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.     

A discrete‐time nonlinear state observer for the anaerobic digestion process

This paper is concerned with the design of an LMI‐based discrete‐time nonlinear state observer for an anaerobic digestion model. In presence of disturbances in both the dynamics of the model and the output measurement signals, the proposed observer robustly estimates all state variables including bacteria concentrations, which are costly and difficult to measure. In the goal to increase applicability of the proposed observer for other systems, we present the theoretical results in a general way. First, due to the use of Young's inequality in a convenient way, we get new sufficient conditions, expressed in terms of bilinear matrix inequalities (BMIs), ensuring the criterion. Then, to render the BMIs convex, two alternative solutions are proposed, where both lead to linear matrix inequality (LMI) conditions. It is shown analytically and numerically that these two solutions provide less conservative LMI conditions compared to the existing methods in the literature. To validate the proposed methodology on a real‐world model, an application to an anaerobic digestion model is given.     

Attractive ellipsoid design for robust sliding‐mode observation error in stochastic nonlinear discrete‐time systems

In this paper the observation process of stochastic discrete‐time nonlinear system is analyzed. The system to be observed is assumed to be uncertain, but fulfilling the global "quasi‐Lipschitz" condition and is subjected to stochastic input and output disturbances of a white noise type. The combination of a traditional Luenberger residual term with a discontinuous one is considered. The designing of the best observer gain matrices is realized by using the Robust Attractive Ellipsoid Method for the analysis of the averaged observation error. The construction of this attractive ellipsoid is based on the numerical solution of some matrix optimization problem under specific constraints of Bilinear and Linear Matrix Inequalities (BMI's and LMI's) type applied to improve the attractiveness zone estimation. Two numerical xamples illustrate the effectiveness of the suggested approach.     

Interval observers for discrete‐time systems

Interval observers are constructed for discrete‐time systems. First, time‐invariant interval observers are proposed for a family of nonlinear systems. Second, it is shown that, for any time‐invariant exponentially stable discrete‐time linear system with additive disturbances, time‐varying exponentially stable discrete‐time interval observers can be constructed. The latter result relies on the design of time‐varying changes of coordinates, which transform a linear system into a nonnegative one. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust discrete‐time nonlinear sliding mode state estimation of uncertain nonlinear systems

In this paper, we propose a discrete‐time nonlinear sliding mode observer for state and unknown input estimations of a class of single‐input/single‐output nonlinear uncertain systems. The uncertainties are characterized by a state‐dependent vector and a scalar disturbance/unknown input. The discrete‐time model is derived through Taylor series expansion together with nonlinear state transformation. A design methodology that combines the discrete‐time sliding mode (DSM) and a nonlinear observer design is adopted, and a strategy is developed to guarantee the convergence of the estimation error to a bound within the specified boundary layer. A relation between sliding mode gain and boundary layer is established for the existence of DSM, and the estimation is made robust to external disturbances and uncertainties. The unknown input or disturbance can also be estimated through the sliding mode. The conditions for the asymptotical stability of the estimation error are analysed. Application to a bioreactor is given and the simulation results demonstrate the effectiveness of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Compensation of time‐varying input delay for discrete‐time nonlinear systems

We consider general discrete‐time nonlinear systems (of arbitrary nonlinear growth) with time‐varying input delays and design an explicit predictor feedback controller to compensate the input delay. Such results have been achieved in continuous time, but only under the restriction that the delay rate is bounded by unity, which ensures that the input signal flow does not get reversed, namely, that old inputs are not felt multiple times by the plant (because on such subsequent occasions, the control input acts as a disturbance). For discrete‐time systems, an analogous restriction would be that the input delay is non‐increasing. In this work, we do not impose such a restriction. We provide a design and a global stability analysis that allow the input delay to be arbitrary (containing intervals of increase, decrease, or stagnation) over an arbitrarily long finite period of time. Unlike in the continuous‐time case, the predictor feedback law in the discrete‐time case is explicit. We specialize the result to linear time‐invariant systems and provide an explicit estimate of the exponential decay rate. Carefully constructed examples are provided to illustrate the design and analytical challenges. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive discrete neural observer design for nonlinear systems with unknown time‐delay

This paper focuses on the adaptive observer design for nonlinear discrete‐time MIMO systems with unknown time‐delay and nonlinear dynamics. The delayed states involved in the system are arguments of a nonlinear function and only the estimated delay is utilized. By constructing an appropriate Lyapunov–Krasovskii function, the delay estimation error is considered in the observer parameter design. The proposed method is then extended to the system with a nonlinear output measurement equation and the delayed dynamics. With the help of a high‐order neural network (HONN), the requirement for a precise system model, the linear‐in‐the‐parameters (LIP) assumption of the delayed states, the Lipschitz or norm‐boundedness assumption of unknown nonlinearities are removed. A novel converse Lyapunov technical lemma is also developed and used to prove the uniform ultimate boundedness of the proposed observer. The effectiveness of the proposed results is verified by simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Fault estimation observer design for discrete‐time systems in finite‐frequency domain

