Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

Limited space heterogeneous queueing problem with an additional special feeding source

This paper considers a limited queue space, single channel, first come first served heterogeneous queueing system. There are two operating states sayE and F. For either state, the time in transition from one state to the other, follows an exponential distribution. In both the states E and F, the arrivals are Poissonian with different parameters and the service times have different exponential distributions. In either state, an additional number M of units join the system whenever the number already there declines to one unit. Thus the channel is busy at all times.     

Suboptimal control of systems with multiple delays

The application of Pontryagin's maximum principle to the optimization of linear systems with time delays results in a system of coupled two-point boundary-value problems involving both delay and advance terms. The exact solution of this system of TPBV problems is extremely difficult, if not impossible. In this paper, a fast-converging iterative approach is developed for obtaining the suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. At each step of the proposed method, a linear nondelay system with an extra perturbing input must be optimized. The procedure can be extended for the optimization of nonlinear systems with multiple time-varying delays, provided that some of the nonlinearities satisfy the Lipschitz condition.     

Piotrowski's Infinite Series of Steiner Quadruple SystemsRevisited

The construction of Bays and deWeck [1] of a SteinerQuadruple System SQS(14) was generalized by Piotrowskiin his dissertation ([7], p. 34) to an SQS(2p), p 7 mod 12 with a group transitive on thepoints. However he gave no proof of his construction and hispresesntation was open to misinterpretation. So Hanfried Lenzsuggested to analyse Piotrowski's construction and to supplyit with a proof. In the following we will present Piotrowski'sideas somewhat differently and will furnish a proof of the construction.     

Control by Interconnection and Energy Shaping of the Timoshenko Beam

In this paper, the dynamical control of a mixed finite and infinite dimensional mechanical system is approached within the framework of port Hamiltonian systems. In particular, a flexible beam, modeled according to the Timoshenko theory and in distributed port Hamiltonian form, with a mass under gravity field connected at a free end, is considered. The control problem is approached by generalization of the concept of structural invariant (Casimir function) to the infinite dimensional case and the so-called control by interconnection technique is extended to the infinite dimensional case. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced.     

Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

Stability for time-varying discrete linear systems in a Banachspace is investigated. On the one hand is established a fairlycomplete collection of necessary and sufficient conditions foruniform asymptotic equistability for input-free systems. Thisincludes uniform and strong power equistability, and uniformand strong l_p-equistability, among other technical conditionswhich also play an essential role in stability theory. On theother hand, it is shown that uniform asymptotic equistabilityfor input-free systems is equivalent to each of the followingconcepts of uniform stability for forced systems: l_p-input l_p-state,e_o-input e_o-state, bounded-input bounded-state, l_p-input bounded-state(with p>1), e_o-input bounded-state, and convergent-inputbounded-state; these are also equivalent to their nonuniformcounterparts. For time-varying convergent systems, the aboveis also equivalent to convergent-input convergent-state stability.The proofs presented here are all ‘lementary’ inthe sense that they are based essentially only on the Banach–Steinhaustheorem.     

时间可逆系统的等时中心问题

若在中心附近的闭轨线都具有相同的周期,则此中心称为等时中心. 时间可逆多项式系统的等时中心问题是一类公开问题. 为了构造性地解决这一难题,讨论一类范围更广的时间可逆解析动力系统, 给出相应横截交换系统的递推公式,此公式可以用于等时中心条件的推导. 在递推公式的基础上,以吴特征集方法为工具,给出一类时间可逆三次系统具有横截交换系统的充要条件.     

Analysis of switched linear systems in the plane,part 1: Local behavior of trajectories and local cycle geometry

This paper investigates the problem of control of switched linear systems evolving inR ². The concept of an opposition point is introduced, and its properties related to the existence of closed trajectories in the phase plane are investigated. The geometry of cycles in a neighborhood of an opposition point is also studied.This research was supported by the Office of Naval Research, ONR Contract No. N0013-80-C-0199, and the United States Department of Energy, DOE Contract No. DE-AC01-79-ET-29363.     

Hybrid systems: Modelling and analysis using emergent dynamics

In this paper, we discuss modelling and analysis of hybrid systems with physical interaction dynamics. Such systems are typically considered complex and they are modelled using abstractions. Abstractions may, however, unintentionally exclude critical details, leading to partial or false results. Therefore, we study here use of a particle system in modelling and analysis. The novelty of the particle system is that it is designed to reveal interaction dynamics as emergent dynamics; thus, supporting analysis of complex and intricate interaction dynamics with acceptable modelling effort. As the main contribution, we formalize the particle system, and use it to model and analyze hybrid systems, both mechanical and biological, with nontrivial interaction dynamics.     

Deterministic and stochastic approaches to supervisory control design for networked systems with time-varying communication delays

This paper proposes a supervisory control structure for networked systems with time-varying delays. The control structure, in which a supervisor triggers the most appropriate controller from a multi-controller unit, aims at improving the closed-loop performance relative to what can be obtained using a single robust controller. Our analysis considers average dwell-time switching and is based on a novel multiple Lyapunov–Krasovskii functional. We develop stability conditions that can be verified by semi-definite programming, and show that the associated state feedback synthesis problem also can be solved using convex optimization tools. Extensions of the analysis and synthesis procedures to the case when the evolution of the delay mode is described by a Markov chain are also developed. Simulations on small and large-scale networked control systems are used to illustrate the effectiveness of our approach.     

Real Lax spectrum implies spectral stability

We consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset of the real line that gives rise to stable eigenvalues is contained in the Lax spectrum. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and it maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples, both in and not in the AKNS hierarchy.     

A note on “Existence and uniqueness of coexistence states for an elliptic system coupled in the linear part”, by Hei Li-jun,Nonlinear Anal. Real World Appl. 5, 2004

In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction–diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counter-examples.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Controllability of nonlinear integrodifferential systems in Banach space

Sufficient conditions for controllability of nonlinear integrodifferential systems in a Banach space are established. The results are obtained using the Schauder fixed-point theorem.The work of the first and third authors was supported by CSIR, New Delhi, India.     

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and the jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.     

Stability analysis of a class of nonlinear positive switched systems with delays

This paper addresses the stability problem of delayed nonlinear positive switched systems whose subsystems are all positive. Both discrete-time systems and continuous-time systems are studied. In our analysis, the delays in systems can be unbounded. Two conditions are established to test the local asymptotic stability of the considered systems. The method to compute the domains of attraction is also proposed provided that the system is locally asymptotically stable. When reduced to general nonlinear positive systems, that is, the considered switched system consists of only one mode, an interesting conclusion follows that the proposed nonlinear positive system is locally asymptotically stable for all admissible delays and nonnegative nonlinearities which satisfy an extra condition at the origin, if and only if the system represented by the linear part is asymptotically stable for all admissible delays. Finally, a numerical example is presented to illustrate the obtained results.