Revisiting the LQR Problem of Singular Systems

Komeil Nosrati; Juri Belikov; Aleksei Tepljakov; Eduard Petlenkov

doi:10.1109/JAS.2024.124665

Volume 11 Issue 11

Nov. 2024

IEEE/CAA Journal of Automatica Sinica

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K. Nosrati, J. Belikov, A. Tepljakov, and E. Petlenkov, “Revisiting the LQR problem of singular systems,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 11, pp. 2236–2252, Nov. 2024. doi: 10.1109/JAS.2024.124665

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K. Nosrati, J. Belikov, A. Tepljakov, and E. Petlenkov, “Revisiting the LQR problem of singular systems,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 11, pp. 2236–2252, Nov. 2024. doi: 10.1109/JAS.2024.124665

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Revisiting the LQR Problem of Singular Systems

doi: 10.1109/JAS.2024.124665

Funds: This work was supported by the European Union’s Horizon Europe research and innovation programme (101120657), project ENFIELD (European Lighthouse to Manifest Trustworthy and Green AI), the Estonian Research Council (PRG658, PRG1463), and the Estonian Centre of Excellence in Energy Efficiency, ENER (TK230) funded by the Estonian Ministry of Education and Research

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Author Bio:
Komeil Nosrati (Senior Member, IEEE) received the B.Sc. degree in computer engineering from Iran University of Science and Technology, the M.Sc. degree in electrical engineering from Ferdowsi University of Mashhad, and the Ph.D. degree in electrical engineering from Amirkabir University of Technology in 2018. He was an R&D Specialist at Tehran Power Distribution Company and was also a Project Manager for power plant and substation automation at DETCO. He is currently a Researcher with the Department of Computer Systems at Tallinn University of Technology, Estonia. His research interests include robust estimation and filtering, nonlinear control theory, and power systems

Juri Belikov (Senior Member, IEEE) received the B.Sc. degree in mathematics from Tallinn University, and the M.Sc. and Ph.D. degrees in computer and systems engineering from Tallinn University of Technology in 2006, 2008, and 2012, respectively. In 2015–2017, he was a Postdoctoral Fellow with the Faculty of Mechanical Engineering and Andrew & Erna Viterbi Faculty of Electrical Engineering, Technion—Israel Institute of Technology. He is currently an Assistant Professor with the Department of Software Science, Tallinn University of Technology. His main research interests lie at the edge of nonlinear control theory and power systems

Aleksei Tepljakov (Senior Member, IEEE) received the Ph.D. degree in information and communication technology from the Tallinn University of Technology in 2015. Since January 2016, he has held a Research Scientist position with the Department of Computer Systems, School of Information Technologies, Tallinn University of Technology. His research interests include fractional-order modeling and control of complex systems and developing efficient mathematical and 3D modeling methods for virtual and augmented reality for educational and industrial applications. He has been a Member of the IEEE Control Systems Society since 2012 and the Education Society since 2018

Eduard Petlenkov (Member, IEEE) received the B.Sc., M.Sc., and Ph.D. degrees in computer and systems engineering from the Tallinn University of Technology, Estonia. He is currently a tenured Full Professor with the Department of Computer Systems, Tallinn University of Technology, and the Head of the Centre for Intelligent Systems. His research interests include the domain of nonlinear control, system analysis, and computational intelligence
Corresponding author: Komeil Nosrati, e-mail: [email protected]
¹ Under indiscernible topological changes in network dynamics, nonregular systems for node dynamics can be common [34]. Additionally, coupling regular singular systems does not necessarily ensure the regularity of the overall system [35].
² Unexpected faults such as power line outages in power grids can cause sudden changes in system layout, leading to different algebraic conditions and constraints [37]. Consequently, the equivalent systems associated with constraints are no longer synchronized with each other.
³ We consider this assumption for stability and convergence analysis. Since regularity is a sufficient condition for the uniqueness of solutions in singular systems, the LQR algorithm derived in this study can also be applied to systems that may not meet regularity requirements.
⁴ Any real symmetric PSD matrix can be diagonalized using an orthogonal matrix. Then, with no loss of generality, let the real symmetric \begin{document}$ Q \succeq 0 $\end{document} be diagonal, i.e., it can be written as \begin{document}$ Q=(Q^{\frac{1}{2}})^\mathsf{T}Q^{\frac{1}{2}} $\end{document}, where \begin{document}$ Q^{\frac{1}{2}} \succeq 0 $\end{document} is a unique square root of Q.
⁵ Note: \begin{document}$ \{T^{-1}\succ0 $\end{document}, \begin{document}$ (T+AS^{-1}A^\mathsf{T})\succ0\} $\end{document} \begin{document}$ \Rightarrow $\end{document} \begin{document}$ \mathbb{P}_s\succ0 $\end{document}.
⁶ Note: \begin{document}$ \{\Psi\succ0 $\end{document}, \begin{document}$ E^\mathsf{T}\Phi^{-1}E\succeq0\} $\end{document} \begin{document}$ \Rightarrow $\end{document} \begin{document}$ \mathbb{Q}_s\succ0 $\end{document}.
Received Date: 2024-06-02
Accepted Date: 2024-07-17

