Orbital stabilization of nonlinear systems via the immersion and invariance technique

Romero, José Guadalupe; Astolfi, Alessandro

doi:10.1002/rnc.4861

Cited by 23 publications

(32 citation statements)

References 29 publications

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“…In this article we solve the PFP formulating it as an orbital stabilization problem similar to the one studied in Reference 6—where an extension to the I&I technique 11 originally applied for equilibria stabilization was proposed. We recall that this problem is to generate stable periodic solutions for the nonlinear system

\dot{x} = f (x) + g (x) u,

with state

x \in ℝ^{2 n}

, input

u \in ℝ^{m}

, with m < 2 n , and g ( x ) being full rank.…”

Section: An Iandi Approach To Orbital Stabilizationmentioning

confidence: 99%

“…The main contributions of the article are 2‐fold. The new reformulation of the PFP is solved by proposing an extension to the recently developed Immersion and Invariance (I&I) orbital stabilization method reported in Reference 6 and using the Mexican sombrero energy‐shaping construction of Reference 7. The resulting controller is a globally defined static state‐feedback that, without the appeal of any guidance laws, guarantees almost global convergence. The result is applied to the benchmark problem of underactuated marine surface vessels.…”

Section: Introductionmentioning

confidence: 99%

“…This assumption is assigned for simplicity to present the main idea, which can be relaxed since the relative degree is unnecessary to stabilize the off-the-manifold coordinate. Moreover, many underactuated mechanical systems satisfy such an assumption 6. For future reference, we defined the vector q and its corresponding partition, with q y the output to be regulated with the remainingcoordinates (q N ,q N )bounded.…”

mentioning

confidence: 99%

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Path following of a class of underactuated mechanical systems via immersion and invariance‐based orbital stabilization

Manchester

Siguerdidjane

2020

Intl J Robust & Nonlinear

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This article aims to provide a new problem formulation of path following for mechanical systems without time parameterization nor guidance laws, namely, we express the control objective as an orbital stabilization problem. It is shown that it is possible to adapt the immersion and invariance technique to design static state-feedback controllers that solve this problem. In particular, we select the target dynamics adopting the recently introduced Mexican sombrero energy assignment method. To demonstrate the effectiveness of the proposed method we apply it to control underactuated marine surface vessels. K E Y W O R D S energy shaping, immersion and invariance, orbital stabilization, path following 1 INTRODUCTION Tracking a geometric path in mechanical systems is a task often encountered in various application fields, for example, robotics, aerospace, and autonomous vehicles. There are two approaches to formulate this problem, depending on whether the predefined path is parameterized by time or not, and they are known as trajectory tracking or path following, respectively. The former approach largely dominates the control literature and it has been extensively studied. Unfortunately, for nonminimum phase systems, which are common in these applications, the achievable performance has a fundamental limitation, first identified in Reference 1. It is possible to circumvent this limitation removing the time

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\dot{x} = f (x) + g (x) u,

with state

x \in ℝ^{2 n}

, input

u \in ℝ^{m}

, with m < 2 n , and g ( x ) being full rank.…”

Section: An Iandi Approach To Orbital Stabilizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Path following of a class of underactuated mechanical systems via immersion and invariance‐based orbital stabilization

Manchester

Siguerdidjane

2020

Intl J Robust & Nonlinear

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“…This is a particular case of the framework adopted in the immersion and invariance (I&I) technique, first reported for equilibrium stabilization in [2], and later extended for observer design and adaptive control in [3]. In [25] it has recently been shown that I&I can also be adapted for orbital stabilization, leading to a procedure that contains, as particular case, the VHC designs. The only modification done to the standard I&I technique is in the definition of the target dynamics that now should be chosen possessing periodic orbits, instead of an equilibrium at the desired point.…”

Section: Introductionmentioning

confidence: 99%

“…The only modification done to the standard I&I technique is in the definition of the target dynamics that now should be chosen possessing periodic orbits, instead of an equilibrium at the desired point. A main drawback in both the VHC and I&I methods is that the steady-state behavior cannot be fixed a priori, but depends on the initial states, see [25,Remark 2] for a discussion on this matter.…”

Section: Introductionmentioning

confidence: 99%

Orbital stabilization of nonlinear systems via Mexican sombrero energy shaping and pumping-and-damping injection

Ortega

et al. 2020

Automatica

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In this paper we show that a slight modification to the widely popular interconnection and damping assignment passivity-based control method-originally proposed for stabilization of equilibria of nonlinear systems-allows us to provide a solution to the more challenging orbital stabilization problem. Two different, though related, ways how this procedure can be applied are proposed. First, the assignment of an energy function that has a minimum in a closed curve, i.e., with the shape of a Mexican sombrero. Second, the use of a damping matrix that changes "sign" according to the position of the state trajectory relative to the desired orbit, that is, pumping or dissipating energy. The proposed methodologies are illustrated with the example of the induction motor and prove that it yields the industry standard field oriented control.

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Robust orbital stabilization: A Floquet theory–based approach

Sætre¹,

Shiriaev

Freidovich

et al. 2021

Intl J Robust & Nonlinear

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The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal, disturbance‐free system, a robustifying feedback extension is designed utilizing the sliding‐mode control (SMC) methodology. The main contribution of the article is to provide a constructive procedure for designing the time‐invariant switching function used in the SMC synthesis. More specifically, its zero‐level set (the sliding manifold) is designed using a real Floquet–Lyapunov transformation to locally correspond to an invariant subspace of the Monodromy matrix of a transverse linearization. This ensures asymptotic stability of the periodic orbit when the system is confined to the sliding manifold, despite any system uncertainties and external disturbances satisfying a matching condition. The challenging task of oscillation control of the underactuated cart–pendulum system subject to both matched‐ and unmatched disturbances/uncertainties demonstrates the efficacy of the proposed scheme.

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Orbital stabilization of nonlinear systems via the immersion and invariance technique

Cited by 23 publications

References 29 publications

Path following of a class of underactuated mechanical systems via immersion and invariance‐based orbital stabilization

Path following of a class of underactuated mechanical systems via immersion and invariance‐based orbital stabilization

Orbital stabilization of nonlinear systems via Mexican sombrero energy shaping and pumping-and-damping injection

Robust orbital stabilization: A Floquet theory–based approach

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