A Novel Method for Stability and Stabilization of Interconnected Descriptor Systems

This paper addresses the issues of stability and stabilization in interconnected descriptor systems. The existing results based on different structures of the Lyapunov equation are not directly applicable to resolving time-varying problems. In this paper, based on the original literature, the stability of the system is re-described and extended, and new results are obtained, these results are extended to encompass time-varying singular systems exhibiting state-space symmetry, which is achieved by introducing a novel Lyapunov functional. The approach in this paper is grounded in the theory of trace inequalities for matrices, establishing a new sufficient condition for the admissibility of interconnected descriptor systems. In the last, an illustrative example demonstrates the feasibility of employing state feedback to render interconnected descriptor systems admissible, affirming the validity of our proposed method.


Introduction
Generalized associated systems represent dynamic systems composed of multiple associated subsystems [1].Analysis of the stability and stabilization of such systems is a fundamental challenge within the broader study of generalized association systems as proposed by Wang S [2].The stability criterion for generalized association systems is typically derived by solving the generalized Lyapunov equation and the generalized Riccati equations.However, this process is known to be intricate, and currently available methods lack a practical computer program for its resolution.Therefore, there is a pressing need for a more convenient, effective, and feasible stability criterion and algorithm.
This paper approaches the stability analysis of generalized association systems by employing matrix trace inequalities.We provide sufficient conditions for system tolerance.Building upon this foundation, we present a design method for a decentralized state feedback controller tailored for the corresponding closed-loop system.Our method overcomes existing constraints associated with requiring the Lyapunov matrix to be a diagonal block matrix in decentralized controller design, resulting in more general outcomes.Additionally, we introduce an algorithm and corresponding optimization procedures for solving such matrix trace inequalities.

Definition and Problem Description
The following notation is introduced.

) (
represents the left half-plane of the complex plane.
In the subsequent content, the following definitions and notations are provided.Definition 1: For the matrix , the sum of its diagonal elements is referred to as the trace, denoted as trA .
With this definition, it is straightforward to confirm that the following properties hold true.Character 1:  (3) For an arbitrary reversible matrix P, , if there exists a matrix, it denoted as  , G is referred to as a generalized inverse matrix of A. A collection of generalized inverses is called {1}, while any element in {1} is denoted as A  .
Considering the generalized systems, (

Ex t Ax t Bu t y t Cx t
inside, E is a n order singular matrix with , and ( ) ( ) x t u t ， denotes singular matrix dimensional state vectors and p dimensional input vectors, respectively.
Character 3: Consider the system (1) in the study by Yang D [3], (a) If the characteristic polynomials of the system det( ) sE A  inconsistency is zero, the systems are called regular; 1) is called stable system; (d) If the system is regular, free, and stable, the system (1) is admissible.If the matrix pair (E, A) is regular, according to Ma Y [4], there always exists a reversible real matrix P and Q, making, while N for power zero array, while X, Y, Z for a matrix of any appropriate dimension.

Lemma 2 If
, the Characteristic values of A are all distributed in this left half of the complex plane.
Lemma 3 Systems (1).The sufficient and necessary condition for no pulse is 0 Consider the generalized association systems, inside, ( )  for the status variable, ( ) rank E r rewrites the system (4) in the following form. Inside, Then the system ( 4) is a N-dimensional linear generalized correlation system.A decentralized state feedback controller is introduced in the Reswarch [5], so ( ) ( ) for the gain matrix.The system (4) obtains the closed-loop systems under the action of the feedback controller.

Main Conclusions
Theorem 1.Consider a regular, pulse-free generalized correlation system (5).If The following inequalities are met The system ( 5) is allowable.Among them, P, Q, satisfy ~1 0 , 0 0 Proof: Under the condition of the generalized correlation system being regular without pulse by• Zhang D Q• [6] and Zhang Y• [7], let's make the first limited equivalent transformation of the generalized correlation system.That is, the reversible array P, Q makes 1 0 0 , 0 0 0 On the other hand, by Lemma 1, we take 0 0 0 By (9), and the above formula, we can be obtained the following expression Thus, we prove the above theorem.
Theorem 2. Consider the generalized association systems (5).If there is , and the dispersion state feedback gain matrix [8] and Chen •[9] meet the following conditions 0 ( ) The closed-loop system is allowed.inside, E   and K  then satisfy the matrix trace inequalities Example.Generalized correlation systems with 2 subsystems are considered.
After calculation, the system is pulse-free and unstable.Using Theorem 2 and the Matlab program, 0.2076 0.2988 0 0 0 0 0.6192 0.7879 from Figure 1, this open loop systems is unstable, and from Figure 2, the closed loop systems is stable.

Conclusion
In this paper, we present sufficient conditions for the admissibility of the generalized association system and the corresponding decentralized state feedback controller.This approach overcomes the limitation of requiring the Lyapunov matrix to be a diagonal block matrix, resulting in more broadly applicable computational outcomes.Additionally, we provide the algorithm and corresponding optimization methods for solving matrix trace inequalities of this nature.Finally, we assess the effectiveness of the proposed method.
If A and B independently m n  and n m  matrix, follow