Interval observers for linear time-invariant systems with disturbances

It is shown that, for any time-invariant exponentially stable linear system with additive disturbances, time-varying exponentially stable interval observers can be constructed. The technique of construction relies on the Jordan canonical form that any real matrix admits and on time-varying changes of coordinates for elementary Jordan blocks which lead to cooperative linear systems. The approach is applied to detectable linear systems.     

Stabilization of linear time varying systems over uncertain channels

In this paper, we study the problem of control of discrete‐time linear time varying systems over uncertain channels. The uncertainty in the channels is modeled as a stochastic random variable. We use exponential mean square stability of the closed‐loop system as a stability criterion. We show that fundamental limitations arise for the mean square exponential stabilization for the closed‐loop system expressed in terms of statistics of channel uncertainty and the positive Lyapunov exponent of the open‐loop uncontrolled system. Our results generalize the existing results known in the case of linear time invariant systems, where Lyapunov exponents are shown to emerge as the generalization of eigenvalues from linear time invariant systems to linear time varying systems. Simulation results are presented to verify the main results of this paper. Copyright © 2012 John Wiley & Sons, Ltd.     

Design of optimal interval observers using set‐theoretic methods for robust state estimation

This article aims to design an optimal interval observer for discrete linear time‐invariant systems. Particularly, the proposed design method first transforms the interval observer into a zonotopic set‐valued observer by establishing an explicit mathematical relationship between the interval observer and the zonoptopic set‐valued observer. Then, based on the established mathematical relationship, a locally optimal observer gain is designed for the interval observer via the equivalent zonotopic set‐valued observer structure and the Frobenious norm‐based size of zonotopes. Third, considering that the dynamics of the optimal interval observer becomes a discrete linear time‐varying system due to the designed time‐varying optimal gain, an optimization problem to obtain a coordinate transformation matrix and the locally optimal observer gain for the interval observer is formulated and handled. Finally, a theoretic comparison on the conservatism of the interval observer and the zonotopic set‐valued observer is made. At the end of this article, a microbial growth bioprocess is used to illustrate the effectiveness of the proposed method.     

Delay‐Dependent Exponential Stability for Discrete‐Time Singular Switched Systems with Time‐Varying Delay

In this paper, the exponential stability problem is investigated for a class of discrete‐time singular switched systems with time‐varying delay. By using a new Lyapunov functional and average dwell time scheme, a delay‐dependent sufficient condition is established in terms of linear matrix inequalities for the considered system to be regular, causal, and exponentially stable. Different from the existing results, in the considered systems the corresponding singular matrices do not need to have the same rank. A numerical example is given to demonstrate the effectiveness of the proposed result.     

Exact Sampling of a Linear Interval Predictor

Recent advances in the design of interval observers have made it possible to ensure the non‐divergence of the computed state bounds from the stability of LTI systems under bounded inputs, with no need for additional monotony assumptions. Time‐varying changes of coordinates can be used to that purpose. Most of the related works result in either continuous‐time or discrete‐time interval dynamics. This paper proposes a constructive algorithm to compute the exact sampled response of a linear interval predictor under bounded inputs, gives a stability equivalence result and discusses the design of interval observers. The exact sampling requires held input bounds but the uncertain input itself needs not to be held. A numerical example exhibiting an oscillatory behavior illustrates the main results.     

Stable Inversion Based Precise Tracking for Periodic Systems

Stable inversion based precise tracking for discrete‐time periodically time‐varying square systems is studied. By means of the lifting technique, the time‐varying system is reformulated equivalently to the time‐invariant lifted system that is analyzed for different cases such as the non‐singular case and the singular case. Combining the stable inversion method of these cases with the optimal state transition method for time‐varying systems, precise tracking is achieved from a limited initial time. The tracking performance of the proposed method is validated through simulations.     

Time‐varying linear controllers for exponential tracking of non‐holonomic systems in chained form

In this paper, the authors address the tracking problem for non‐holonomic systems in chained form with target signals that may exponentially decay to zero. By introducing a time‐varying co‐ordinate transformation and using the cascade‐design approach, smooth time‐varying controllers are constructed, which render the tracking‐error dynamics globally ??‐exponentially stable. The result shows that the popular condition of persistent excitation or not converging to zero for the reference signals is not necessary even for the globally ??‐exponential tracking of the chained‐form system. The effectiveness of the proposed controller is validated by simulation of two benchmark mechanical systems under non‐holonomic constraints. Copyright © 2006 John Wiley & Sons, Ltd.     

