# Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Formulation

#### 2.1. System Description

#### 2.2. Stability Conditions

**Lemma**

**1.**

**Proof.**

**Definition**

**1.**

- ${f}_{ij}=1$, when an interaction exists between subsystems i and j;
- ${f}_{ij}=0$, when no interaction exists between subsystems i and j.

## 3. Main Steps for a Decentralized Stabilization Controller Design

- Transformation of the system to the form with an output block-diagonal matrix.
- Robust stability conditions for a descriptor system.
- Robust stability conditions of a large-scale system under decentralized control.
- Subsystem model augmentation for PID controllers designed at the subsystem level.

#### 3.1. Transformation of the Output Matrix to a Block-Diagonal Form

#### 3.2. Robust Stability of Uncertain Descriptor System

**Definition**

**3**

**.**The linear descriptor system

**Definition**

**4**

**Lemma**

**2**

**.**Linear discrete descriptor system (10) is regular, causal, and asymptotically stable if and only if there exists a generalized Lyapunov function $V\left(Ex\left(t\right)\right)=x{\left(t\right)}^{T}{E}^{T}PEx\left(t\right)$ satisfying

**Theorem**

**1.**

**Proof.**

#### 3.3. Robust Stability of Large-Scale System with Decentralized Control

- If the obtained $\alpha \ge 1$, the uncertain descriptor complex system is impulse-free and asymptotically stable.
- If the obtained $\alpha <1$, the uncertain complex system is not stable.

**Remark**

**1.**

#### 3.4. PID Controller as a Static Output Feedback

## 4. Robust Decentralized Controller Design

#### 4.1. Regional Pole Placement Approach to Descriptor Systems

#### 4.2. Robust Decentralized Control for Descriptor Systems

- Transform a system into the fully decentralized form using an auxiliary descriptor system (8).
- Calculate the robust stability boundary ${S}_{c}$ as in (16).
- Solve BMI matrix inequality (26) for each subsystem considering the ${D}_{R}$ region complying with the value of ${S}_{c}$ (disc region with radius ${S}_{c}$).

## 5. Example

- Case A
- (interactions $\overline{A}m$)

- Case B
- (uncertain interactions $\overline{A}m$ to $3\overline{A}m$)

## 6. Conclusions

- 1.
- A novel transformation method for discrete-time dynamic systems is proposed, enabling the transformation of a linear state-space system into an uncertain descriptor system with a decentralized structure of input and output matrices appropriate for a decentralized controller design.
- 2.
- Derivation of conditions for determining the stability boundaries of complex descriptor systems and their use in subsystem controller design [20].
- 3.
- Validation of the proposed discrete-time system transformation method using the recently developed decentralized control design approach.
- 4.
- Modification of the regional pole placement method to accommodate descriptor systems.
- 5.
- Demonstration of the effectiveness of the proposed decentralized control design procedure through a practical example.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

