Underactuated Nonlinear Systems with Non-Stabilizable Linear Approximation: A Case Study

1998 ◽  
Vol 31 (27) ◽  
pp. 213-218
Author(s):  
M.C. Laiou ◽  
A. Astolfi
Keyword(s):  
Nonlinear Systems ◽  
Linear Approximation

scholarly journals Approximation of Linearized Systems to a Class of Nonlinear Systems Based on Dynamic Linearization

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 854
Author(s):  
Raquel S. Rodríguez ◽  
Gilberto Gonzalez Avalos ◽  
Noe Barrera Gallegos ◽  
Gerardo Ayala-Jaimes ◽  
Aaron Padilla Garcia
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Nonlinear Dynamic ◽  
Bond Graph ◽  
Physical Domain ◽  
Graph Models ◽  
Simple Task ◽  
Analysis And Synthesis ◽  
Simulation Results

An alternative method to analyze a class of nonlinear systems in a bond graph approach is proposed. It is well known that the analysis and synthesis of nonlinear systems is not a simple task. Hence, a first step can be to linearize this nonlinear system on an operation point. A methodology to obtain linearization for consecutive points along a trajectory in the physical domain is proposed. This type of linearization determines a group of linearized systems, which is an approximation close enough to original nonlinear dynamic and in this paper is called dynamic linearization. Dynamic linearization through a lemma and a procedure is established. Therefore, linearized bond graph models can be considered symmetric with respect to nonlinear system models. The proposed methodology is applied to a DC motor as a case study. In order to show the effectiveness of the dynamic linearization, simulation results are shown.

Relative Stability Via the Direct Method of Lyapunov

1964 ◽  
Vol 86 (1) ◽  
pp. 87-90 ◽  
Author(s):  
W. G. Vogt
Keyword(s):  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Control Theory ◽  
Linear Approximation ◽  
Relative Stability ◽  
Direct Method ◽  
Positive Definite ◽  
Error Criterion ◽  
Definition Of ◽  
Classical Control

A mathematically rigorous concept of relative stability based on the v-functions of the direct method of Lyapunov is introduced. Two systems of the type representable by x˙ = f(x) are considered, where under the proper restrictions on f(x), a Lyapunov function, v(x) is uniquely determined by a positive definite error criterion r(x) and the equation v˙(x) = −r(x). The definition of the relative stability proposed, eventually leads to conditions on the linear approximation systems which are sufficient to assure the relative stability of the nonlinear systems. This leads to conditions on the eigenvalues of the linear approximation system which are necessary but not sufficient for relative stability. Additional conditions on the choice of the error criteria are needed. The present definition permits the gap between concepts of stability in classical control theory and that due to the direct method of Lyapunov to be at least partially bridged.

Stabilization of Nonlinear Systems Using Linear Observers

1974 ◽  
Vol 96 (1) ◽  
pp. 55-60 ◽  
Author(s):  
R. E. Strane ◽  
W. G. Vogt
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Nonlinear Equations ◽  
Linear Approximation ◽  
Asymptotically Stable ◽  
State Variables ◽  
Finite Region ◽  
Linear Portion ◽  
Linear Observer

In this paper, it is shown that a linear observer can always be designed to stabilize a nonlinear system which contains a Lur’e type nonlinearity in the sector [0, k], where k is finite, if both the output of the nonlinearity and a completely observable output of the linear portion are available as inputs to the observer. In case a completely observable output is not available from the linear portion, stabilization is shown to be possible if the original linear approximation of the system is asymptotically stable or those state variables corresponding to the unstable eigenvalues are available. It is also established that a linear observer can be used to guarantee that a finite region of asymptotic stability exists for a plant described by a more general set of nonlinear equations, and in some cases the domain of asymptotic stability can be made as large as desired.

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