Mathematics of Control Signals and Systems最新文献

Among system uncertainties, unknown control direction is a rather essential one, whose compensation should entail the so-called Nussbaum function. Recently, strategies based on multiple Nussbaum functions (MNFs) have been proposed to make possible continuous adaptive control for the systems with nonidentical unknown control directions, such as large-scale systems and multi-agent systems, which cannot be addressed by a single Nussbaum function. But the existing stability theorems required: MNFs have explicit expressions; MNFs are all odd or all even. This paper aims to overcome the limitations of the theorems. We first delineate MNFs by two aspects of basic properties, replacing their explicit expressions as in the literature. Specifically, MNFs have the bivariate-product form, where one variate delineates frequent changes of signs and persistent intensity of MNFs, and the other delineates the rapid growth of MNFs. Such a delineation can capture the essence of MNFs and cover as many MNFs as possible. Based on the delineated MNFs, we present two stability theorems overcoming the aforementioned limitations. Notably, the two theorems do not require MNFs to have explicit expressions. Specifically, Theorem 1 allows nonmonotonic dynamic variables of MNFs while requiring MNFs to be odd, avoiding too large gains and excessive control cost. Theorem 4 does not require all MNFs to be odd while requiring dynamic variables of MNFs to be monotonic, giving users more freedom in selecting MNFs. The two stability theorems are applied to a large-scale system and a nonlinear system with parameterized uncertainties, both with nonidentical unknown control directions, to avoid monotonicity of dynamic gains and overparameterization, respectively, in control design.

This paper develops a method of stability analysis of linear time-delay systems with commensurate delays and delay-dependent coefficients. The method is based on a D-decomposition formulation that consists of identifying the critical pairs of delay and frequency, and determining the corresponding crossing directions. The process of identifying the critical pairs consists of a magnitude condition and a phase condition. The magnitude condition utilizes the Orlando’s formula, and generates frequency curves within the delay interval of interest. Such frequency curves correspond to the delay-frequency pairs such that the decomposition equation has at least one solution on the unit circle. The delay interval of interest is divided into continuous frequency curve intervals (CFCIs). Under some nondegeneracy assumptions, the number of frequency curves remains constant within each CFCI, and the associated decomposition equation has one and only one solution on the unit circle at any point on a frequency curve. By traversing through the frequency curves, all the crossing points can be identified. The crossing direction is related to the sign of the lowest-order nonzero derivative of the phase angle with respect to the delay, which is a generalization of the existing literature even for the case with single delay. This conclusion allows one to determine the crossing direction by examining the phase angle vs delay diagram. An example is presented to illustrate how a stability analysis can be conducted if some nondegeneracy assumptions are violated.

We consider the controllability of a fluid–structure interaction system, where the fluid is modeled by the Navier–Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bidimensional, whereas the deformation of the structure is one-dimensional, and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequency solutions and in the case where the observation domain is an horizontal strip. Then, we conclude this observability result by using a Lebeau–Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequency solutions. Finally, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.

Several recent results on spectrum-based analysis and control of linear time-invariant time-delay system concern the characterization and exploitation of situations where the so-called multiplicity-induced dominancy property holds, that is, the higher multiplicity of a characteristic roots implies that it is a rightmost root. This direction of research is inspired by observed multiple roots after minimizing the spectral abscissa as a function of controller parameters. However, unlike the relation between multiple roots and rightmost roots, barely theoretical results about the relation of the former with minimizers of the spectral abscissa are available. Consequently, in the first part of the paper the characterization of rightmost roots in such minimizers is briefly revisited for all second-order systems with input delay, controlled with state feedback. As the main theoretical results, the governing multiple root configurations are proved to correspond not only to rightmost roots, but also to global minimizers of the spectrum abscissa function. The proofs rely on perturbation theory of nonlinear eigenvalue problems and exploit the quasi-convexity of the spectral abscissa function. In the second part, a computational characterization of minima of the spectral abscissa is made for output feedback, yielding a more complex picture, which includes configurations with both multiple and simple rightmost roots. In the analysis, the pivotal role of the invariant zeros is highlighted, which translate into restrictions on the tunable parameters in the closed-loop quasi-polynomial.

We investigate the local (in time) description of incrementally scattering passive nonlinear systems. We show that these systems can be defined by a differential inclusion and a function that gives the current output in term of the current state and the current input. Our approach uses the theory of maximal monotone operators and Lax–Phillips-type nonlinear semigroups.

In this paper, we show that for a linear control system on a nilpotent Lie group, the Lie algebra rank condition is enough to assure the existence of a control set with a nonempty interior, as soon as one impose a compactness assumption on the generalized kernel of the drift. Moreover, this control set is unique and contains the singularities of the drift in its closure.

In this paper, the convergence proof of the second-order linear tracking differentiator, proposed by Han, is performed for signals with Kurzweil–Henstock integrable derivatives. Numerical simulations of some examples are also presented to validate the convergence of the tracking differentiator.

This paper deals with the modelling and stabilization of a flexible clamped beam controlled with a piezoelectric actuator in the self-sensing configuration. We derive the model starting from general principles, using the general laws of piezoelectricity. The obtained model is composed by a PDE, describing the flexible deformations dynamics, interconnected with an ODE describing the electric charge dynamics. Firstly, we show that the derived linear model is well-posed and the origin is globally asymptotically stable when a voltage control law, containing the terms estimated in the self-sensing configuration, is applied. Secondly, we make the more realistic assumption of the presence of hysteresis in the electrical domain. Applying a passive control law, we show the well-posedness and the origin’s global asymptotic stability of the nonlinear closed-loop system.

Let G be a noncompact connected semisimple Lie group, with finite center. In this paper we study the action of semigroups S of G, with nonempty interior, acting on maximal compact connected subgroups K of G. When S is connected, it is well known that the invariant control sets on K are the connected components of (pi ^{-1}(C)), where (pi ) is the canonical projection of K onto F, F is the flag type of S and C is the only invariant control set for S on F. One of the main results of the present paper describes the set of transitivity of a control set, not necessarily invariant, of a semigroup S, not necessarily connected, acting on K, as fixed points of regular elements in S. Furthermore, we show that the number of control sets on K is the product of the number of control sets on the maximal flag manifold of G by the number of invariant control sets on K.

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.