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.     

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.     

Observer design for nonlinear discrete‐time systems: Immersion and dynamic observer error linearization techniques

This paper focuses on the observer design for nonlinear discrete‐time systems by means of nonlinear observer canonical form. At first, sufficient and necessary conditions are obtained for a class of autonomous nonlinear discrete‐time systems to be immersible into higher dimensional observer canonical form. Then a method called dynamic observer error linearization is developed. By introducing a dynamic auxiliary system, the augmented system is shown to be locally equivalent to the generalized observer form, whose nonlinear terms contain auxiliary states and output of the system. A constructive algorithm is also provided to obtain the state coordinate transformation. These results are an extension of their counterparts of nonlinear continuous‐time systems to nonlinear discrete‐time systems (Syst. Control Lett. 1986; 7 :133–142; SIAM. J. Control Optim. 2003; 41 :1756–1778; Int. J. Control 2004; 77 :723–734; Automatica 2006; 42 :321–328; IEEE Trans. Automat. Control 2007; 52 :83–88; IEEE Trans. Automat. Control 2004; 49 :1746–1750; Automatica 2006; 42 :2195–2200; IEEE Trans. Automat. Control 1996; 41 :598–603; Syst. Control Lett. 1997; 31 :115–128). Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear controls for a class of discrete‐time bilinear systems

For a discrete‐time neutrally stable bilinear system, a nonlinear state feedback control based on the passivity design has been proposed to stabilize the system globally and asymptotically. This paper shows that the decay rate resulting from the passivity control is not exponential, and the system's response speed becomes very sluggish asymptotically. A ‘normalized’ nonlinear control is therefore proposed to achieve exponential stability. The new exponentially stabilizing control not only improves the system's response speed, but also enhances the system's robustness against small parametric perturbations. Copyright © 2003 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.     

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     

Robust Kalman filtering for uncertain discrete‐time linear systems

This paper presents a steady‐state robust state estimator for a class of uncertain discrete‐time linear systems with norm‐bounded uncertainty. It is shown that if the system satisfies some particular structural conditions and if the uncertainty has a specific structure, the gain of the robust estimator (which assures a guaranteed cost) can be calculated using a formula only involving the original system matrices. Among the conditions the system has to satisfy, the strongest one relies on a minimum phase argument. It is also shown that under the assumptions considered, the robust estimator is in fact the Kalman filter for the nominal system. Copyright © 2003 John Wiley & Sons, Ltd.     

Using set invariance to design robust interval observers for discrete‐time linear systems

Based on interval and invariant set computation, an interval version of the Luenberger state observer for uncertain discrete‐time linear systems is proposed in this work. This new interval observer provides a punctual estimation of the state vector and guaranteed bounds on the estimation error. An off‐line and an on‐line approach to characterize, in a guaranteed way, the estimation error are introduced. Compared with the existing approaches, the proposed interval observer design method is not restrictive in terms of required assumptions, complexity, and on‐line computation time. Furthermore, the convergence issue of the estimation error is well established and to reduce the conservatism of the estimated state enclosure induced by the bounded additive state disturbance and noise measurement, an H_∞ method to compute the optimal observer gain is proposed. The performance of the proposed state estimation approach are highlighted on different illustrative examples.     

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.     

IQC‐based robustness analysis of discrete‐time linear time‐varying systems

The paper investigates the robust stability and performance of uncertain linear time‐varying (LTV) systems using an integral quadratic constraint (IQC) based analysis approach. Specifically, previous theoretical work on IQC‐based robustness analysis of linear time‐invariant (LTI) systems is extended to discrete‐time LTV systems. In the case of a general LTV nominal system, the analysis solution is provided in terms of an infinite‐dimensional convex optimization problem. This optimization problem reduces into a finite‐dimensional semidefinite program when the nominal system in question is finite horizon, periodic, or, more generally, eventually periodic. Finally, the results are applied to an unmanned aircraft control system executing an aggressive maneuver, where the developed techniques are used to find the region in which the aircraft is guaranteed to reside at the end of its planned trajectory. Copyright © 2017 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.     

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.     

