Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

2011 ◽  
Vol 60 (9) ◽  
pp. 778-785 ◽  
Author(s):  
Dong Hwan Lee ◽  
Jin Bae Park ◽  
Young Hoon Joo
Keyword(s):  
Lyapunov Functions ◽  
Linear Time ◽  
Higher Order ◽  
Stability Conditions ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant ◽  
Derivatives Of

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Active Control of Linear Time-Varying Structures by Confinement of Vibrations

2005 ◽  
Vol 11 (1) ◽  
pp. 89-102 ◽  
Author(s):  
S. Choura ◽  
A. S. Yigit
Keyword(s):  
Control Strategy ◽  
Linear Time ◽  
Steady States ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Slowing Down ◽  
Rapid Removal ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

We propose a control strategy for the simultaneous suppression and confinement of vibrations in linear time-varying structures. The proposed controller has time-varying gains and can also be used for linear time-invariant systems. The key idea is to alter the original modes by appropriate feedback forces to allow parts of the structure reach their steady states at faster rates. It is demonstrated that the convergence of these parts to zero is improved at the expense of slowing down the settling of the remaining parts to their steady states. The proposed control strategy can be applied for the rapid removal of vibration energy in sensitive parts of a flexible structure for safety or performance reasons. The stability of the closed-loop system is proven through a Lyapunov approach. An illustrative example of a five-link manipulator with a periodic follower force is given to demonstrate the effectiveness of the method for time-varying as well as time-invariant systems.

Stability Analysis of LTI Systems With Three Independent Delays—A Computationally Efficient Procedure

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Rifat Sipahi ◽  
Hassan Fazelinia ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Numerical Procedure ◽  
Computationally Efficient ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Operation Results ◽  
Time Invariant

A practical numerical procedure is introduced for determining the stability robustness map of a general class of higher order linear time invariant systems with three independent delays, against uncertainties in the delays. The procedure is based on an efficient and exhaustive frequency-sweeping technique within a single loop. This operation results in determination of the complete description of the kernel and the offspring hypersurfaces, which constitute exhaustively the potential stability switching loci in the space of the delays. The new numerical procedure corresponds to the first step in the overarching framework, called the cluster treatment of characteristic roots. The results of this treatment can also be represented in another domain (called the spectral delay space) within a finite dimensional cube called the building block, which is much simpler to view and analyze. The paper also offers several case studies to demonstrate the practicality of the new numerical methodology.

scholarly journals Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Dynamic Systems ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Linear Time Invariant ◽  
Nonnegative Solutions ◽  
Fractional Differential ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant ◽  
Point Delays

This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained results are independent of the sizes of the delays.

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