Adaptive generation of limit cycles in a class of nonlinear systems with unknown parameters and dead-zone nonlinearity

Circuits Syst Signal Process

2021

Self Cite

This paper studies the adaptive limit cycle controller design for shaping stable output oscillations through creating stable limit cycles. For this purpose, a class of time-delay nonlinear systems with multiple time delays and external disturbances is considered. The desirable limit cycle is shaped in the first step by using an extended Lyapunov theorem for stability analysis of positive limit sets. The Lyapunov-Krasovskii theorem is also employed to handle the difficulties raised by multiple time delays. Besides, an adaptive approach is developed to deal with unknown but bounded external disturbances. Unlike the classic approaches, the upper bounds of disturbances are not required to be known in advance. The method is discussed for the second-order system, firstly and then, developed for higher-order systems by the recursive procedure of the backstepping technique. It will be shown that the closedloop system is practically stable and the phase trajectories of the system converge to an arbitrarily small neighborhood of the target limit cycle. Simulation results are provided to show the effectiveness of the proposed approach.

Section: Resultsmentioning

confidence: 99%

“…To prove the stability of the limit cycle S in the controlled subsystem (5), let define the invariant set…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Robust Adaptive Limit Cycle Controller Design for Nonlinear Time-delay Systems

Hakimi

Circuits Syst Signal Process

2021

Self Cite

“…Furthermore, this approach is extended in [40] for MIMO nonlinear systems. The authors of [41] investigated this problem for the non-linear systems with dead-zone non-linearity. In [42,43] a backstepping method is proposed for limit cycle control of non-linear time-delay systems.…”

Section: Introductionmentioning

confidence: 99%

Robust adaptive finite‐time control design for the generation of stable oscillations in the strict‐feedback form non‐linear systems with input dead‐zone non‐linearity

Azhdari

IET Control Theory & Appl

2022

This paper presents a novel finite‐time limit cycle control for a class of strict‐feedback non‐linear systems with input dead‐zone non‐linearity, parameter uncertainties, and unknown disturbances. Creating attractive limit cycles in the phase plane of dynamical systems leads to generating sustained oscillations in the system's output. Achieving the finite‐time limit cycle is an extremely challenging problem because existing finite‐time theorems cannot be applied directly. To solve this difficulty, a novel structure of the sliding surface is introduced concerning the shape of the desired limit cycle. Then, to achieve the control objective, the sliding mode and backstepping control techniques are employed throughout the finite‐time stability concept. Simultaneously, to deal with the parameter uncertainties in the system's dynamic, the adaptive scheme is utilized. Moreover, a finite‐time command filter is introduced to cope with the trouble of the explosion of complexity in the backstepping method. The presented method rigorously ensures that the trajectories of the closed‐loop system converge to a small boundary region of the desired stable limit cycle within a finite time and equivalently the system's output reaches the desired periodic oscillation after a finite time. The applicability and effectiveness of the proposed method are validated by providing the comparative simulation results.

“…[1][2][3][4] For example, in a walking robot, the foot motion follows a repetitive pattern. This behavior can be described by an attractive limit cycle in the phase trajectories of the dynamical system [5][6][7][8] The limit cycle is an isolated periodic orbit in the phase plane, which is a very rich dynamical behavior in the nonlinear dynamical systems. The authors of 9,10 have shown that walking and running can also be addressed as periodic motions, which are equal to specified limit cycles in the phase trajectories of the robotic system.…”

Section: Introductionmentioning

confidence: 99%

Robust Stable Limit Cycle Generation in Multi-Input Mechanical Systems

Karimi

2021

Robotica

SUMMARY This paper proposes a robust controller for the generation of stable limit cycles in multi-input mechanical systems subjected to model uncertainties. The proposed idea is based on Port-Controlled Hamiltonian (PCH) model and energy-based control by considering the Hamiltonian function as the Lyapunov function. For this purpose, first, a nominal controller is designed by shaping the energy function of the system according to the structure of the desired limit cycle. Then, an additional robustifying control term is designed based on the integral sliding mode method with the selection of an appropriate sliding surface. Finally, computer simulations for two practical case studies are provided to confirm the effectiveness of the proposed controller in the generation of stable limit cycles in the presence of uncertainties.