Reachable set estimation for discrete‐time linear systems with time delays

This paper is concerned with the reachable set estimation problem for discrete‐time linear systems with multiple constant delays and bounded peak inputs. The objective is to check whether there exists a bounded set that contains all the system states under zero initial conditions. First, delay‐dependent conditions for the solvability of the addressed problem are derived by employing a novel Lyapunov–Krasovskii functional. The obtained conditions are expressed in terms of matrix inequalities, which are linear when only one scalar variable is fixed. On the basis of these conditions, an ellipsoid containing the reachable set of the considered system is obtained. An approach for determining the smallest ellipsoid is also provided. Second, the approach and results developed in the first stage are generalized to the case of systems with polytopic parameter uncertainties, and delay‐dependent conditions are given in the form of relaxed matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Composite stratified anti‐disturbance control for a class of MIMO discrete‐time systems with nonlinearity

Anti‐disturbance control and estimation problem are investigated for nonlinear system subject to multi‐source disturbances. The disturbances classified model is proposed based on the error and noise analysis of priori knowledge. The disturbance observers are constructed separately from the controller design to estimate the disturbance with partial known information. By integrating disturbance‐observer‐based control with discrete‐time sliding‐mode control (DSMC), a novel type of composite stratified anti‐disturbance control scheme is presented for a class of multiple‐input–multiple‐output discrete‐time systems with known and unknown nonlinear dynamics, respectively. Simulations for a flight control system are given to demonstrate the effectiveness of the results compared with the previous schemes. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐uniform in time robust global asymptotic output stability for discrete‐time systems

In this paper the notions of non‐uniform in time robust global asymptotic output stability (RGAOS) and input‐to‐output stability (IOS) for discrete‐time systems are studied. Characterizations as well as links between these notions are provided. Particularly, it is shown that a discrete‐time system with continuous dynamics satisfies the non‐uniform in time IOS property if and only if the corresponding unforced system is non‐uniformly in time RGAOS. Necessary and sufficient conditions for the solvability of the robust output feedback stabilization (ROFS) problem are also given. Copyright © 2005 John Wiley & Sons, Ltd.     

Scaled consensus for asynchronous high‐order discrete‐time multiagent systems

This paper investigates the distributed scaled consensus problem of multiple agents with high‐order dynamics under the asynchronous setting, where each agent measures the neighbors' information at certain discrete time instants according to its own clock rather than the whole discrete process and all agents' clocks are independent of each other. Assume that the communication topology can be arbitrarily switched and the information transfer between agents has a time‐varying delay. Under the designed asynchronous distributed control protocol, it is shown that the agents with the same scale will reach a common final state, while the agents with different scales will reach different final states. Moreover, an effective parameters selection strategy is presented for a large number of gain parameters in high‐order multiagent systems based on novel model transformation techniques. Simulation examples are provided to demonstrate the high‐order scaled consensus performances for the agents in the presence of asynchronous setting.     

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.     

Gain scheduled control for discrete‐time systems depending on bounded rate parameters

In this paper we consider a linear, discrete‐time system depending multi‐affinely on uncertain, real time‐varying parameters. A new sufficient condition for the stability of this class of systems, in terms of a feasibility problem involving linear matrix inequalities (LMIs), is obtained under the hypothesis that a bound on the rate of variation of the parameters is known. This condition, obtained by the aid of parameter dependent Lyapunov functions, obviously turns out to be less restrictive than that one obtained via the classical quadratic stability (QS) approach, which guarantees stability in presence of arbitrary time‐varying parameters. An important point is that the methodology proposed in this paper may result in being less conservative than the classical QS approach even in the absence of an explicit bound on the parameters rate of variation. Concerning the synthesis context, the design of a gain scheduled compensator based on the above approach is also proposed. It is shown that, if a suitable LMI problem is feasible, the solution of such problem allows to design an output feedback gain scheduled dynamic compensator in a controller‐observer form stabilizing the class of systems which is dealt with. The stability conditions are then extended to take into account L₂ performance requirements. Some numerical examples are carried out to show the effectiveness and to investigate the computational burden required by the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Robust consensus for uncertain multi‐agent systems with discrete‐time dynamics

This paper investigates robust consensus for multi‐agent systems with discrete‐time dynamics affected by uncertainty. In particular, the paper considers multi‐agent systems with single and double integrators, where the weighted adjacency matrix is a polynomial function of uncertain parameters constrained into a semialgebraic set. Firstly, necessary and sufficient conditions are provided for robust consensus based on the existence of a Lyapunov function polynomially dependent on the uncertainty. In particular, an upper bound on the degree required for achieving necessity is provided. Secondly, a necessary and sufficient condition is provided for robust consensus with single integrator and nonnegative weighted adjacency matrices based on the zeros of a polynomial. Lastly, it is shown how these conditions can be investigated through convex programming by exploiting linear matrix inequalities and sums of squares of polynomials. Some numerical examples illustrate the proposed results. Copyright © 2013 John Wiley & Sons, Ltd.