Available Online: 2024-08-23

Abstract

Abstract

In the development of linear quadratic regulator (LQR) algorithms, the Riccati equation approach offers two important characteristics—it is recursive and readily meets the existence condition. However, these attributes are applicable only to transformed singular systems, and the efficiency of the regulator may be undermined if constraints are violated in nonsingular versions. To address this gap, we introduce a direct approach to the LQR problem for linear singular systems, avoiding the need for any transformations and eliminating the need for regularity assumptions. To achieve this goal, we begin by formulating a quadratic cost function to derive the LQR algorithm through a penalized and weighted regression framework and then connect it to a constrained minimization problem using the Bellman’s criterion. Then, we employ a dynamic programming strategy in a backward approach within a finite horizon to develop an LQR algorithm for the original system. To accomplish this, we address the stability and convergence analysis under the reachability and observability assumptions of a hypothetical system constructed by the pencil of augmented matrices and connected using the Hamiltonian diagonalization technique.
- DC motor,
- optimal control,
- penalized weighted regression,
- power system,
- quadratic regulator,
- singular system

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¹ Under indiscernible topological changes in network dynamics, nonregular systems for node dynamics can be common [34]. Additionally, coupling regular singular systems does not necessarily ensure the regularity of the overall system [35].
² Unexpected faults such as power line outages in power grids can cause sudden changes in system layout, leading to different algebraic conditions and constraints [37]. Consequently, the equivalent systems associated with constraints are no longer synchronized with each other.
³ We consider this assumption for stability and convergence analysis. Since regularity is a sufficient condition for the uniqueness of solutions in singular systems, the LQR algorithm derived in this study can also be applied to systems that may not meet regularity requirements.
⁴ Any real symmetric PSD matrix can be diagonalized using an orthogonal matrix. Then, with no loss of generality, let the real symmetric $ Q \succeq 0 $ be diagonal, i.e., it can be written as $ Q=(Q^{\frac{1}{2}})^\mathsf{T}Q^{\frac{1}{2}} $, where $ Q^{\frac{1}{2}} \succeq 0 $ is a unique square root of Q.
⁵ Note: $ \{T^{-1}\succ0 $, $ (T+AS^{-1}A^\mathsf{T})\succ0\} $ $ \Rightarrow $ $ \mathbb{P}_s\succ0 $.
⁶ Note: $ \{\Psi\succ0 $, $ E^\mathsf{T}\Phi^{-1}E\succeq0\} $ $ \Rightarrow $ $ \mathbb{Q}_s\succ0 $.

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Examine the conditions for the existence of the LQR algorithm for discrete singular systems
Derive LQR algorithm via dynamic programming and penalized LSs over a finite horizon
Link the problem to a system using Hamiltonian diagonalization for steady-state analysis
Provide sufficient conditions for a unique PD solution to the Riccati equation
Ensure the stability of the derived algorithm by contradiction and under the given assumptions

Revisiting the LQR Problem of Singular Systems

doi: 10.1109/JAS.2024.124665

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通讯作者: 陈斌, [email protected]

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