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.     

High-order sliding-mode observation for linear systems with unknown inputs

A high-order sliding-mode observer is designed for linear systems with unknown inputs. The concepts of strong observability and strong detectability are studied and their relation with the relative degree of a plant is established. High-order sliding-mode-based observers for linear time-invariant systems with unknown inputs satisfying the condition of strong observability or strong detectability are developed.     

Finite-time L1 control for positive switched linear systems with time-varying delay

This paper is concerned with the problem of finite-time L₁ control for a class of positive switched linear systems with time-varying delay. Firstly, by using the average dwell time approach, sufficient conditions which can guarantee the L₁ finite-time boundedness of the underlying system are given. Then, in virtue of the results obtained, a state feedback controller is designed to ensure that the resulting closed-loop system is finite-time bounded with L₁-gain performance. All the obtained results are formulated in terms of linear matrix inequalities (LMIs), which can be solved conveniently. Finally, an example is given to illustrate the efficiency of the proposed method.     

On the observability of linear,time-invariant systems with unknown inputs

This paper considers the observation of linear, time-invariant dynamical systems in the general case in which some of the input functions are unknown. By arguments based on the concepts of controlled and conditioned invariance, a convenient expression for the observability subspace is found which includes the well-known expression for the case in which the input functions are given.     

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.     

Robust H∞ control for linear Markovian jump systems with unknown nonlinearities

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H_∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique.     

Hybrid observer design for linear switched system via Differential Petri Nets

The aim of this paper is to propose a hybrid observer design for linear switched systems modelled either via Differential Petri Nets (DPN) or via Timed Differential Petri Nets (TDPN). The switched systems, herein, considered are characterized by switching laws that can depend on the continuous states or on both of a given dwell time and the continuous states. In addition, the structure of the proposed observers is based on a discrete observer and a continuous observer on interaction. The discrete observer reconstructs the discrete mode, by estimating both of the discrete marking and the firing vector. Once, the active mode is obtained, the continuous states are estimated. Finally, the outputs of the continuous observer are used to update the marking and the firing vector. At the end of the paper, several simulation results are presented to illustrate the proposed approach.     

On the representation of switched systems with inputs by perturbed control systems

This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.     

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.     

Finite-time stabilization of impulsive dynamical linear systems

Finite-time stabilization of a special class of hybrid systems, namely impulsive dynamical linear systems (IDLS), is tackled in this paper. IDLS exhibit jumps in the state trajectory which can be either time-driven (time-dependent IDLS) or subordinate to specific state values (state-dependent IDLS). Sufficient conditions for finite-time stabilization of IDLS are provided. Such results require solving feasibility problems which involve Differential–Difference Linear Matrix Inequalities (D/DLMIs), which can be numerically solved in an efficient way, as illustrated by the proposed examples.     

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.     

Periodical stabilization of switched linear systems

Periodical stabilization problems for switched linear systems are investigated in this paper. For autonomous switched systems, if there exists a stable convex combination of the subsystems, then a periodically switching signal can be constructed such that the overall system is asymptotically stable. Based on this fact, for switched control systems, corresponding sufficient conditions are presented under which constant/switching direct/observer-based state feedback controller can be designed such that the corresponding closed-loop systems are asymptotically stable under some periodically switching signal. Some numerical examples are given to illustrate our results.     

Relaxation methods for solving linear inequality systems: converging results

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. In González-Gutiérrez and Todorov (Optim. Lett. doi:10.1007/s11590-010-0244-4, 2011), an algorithm, called extended relaxation method, for solving the feasibility problem has been proposed by the authors. Convergence of the algorithm has been proven. In this paper, we consider a class of extended relaxation methods depending on a parameter and prove their convergence. Numerical experiments have been provided, as well.     

Stabilization of switched linear systems with mode-dependent time-varying delays

In this paper, we investigate the problem of stabilization via state feedback and/or state-based switching for switched linear systems with mode-dependent time-varying delays. By using the multiple Lyapunov functional method, we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and/or switching under time-varying delays with appropriate upper bounds. The main results are presented in terms of linear matrix inequalities (LMIs) which generalize some known results and can be easily tested by using the Matlab’s LMI Tool-box.     

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.     

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.