Model falsification using set‐valued observers for a class of discrete‐time dynamic systems: a coprime factorization approach

This article introduces a new method for model falsification using set‐valued observers, which can be applied to a class of discrete linear time‐invariant dynamic systems with time‐varying model uncertainties. In comparison with previous results, the main advantages of this approach are as follows: The computation of the convex hull of the set‐valued estimates of the state can be avoided under certain circumstances; to guarantee convergence of the set‐valued estimates of the state, the required number of previous steps is at most as large as the number of states of the nominal plant; and it provides a straightforward nonconservative method to falsify uncertain models of dynamic systems, including open‐loop unstable plants. The results obtained are illustrated in simulation, emphasizing the advantages and shortcomings of the suggested method. Copyright © 2013 John Wiley & Sons, Ltd.     

A New Approach For Stability Analysis Of Time‐Dependent Switched Continuous‐Time Linear Systems

In this paper, a new approach for stability analysis of time‐dependent switched linear systems is proposed. System equivalence is the main idea in this new approach, which derives a switched discrete linear parameter‐varying system from the switched continuous‐time linear switched system with interval dwell time, and the stability properties of the two corresponding systems are proved to be equivalent. Then, by applying a quadratic Lyapunov function approach for the equivalent switched discrete system, the stability of the switched continuous‐time linear system can be established without checking any average dwell time condition. Finally the computation complexity is analyzed, and mode incidence matrix is introduced to reduce the computation cost.     

Observer‐based Reliable Control for Discrete Time Systems: An Average Dwell Time Approach

This paper deals with the problem of reliable control for discrete time systems with actuator failures. The actuator is assumed to fail occasionally and can recover over a time interval. During the time of suffering failures, the considered closed‐loop system is assumed unstable. Using an average dwell time method and under the condition that the activation time ratio between the system without actuator failures and the system with actuator failures is not less than a specified constant, an observer‐based feedback controller is developed in terms of linear matrix inequalities such that the resulting closed‐loop system is exponentially stable. An example is included to demonstrate the effectiveness of the proposed approach.     

Feedback stabilization of discrete time singularly perturbed systems with communication constraints

In this paper, the quantized feedback problem for a class of discrete time singularly perturbed systems with information constraints is considered. First, a proper coder‐decoder pair is presented so that the transmission errors tend to zero exponentially under information constraints. Next, linear matrix inequalities are constructed such that the resulting closed‐loop system is input‐to‐state stable (ISS) with respect to the transmission error, and the asymptotic stability of the closed‐loop system also can be guaranteed. The theoretical results have shown that the presented method is simple and easy to operate. Moreover, the upper bound of the small perturbed parameter of the stable system can be explicitly estimated using this feasible method. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.     

Tracking control of second‐order chained form systems by cascaded backstepping

A design methodology is presented for tracking control of second‐order chained form systems. The methodology separates the tracking‐error dynamics, which are in cascade form, into two parts: a linear subsystem and a linear time‐varying subsystem. The linear time‐varying subsystem, after the first subsystem has converged, can be treated as a chain of integrators for the purposes of a backstepping controller. The two controllers are designed separately and the proof of stability is given by using a result for cascade systems. The method consists of three steps. In the first step we apply a stabilizing linear state feedback to the linear subsystem. In the second step the second subsystem is exponentially stabilized by applying a backstepping procedure. In the final step it is shown that the closed‐loop tracking dynamics of the second‐order chained form system are globally exponentially stable under a persistence of excitation condition on the reference trajectory. The control design methodology is illustrated by application to a second‐order non‐holonomic system. This planar manipulator with two translational and one rotational joint (PPR) is a special case of a second‐order non‐holonomic system. The simulation results show the effectiveness of our approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Robust H∞ control of uncertain linear impulsive stochastic systems

This paper develops robust stability theorems and robust H_∞ control theory for uncertain impulsive stochastic systems. The parametric uncertainties are assumed to be time varying and norm bounded. Impulsive stochastic systems can be divided into three cases, namely, the systems with stable/stabilizable continuous‐time stochastic dynamics and unstable/unstabilizable discrete‐time dynamics, the systems with unstable/unstabilizable continuous dynamics and stable/stabilizable discrete‐time dynamics, and the systems in which both the continuous‐time stochastic dynamics and the discrete‐time dynamics are stable/stabilizable. Sufficient conditions for robust exponential stability and robust stabilization for uncertain impulsive stochastic systems are derived in terms of an average dwell‐time condition. Then, a linear matrix inequality‐based approach to the design of a robust H_∞ controller for each system is presented. Finally, the numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Global Exponential Stability Analysis of Discrete‐Time Genetic Regulatory Networks with Time Delays

This paper focuses on the study of global robust exponential stability of discrete‐time genetic regulatory networks (GRNs) with time‐invariant/time‐varying delays and parameter uncertainties. Many existing results on this problem are based on the linear matrix inequality (LMI) approach, which needs to verify whether there exists a feasible solution of a set of LMIs. Along with the increase in the number of genes, dimensions of LMIs will increase accordingly, which will lead to a large amount of calculation. Based on M‐matrix theory, sufficient conditions ensuring the global robust exponential stability of a class of discrete‐time GRNs with time‐invariant/time‐varying delays and parameter uncertainties are presented. These given conditions are to check whether a constructed constant matrix is a nonsingular M‐matrix. Simulation results of several examples are given to demonstrate the validity of the proposed method.     

