Bounded-input Bounded-state Research Articles

Letter on Convergence of In-Parameter-Linear Nonlinear Neural Architectures With Gradient Learnings.

This letter summarizes and proves the concept of bounded-input bounded-state (BIBS) stability for weight convergence of a broad family of in-parameter-linear nonlinear neural architectures (IPLNAs) as it generally applies to a broad family of incremental gradient learning algorithms. A practical BIBS convergence condition results from the derived proofs for every individual learning point or batches for real-time applications.

Frequency-domain stability methods for reset control systems

Reset controllers have the potential to enhance the performance of high-precision industrial motion systems. However, similar to other non-linear controllers, the stability analysis for these controllers is complex and often requires parametric model of the system, which may hinder their applicability. In this paper a frequency-domain approach for assessing stability properties of control systems with first and second order reset elements is developed. The proposed approach is also able to determine uniformly bounded-input bounded-state (UBIBS) property for reset control systems in the case of resetting to non-zero values. An illustrative example to demonstrate the effectiveness of the proposed approach in using frequency response measurements to assess stability properties of reset control systems is presented.

State estimation for a class of uncertain nonlinear systems: a finite-time observer approach

ABSTRACT In this paper, the problem of state estimation for a class of uncertain, and possibly unbounded, nonlinear systems is tackled. In order to deal with such a problem, a nonlinear observer is proposed based on a continuous homogeneous and Levant's nth order differentiators. The proposed nonlinear observer provides finite-time convergence and the observer synthesis is formulated in terms of some linear matrix inequalities (LMIs). Moreover, contrary to what the literature shows, the finite-time observer does not require any type of Bounded-Input Bounded-State (BIBS) condition. Some simulation results illustrate the effectiveness of the proposed finite-time observer applied to an unstable nonlinear system.

Dynamic positioning for underactuated surface vessel via L1 adaptive backstepping control

This paper is concerned with the problem that fast-transient response and excellent robustness cannot be satisfied simultaneously in the process of dynamic positioning (DP) for underactuated surface vessel (USV) in shallow water. By combing the improved L1 adaptive control with backstepping method, a novel control scheme is designed, which can ensure a fast adaptation with a guaranteed smooth transient response without any overshoot and chattering phenomenon. System uncertainties and disturbances are estimated by the nonlinear observer. Moreover, the optimized extremum seeking control (ESC) is employed to reduce energy consumption under environmental disturbances. Rigorous theoretical analysis shows that all closed-loop signals are bounded-input bounded-state. Simulation and sea test results are presented to illustrate the effectiveness and the robustness of the proposed strategy under the condition of external disturbances and parametric uncertainties.

Mittag-Leffler input stability of fractional differential equations and its applications

This paper addresses the Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. We continuous the first note. We discuss three properties of the Mittag-Leffler input stability: converging-input converging-state, bounded-input bounded-state, and Mittag-Leffler stability of the unforced fractional differential equation. We present the Lyapunov characterization of the Mittag-Leffler input stability, and conclude by introducing the fractional input stability for delay fractional differential equations, and we provide its Lyapunov-Krasovskii characterization. Several examples are treated to highlight the Mittag-Leffler input stability.

An Inverse Neural Controller Based on the Applicability Domain of RBF Network Models.

This paper presents a novel methodology of generic nature for controlling nonlinear systems, using inverse radial basis function neural network models, which may combine diverse data originating from various sources. The algorithm starts by applying the particle swarm optimization-based non-symmetric variant of the fuzzy means (PSO-NSFM) algorithm so that an approximation of the inverse system dynamics is obtained. PSO-NSFM offers models of high accuracy combined with small network structures. Next, the applicability domain concept is suitably tailored and embedded into the proposed control structure in order to ensure that extrapolation is avoided in the controller predictions. Finally, an error correction term, estimating the error produced by the unmodeled dynamics and/or unmeasured external disturbances, is included to the control scheme to increase robustness. The resulting controller guarantees bounded input-bounded state (BIBS) stability for the closed loop system when the open loop system is BIBS stable. The proposed methodology is evaluated on two different control problems, namely, the control of an experimental armature-controlled direct current (DC) motor and the stabilization of a highly nonlinear simulated inverted pendulum. For each one of these problems, appropriate case studies are tested, in which a conventional neural controller employing inverse models and a PID controller are also applied. The results reveal the ability of the proposed control scheme to handle and manipulate diverse data through a data fusion approach and illustrate the superiority of the method in terms of faster and less oscillatory responses.