This paper proposes a framework of fault estimation observer design in finite‐frequency domain for discrete‐time systems. First, under the multiconstrained idea, a full‐order fault estimation observer in finite‐frequency domain is designed to achieve fault estimation by using the generalized Kalman–Yakubovich–Popov lemma to reduce conservatism generated by the entire frequency domain. Then, a reduced‐order fault estimation observer is constructed, which results in a new fault estimator to realize fault estimation using current output information. Furthermore, by introducing slack variables, improved results on full‐order fault estimation observer and reduced‐order fault estimation observer design with finite‐frequency specifications are obtained such that different Lyapunov matrices can be separately designed for each constraint. Simulation results are presented to illustrate the advantages of the theoretic results obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive event‐triggered control for nonlinear discrete‐time systems

This paper focuses on the analysis and the design of event‐triggering scheme for discrete‐time systems. Both static event‐triggering scheme (SETS) and adaptive event‐triggering scheme (AETS) are presented for discrete‐time nonlinear and linear systems. What makes AETS different from SETS is that an auxiliary dynamic variable satisfying a certain difference equation is incorporated into the event‐triggering condition. The sufficient conditions of asymptotic stability of the closed‐loop event‐triggered control systems under both two triggering schemes are given. Especially, for the linear systems case, the minimum time between two consecutive control updates is discussed. Also, the quantitative relation among the system parameters, the preselected triggering parameters in AETS, and a quadratic performance index are established. Finally, the effectiveness and respective advantage of the proposed event‐triggering schemes are illustrated on a practical example. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and stabilization of nonlinear discrete‐time stochastic systems

This paper mainly studies the locally/globally asymptotic stability and stabilization in probability for nonlinear discrete‐time stochastic systems. Firstly, for more general stochastic difference systems, two very useful results on locally and globally asymptotic stability in probability are obtained, which can be viewed as the discrete versions of continuous‐time Itô systems. Then, for a class of quasi‐linear discrete‐time stochastic control systems, both state‐ and output‐feedback asymptotic stabilization are studied, for which, sufficient conditions are presented in terms of linear matrix inequalities. Two simulation examples are given to illustrate the effectiveness of our main results.     

H ∞ observer design for uncertain nonlinear discrete‐time systems with time‐delay: LMI optimization approach

We present a robust H _∞ observer for a class of nonlinear discrete‐time systems. The class under study includes an unknown time‐varying delay limited by upper and lower bounds, as well as time‐varying parametric uncertainties. We design a nonlinear H _∞ observer, by using the upper and lower bounds of the delay, that guarantees asymptotic stability of the estimation error dynamics and is also robust against time‐varying parametric uncertainties. The described problem is converted to a standard optimization problem, which can be solved in terms of linear matrix inequalities (LMIs). Then, we expand the problem to a multi‐objective optimization problem in which the maximum admissible Lipschitz constant and the minimum disturbance attenuation level are the problem objectives. Finally, the proposed observer is illustrated with two examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust output regulation for discrete‐time nonlinear systems

In this paper, we will establish a framework that can convert the robust output regulation problem for discrete‐time nonlinear systems into a robust stabilization problem for an appropriately augmented system consisting of the given plant and a specific dynamic system called internal model. We then apply this framework to solve the local robust output regulation problem for a general class of discrete‐time nonlinear systems. The results of this paper gives a discrete‐time counterpart of the recent results on the continuous‐time robust output regulation problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Observer‐based controller design of discrete‐time piecewise affine systems

This paper presents a novel observer‐based controller design method for discrete‐time piecewise affine (PWA) systems. The basic idea is as follows: at first, a piecewise linear (without affine terms) state feedback controller and a PWA observer are designed separately, and then it is proved that the output feedback controller constructed by the resulting observer and state feedback controller gains can guarantee the stability of the closed‐loop system. During the controller design, the piecewise‐quadratic Lyapunov function technique is used. Moreover, the region information is taken into account to treat the affine terms, so the controller gains can be obtained by solving a set of linear matrix inequalities, which are numerically feasible with commercially available software. Three simulation examples are given finally to verify the proposed theoretical results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Iterative learning control design with high‐order internal model for discrete‐time nonlinear systems

In this paper, a high‐order internal model (HOIM)‐based iterative learning control (ILC) scheme is proposed for discrete‐time nonlinear systems to tackle the tracking problem under iteration‐varying desired trajectories. By incorporating the HOIM that is utilized to describe the variation of desired trajectories in the iteration domain into the ILC design, it is shown that the system output can converge to the desired trajectory along the iteration axis within arbitrarily small error. Furthermore, the learning property in the presence of state disturbances and output noise is discussed under HOIM‐based ILC with an integrator in the iteration axis. Two simulation examples are given to demonstrate the effectiveness of the proposed control method. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis and design of chattering‐free discrete‐time sliding mode control

