On the stabilization of the homogeneous bilinear systems

The problem of stabilizing the single-input homogeneous bilinear systems is considered. A special class of these systems is considered: the matrix of the linear (i.e., control independent) term of the right-hand side is semitable while the matrix of the bilinear term is skew-symmetric. Two types of stabilization are investigated: constant feedback global asymptotic stabilization and practical stabilization by a family of the linear feedbacks. It will be shown for the planar case that the above system is always practically stabilizable by a family of the linear feedbacks. For n = 3, the solution of both stabilization problems depends on the mutual position of the expanding direction of the linear part and the direction of the infinitesimal rotation defined by the bilinear term.     

Continuous stabilization controllers for singular bilinear systems: The state feedback case

Global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper. New approaches are developed by means of the LaSalle invariant principle for nonlinear systems. A new set of sufficient condition is first derived via the continuous static state feedback, the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system.     

Remark on local and global stabilization of homogeneous bilinear systems

In this paper, we deal with the problem of stabilization of homogeneous bilinear systems. The aim is to clarify some results on stabilizability of these systems.     

Asymptotic stabilization of planar systems

Sufficient conditions for the existence of a C¹ asymptotically stabilizing feedback control for planar, affine systems are given. The methods are constructive and yield homogeneous feedback controls for homogeneous systems which allow the results to be extended to more general systems.     

Global feedback stabilization of new class of bilinear systems

In this paper, we consider some new classes of bilinear systems. We give necessary and sufficient conditions for the asymptotic stabilization by using homogeneous feedbacks.     

Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.     

Characterization of equilibrium sets for bilinear systems with feedback control

Simple analytical methods for the characterization of the equilibrium sets of bilinear systems with linear feedback control are developed based on elimination theory. It is shown that the number of equilibrium points can be determined as a function of the feedback gains, without having to solve for the locations of the equilibrium points. Necessary and sufficient conditions for the origin to be the unique equilibrium point are also presented. These results provide the basis for computing the equilibrium points via a straightforward procedure, which eliminates the convergence, stability and uncertainty problems that can occur when iterative root finding techniques are used. Several illustrative examples are included. The methods developed in this paper can also be used to characterize the equilibrium sets of other classes of nonlinear systems such as quadratic systems.     

Finite-time stabilization of output-constrained planar switched systems via output feedback

Finite-time stabilization (FTS) problem of output-constrained planar switched systems via output feedback is investigated in this paper. State feedback control laws are constructed in a systematic way by combining the revamped adding a power integrator technique (APIT) with the elaborately designed logarithm-type barrier Lyapunov function (BLF). By merging the constructed variable-gain switched observers, finite-time output-feedback stabilization, then, is achieved with the output constraint meets too. At the end, simulations are presented to show the effectiveness of the proposed method.     

Asymptotic stabilization via output feedback for nonlinear systems with delayed output

We propose a novel and simple design scheme of output feedback controller for a class of nonlinear systems with delayed output. The nonlinear systems considered here are more general than feedforward systems (upper triangular systems). By constructing an appropriate Lyapunov–Krasovskii functional (LKF) and solving linear matrix inequalities (LMIs), the delay-dependent controller making the closed-loop system globally asymptotically stable (GAS) is explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

It is known that if a system can be (robustly) globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable (UCO), then this system can be practically semiglobally stabilized by means of (possibly dynamic) output feedback. This papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by UCO functions is guaranteed. The class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback. In particular, the hypothesis does not require the system to be minimum phase.     

切换自治系统的动态输出反馈镇定

本文研究离散时间切换线性自治系统的输出反馈镇定问题. 在切换系统可观测的假设下, 设计具有多线性 时变增益的动态观测器, 实现有限时间状态估计. 在此基础上, 设计多路径动态输出反馈切换策略, 实现闭环系统指 数收敛.     

Output feedback stabilization of MIMO non-linear systems via dynamic sliding mode

Previous work on asymptotic stabilization of MIMO non-linear systems using dynamic sliding mode control to produce dynamic state feedback has been generalized to dynamic output feedback. All the states in the feedback controller are replaced with estimated states which come from a semi-high-gain observer. The bound on the observer gain is explicitly given, which depends on some previously chosen design parameters. If the given differential input–output (I–O) system is minimum phase and proper, local uniform asymptotic output feedback stabilization can be achieved. In addition, the restriction on the stability of the zero dynamics has been relaxed to weakly minimum phase and semi-global results are obtained under some mild conditions. The stability of the closed-loop system is based on some stability results of triangular systems when continuous or discontinuous controllers are adopted. Two pertinent examples are given to illustrate the design method. Copyright © 1999 John Wiley & Sons, Ltd.     

Output stabilization of distributed bilinear systems

This paper studies regional stabilization of a distributed bilinear system evolving on a spatial domain $\varOmega$. Sufficient conditions for regional weak, strong and exponential stabilization are given. Also we discuss a regional optimal stabilization problem. The obtained results are illustrated by examples and simulations.     

Asymptotic stabilization via control by interconnection of port-Hamiltonian systems

We study the asymptotic properties of control by interconnection, a passivity-based controller design methodology for stabilization of port-Hamiltonian systems. It is well-known that the method, in its basic form, imposes some unnatural controller initialization to yield asymptotic stability of the desired equilibrium. We propose two different ways to overcome this restriction, one based on adaptation ideas, and the other one adding an extra damping injection to the controller. The analysis and design principles are illustrated through an academic example.     

Dynamic output feedback linearization and global stabilization

We present a class of single-input single-output nonlinear systems which are globally transformable by a dynamic output feedback control and a time-varying state space transformation into a linear, observable and minimum phase system. We then show how those systems can be globally stabilized by a dynamic output feedback nonlinear control and how global output tracking can be achieved as well.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

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, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

Equivalence of robust stabilization and robust performance via feedback

One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified L ²-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to linear matrix inequality conditions.     

Global stabilization of linear discrete-time systems with bounded feedback

This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less than or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (“single hidden layer neural networks”) of simple saturation functions.     

A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback

In this paper, we introduce a generalized framework for global output feedback stabilization of a class of uncertain, inherently nonlinear systems of a particularly complex nature since their linearization around the equilibrium is not guaranteed to be either controllable or observable. Based on a new observer/controller construction and a homogeneous domination design, this framework not only unifies the existing output feedback stabilization results, but also leads to more general results which have been never achieved before, establishing this methodology as a universal tool for the global output feedback stabilization of inherently nonlinear systems. Copyright © 2006 John Wiley & Sons, Ltd.     

A unifying point of view on output feedback designs for global asymptotic stabilization

The design of output feedback for ensuring global asymptotic stability is a difficult task which has attracted the attention of many researchers with very different approaches. We propose a unifying point of view aiming at covering most of these contributions.We start with a necessary condition on the structure of the Lyapunov functions for the closed loop system. This motivates the distinction of two classes of designs:-the direct approach, also called control error model analysis, in which the attention is focused on directly estimating a stabilizer, and-the indirect approach, also called dynamic error model analysis, in which the stabilization task is fulfilled for an estimated model of the system and not directly for the system itself.We show how most available results on this topic can be reinterpreted along these lines.