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.     

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.     

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     

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.     

Distributed identification and control of spatially interconnected systems with application to an actuated beam

This paper presents a case study on modelling and control of spatially interconnected systems. Considered is a vibration control problem, with experimental results on a flexible beam that is equipped with an array of piezo sensors and actuators. The sensor–actuator array induces a spatial discretization of the beam into an array of interconnected subsystems. Models are experimentally identified that have the structure of spatially interconnected systems. Based on the identified models, distributed control schemes are designed by solving a linear matrix inequality (LMI) problem that has the size of a single subsystem. Modelling and control is considered for both spatially invariant and spatially varying systems; in the latter case the system is represented as linear parameter-varying (LPV) system that is scheduled not over time but over space. Simulation and experimental closed-loop results demonstrate the practicality and efficiency of the underlying framework.     

Exact observability of diagonal systems with a finite-dimensional output operator

In this paper equivalent conditions for exact observability of diagonal systems with a finite-dimensional output operator are given. One of these equivalent conditions is the conjecture of Russell and Weiss (SIAM J. Control Opt. 32(1) (1994) 1–23). The other conditions are in terms of the eigenvalues and the Lyapunov solutions of finite-dimensional subsystems.     

Stabilization of discrete-time systems with stochastic parameters

This letter presents a solution to the problem of stabilization by a constant feedback of a linear system whose system and control matrices are multiplied by scalar correlated noise sequences.     

Stability of switching infinite-dimensional systems

In this note, we generalize the results from Narendra and Balakrishnan (IEEE Trans. Automatic Control 39 (1994) 2469) to the infinite-dimensional system theoretic setting. The paper gives results on the stability of a switching system of the form , i∈{1,2}, when the infinitesimal generators A₁ and A₂ commute. In addition, the existence of a common quadratic Lyapunov function is demonstrated.     

A local stabilization theorem for interconnected systems

We prove that if a nonlinear control system is locally asymptotically stabilizable at its equilibrium by means of a continuous feedback, then adding an integrator the resulting system is also locally asymptotically stabilizable by means of a feedback law, which is smooth except possibly at the equilibrium. Our approach is based on previous works of Arstein, Sontag and the author.