In this paper, the disturbance observer–based chattering‐free discrete‐time sliding mode control (DSMC) approach is proposed for systems with external disturbances. The proposed disturbance observer, which makes full use of the state and input information at the current and last steps, improves the estimation accuracy and achieves accurate compensation for disturbances. Then, with the help of disturbance observer, a new reaching law, which contains not only a nonsmooth term with a dynamically adjusted gain parameter but also a second order difference of the disturbance, is proposed to reduce the range of the quasi‐sliding mode band and eliminate chattering. The proposed DSMC approach realizes the active disturbance rejection and strong robustness. Finally, a simulation example is presented to verify the effectiveness of the proposed method.     

Observer‐based controller for constrained uncertain stochastic nonlinear discrete‐time systems

In this paper, an observer‐based control approach is proposed for uncertain stochastic nonlinear discrete‐time systems with input constraints. The widely used extended Kalman filter (EKF) is well known to be inadequate for estimating the states of uncertain nonlinear dynamical systems with strong nonlinearities especially if the time horizon of the estimation process is relatively long. Instead, a modified version of the EKF with improved stability and robustness is proposed for estimating the states of such systems. A constrained observer‐based controller is then developed using the state‐dependent Riccati equation approach. Rigorous analysis of the stability of the developed stochastically controlled system is presented. The developed approach is applied to control the performance of a synchronous generator connected to an infinite bus and chaos in permanent magnet synchronous motor. Simulation results of the synchronous generator show that the estimated states resulting from the proposed estimator are stable, whereas those resulting from the EKF diverge. Moreover, satisfactory performance is achieved by applying the developed observer‐based control strategy on the two practical problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonlinear minimization approach to multiobjective and structured controls for discrete‐time systems

In this paper, a nonlinear minimization approach is proposed for multiobjective and structured controls for discrete‐time systems. The problem of finding multiobjective and structured controls for discrete‐time systems is represented as a quadratic matrix inequality problem. It is shown that the problem is reduced to a nonlinear minimization problem that has a concave objective function and linear matrix inequality constraints. An algorithm for the nonlinear minimization problem is proposed, which is easily implemented with existing semidefinite programming algorithms. The validity of the proposed algorithm is illustrated by comparisons with existing methods. In addition, applications of this work are demonstrated via numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonlinear H∞ observer design for one‐sided Lipschitz discrete‐time singular systems with time‐varying delay

This paper investigates the H_∞ observer design problem for a class of nonlinear discrete‐time singular systems with time‐varying delays and disturbance inputs. The nonlinear systems can be rectangular and the nonlinearities satisfy the one‐sided Lipschitz condition and quadratically inner‐bounded condition, which are more general than the traditional Lipschitz condition. By appropriately dealing with these two conditions and applying several important inequalities, a linear matrix inequality–based approach for the nonlinear observer design is proposed. The resulting nonlinear H_∞ observer guarantees asymptotic stability of the estimation error dynamics with a prescribed performance γ. The synthesis condition of H_∞ observer design for nonlinear discrete‐time singular systems without time delays is also presented. The design is first addressed for one‐sided Lipschitz discrete‐time singular systems. Finally, two numerical examples are given to show the effectiveness of the present approach.     

Combining state estimator and disturbance observer in discrete‐time sliding mode controller design

In response to a multiple input/multiple output discrete‐time linear system with mismatched disturbances, an algorithm capable of performing estimated system states and unknown disturbances is proposed first, and then followed with the design of the controller. Attributed to the fact that both system states and disturbances can be estimated simultaneously with our proposed method, the estimation error is constrained at less than O(T) as the disturbance between the two sampling points is insignificant. In addition, the estimated system states and disturbances are then to be used in the controller when implementing our algorithm in a non‐minimum phase system (with respect to the relation between the output and the disturbance). The tracking error is constrained in a small bounded region and the system stability is guaranteed. Finally, a numerical example is presented to demonstrate the applicability of the proposed control scheme. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Constructive algorithm for dynamic observer error linearization via integrators: single output case

Dynamic observer error linearization which has been introduced recently is a new framework for observer design. Although this approach unifies several existing results on the problem and extends the class of systems that can be transformed into an observable linear system with an injection term of known signals, constructive algorithms to check the applicability are not available yet. In this paper, a constructive algorithm is proposed to solve the problem under some restrictions on the system structure and on the auxiliary dynamics introduced in the problem. The algorithm is constructive in the sense that the components of the transformation can be obtained step‐by‐step. Copyright © 2006 John Wiley & Sons, Ltd.