)
- When $ker(V)=0$, V is a column full rank matrix. Since V is not a negative definite matrix, V must have a positive eigenvalue.
- (ii)
- When $ker(V)\ne 0$, for any non-zero ${x}_{0}\in ker(V)$, pre-multiplying ${x}_{0}^{\prime}$ and post-multiplying ${x}_{0}$ on both sides of (36), we have$$\begin{array}{cc}\hfill & {x}_{0}^{\prime}(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)V{(\frac{A}{\sqrt{\alpha}}+\frac{B}{\sqrt{\alpha}}K)}^{\prime}{x}_{0}+{x}_{0}^{\prime}{\sigma}^{2}({(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}K)V{({(\frac{A}{\sqrt{\alpha}})}^{d}\frac{C}{\sqrt{\alpha}}K)}^{\prime}{x}_{0}\hfill \\ \hfill & ={x}_{0}^{\prime}W{x}_{0}.\hfill \end{array}$$

## 6. Simulation

**Example**

**1.**

**Example**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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A | B | C | |
---|---|---|---|

1 | $\left[\begin{array}{ccccc}\frac{1}{3}& 0& ;& 0& -\frac{1}{5}\end{array}\right]$ | $\left[\begin{array}{ccccc}1& \frac{1}{5}& ;& 0& -\frac{1}{3}\end{array}\right]$ | $\left[\begin{array}{ccccc}-\frac{1}{5}& \frac{1}{3}& ;& 0& \frac{1}{5}\end{array}\right]$ |

2 | $\left[\begin{array}{ccccc}1& -\frac{1}{5}& ;& 1& -\frac{1}{7}\end{array}\right]$ | $\left[\begin{array}{ccccc}1& \frac{1}{4}& ;& 2& \frac{1}{4}\end{array}\right]$ | $\left[\begin{array}{ccccc}\frac{1}{3}& \frac{1}{5}& ;& 0& 2\end{array}\right]$ |

3 | $\left[\begin{array}{ccccc}1& -\frac{1}{4}& ;& 1& \frac{1}{7}\end{array}\right]$ | $\left[\begin{array}{ccccc}2& -\frac{1}{3}& ;& 1& \frac{1}{2}\end{array}\right]$ | $\left[\begin{array}{ccccc}1& -\frac{1}{2}& ;& \frac{1}{3}& \frac{1}{4}\end{array}\right]$ |

H | Z | ${\mathit{P}}_{0}$ | |
---|---|---|---|

1 | $\left[\begin{array}{cc}-174.5693& 52.8967\\ -11.6459& -276.5355\end{array}\right]$ | $\left[\begin{array}{cc}532.7641& 0.3806\\ 0.3806& 531.4675\end{array}\right]$ | $\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$ |

2 | $\left[\begin{array}{cc}-44.0803& 6.0853\\ -150.2109& 29.4270\end{array}\right]$ | $\left[\begin{array}{cc}109.7799& 1.6274\\ 1.6274& 115.1478\end{array}\right]$ | $\left[\begin{array}{cc}4& 6\\ 6& 9\end{array}\right]$ |

3 | $\left[\begin{array}{cc}-69.6452& 1.8577\\ -113.4397& -18.3662\end{array}\right]$ | $\left[\begin{array}{cc}117.1685& 2.5141\\ 2.5141& 117.3061\end{array}\right]$ | $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ |

$lim}_{\mathit{x}\to \mathit{\infty}}{\mathit{P}}_{\mathit{k}$ | N | |
---|---|---|

1 | $\left[\begin{array}{cc}1.0304& 0.9834\\ 0.9834& 1.0120\end{array}\right]$ > 0 | 11 |

2 | $\left[\begin{array}{cc}1.6470& 0.8792\\ 0.8792& 1.0231\end{array}\right]$ > 0 | 27 |

3 | $\left[\begin{array}{cc}2.6428& 0.8405\\ 0.8405& 1.0700\end{array}\right]$ > 0 | 19 |

K | ${10}^{-8}\mathit{W}$ | ${10}^{-6}\mathit{V}$ | ${10}^{-5}{\mathit{\lambda}}_{\mathit{V}}$ | |
---|---|---|---|---|

1 | $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ | $\left[\begin{array}{cc}0.3936& 0.0582\\ 0.0582& 1.8208\end{array}\right]$ | $\left[\begin{array}{cc}6.5492& 6.8033\\ 6.8033& 6.8989\end{array}\right]$ | $\left[\begin{array}{cc}-0.8200& 135.00\end{array}\right]$ |

2 | $\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$ | $\left[\begin{array}{cc}0.6375& -0.2810\\ -0.2810& 1.8007\end{array}\right]$ | $\left[\begin{array}{cc}0.5420& 0.48857\\ 0.48857& 0.476510\end{array}\right]$ | $\left[\begin{array}{cc}0.0398& 9.8566\end{array}\right]$ |

3 | $\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]$ | $\left[\begin{array}{cc}0.6467& 0.5164\\ 0.5164& 2.5623\end{array}\right]$ | $\left[\begin{array}{cc}6.0390& 22.4000\\ 22.4000& 31.8010\end{array}\right]$ | $\left[\begin{array}{cc}-692.0000& 447.5900\end{array}\right]$ |

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**MDPI and ACS Style**

Tan, C.; Di, J.; Xiang, M.; Chen, Z.; Zhu, B.
Stochastic Stabilization for Discrete-Time System with Input Delay and Multiplicative Noise in Control Variable. *Processes* **2022**, *10*, 989.
https://doi.org/10.3390/pr10050989

**AMA Style**

Tan C, Di J, Xiang M, Chen Z, Zhu B.
Stochastic Stabilization for Discrete-Time System with Input Delay and Multiplicative Noise in Control Variable. *Processes*. 2022; 10(5):989.
https://doi.org/10.3390/pr10050989

**Chicago/Turabian Style**

Tan, Cheng, Jianying Di, Mingyue Xiang, Ziran Chen, and Binlian Zhu.
2022. "Stochastic Stabilization for Discrete-Time System with Input Delay and Multiplicative Noise in Control Variable" *Processes* 10, no. 5: 989.
https://doi.org/10.3390/pr10050989