Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

Lyapunov functions enable analyzing the stability of dynamic systems described by ordinary differential equations without finding the solution of such equations. For nonlinear systems, devising a Lyapunov function is not an easy task to solve in general. In this paper, we present an approach to the construction of Lyapunov funtions to prove stability in estimation problems. To this end, we motivate the adoption of input-to-state stability (ISS) to deal with the estimation error involved by state observers in performing state estimation for nonlinear continuous-time systems. Such stability properties are ensured by means of ISS Lyapunov functions that satisfy Hamilton–Jacobi inequalities. Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions.

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Stability of fractional positive nonlinear systems

Archives of Control Sciences ◽

10.1515/acsc-2015-0031 ◽

2015 ◽

Vol 25 (4) ◽

pp. 491-496 ◽

Cited By ~ 13

Author(s):

Tadeusz Kaczorek

Keyword(s):

Nonlinear Systems ◽

Continuous Time ◽

Lyapunov Method ◽

Linear Part ◽

Stability Conditions ◽

Continuous Time Systems ◽

Nonlinear Vector ◽

The Stability ◽

Metzler Matrix ◽

Time Systems

AbstractThe conditions for positivity and stability of a class of fractional nonlinear continuous-time systems are established. It is assumed that the nonlinear vector function is continuous, satisfies the Lipschitz condition and the linear part is described by a Metzler matrix. The stability conditions are established by the use of an extension of the Lyapunov method to fractional positive nonlinear systems.

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State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽

10.3390/math8091424 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1424 ◽

Cited By ~ 2

Author(s):

Angelo Alessandri ◽

Patrizia Bagnerini ◽

Roberto Cianci

Keyword(s):

Dynamical Systems ◽

Asymptotic Stability ◽

Lyapunov Functions ◽

Nonlinear Dynamical Systems ◽

Estimation Error ◽

Stability Conditions ◽

Nonlinear Dynamical ◽

State Observation ◽

State Observers ◽

The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

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