LSS | Large-Scale System |

PID | Proportional Integral Derivative |

LMI | Linear Matrix Inequality |

BMI | Bilinear Matrix Inequality |

VSS | Variable-Structure Systems |

## References

- Axelson-Fisk, M.; Knorn, S. Aspects of Fairness in Robust, Distributed Control of Interconnected Systems. In Proceedings of the IEEE Conference on Decision and Control, Jeju, Republic of Korea, 14–18 December 2020; pp. 3084–3089. [Google Scholar]
- Ghanati, G.; Azadi, S. Decentralized robust control of a vehicle’s interior sound field. J. Vib. Control
**2020**, 26, 1815–18231. [Google Scholar] [CrossRef] - Mahmud, M.A.; Roy, T.K.; Saha, S.; Enamul Haque, M.; Pota, H.R. Robust Nonlinear Adaptive Feedback Linearizing Decentralized Controller Design for Islanded DC Microgrids. IEEE Trans. Ind. Appl.
**2019**, 55, 5343–5352. [Google Scholar] [CrossRef] - Wang, D.; Mu, C. Overview of robust adaptive critic control design, Studies in Systems. Decis. Control
**2019**, 167, 1–43. [Google Scholar] - Kant, P.; Singhal, P.; Mahto, M.K.; Jain, D. Control strategies for DC Microgrids: An overview. In Proceedings of the 2022 2nd International Conference on Power Electronics & IoT Applications in Renewable Energy and Its Control (PARC), Mathura, India, 21–22 January 2022; pp. 1–6. [Google Scholar] [CrossRef]
- Dorato, P. Case studies in robust control design: An overview. In Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, USA, 5–7 December 1990; Volume 4, pp. 2030–2032. [Google Scholar] [CrossRef]
- Siljak, D.D. Large Scale Dynamic Systems, Stability and Structure; Dover Publications: New York, NY, USA, 1978. [Google Scholar]
- Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrish, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Yuan, L.; Chen, S.; Zhang, C.; Yang, G. Structured controller synthesis through block-diagonal factorization and parameter space optimization. Automatica
**2023**, 147, 110709. [Google Scholar] [CrossRef] - Peaucelle, D.; Alzelier, D.; Bachelier, O.; Bernussou, J. A new robust D stability Condition for real Convex polytopic Uncertainty. Syst. Control. Lett.
**2000**, 40, 21–30. [Google Scholar] [CrossRef] - Oliveira, M.C. A Robust Version of the Elimination Lemma. In Proceedings of the 16th Triennial IFAC World Congress, Prague, Czech Republic, 3–8 July 2005. [Google Scholar]
- Vesely, V.; Rosinova, D. Robust PID-PSD controller design: BMI approach. Asian J. Control
**2013**, 15, 469–478. [Google Scholar] [CrossRef] - Fradkov, A.L. Adaptive Control of Complex Systems. Nauka, Moscow, 1990. Available online: https://www.ipme.ru/ipme/labs/ccs/history.htm (accessed on 11 September 2023).
- Gavel, D.; Siljak, D.D. Decentralized Adaptive Control Structural Conditions for Stability. In Proceedings of the 1988 American Control Conference, Atlanta, GA, USA, 15–17 June 1988. [Google Scholar]
- Hovd, M.; Skogestad, S. Sequantial design of decentralized controllers. Automatica
**1994**, 30, 1601–1607. [Google Scholar] [CrossRef] - Kozakova, A.; Vesely, V.; Kucera, V. Robust Decentralized Controller Design Based on Equivalent Subsystems. Automatica
**2019**, 107, 29–35. [Google Scholar] [CrossRef] - Morgan, R.G.; Ozguner, U. A decentralized variable structure control algorithm for robotic manipulators. IEEE Trans. Robot. Automat.
**1985**, 1, 57–65. [Google Scholar] [CrossRef] - Bakule, L. Decentralized Control: An Overview. Annu. Rev. Control
**2008**, 32, 87–98. [Google Scholar] [CrossRef] - Davidson, E.J. Decentralized Control of Large-Scale Systems; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Vesely, V.; Paulusova, J.; Kucera, V. Decentralized controller design for a large-scale linear discrete-time polytopic uncertain systems. Int. J. Syst. Sci.
**2022**, 53, 3496–3507. [Google Scholar] [CrossRef] - Veselý, V.; Körösi, L. Decentralized Controller Design for Large-Scale Uncertain Linear Systems with No Block Diagonal Output Matrix. Int. J. Innov. Comput. Inf. Control
**2023**, 19, 1323–1336. [Google Scholar] [CrossRef] - Wang, S.H.; Davison, E. On the stabilization of decentralized Control Systems. IEEE Trans. Autom. Control
**1973**, 18, 473–478. [Google Scholar] [CrossRef] - Jayanthi, R.; Chidambaran, I.A.; Banusri, C. Decentralized controller gain scheduling using PSO for power system restoration assessment in a two-area interconnected power system. Int. J. Eng. Sci. Technol.
**2011**, 3, 14–20. [Google Scholar] [CrossRef] - Debelkovic, D.L.; Visnjic, N.; Pjascic, M. The stability of linear continuous Singular systems in the sense of Lyapunov: An Overview. Sci. Tech. Rev.
**2007**, LVII, 51–65. [Google Scholar] - Hovd, M.; Skogestad, S. Improved independent design of robust decentralized control. J. Process. Control
**1993**, 3, 43–51. [Google Scholar] [CrossRef] - Davison, E.J.; Chang, T.N. Decentralized Stabilization and pole assignment for general improper systems. In Proceedings of the 1987 American Control Conference, Minneapolis, MN, USA, 10–12 June 1987; pp. 1669–1975. [Google Scholar]
- Matrosov, V.M. On the theory of stability motion. Prikl. Mat. Mekhanika
**1962**, 26, 992–1002. [Google Scholar] [CrossRef] - Debeljkovic, D.L.I.; Buzurovic, L.M.; Simenuoviic, G.V. Stability of Linear discrete descriptor systems in the sense of Ljapunov. Int. Inf. Syst. Sci.
**2012**, 7, 302–322. [Google Scholar]

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**MDPI and ACS Style**

Rosinová, D.; Körösi, L.; Veselý, V.
Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix. *Electronics* **2023**, *12*, 4358.
https://doi.org/10.3390/electronics12204358

**AMA Style**

Rosinová D, Körösi L, Veselý V.
Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix. *Electronics*. 2023; 12(20):4358.
https://doi.org/10.3390/electronics12204358

**Chicago/Turabian Style**

Rosinová, Danica, Ladislav Körösi, and Vojtech Veselý.
2023. "Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix" *Electronics* 12, no. 20: 4358.
https://doi.org/10.3390/electronics12204358