Robust output feedback model predictive control of constrained linear systems: Time varying case

The problem of output feedback model predictive control of discrete time systems in the presence of additive but bounded state and output disturbances is considered. The overall controller consists of two components, a stable state estimator and a tube based, robustly stabilizing model predictive controller. Earlier results are extended by allowing the estimator to be time varying. The proposed robust output feedback controller requires the online solution of a standard quadratic program. The closed loop system renders a specified invariant set robustly exponentially stable.     

Exponential H∞ filtering for switched linear systems with interval time‐varying delay

This paper deals with the problem of exponential H_∞ filtering for a class of continuous‐time switched linear system with interval time‐varying delay. The time delay under consideration includes two cases: one is that the time delay is differentiable and bounded with a constant delay‐derivative bound, whereas the other is that the time delay is continuous and bounded. Switched linear filters are designed to ensure that the filtering error systems under switching signal with average dwell time are exponentially stable with a prescribed H_∞ noise attenuation level. Based on the free‐weighting matrix approach and the average dwell technology, delay‐dependent sufficient conditions for the existence of such a filter are derived and formulated in terms of linear matrix inequalities (LMIs). By solving that corresponding LMIs, the desired filter parameterized matrices and the minimal average dwell time are obtained. Finally, two numerical examples are presented to demonstrate the effectiveness of the developed results. Copyright © 2008 John Wiley & Sons, Ltd.     

Dissipative interval observer design for discrete‐time nonlinear systems

This paper studies the problem of designing interval observers for a family of discrete‐time nonlinear systems subject to parametric uncertainties and external disturbances. The design approach states that the interval observers are constituted by a couple of preserving order observers, one providing an upper estimation of the state while the other provides a lower one. The design aim is to apply the cooperative and dissipative properties to the discrete‐time estimation error dynamics in order to guarantee that the upper and lower estimations are always above and below the true state trajectory for all times, while both estimations asymptotically converge towards a neighborhood of the true state values. The approach represents an extension to the original method proposed by the authors, which focuses on the continuous‐time nonlinear systems. In some situations, the design conditions can be formulated as bilinear matrix inequalities (BMIs) and/or linear matrix inequalities (LMIs). Two simulation examples are provided to show the effectiveness of the design approach.     

Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time‐varying delay

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time‐varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay‐dependent stability criterion. By applying free‐weighting matrix technique and by equivalently eliminating time‐varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of ε‐dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for ε=0 are feasible then the system is exponentially stable in mean square for sufficiently small ε?0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for ε=0, then the system is ε‐uniformly exponentially stable for all sufficiently small ε?0. Based on the stability criteria, an ε‐independent state‐feedback controller that stabilizes the system for sufficiently small ε?0 is derived. Finally, numerical examples are presented, which show our results are effective and useful. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust ℋ︁∞ sliding mode control for nonlinear stochastic systems with multiple data packet losses

In this paper, an ??_∞ sliding mode control (SMC) problem is studied for a class of discrete‐time nonlinear stochastic systems with multiple data packet losses. The phenomenon of data packet losses, which is assumed to occur in a random way, is taken into consideration in the process of data transmission through both the state‐feedback loop and the measurement output. The probability for the data packet loss for each individual state variable is governed by a corresponding individual random variable satisfying a certain probabilistic distribution over the interval [0 1]. The discrete‐time system considered is also subject to norm‐bounded parameter uncertainties and external nonlinear disturbances, which enter the system state equation in both matched and unmatched ways. A novel stochastic discrete‐time switching function is proposed to facilitate the sliding mode controller design. Sufficient conditions are derived by means of the linear matrix inequality (LMI) approach. It is shown that the system dynamics in the specified sliding surface is exponentially stable in the mean square with a prescribed ??_∞ noise attenuation level if an LMI with an equality constraint is feasible. A discrete‐time SMC controller is designed capable of guaranteeing the discrete‐time sliding mode reaching condition of the specified sliding surface with probability 1. Finally, a simulation example is given to show the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

On the almost sure and mean-square exponential convergence of somestochastic observers

It is pointed out that linear observers used for estimating the state of the discrete-time stochastic-parameter systems are both almost surely and mean-square (MS) exponentially convergent under the same conditions guaranteeing mean-square convergence. In addition to the mean-square convergence properties of linear observers constructed for mean-square stable stochastic-parameter systems, they also possess an almost-sure exponential convergence property, and the rate of MS convergence is exponential. This rate depends on the parameters used in the design