Stabilization and the Design of Switching Laws of a Class of Switched Singularly Perturbed Systems via the Composite Control

Abstract

This paper proves that the controller design for switched singularly perturbed systems can be synthesized from the controllers of individual slow–fast subsystems. Under the switching rules of individual slow–fast subsystems, switched singularly perturbed systems can be stabilized under a small value of ε. The switching rule is designed on the basis of state transformation of the individual subsystems.

Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

“…( 2) ] . First, according to the Lyapunov Equation (4), we can obtain ] for i=1, 2, and the four intersections, ̄1 = 0.0927, ̄2 = 0.1129, ̂1 = 0.2361 and ̂2 = 0.1642 can be calculated from (7). Therefore, we conclude that system (17) can be stabilized by the following feedback control law for all ε(0, ):…”

“…If k 1 and k 2 are selected inside the region of the convex polygon as shown in Figure 2, then the observer-based feedback (10) stabilizes the system (1) in the entire state space for all ε∈(0,∞), where the six intersections of convex polygon with k 1 -axis and with k 2 -axis are described as Equation ( 7), and (10), where is designed such that is stable. If 1 2 are selected inside the region of the convex polygon as shown in Figure 2, then the observer-based feedback (10) stabilizes the system (1) in the entire state space for all ε(0, ), where the six intersections of convex polygon with k 1axis and with k 2 -axis are described as Equation (7), and Proof. The proof can be also divided into two parts:…”

“…Due to the high-order dimensions of these factories, a large amount of computer memory and considerable operating time are required to process such a large-scale system. The singular perturbation method [5][6][7][8][9][10][11][12][13][14][15][16] is very useful for solving the problems of high dimensionality and stiffness when dealing with large-scale systems. Although the singularly perturbed systems have so many applications and combinations with other methodologies, apparently many basic issues remain to further address.…”

Is exponential stability achievable in singular perturbed delayed systems with time-varying parameters? A comprehensive analysis

“…The stability properties of singular perturbed systems and the consequences of delayed dynamics have been the subject of extensive investigation in numerous studies [ [51] , [52] , [53] , [54] , [55] ]. Traditional methodologies employed in these studies have commonly assumed exponential stability in the fast subsystem, yielding valuable insights across diverse contexts [ [56] , [57] , [58] ]. Notably, the emergence of LMI methods has revolutionized the analysis of singularly perturbed systems, offering efficient techniques for stability analysis and providing a deeper understanding of their dynamic behavior [ [59] , [60] , [61] ].…”