Reduction theorems for stability of closed sets with application to backstepping control design

El-Hawwary, Mohamed I.; Maggiore, Manfredi

doi:10.1016/j.automatica.2012.09.013

Cited by 67 publications

(75 citation statements)

References 18 publications

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“…3, the set Γ i was asymptotically stable relative to Γ i+1 . Consequently, according to Proposition 14 in [29], the set Γ 1 is asymptotically stable for the controlled system. This implies that all the control objectives I-IV will be achieved, and all the solutions of the controlled system remain uniformly bounded.…”

Section: Maneuvering Control: the Geometric Taskmentioning

confidence: 99%

“…Note that we could have chosenλ to be given by the compensator in (29). However, since (29) represents the frequency of the lateral undulation motion given in (10) it is desirable from a practical implementation point of view to smooth the frequency function.…”

Section: Maneuvering Control: the Dynamic Taskmentioning

confidence: 99%

“…However, since (29) represents the frequency of the lateral undulation motion given in (10) it is desirable from a practical implementation point of view to smooth the frequency function. We thus introduce an extra step of backstepping, and define the second error variable as z 2 =λ − δ 2 which we aim to drive to zero.…”

Section: Maneuvering Control: the Dynamic Taskmentioning

confidence: 99%

“…The approach that we present here uses the theoretical results of [29] by application of reduction theorems for asymptotic stability of closed sets with application to backstepping control design. Path following and maneuvering control approaches for a single snake robot have previously been considered in e.g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Formation control of underactuated bio-inspired snake robots

Rezapour¹,

Pettersen

Gravdahl

et al. 2016

Artif Life Robotics

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This paper considers formation control of snake robots. In particular, based on a simplified locomotion model, and using the method of virtual holonomic constraints, we control the body shape of the robot to a desired gait pattern defined by some prespecified constraint functions. These functions are dynamic in that they depend on the state variables of two compensators which are used to control the orientation and planar position of the robot, making this a dynamic maneuvering control strategy. Furthermore, using a formation control strategy we make the multiagent system converge to and keep a desired geometric formation, and enforce the formation follow a desired straight line path with a given speed profile. Specifically, we use the proposed maneuvering controller to solve the formation control problem for a group of snake robots by synchronizing the commanded velocities of the robots. Simulation results are presented which illustrate the successful performance of the theoretical approach.

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Section: Maneuvering Control: the Geometric Taskmentioning

confidence: 99%

Section: Maneuvering Control: the Dynamic Taskmentioning

confidence: 99%

Section: Maneuvering Control: the Dynamic Taskmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Formation control of underactuated bio-inspired snake robots

Rezapour¹,

Pettersen

Gravdahl

et al. 2016

Artif Life Robotics

View full text Add to dashboard Cite

show abstract

“…Other than methods such as central pattern generators, using VHCs to encode the robotic gait has the advantage that it makes the control design amenable to hierarchical synthesis, where the gait is enforced at the lowest level and velocity and orientation control are done at a higher level. To this end, we will employ the hierarchical control design methodology from [13], that was used for control design for ships in [14] and terrestrial snake robots in [15]. In order to make the feedback independent of the unknown relative velocities that enter the dynamic equations via the fluid drag forces, we will make use of adaptive backstepping control [16].…”

Section: Introductionmentioning

confidence: 99%

Velocity and orientation control of underwater snake robots using absolute velocity feedback

Kohl

Pettersen

Gravdahl

2017

2017 IEEE Conference on Control Technology and Applications (CCTA)

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Abstract-This article presents a control system for velocity and orientation control of underwater snake robots using absolute velocity feedback. The control system is structured in a hierarchical way, where the highest priority is to enforce virtual constraints encoding a planar gait on the body shape of the robot. To this end, we propose an adaptive joint controller and show that it asymptotically stabilizes the constraint manifold. The virtual constraints are parametrized by the states of two dynamic compensators, which can be used to control the velocity and the orientation of the robot, the second and third control priority. We design an adaptive controller that asymptotically stabilizes the forward velocity to a reference and an orientation controller that utilizes the current estimate of the velocity controller. It is shown that the zero dynamics of the closed-loop system remains bounded, and we present simulation results that demonstrate the performance of the controller.

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Comments on recent claims about trajectories of control systems valid for particular initial conditions

Ortega

2021

Asian Journal of Control

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In recent years, we have witnessed the appearance in the control literature of claims regarding the behavior of the trajectories of closed-loop systems, which are valid only for a specific set of initial conditions (ICs). Being these "trajectory-dependent" claims, a natural question that arises is whether these claims are robust, in some well-defined sense. Regarding Lyapunov stability claims, it is well-known that this property is equivalent to a form of continuity of solutions with respect to the ICs. For linear time-invariant (LTI) systems characterized by transfer matrices, the property of internal stability is widely adopted as a necessary condition to ensure robustness. However, to the best of our knowledge, this question has not been addressed for claims-different from stability-concerning the behavior of trajectories of nonlinear time-varying (NLTV) systems, which is the scenario in the aforementioned claims. The main objective of this note is to propose a framework for the characterization of robustness or fragility (with respect to ICs) of claims of this nature for NLTV systems.

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Reduction theorems for stability of closed sets with application to backstepping control design

Cited by 67 publications

References 18 publications

Formation control of underactuated bio-inspired snake robots

Formation control of underactuated bio-inspired snake robots

Velocity and orientation control of underwater snake robots using absolute velocity feedback

Comments on recent claims about trajectories of control systems valid for particular initial conditions

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