Abstract
This paper is concerned with the stability of second-order linear time-varying systems. By utilizing the Lyapunov approach, a generally uniformly asymptotic stability criterion is obtained by adding the system matrices into the quadratic Lyapunov candidate function. In the case of the derivative of the Lyapunov candidate function is semi-positive definite, the stability criterion is also efficient. Based on the proposed results, the systems with uncertain disturbances such as structured independent and structured dependent perturbations are considered. Using the matrix measure and the singular value theory, the bounds of the uncertainties are obtained that guarantee the system uniformly asymptotically stable, while the bounds of state feedback control input are also derived to stabilize the second-order linear time-varying systems. Finally, several numerical examples are given to prove the validity and correctness of the proposed criteria with existing ones.
Highlights
Second-order systems can represent many practical processes in nature, such as RLC circuit networks, armature-controlled DC motors, and spring-massdamper systems, etc.[1,2,3] For example, one of the springmass-damper systems is the buffer
Different from linear time-invariant systems, there are several different definitions for linear time-varying (LTV) systems, including stability, asymptotic stability, and exponential stability, which can distinguish uniform stability from non-uniform stability according to the relationship with the initial time of the system
For time-invariant systems, researchers generally select the quadratic matrix of the derivative of Lyapunov function as an identity matrix, obtain the condition by employing Linear Matrix Inequalities (LMIs).[4,5,6,7,8]
Summary
Second-order systems can represent many practical processes in nature, such as RLC circuit networks, armature-controlled DC motors, and spring-massdamper systems, etc.[1,2,3] For example, one of the springmass-damper systems is the buffer. Uncertain disturbances such as structured independent and structured dependent perturbations are considered in Sections 5 and 6. Consider a class of SLTV systems as follow: M(t)€x(t) + N (t)x_(t) + K(t)x(t) = 0, t 2 J , ð1Þ where M(t) = MT(t) 2 Rn 3 n, N (t) 2 Rn 3 n and K(t) 2 Rn 3 n are all piece-wisely differentiable with respect to t and uniformly bounded time-varying symmetric coefficient matrices, x(t) 2 Rn is the state variable, and J = 1⁄2tÃ, ‘Þ, tà is a finite number.
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