# Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Theorem**

**1**

**.**The output ${y}_{*}={h}_{*}(x)$ can be decoupled from the unknown function $w(k)$ by compensator (2) if and only if there exist $(h,f)$-invariant function $\mathsf{\alpha}$ and a controlled invariant function $\mathsf{\chi}$ such that

## 3. Logic-Dynamic Approach

**Step 1.**The nonlinear term is removed from the initial nonlinear system (4).

**Step 2.**The problem under consideration is solved for the linear part, obtained in Step 1, under some linear limitation. Such a limitation is used to find out whether or not the nonlinear term is designed on the basis of the linear solution obtained in this step.

**Step 3.**The solution, obtained in Step 2, is supplemented by the transformed nonlinear term.

## 4. Problem Solution

#### 4.1. Disturbance Decoupling for the Linear Part of a System

#### 4.2. Dynamic Part of the Compensator Design

**Theorem**

**2**

**.**Let ${\mathsf{\Phi}}^{(1)}$, …, ${\mathsf{\Phi}}^{(N)}$ be matrices calculated on the basis of (10) and (11) and satisfying the condition (7). Then the linear combination of rows (13) with some coefficients ${v}_{1}$, …, ${v}_{N}$ yields the matrix $\mathsf{\Phi}={v}_{1}{\mathsf{\Phi}}^{(1)}+\dots +{v}_{N}{\mathsf{\Phi}}^{(N)}$, satisfying the condition (7) as well.

#### 4.3. Function $\mathsf{\chi}$ Design

**Assumption**

**1**

**.**${w}_{i}>{r}_{i}$ and ${w}_{i}>{r}_{i}^{\prime}$ for all $i=1,\dots ,L$, otherwise the DDDP is not solved.

**Theorem**

**3**

**.**Set

**Assumption**

**3.**

#### 4.4. Discussion

## 5. Example

## 6. Conclusions

## Funding

## Conflicts of Interest

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Zhirabok, A.
Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution. *Symmetry* **2019**, *11*, 555.
https://doi.org/10.3390/sym11040555

**AMA Style**

Zhirabok A.
Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution. *Symmetry*. 2019; 11(4):555.
https://doi.org/10.3390/sym11040555

**Chicago/Turabian Style**

Zhirabok, Alexey.
2019. "Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution" *Symmetry* 11, no. 4: 555.
https://doi.org/10.3390/sym11040555