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.     

Linear stabilizability of planar nonlinear systems

This paper considers scalar-input two-dimensional nonlinear systems for which the linearization, has a simple zero uncontrollable eigenvalue. The existence of linear stabilizing feedback laws is investigated, using center manifold techniques. This work was partially supported by the Ministero della Pubblica Istruzione, Italia (Progetti di ricerca, di interesse nazionale).     

A manifold-like characterization of asymptotic stabilizability of homogeneous systems

In this paper, we present a version of Artstein's theorem for homogeneous systems and we drive sufficient Manifold-like conditions for stabilization of single-input homogeneous systems by means of a homogeneous feedback law.     

Asymptotic stabilizability of homogeneous polynomial systems of odd degree

We give some tools for the construction of the homogeneous feedback witch stabilizes a generic class of single input, two dimensional, homogeneous systems.     

A full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs

This paper studies open-loop stabilization problem for bimodal systems with continuous vector field. It is based on the earlier work of the authors on the controllability problem for the same class of systems. A full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions. It is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case.     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstract second-order infinite-dimensional dynamical systems is considered. It is proved using the frequency-domain method, that approximate controllability of second-order system can be verified by the approximate controllability conditions for the corresponding simplified first-order system. General results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends earlier results on approximate controllability of second-order abstract dynamical systems.     

Finite-dimensional regulators for a class of infinite-dimensional systems

We show that the ‘direct approach’, developed by the author for stabilization of certain classes of distributed parameter systems, can be extended to cover regulation problems as well. An iterative design algorithm is presented, together with proof that the algorithm will converge after a finite number of steps. The procedure is illustrated with an example of a constant disturbance acting on a delay system.     

Equivalent characterizations of detectability and stabilizability for a class of linear time-varying systems

This paper addresses four characterizations of detectability and stabilizability for a broad class of linear continuous-time time-varying systems. It is shown that for the class of constant-rank systems all these notions of detectability and stabilizability are actually equivalent to each other.     

Normalized doubly coprime factorizations for infinite-dimensional linear systems

We obtain explicit formulas for normalized doubly coprime factorizations of the transfer functions of the following class of linear systems: the input and output operators are vector-valued, but bounded, and the system is input and output stabilizable. Moreover, we give explicit formulas for the Bezout factors. Using a reciprocal approach, we extend our results to a larger class where the input and output operators are allowed to be unbounded. This class is much larger than the class of well-posed linear systems.     

Stability radii of infinite-dimensional positive systems

We show that for infinite-dimensional discrete-time positive systems the complex and real stability radii coincide. Furthermore, we provide a simple formula for the complex stability radius of positive systems by the associated transfer function. We illustrate our results with an example dealing with a simple type of differential-difference equations.The author would like to thank the Deutsche Forschungsgemeinschaft for its support during this work.     

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on the set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover, a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov–Metzler condition for stabilizability does not hold.     

Controllability and stabilizability of switched linear-systems

In this paper, we study the conjecture made in (IEEE Trans. Automat. Control 47 (8) (2002) 1401) concerning the existence of basic switching sequence for switched systems, and give it an affirmative answer and a construction method. This paper proves that for switched linear system, there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system. Hence, the controllability can be realized by using only one switching sequence. Then, the result is extended to the systems with time-delay in controls. Furthermore, we define the stabilizability of switched linear systems, and based on certain transformation of state coordinate, we give a necessary and sufficient condition for the stabilizability. Two examples are presented to illustrate the obtained results.     

On stabilizability and exact observability of stochastic systems with their applications

This paper mainly studies the stabilizability and exact observability of stochastic linear controlled systems and their applications. With the aid of the operator spectrum, a necessary and sufficient condition is given for the stabilizability of stochastic systems. Some new concepts such as unremovable spectrum and strong solution are introduced. An unremovable spectral theorem and a stochastic Popov-Belevith-Hautus Criterion for exact observability are presented. As applications, a comparison theorem for stochastic algebraic Riccati equations and a result on Lyapunov-type equations are obtained.     

Observability of polynomially stable systems

For finite-dimensional systems the Hautus test is a well known and easily checkable condition for observability. Russell and Weiss [A general necessary condition for exact observability, SIAM J. Control Optim. 32 (1994) 1–23] suggested an infinite-dimensional version of the Hautus test, which is necessary for exact infinite-time observability and sufficient for approximate infinite-time observability of exponentially stable systems. In this paper the notion of observability is studied for polynomially stable systems. Several known results for exponentially stable systems are extended to the setting of polynomially stable systems. By means of an example the obtained results are illustrated.     

Global adaptive stabilization of infinite-dimensional systems

The purpose of this paper is to show that a first-order adaptive regulator globally stabilizes a class of infinite-dimensional systems having no right half plane zeros. Knowledge of the spectrum of the model's high-frequency gain is not required.     

Stabilizability by state feedback implies stabilizability by encoded state feedback

Encoded state feedback is a term which refers to the situation in which the state feedback signal is sampled every T units of time and converted (encoded) into a binary representation. In this note stabilization of nonlinear systems by encoded state feedback is studied. It is shown that any nonlinear control system which can be globally asymptotically stabilized by “standard” (i.e. with no encoding) state feedback can also be globally asymptotically stabilized by encoded state feedback, provided that the number of bits used to encode the samples is not less than an explicitly determined lower bound. By means of this bound, we are able to establish a direct relationship between the size of the expected region of attraction and the data rate, under the stabilizability assumption only, a result which—to the best of our knowledge—does not have any precedent in the literature.     

Practical stabilizability of dyadic homogeneous bilinear systems

This paper considers a class of homogeneous bilinear systems in which the rank of the B matrix is equal to one. It uses a transformation to convert the system into a form similar to the companion from. It then considers feedback control laws in the form of variable structure. Results indicate that it is possible to obtain practical stability for a class of dyadic systems. The results are particularly useful for low dimensional systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Spreadability and vulnerability of distributed parameter systems

In this article we focus on the mathematical development of vulnerability concept and vulnerability index and analyze the dependence of the vulnerability concept definition on that of spreadability [A. El Jai and K. Kassara, “Spreadable distributed systems”, Mathem. and Comp. model., 20, pp. 47--64, 1994; A. Bernoussi and A. El Jai, “New approach of spreadability”, Journal of Mathematical and Computer Modelling 31, pp 93--109, 2000; A. Bernoussi, A. El Jai and A. J. Pritchard, “Spreadability and evolving interfaces”, Inter. J. Syst. Sci., 32, pp. 1217--1232, 2001]. It is found that the mathematical approach of the vulnerability concept is rather attractive and can be applied to many engineering problems. To illustrate this, some examples and applications are considered.     

Necessary and sufficient conditions for strong stability of distributed parameter systems

In this paper we propose definitions for strong stabilizability and strong detectability of infinite-dimensional control systems. We show that these definitions are appropriate by showing that they can be used to give necessary and sufficient conditions for the strong stability of a system in terms of its input-output stability. As an application, we discuss the strong stability of the feedback interconnection of two strongly stabilizable and strongly detectable systems.