Stabilization via homogeneous feedback controls

In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for an affine control system and satisfies an homogeneous condition. We use a modified version of the Sontag formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous affine system leads to an homogeneous closed-loop system by using the previous feedback control.     

On the stabilization of a class of bilinear systems

In this paper the problem of the stabilization of a class of continuous time bilinear systems, containing the class of the homogeneous ones, is studied, and simple stabiliza-bility conditions are derived, expressed in terms of the stability or stabilizability of some linear or homogeneous bilinear subsystems of the given system. On this basis, it is shown that this problem can be decomposed into subproblems and reduced to the stabilization of homogeneous subsystems of the system, obtained through successive decompositions of it. A sufficient condition for the stabilization of a homogeneous bilinear system is also given.     

一种求解黑白球均匀排列问题的算法

主要研究黑白球的均匀排列问题,在简要分析了不同情况下如何做到“尽量均匀”后,给出一个基于累加的算法,并对该算法进行了简单的分析。最后,通过实验验证该算法的有效性。从实验结果来看,算法排出的白球和黑球相当规整,而且也最大限度地满足了“尽量均匀”的要求。此问题是一个很具有实用价值的模型,经过扩展后的算法可用于多维协调控制问题的求解。     

Global finite-time control for a class of switched nonlinear systems with different powers via output feedback

This paper is concerned with the problem of global finite-time control for switched nonlinear systems whose nonlinear terms satisfy homogenous growth conditions. At first, we design homogenous output feedback controllers for nominal switched nonlinear systems by adding a power integrator method. Then, we employ the homogeneous domination approach to scale homogeneous observers and controllers to render switched nonlinear systems with lower-triangular homogenous growth condition globally finite-time stable. Finally, the proposed control method can be extended to switched nonlinear systems satisfying with upper-triangular homogeneous growth condition. Two examples are provided to demonstrate the effectiveness of the proposed control scheme.     

On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems

It is known that robust analysis of polytopic systems, i.e. linear system with polytopic state space uncertainty, is a difficult problem. Recently we have proposed for this problem, in both continuous-time and discrete-time cases, the use of homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs) by introducing a novel LMI relaxation which provides a sufficient condition via a convex optimization without requiring the use of auxiliary multipliers of unknown degree. In this paper we show that this condition is not only sufficient but it is also necessary.     

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF.     

具有多项式型非线性项的大规模随机非线性系统的分散输出反馈镇定

基于齐次占优方法研究 具有多项式型非线性项的大规模随机非线性系统的分散输出反馈镇定问题. 本文的主要贡献在于利用高增益齐次占优方法来解决大规模随机非线性系统的分散控制问题. 这种方法能彻底地去掉传统结果中所要求的线性增长条件, 不仅推广了以前的结果还得到了一些新的结果. 仿真验证了输出反馈控制器的有效性.     

一类Holling-Tanner捕食模型的全局稳定性

研究了一类具有扩散项的Holling-Tanner捕食食饵模型,讨论了该模型在齐次Newmann边界条件下正常数平衡解的全局稳定性;并利用比较原理构造上下解的方法,得到了正常数平衡态解全局渐近稳定的条件。     

Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach

In this note, robust stability of state-space models with respect to real parametric uncertainty is considered. Specifically, a new class of parameter-dependent quadratic Lyapunov functions for establishing stability of a polytope of matrices is introduced, i.e., the homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs). The choice of this class, which contains parameter-dependent quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form, is motivated by the property that a polytope of matrices is stable if and only there exists an HPD-QLF. The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricial forms and the Lyapunov matrix equation. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.     

Homogeneous Lyapunov functions for systems with structured uncertainties

The problem addressed in this paper is the construction of homogeneous polynomial Lyapunov functions (HPLFs) for linear systems with time-varying structured uncertainties. A sufficient condition for the existence of an HPLF of given degree is formulated in terms of a linear matrix inequalities (LMI) feasibility problem. This condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the Lyapunov function. The maximum ?_∞ norm of the parametric uncertainty for which there exists a homogeneous polynomial Lyapunov function is computed by solving a generalized eigenvalue problem. The construction of such Lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in LMI form. Comparisons with other classes of Lyapunov functions through numerical examples taken from the literature show that HPLFs are a powerful tool for robustness analysis.     

Stability of switched positive nonlinear systems

This paper considers the stability problem for a class of switched positive nonlinear systems (SPNSs), which includes switched positive linear systems as a special case. We derive a necessary and sufficient condition for stability of continuous‐time SPNSs defined by homogeneous and cooperative vector fields under average dwell time switching. A corresponding necessary and sufficient condition is also given for stability of discrete‐time SPNSs defined by homogeneous and order‐preserving vector fields under average dwell time switching. The stability results for switched positive linear systems, which have been studied in the literature, can be easily obtained. We also extend the results to general switched linear systems. Finally, a numerical example is provided to illustrate the effectiveness of our results. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive steady states in an epidemic model with nonlinear incidence rate

In this paper, we investigate a diffusive epidemic model with nonlinear incidence rate under homogeneous Neumann boundary condition. The value of this study lies in two aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady states. And epidemiologically, we find that the system exhibits Turing patterns controlled by diffusion.     

Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization

This paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization. To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the nonseparation principle based design method, we develop a robust output feedback control scheme for uncertain nonlinear systems in the p-normal form, under a homogeneous growth condition. The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself. The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure. Examples are provided to illustrate the results of the paper.     

Global output feedback stabilization for a class of nonlinear systems with quantized input and output

In this paper, we aim at addressing the problem of global output feedback stabilization for a class of uncertain nonlinear systems with quantized input and output. The nonlinear functions of the system are assumed to satisfy high‐order growth condition on the unmeasurable states. Based on the homogeneous domination approach and sector bound approach, a homogeneous quantized controller computed from quantized output is constructed, and a guideline is derived to select the parameters of the quantizers. Further, it is proved that, with the proposed scheme, the closed‐loop system is globally asymptotically stable. Copyright © 2016 John Wiley & Sons, Ltd.     

Semi-Inner-Products for Convex Functionals and Their Use in Image Decomposition

Semi-inner-products in the sense of Lumer are extended to convex functionals. This yields a Hilbert-space like structure to convex functionals in Banach spaces. In particular, a general expression for semi-inner-products with respect to one homogeneous functionals is given. Thus one can use the new operator for the analysis of total variation and higher order functionals like total-generalized-variation. Having a semi-inner-product, an angle between functions can be defined in a straightforward manner. It is shown that in the one homogeneous case the Bregman distance can be expressed in terms of this newly defined angle. In addition, properties of the semi-inner-product of nonlinear eigenfunctions induced by the functional are derived. We use this construction to state a sufficient condition for a perfect decomposition of two signals and suggest numerical measures which indicate when those conditions are approximately met.     

Problem of pursuit of a group of coordinated evaders in linear recurrent differential games

For a linear nonstationary problem of pursuit of a group of evaders by a group of pursuers under the condition that the fundamental matrix of the homogeneous system is a recurrent function and all the evaders use the same control, sufficient conditions for capturing at least one evader are found.     

Non-reflecting artificial boundaries for modelling scalar wave propagation problems in two-dimensional half space

Using the operator splitting method, the non-reflecting artificial boundary condition, which has the same form as those originally presented by Hagstrom and Hariharan [Appl. Numer. Math. 27 (1998) 403] as well as Thompson et al. [Comput. Meth. Appl. Mech. Engrg. 191 (2001) 311], is re-derived in this paper to deal with transient scalar wave propagation problems in a two-dimensional homogeneous half space. In particular, the original non-reflecting artificial boundary condition is extended to approximately solve transient scalar wave propagation problems in a two-dimensional layered half space, which has a far field consisting of non-homogeneous materials. The proposed non-reflecting artificial boundary condition can be decomposed into three types of elements: physically meaningful springs and dashpots as well as the generalized energy absorbers, which represent the effects of incident waves at previous time instants on the truncated artificial boundary. Due to the meaningfulness in physics, the proposed non-reflecting artificial boundary is straightforwardly applied to the finite element analysis of wave propagation problems in infinite media. The related numerical results from three application examples have demonstrated the effectiveness, efficiency and usefulness of the proposed non-reflecting artificial boundary in dealing with scalar wave propagation problems in two-dimensional homogeneous and layered half spaces.     

Design of sliding mode controllers for bilinear systems with time varying uncertainties.

Sliding mode controllers for the bilinear systems with time varying uncertainties are developed in this paper. The bilinear coefficient matching condition which is similar to the traditional matching condition for linear system is defined for the homogeneous bilinear systems. It can be seen that the bilinear coefficient matching condition is very limited and is not generally applicable to the nonhomogeneous bilinear system. Thus, the sliding coefficient matching condition is also considered for the bilinear systems with time varying uncertainties. Then, the sufficient conditions are provided for the reaching mode of the time varying uncertain bilinear systems to be guaranteed by the designed sliding mode controllers. Moreover, the stability of the uncertain bilinear systems with the sliding mode controller is discussed. Simulation results are included to illustrate the effectiveness of the proposed sliding mode controllers.     

Global output feedback stabilization of upper‐triangular nonlinear systems using a homogeneous domination approach

This paper addresses the problem of global output feedback stabilization for a class of upper‐triangular systems with perturbing nonlinearities that are higher‐order in the unmeasurable states. A new design method based on the homogeneous domination approach and finite‐time stabilization technique is developed, which leads to global output feedback stabilizers for the upper‐triangular nonlinear systems under a homogeneous growth condition. A new perspective shown in this paper is that the finite‐time stabilization, in addition to its faster convergence rate, can also be utilized to handle control problems that were previously unresolved under asymptotic stabilization. Copyright © 2006 John Wiley & Sons, Ltd.     

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.