Control of Dynamic Systems with Time-Periodic Coefficients via the Lyapunov-Floquet Transformation and Backstepping Technique

2004 ◽  
Vol 10 (10) ◽  
pp. 1517-1533 ◽  
Author(s):  
V. S. Deshmukh ◽  
S. C. Sinha
Keyword(s):  
Asymptotic Stability ◽  
Closed Loop ◽  
Nonlinear Dynamical Systems ◽  
Axial Direction ◽  
Backstepping Technique ◽  
Periodic Force ◽  
Periodic Coefficients ◽  
Linearized System ◽  
Linear And Nonlinear ◽  
Time Periodic

We address the problem of designing controllers that guarantee asymptotic stability of a class of linear as well as nonlinear dynamical systems with time-periodic coefficients. Using a repeated procedure consisting of the Lyapunov-Floquet transformation, the backstepping technique, and Floquet theory, the asymptotic stability of the closed-loop linearized system is guaranteed. Further, a Lyapunov matrix for the closed-loop asymptotically stable linearized system is constructed. This Lyapunov function is then used to design a combination of linear and nonlinear controllers in order to guarantee the asymptotic stability of the nonlinear system. The methodology is illustrated by designing linear and nonlinear control laws for a system consisting of two statically coupled pendula, each subjected to a time-periodic force acting in the axial direction.

Quasi-Periodic Motions of Articulated Pipes Conveying Flowing Fluid

2001 ◽  
Author(s):  
Zsolt Szabó
Keyword(s):  
Autonomous System ◽  
Dynamic Behaviour ◽  
Periodic Motions ◽  
Flowing Fluid ◽  
Nonlinear Part ◽  
Linearized System ◽  
Linear And Nonlinear ◽  
A Chain ◽  
The Stability ◽  
Time Periodic

Abstract In this paper two nice examples are investigated where a ‘chain’ of n = (1, 2) pieces of rigid pipes contains incompressible and frictionless flowing fluid. We give an overview about the linear and nonlinear analysis of the autonomous system, i.e. when the pipes contain steady flow. Assuming pulsatile flow, the system becomes time-periodic. The stability charts of the linearized system are generated applying a numerical method based on Chebyshev polynomials. Finally, we analyze the effect of the nonlinear part in some critical points of the obtained stability charts and the dynamic behaviour of the original nonlinear periodic system is simulated numerically. The results are shown in Poincaré maps and bifurcation diagrams.

Stabilization of Nonlinear Strict-Feedback Systems With Time-Varying Delayed Integrators

2011 ◽  
Author(s):  
Nikolaos Bekiaris-Liberis ◽  
Miroslav Krstic
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Closed Loop ◽  
Loop System ◽  
Time Varying ◽  
Feedback Systems ◽  
Closed Loop System ◽  
Feedback Form ◽  
Globally Asymptotically Stable ◽  
Strict Feedback

We consider nonlinear systems in the strict-feedback form with simultaneous time-varying input and state delays, for which we design a predictor-based feedback controller. Our design is based on time-varying, infinite-dimensional backstepping transformations that we introduce, to convert the system to a globally asymptotically stable system. The solutions of the closed-loop system in the transformed variables can be found explicitly, which allows us to establish its global asymptotic stability. Based on the invertibility of the backstepping transformation, we prove global asymptotic stability of the closed-loop system in the original variables. Our design is illustrated by a numerical example.

scholarly journals Nonlinear feedback controller for the synchronization of (chaotic) systems with known parameters

2020 ◽  
Vol 23 (02) ◽  
pp. 124-135
Author(s):  
Muhammad Haris ◽  
Muhammad Shafiq ◽  
Adyda Ibrahim ◽  
Masnita Misiran
Keyword(s):  
Closed Loop ◽  
Lyapunov Stability Theory ◽  
Feedback Controller ◽  
Control Signal ◽  
Stability And Convergence ◽  
Nonlinear Feedback ◽  
Nonlinear Part ◽  
Synchronization Errors ◽  
Novel Structure ◽  
Linear And Nonlinear

This paper proposes, designs, and analyses a novel nonlinear feedback controller that realizes fast, and oscillation free convergence of the synchronization error to the equilibrium point. Oscillation free convergence lowers the failure chances of a closed-loop system due to the reduced chattering phenomenon in the actuator motion, which is a consequence of low energy sm ooth control signal. The proposed controller has a novel structure. This controller does not cancel nonlinear terms of the plant in the closed-loop; this attribute improves the robustness of the loop. The controller consists of linear and nonlinear parts; each part executes a specific task. The linear term in the controller keeps the closed-loop stable, while the nonlinear part of the controller facilitates the fast convergence of the error signal to the vicinity of the origin. Then the linear controller synthesizes a smooth control signal that moves the error signals to zero without oscillations. The nonlinear term of the controller does not contribute to this synthesis. The collaborative combination of linear and nonlinear controllers that drive the synchronization errors to zero is innovative. The paper establishes proof of global stability and convergence behavior by describing a detailed analysis based on the Lyapunov stability theory. Computer simulation results of two numerical examples verify the performance of the proposed controller approach. The paper also provides a comparative study with state-of-the-art controllers.

scholarly journals The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jifeng Chu ◽  
Ting Xia
Keyword(s):  
Periodic Function ◽  
Lyapunov Stability ◽  
Stability Criterion ◽  
Normal Forms ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Time Periodic ◽  
Damped Oscillator

Leta(t),b(t)be continuousT-periodic functions with∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillatorx′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillatorx′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, wheren≥2,c(t)is a continuousT-periodic function,e(t,x)is continuousT-periodic intand dominated by the powerx2nin a neighborhood ofx=0.

A PREY-PREDATOR MODEL WITH DIFFUSION AND A SUPPLEMENTARY RESOURCE FOR THE PREY IN A TWO‐PATCH ENVIRONMENT

2005 ◽  
Vol 9 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J. Dhar
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Prey Species ◽  
Steady State Solution ◽  
State Solution ◽  
Prey Population ◽  
Predator Species ◽  
Additional Resource ◽  
Linear And Nonlinear ◽  
Predator Model

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases. Šiame straipsnyje modeliuojama ir analizuojama plešr‐unu ir auku dinamika, laikant, kad plešr-unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas. Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

scholarly journals ON STABILITY CRITERIA OF FRACTAL DIFFERENTIAL SYSTEMS OF CONFORMABLE TYPE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040009
Author(s):  
AWAIS YOUNUS ◽  
THABET ABDELJAWAD ◽  
TAZEEN GUL
Keyword(s):  
Asymptotic Stability ◽  
Control Theory ◽  
Stability Criteria ◽  
Differential Systems ◽  
Central Concern ◽  
Finite Dimensional ◽  
Linear And Nonlinear ◽  
Stability Results

In this paper, stability results of central concern for control theory are given for finite-dimensional linear and nonlinear local fractional or fractal differential systems. The main purpose of this paper is to provide some results on stability and asymptotic stability of conformable order systems, together with some illustrating examples.

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