How to have narrow-stripe semiconductor lasers self-pulsate

The rate equations of narrow-stripe semiconductor lasers are considered. These equations represent the dynamics of the photon number and the electron densities in the active and absorbing regions. Having the rate equations, (i) it is shown that the laser is the bounded-input bounded-state (BIBS) and the bounded-input bounded-output (BIBO) stable; (ii) the amplitudes of step inputs are determined for which all equilibrium points of the laser are unstable. The boundedness of the laser output and the instability of its equilibrium points imply that the laser can have a periodic, a quasi-periodic, or a chaotic output. When the output is periodic, the laser is self-pulsating, which is the desirable behavior of the laser. Moreover, a procedure is given to determine the values of laser parameters for which the laser self-pulsates.

Generation of self-pulsation in passively Q-switched lasers

The rate equations of passively Q-switched lasers are considered. These equations represent the dynamics of the photon number in the laser cavity, the population inversion of the laser, and the ground-state population of the saturable absorber, when excited by an input (pumping rate). Having the rate equations, (i) it is shown that the laser is the bounded-input bounded-state (BIBS) and the bounded-input bounded-output (BIBO) stable; (ii) the amplitudes of step inputs are determined for which all equilibrium points of the laser are unstable. The boundedness of the laser output and the instability of its equilibrium points imply that the laser can have a periodic, a quasi-periodic, or a chaotic output. When the output is periodic, the laser is self-pulsating, which is the desirable behavior of the laser.

A system theoretic approach to the self-pulsation of passively Q-switched lasers

The rate equations of passively Q-switched lasers are considered. These equations represent the dynamics of the photon number in the laser cavity, the population inversion of the laser, and the ground-state population of the saturable absorber, when excited by an input (pumping rate). Having the rate equations, (1) it is shown that the laser is the bounded input-bounded state (BIBS) and the bounded input-bounded output (BIBO) stable and (2) the amplitudes of step inputs are determined for which all equilibrium points of the laser are unstable. The boundedness of the laser output and the instability of its equilibrium points imply that the laser can have a periodic, a quasiperiodic, or a chaotic output. When the output is periodic, the laser is self-pulsating, which is the desirable behavior of the laser.

A Necessary and Sufficient Condition for Bounded-Input Bounded-State Stability of Nonlinear Systems

The main purpose of this work is to prove a converse theorem for bounded-input bounded-state stability of nonlinear systems in the framework of Liapunov's second method. The construction gives rise to an upper semicontinuous time-dependent Liapunov function. In certain cases, the monotonicity conditions can be checked by means of contingent directional derivatives.

Necessary and Sufficient Conditions for Bounded-Input/Bounded-state Stability of Discrete-Time, Bilinear Systems

In this note, bounded-input/bounded state (BIBS) stability is studied for discrete-time multi-input, bilinear systems. The main result gives generic necessary and sufficient conditions for BIBS stability of such systems. These conditions comprise the existence of a single similarity transformation that takes to triangular form all the matrices defining a multi-input bilinear system, as well as constraints on their spectral radiuses.

Sufficient excitation and stable reduced-order adaptive IIR filtering

A perturbed error system is used to describe the parameter and output error behaviors of reduced-order application of adaptive identifier/filters such as the hyperstable adaptive recursive filter (HARF). Given satisfaction of a sufficient excitation condition, this error system is shown to exhibit a bounded-input, bounded-state (BIBS) property. This implies that, despite order insufficiency, the output and parameter estimates of HARF (and similar adaptive identifier/filter algorithms) remain bounded.

Singular perturbation and bounded-input bounded-state stability

Re represents a linear time-invariant system of order n + h. When e → 0, Re, reduces to R0 of order n ≤ n + h. Conditions are given under which the bounded-input bounded-state stability of R0 implies that of Re and we show that, under these conditions, as e → 0, the state trajectories of Re approach those of R0 for any continuous input.