scholarly journals Time-Varying Vector Norm and Lower and Upper Bounds on the Solutions of Uniformly Asymptotically Stable Linear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 915
Author(s):  
Robert Vrabel
Keyword(s):  
Linear Systems ◽  
Linear Time ◽  
Upper Bounds ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Lower And Upper Bounds ◽  
Linear Time Invariant ◽  
Vector Norm ◽  
Linear Time Invariant Systems ◽  
Stable Linear

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space R n we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant systems to the linear time-varying ones.

Linear Approximations for Swing Leg Motion During Gait

1984 ◽  
Vol 106 (2) ◽  
pp. 137-143 ◽  
Author(s):  
W. H. Lee ◽  
J. M. Mansour
Keyword(s):  
Linear Systems ◽  
System Analysis ◽  
Linear Time ◽  
Time Varying ◽  
Two Dimensional ◽  
State Variables ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Leg Motion ◽  
Time Invariant

The applicability of a linear systems analysis of two-dimensional swing leg motion was investigated. Two different linear systems were developed. A linear time-varying system was developed by linearizing the nonlinear equations describing swing leg motion about a set of nominal system and control trajectories. Linear time invariant systems were developed by linearizing about three different fixed limb positions. Simulations of swing leg motion were performed with each of these linear systems. These simulations were compared to previously performed nonlinear simulations of two-dimensional swing leg motion and the actual subject motion. Additionally, a linear system analysis was used to gain some insight into the interdependency of the state variables and controls. It was shown that the linear time varying approximation yielded an accurate representation of limb motion for the thigh and shank but with diminished accuracy for the foot. In contrast, all the linear time invariant systems, if used to simulate more than a quarter of the swing phase, yielded generally inaccurate results for thigh shank and foot motion.

scholarly journals FE Analysis of Communications Systems for Drive-Thru Restaurants in a Business Dispute Over Specifications and Design Process

2021 ◽  
Vol 38 (1) ◽  
Author(s):  
Robert Peruzzi
Keyword(s):  
Communication System ◽  
Communication Systems ◽  
Linear Time ◽  
Forensic Analysis ◽  
Time Varying ◽  
Audio Quality ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Quick Service ◽  
Time Invariant

Forensic analysis in this case involves the design of a communication system intended for use in Quick Service Restaurant (QSR) drive-thru lanes. This paper provides an overview of QSR communication system components and operation and introduces communication systems and channels. This paper provides an overview of non-linear, time-varying system design as contrasted with linear, time-invariant systems and discusses best design practices. It also provides the details of how audio quality was defined and compared for two potentially competing systems. Conclusions include that one of the systems was clearly inferior to the other — mainly due to not following design techniques that were available at the time of the project.

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.

Emulating time varying nonlinear uncertainties and disturbances in linear time invariant systems

SIMULATION ◽  
2012 ◽  
Vol 88 (12) ◽  
pp. 1499-1507 ◽  
Author(s):  
Y Horen ◽  
A Kuperman ◽  
Z Vainer ◽  
S Tapuchi ◽  
M Averbukh
Keyword(s):  
Linear Time ◽  
Time Varying ◽  
Parameter Uncertainties ◽  
External Disturbances ◽  
Nominal System ◽  
Linear Time Invariant ◽  
Nonlinear Uncertainties ◽  
System Output ◽  
Actual Field ◽  
Linear Time Invariant Systems

An approach allowing the creation of parameter uncertainties and external disturbances without any hardware parts supplementary to the nominal system is proposed in this manuscript. The emulating signal, reflecting the plant variations, essential for testing of controllers, is created in software and added to the plant input, forcing the nominal system output to resemble the output of a system with actual uncertainties and disturbances, thus allowing us to test the controller’s robustness prior to an actual field test. In addition, the full state vector of the emulated system may be reconstructed and fed back to the controller, if necessary. The proposed methods allow simultaneous emulation of any combination of time-varying parameter variations and external disturbances. The method can be related to a class of enhanced hardware-in-the-loop simulations, since the nominal hardware is present in the setup in addition to the controller under test. The proposed techniques can be used to test the performance of advanced control algorithms before their mass production. Extended simulation results are reported to confirm the feasibility of the proposed approaches.

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