Digital stabilizer design for a switched linear system

We consider the problem of designing a digital controller stabilizing a continuoustime switched linear system. Our approach to stabilization includes the construction of a continuous-discrete time closed-loop system, the passage to its discrete-time model, and the subsequent discrete-time controller design based on simultaneous stabilization methods.     

Finite-time converging jump observer for switched linear systems with unknown inputs

This work addresses the state observation problem for a class of switched linear systems with unknown inputs. The proposed high-order sliding-mode observer provides a finite-time converging estimate of the continuous system’s state vector in spite of the presence of unknown inputs. The design procedure, which assumes knowledge of the discrete state of the switched system, is based on the principles of disturbance decoupling and hierarchical observer output injection. In order to cope with the switching nature of the plant under observation, jumps in the observer state space are enforced intentionally. The implementation of an additional observer allows for the reconstruction of the unknown inputs, which may be important in the framework of fault detection. Numerical examples illustrate the effectiveness of the suggested technique.     

Robust observer design for nonlinear uncertain switched systems under asynchronous switching

Switching between the system and the associated observer or controller is in fact asynchronous in switched control systems. However, many times we assume it synchronous, for simplicity. In this paper, the robust observer design problems for a class of nonlinear uncertain switched systems for synchronous and asynchronous switching are addressed. At first, a robust observer under synchronous switching is proposed based on average dwell time approach. After that, the results are extended to robust observer design in the asynchronous case. In this case, two working modes are adopted to facilitate the studies on the issue. Finally, an extension case covering more practical applications is investigated under asynchronous switching. The designed observer cannot maintain the asymptotical stability of error state, but the eventual boundness is guaranteed. At the end, a numerical design example is given to illustrate our results.     

On uniform controller design for linear switched systems

Traditional method to design a controller for each subsystem of switched systems will increase the complexity of the controller’s realization. Hence this paper gives sufficient conditions for designing a uniform output feedback controller for linear switched systems, and this common controller can be used for all subsystems of the switched systems. Then the output stabilization problem for a particular class of linear switched systems under this uniform output feedback controller has been studied. An illustrative example is given in order to highlight the proposed method.     

Commuting and stable feedback design for switched linear systems

In the paper, commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching.     

Complements to full order observer design for linear systems with unknown inputs

This note completes the results obtained in [M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Automat. Control., AC-39 (3) (1994) 606–609]. It presents a full order observer for linear systems with unknown inputs in the state and in the measurement equations. It gives a more general approach for the observer’s design than that presented in the above mentioned reference and it shows that all the obtained results are independent of the choice of the generalized inverses considered in the observer’s design. Continuous- and discrete-time systems are considered.     

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.     

Linear system identification via an asymptotically stable observer

This paper presents a formulation for identification of linear multivariable systems from single or multiple sets of input-output data. The system input-output relationship is expressed in terms of an observer, which is made asymptotically stable by an embedded eigenvalue assignment procedure. The prescribed eigenvalues for the observer may be real, complex, mixed real and complex, or zero corresponding to a deadbeat observer. In this formulation, the Markov parameters of the observer are first identified from input-output data. The Markov parameters of the actual system are then recovered from those of the observer and used to realize a state space model of the system. The basic mathematical formulation is derived, and numerical examples are presented to illustrate the proposed method.     

Methods of stabilizing or destabilizing a switched linear system

This paper shows that the stability or nonstability of switched systems does not depend on the eigenvalues of the matrices. The result is obtained by giving examples of no stable (resp. no unstable) switched linear systems consisting of stable (resp. unstable) matrices. Moreover, for the first time all kinds of eigenvalues are considered, as well as two general results for establishing the stability or nonstability of switched linear systems are presented.     

Differential inequalities and flow-invariance via linear functionals

This paper deals with flow-invariance properties with respect to x′ ? f(t, x) and resulting comparison inequalities. Both the order relation “?” and the regularity conditions on x are defined, in a general manner, in terms of a specified set of linear functionals, S. This permits greater flexibility for applications e.g., a proper choice of S gives the PDE operators of the form f(t, x) = x_ss + qx_s where p ? 0, an appropriate quasimonotonicity property.     

On stability for switched linear positive systems

This paper addresses the stability properties of switched linear positive systems in continuous-time as well as in discrete-time. In the discrete-time case, some sufficient and necessary conditions for asymptotic stability are derived for pairs of second order systems. Similar conditions are also established for a finite number of second order systems. Furthermore, for higher order systems, some results on stability are provided in a similar manner. In particular, in this case, a common linear Lyapunov function guaranteeing the stability of the switched positive systems can be easily located by means of geometry properties. In the continuous-time case, a finite number of second order systems are considered. Some equivalent conditions for stability of such systems are developed.     

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

Global observer design for nonlinear systems

This paper is a geometric study of the global observer design for nonlinear systems. Using the theory of foliations, we derive necessary and sufficient conditions for global exponential observers for nonlinear systems under some assumptions. Our proof for these necessary and sufficient conditions for global exponential observers if via defining two equivalence relations known as horizontal and vertical equivalence relations, and constructing two foliations known as horizontal and vertical foliations from these equivalence relations. Finally, as a corollary of our global theorem, we derive necessary and sufficient conditions for local exponential observers of critically Lyapunov stable nonlinear systems.     

基于行为分析的AS/RS有色赋时Petri网模型研究

为了研究如何提高AS/RS系统的性能,有必要构建其模型.提出了库所双重着色的有色赋时Petri网方法,分析了AS/RS系统活动资源的行为特点和要求,并详细阐述了使用有色赋时Petri网分别构建这些行为模型的过程,从而实现整个AS/RS系统框架模型.采用visual c++软件仿真表明,该方法在构建面向资源的离散系统模型时是有效的.     

Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Asymptotic disturbance attenuation properties for uncertain switched linear systems

In this paper, the disturbance attenuation properties in the sense of uniformly ultimate boundedness are investigated for a class of switched linear systems with parametric uncertainties and exterior disturbances. The aim is to characterize the conditions under which the switched system can achieve a finite disturbance attenuation level. First, arbitrary switching signals are considered, and a necessary and sufficient condition is given. Secondly, conditions on how to restrict the switching signals to achieve finite disturbance attenuation levels are investigated. Two cases are considered here that depend on whether all the subsystems are uniformly ultimately bounded or not. Both discrete-time and continuous-time switched systems are considered, and the techniques are based on multiple polyhedral Lyapunov functions and their extensions.     

Conditions for distinguishability and observability of switched linear systems

A characterization for weak distinguishability of two linear controlled systems are derived. In addition, the observability of a special switched linear controlled system is considered, and necessary and sufficient conditions for observability are obtained.