35,839 research outputs found
Stabilization via generalized homogeneous approximations
We introduce a notion of generalized homogeneous approximation at the origin and at infinity which extends the classical notions and captures a large class of nonlinear systems, including (lower and upper) triangular systems. Exploiting this extension and although this extension does not preserve the basic properties of the classical notion, we give basic results concerning stabilization and robustness of nonlinear systems, by designing a homogeneous (in the generalized sense) feedback controller which globally asymptotically stabilizes a chain of power integrators and makes it the dominant part at infinity and at the origin (in the generalized sense) of the dynamics. Stability against nonlinear perturbation follows from domination arguments
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
Global Output Feedback Stabilization of a Class of Nonlinear Systems With Multiple Output
This paper considers global output feedback stabilization of a class of upper-triangular nonlinear systems with multiple outputs. By coupling a finite-time convergent observer and a saturated homogeneous stabilizer, the global output feedback stabilization can be achieved without the homogeneous growth condition. The proposed techniques are also extended to more general complex nonlinear systems. Various examples, including a ball-and-beam mechanical system and a planar vertical takeoff and landing aircraft, are presented to illustrate the design
Finite-time stabilization of homogeneous non-Lipschitz systems
This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Holder, non-Lipschitz state feedback, which renders the non-Lipschitz systems in the
special case dominated by a lower-triangular nonlinear system finite-time stable. The proof is based
on a recursive design algorithm developed recently to construct the virtual Holder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz nonlinear systems
Exponential stabilization of driftless nonlinear control systems via time-varying, homogeneous feedback
This paper brings together results from a number of different areas in control theory to provide an algorithm for the synthesis of locally exponentially stabilizing control laws for a large class of driftless nonlinear control systems. The stability is defined with respect to a nonstandard dilation and is termed "ÎŽ-exponential" stability. The ÎŽ-exponential stabilization relies on the use of feedbacks which render the closed loop vector field homogeneous with respect to a dilation. These feedbacks are generated from a modification of Pomet's algorithm (1992) for smooth feedbacks. Converse Lyapunov theorems for time-periodic homogeneous vector fields guarantee that local exponential stability is maintained in the presence of higher order (with respect to the dilation) perturbing terms
Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space
We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions
Homogeneous Approximation, Recursive Observer Design, and Output Feedback
We introduce two new tools that can be useful in nonlinear observer and
output feedback design. The first one is a simple extension of the notion of
homogeneous approximation to make it valid both at the origin and at infinity
(homogeneity in the bi-limit). Exploiting this extension, we give several
results concerning stability and robustness for a homogeneous in the bi-limit
vector field. The second tool is a new recursive observer design procedure for
a chain of integrator. Combining these two tools, we propose a new global
asymptotic stabilization result by output feedback for feedback and feedforward
systems
Taming of Modulation Instability by Spatio-Temporal Modulation of the Potential
Spontaneous pattern formation in a variety of spatially extended nonlinear
system always occurs through a modulation instability: homogeneous state of the
system becomes unstable with respect to growing modulation modes. Therefore,
the manipulation of the modulation instability is of primary importance in
controlling and manipulating the character of spatial patterns initiated by
that instability. We show that the spatio-temporal periodic modulation of the
potential of the spatially extended system results in a modification of its
pattern forming instability. Depending on the modulation character the
instability can be partially suppressed, can change its spectrum (for instance
the long wave instability can transform into short wave instability), can split
into two, or can be completely eliminated. The latter result is of especial
practical interest, as can be used to stabilize the intrinsically unstable
system. The result bears general character, as it is shown here on a universal
model of Complex Ginzburg-Landau equations in one and two spatial dimension
(and time). The physical mechanism of instability suppression can be applied to
a variety of intrinsically unstable dissipative systems, like self-focusing
lasers, reaction-diffusion systems, as well as in unstable conservative
systems, like attractive Bose Einstein condensates.Comment: 5 pages, 4 figures, 1 supplementary video fil
Nonlinear Rescaling of Control Laws with Application to Stabilization in the Presence of Magnitude Saturation
Motivated by some recent results on the stabilization of homogeneous systems, we present a gain-scheduling approach for the stabilization of non-linear systems. Given
a one-parameter family of stabilizing feedbacks and associated Lyapunov functions, we show how the parameter can be rescaled as a function of the state to give a new
stabilizing controller. In the case of homogeneous systems, we obtain generalizations of some existing results. We show that this approach can also be applied to nonhomogeneous
systems. In particular, the main application considered in this paper is to the problem of stabilization with magnitude limitations. For this problem, we develop a design method for single-input controllable systems with eigenvalues in the left closed plane
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