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.     

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.     

Adaptive stabilization of a class of nonlinear systems with unstable internal dynamics

This note considers global stabilization of a class of uncertain nonlinear output feedback systems with unstable internal dynamics. The coefficients which characterize the internal dynamics are allowed to be functions of system output, and the uncertainty of the system is characterized by an unknown constant parameter vector. The key step in the proposed control design is to estimate the internal state variables and impose control over them. The control design is presented first for systems without unknown parameters, and then for the systems with unknown parameters using adaptive control techniques.     

Quadratic stabilization of bilinear control systems

In this paper, a stabilization problem of bilinear control systems is considered. Using the linear matrix inequality technique and quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system emanating from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear systems.     

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.     

The output stabilization of SISO bilinear systems

A quite general approach to the problem of stabilizing a SISO (single-input single-output) bilinear system by output feedback is presented. Following a Lyapunov-like method, a family of admissible controls is defined and an easy way for deriving its general features is proposed. Previous results can be obtained as special cases of this approach. Numerical examples are included to show the feasibility of this algorithm in determining simple control laws     

Stabilization of a class of generalized bilinear systems

This paper studies local stabilization of a class of analytic nonlinear systems in terms of, which includes ordinary bilinear systems as its subset, ż=f(z)+g(z)u, f(0)=0, g(0)=0, z∈R² which can be achieved via a feedback control law u=u(z) with u(0)=0. Following the theoretical results a potential application, stabilization of non-minimum phase systems, is investigated. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

一类齐次离散双线性系统可控性分析

在系统分析中,可控性是系统的一个重要特性.在工程实际操作中,往往需要对一个连续系统进行离散化处理,人们希望系统在离散化后能保留原系统的重要系统特征,比如可控性.对于线性系统,我们有成熟的判断方法.然而,对于非线性系统则无统一的判别方法.Elliott在2005年给出了一个二阶双线性系统经过离散化后,可控性发生变化的例子.它表明一个系统在离散化前后,它的可控性可能会发生改变.本文旨在给出一类二阶离散化双线性系统可控性充分条件,并和已有结果作比较,表明本文结果更具有一般性.另外,本文对于3阶及以上的这类系统可控性做出了不可控的判断.     

Stabilization of a class of singularly perturbed bilinear systems

The Lyapunoy stability of a form of singularly perturbed bilinear system is investigated. A theorem is proposed to show how to calculate the stabilizing controller for the whole system from a knowledge of the stability of the reduced-order model. A numerical example is given.     

Controllability of a class of discrete time bilinear systems

In this paper the controllability of a class of discrete time bilinear systems is discussed. Necessary and sufficient conditions for complete controllability are derived. In addition a complete characterization of the controllable regions of this class of systems, when not completely controllable, is made.     

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.     

On the controllability of a class of discrete bilinear systems

The subject of this paper is the controllability of a class of time invariant discrete bilinear systems. Bilinear systems are classified into two categories: homogeneous and inhomogeneous. Necessity as well as the sufficiency results are obtained by means of decomposing the bilinear system into a linear system and a multiplicative feedback. Byproducts of the decomposition are the notions of multiplicative feedback compensation and the multiplicative feedback with bias compensation which may prove to be alternatives for the classical linear feedback compensation.     

Optimal control of a class of unconstrained bilinear systems†

The optimal control of a class of control unconstrained bilinear systems is considered. It is demonstrated that mere application of known necessary conditions of optimality is insufficient to solve the problem numerically. A technique overcoming this difficulty is proposed. It is demonstrated in an example involving optimal shut-down control of a nuclear reactor under xenon poisoning. The TPBVP is solved numerically using the variation of extremals method.     

On a class of suboptimal controls for infinite-dimensional bilinear systems

The suboptimal control of a bilinear system is considered with respect to a quadratic cost criterion. The feedback control is in the space of formal power series on a Hilbert space.     

Stability and asynchronous stabilization for a class of discrete-time switched nonlinear systems with stable and unstable subsystems

The stability analysis and asynchronous stabilization problems for a class of discrete-time switched nonlinear systems with stable and unstable subsystems are investigated in this paper. The Takagi-Sugeno (T-S) fuzzy model is used to represent each nonlinear subsystem. Through using the T-S fuzzy model, the studied systems are modeled into the switched T-S fuzzy systems. By using the switching fuzzy-basis-dependent Lyapunov functions (FLFs) approach and mode-dependent average dwell time (MDADT) technique, the stability conditions for the open-loop switched T-S fuzzy systems with unstable subsystems and asynchronous stabilization conditions for the closed-loop switched T-S fuzzy systems with unstable subsystems are obtained. Both the stability results and asynchronous stabilization results are derived in terms of linear matrix inequalities (LMIs). Finally two numerical examples are provided to illustrate the effectiveness of the results obtained.     

Exponential stabilization for nonlinear systems with applications to nonholonomic systems

The paper proposes a general framework to study the exponential stabilization problem for a wide class of nonlinear systems. By combining the concept of dilation with the method of σ-processing, a simple stability criterion is given in terms of the stability property of an augmented system. Moreover, the global ρ-exponential stability of the closed-loop system can be achieved by employing continuous controllers. A well-known nonholonomic system is given to illustrate the effectiveness of the proposed results.     

On near-controllability and stabilizability of a class of discrete-time bilinear systems

In this paper, the near-controllability and stabilizability of a class of discrete-time bilinear systems are studied. A necessary and sufficient condition for the near-controllability of the systems is first derived. Then, on the basis of the corresponding near-controllability criterion, the stabilizability problem of the systems is solved in turn.     

A class of µ-commutative bilinear control systems

A class of μ-commutative bilinear control systems is defined and the properties of coefficients of a μ-commutative system are studied. A μ-commutative scalar control system is studied and an explicit representation for its input-output mapping is derived. This mapping is used to study the properties of a system, particularly the geometry of its reachable set. Original Russian Text ? M.V. Topunov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 113–125. This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board     

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.