Class Of Planar Nonlinear Systems Research Articles

Finite-time integral control for a class of nonlinear planar systems with non-vanishing uncertainties

This paper considers the problem of globally regulating a class of planar nonlinear systems perturbed by various non-vanishing uncertainties including constant step disturbances, exogenous time-varying disturbances with unknown magnitudes, and modeling uncertainties with unknown system parameters. A new integral controller consisting of a nonlinear integral dynamic and a semi-linear control law is constructed to drive the states of the uncertain systems to the origin in a finite time. This is achieved by three major mechanisms: (i) for the purpose of finite-time convergence, a lower-order integral dynamic is first constructed; (ii) by revamping the technique of adding a power integrator, a semi-linear control law containing a linear corrective term is proposed to handle the various forms of uncertainties; and (iii) a new inequality is established to provide an effective estimating tool for the selection of a suitable control gain to guarantee finite-time stability.

Finite-Time Controllers for a Class of Planar Nonlinear Systems With Mismatched Disturbances

This letter considers the problem of globally finite-time stabilizing a planar nonlinear system with a mismatched unknown disturbance. First, a nonlinear integral dynamic is constructed to compensate the disturbance. For finite-time convergence, we use the system states and integral state to construct a new controller which not only is homogeneous with a negative degree but also contains a linear combination of states to handle the unmatched disturbance. The finite-time integral controller originally obtained for a linear planar system is appropriately scaled to regulate a nonlinear planar system within the framework of homogeneous domination. In addition, for a faster convergence speed for any initial condition, a dual-mode integral controller is introduced with better performances demonstrated by simulation studies.

A necessary and sufficient condition for stability of a class of planar nonlinear systems

In this paper, the stability problem of a class of planar nonlinear systems is investigated. Motivated by the Routh–Hurwitz stability criterion for planar linear systems, a necessary and sufficient condition for stability of a class of planar nonlinear systems is established. To prove the necessity, a new method called particular solution method is provided to justify instability. In addition, a sufficient condition for existence of oscillations is introduced. The necessary and sufficient condition can provide a simple way to design controllers by adjusting control parameters for some planar nonlinear systems.

Global Output-feedback Stabilization for Nonlinear Time-delay Systems with Unknown Control Coefficients

This paper investigates the problem of global stabilization by output-feedback for a class of planar nonlinear systems with unknown time-delay and control coefficients. Compared with the closely related works, the highlight of the paper is that it is not required that the upper and lower bounds of unknown control coefficients are known a prior, and meanwhile, the derivative of unknown time-varying time-delay is also not required to have known upper bound. This makes the problem essentially different, and much more difficult to solve. Motivated by our recent works, by combining the method of universal control with backstepping technique, and skillfully constructing a new Lyapunov-Krasoviskii functional, we design a new adaptive stabilizing controller with the suitable design parameters which guarantees that the resulting closed-loop system state is globally bounded while the state of the original system converges to the origin after a finite time. A numerical example is given to illustrate the effectiveness of the theoretical results.

Global Output-Feedback Adaptive Stabilization for Planar Nonlinear Systems with Unknown Growth Rate and Output Function

This paper studies the problem of global output-feedback stabilization by adaptive method for a class of planar nonlinear systems with unmeasurable state dependent growth and unknown output function. It is worth emphasizing that not only the growth rate, but also the upper and lower bounds of the derivative of output function are not required to be known in the paper. To solve the problem, we propose a new adaptive output-feedback control scheme based on only one dynamic high-gain, by skillfully constructing the new Lyapunov function, and flexibly using the ideas of universal control and backstepping. It is shown that the state of the closed-loop system is bounded while global asymptotic stability can be achieved. Two examples including a practical example are given to demonstrate the effectiveness of the theoretical results.

A dual‐observer approach for global output feedback stabilization of planar nonlinear systems with output‐dependent growth rates

SummaryIn this paper, we investigate the problem of global output feedback stabilization for a class of planar nonlinear systems under a more general growth condition, which encompasses both lower‐order and higher‐order state growths with output‐dependent rates. For more accurate estimation, two new observers with nonlinear gains are constructed to estimate the states on the lower‐order and higher‐order scales, respectively. The estimates produced from the dual‐observer are used delicately in the output feedback control law with both lower‐order and higher‐order modes. The overall stability of the system is guaranteed by rigorously choosing these nonlinear gains in the control law and the dual‐observer.

Semi-global finite-time stabilisation via output feedback of planar non-linear systems with application to MPPT in photovoltaic systems

This paper investigates the problem of using output feedback to semi-globally stabilise a class of planar non-linear systems with unknown control coefficients. For any given bounded set of initial condition, a non-smooth output feedback controller using a reduced-order observer is explicitly constructed to stabilise the planar system in a finite time. We also show that the proposed non-smooth output feedback controller can be used to solve the problem of Maximum Power Point Tracking (MPPT) in Photovoltaic (PV) arrays for certain solar radiation and temperature.

Stability analysis of planar systems with nilpotent (non-zero) linear part

A simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization, which is nilpotent and not zero. The condition is based on the use of Darboux polynomials, to derive a Lyapunov function for the first approximation of the system with respect to a vector function representing a dilation. Results by Andreev (1958), on the qualitative behaviour of equilibrium points, and the Belitskii normal form are used in the proofs. The proposed condition is applied to the asymptotic stabilization of the origin, for a class of planar systems whose linearization cannot be stabilized.

Adaptive output feedback control for a class of planar nonlinear systems

AbstractThis paper is concerned with the problem of global adaptive stabilization by output feedback for a class of planar nonlinear systems with uncertain control coefficient and unknown growth rate. The control coefficient is not supposed to have known upper bound, and this relaxes the corresponding requirement in the existing literature (see e.g. 1, 2. First, by the universal control method, an observer is constructed based on the dynamic high‐gain K‐filters. Then, the control design procedure is developed to obtain the stabilizing controller and dynamic compensator for the uncertainties in the control coefficient. It is shown that the global stability of the closed‐loop system can be guaranteed by the appropriate choice of the design parameters. A simulation example is also provided to illustrate the correctness of the theoretical results. © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society.