Stabilization of nonlinear systems in the plane

We consider local feedback equivalence of control-affine systems in the plane. We present a general normal form for such systems with scalar control. In this form a polynomial of one variable appears, with coefficients which are functions of another variable. We show that these coefficients (called functional parameters) are invariants. This gives a complete feedback classification of planar systems. The classification results are stated separately for nonequilibrium and equilibrium points. The functional invariants which appear in the normal form have a direct interpretation in terms of singular, time-extremal trajectories of the systems. We show that, roughly, the singular extremals determine the functional invariants of the system and vice versa.

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“…We will only mention deep studies concerning the structure of time-optimal trajectories in the plane (Sussmann [11], [12], [13] [10]). …”

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confidence: 99%

Feedback Classification of Analytic Control Systems in the Plane

Jakubczyk

Respondek

1991

Analysis of Controlled Dynamical Systems

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“…Various sufficient conditions for forward uniqueness can be found in [23]. Let us consider the system (1) where f is continuous on R n and where f has unique solutions in forward time.…”

Section: Finite-time Stabilitymentioning

confidence: 99%

Global finite-time observers for non linear systems

Ménard

Moulay

Perruquetti

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

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“…Earlier results on asymptotic stabilization [18,31] of continuous homogeneous systems are based on the definition of a class of dilations where each state is dilated with a different weight [18]. The notion of geometric homogeneity and its application to stabilization were developed in [19,20].…”

Section: Introductionmentioning

confidence: 99%

“…The motivation also lies in proving robustness to discontinuous disturbances, which is a stronger property than the existing methods for continuous disturbances which utilize the link between homogeneity [18,31] and finite time stability [9]. The motivation also lies in computing the settling time for the class of homogeneous controllers [6,Example 2] in the presence of nonhomogeneous perturbations.…”

Section: Introductionmentioning

confidence: 99%

“…The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers. Earlier results on asymptotic stabilization [18,31] of continuous homogeneous systems are based on the definition of a class of dilations where each state is dilated with a different weight [18]. The notion of geometric homogeneity and its application to stabilization were developed in [19,20].…”

mentioning

confidence: 99%

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Continuous Uniform Finite Time Stabilization of Planar Controllable Systems

Oza¹,

Orlov²,

Spurgeon³

2015

SIAM J. Control Optim.

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Abstract. Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong C 1 Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piecewise continuous nonhomogeneous disturbances. A finite upper bound on the settling time is also computed. The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers. Earlier results on asymptotic stabilization [18,31] of continuous homogeneous systems are based on the definition of a class of dilations where each state is dilated with a different weight [18]. The notion of geometric homogeneity and its application to stabilization were developed in [19,20]. A detailed literature review on the topic of geometric homogeneity is presented in [9], where it is established that geometric homogeneity leads to finite time stability if the homogeneity degree of the asymptotically stable continuous homogeneous system is negative. A result on output feedback synthesis which combines a continuous finite time observer with a continuous finite time controller can be found in [15]. More recently, homogeneous approximations have been studied [3] that led to the development of tools to establish global asymptotic (and in some cases finite time) stability of nonlinear systems. This result used previous results on the so-called homogeneous domination approach (see [30], [3, section 5], and references therein for a detailed literature review).

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Stabilization of nonlinear systems in the plane

Cited by 264 publications

References 8 publications

Feedback Classification of Analytic Control Systems in the Plane

Feedback Classification of Analytic Control Systems in the Plane

Global finite-time observers for non linear systems

Continuous Uniform Finite Time Stabilization of Planar Controllable Systems

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