Preserving order observers for nonlinear systems

Preserving Order Observers provide an estimation that is always above or below the true variable, and in the absence of uncertainties/perturbations, the estimation converges asymptotically to the true value of the variable. In this paper, we propose a novel methodology to design preserving order observers for a class of nonlinear systems in the nominal case or when perturbations/uncertainties are present. This objective is achieved by combining two important systemic properties: dissipativity and cooperativity. Dissipativity is used to guarantee the convergence of the estimation error dynamics, whereas cooperativity of the error dynamics assures the order‐preserving properties of the observer. The use of dissipativity for observer design offers a big flexibility in the class of nonlinearities that can be considered while keeping the design simple: it leads in many situations to the solution of a linear matrix inequality (LMI). Cooperativity of the observer leads to an LMI. When both properties are considered simultaneously, the design of the observer can be reduced, in most cases, to the solution of both a bilinear matrix inequality and an LMI. Because a couple of preserving order observers, one above and one below, provide an interval observer, the proposed methodology unifies several interval observers design methods. The design methodology has been validated experimentally in a three‐tanks system, and it has also been tested numerically and compared with an example from the literature.Copyright © 2013 John Wiley & Sons, Ltd.     

A robust estimation approach for uncertain systems with perturbed measurements

This paper deals with state estimation problem for uncertain continuous‐time systems. A numerical treatment is proposed for designing interval observers that ensures guaranteed upper and lower bounds on the estimated states. In order to take into account possible perturbations on the system and its outputs, a new type of interval observers is introduced. Such interval observers consist of two coupled general Luenberger‐type observers that involve dilatation functions. In addition, we provide an optimality criterion in order to find optimal interval observers that lead to tight interval error estimation. The proposed existence and optimality conditions are expressed in terms of linear programming. Also, some illustrative examples are given. Copyright © 2015 John Wiley & Sons, Ltd.     

Observer design with guaranteed RMS gain for discrete‐time LPV systems with Markovian jumps

In this paper we consider the problem of designing state observers with guaranteed power‐to‐power (RMS) gain for a class of stochastic discrete‐time linear systems that possess both measurable parameter variations and Markovian jumps in their dynamics. It is shown in the paper that an upper bound on the RMS gain of the observer can be characterized in terms of feasibility of a family of parameter‐dependent linear matrix inequalities (LMIs). Any feasible solution to these LMIs can then be used to explicitly construct a parameter‐varying jump observer that guarantees the desired performance level. This design framework is then specialized to a problem of state estimation for a linear parameter‐varying plant whose state measurements are available through a lossy Bernoulli channel. Two numerical examples illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinear observer design for one‐sided Lipschitz systems with time‐varying delay and uncertainties

This paper investigates the problem of state observer design for a class of nonlinear uncertain dynamical systems with interval time‐varying delay and the one‐sided Lipschitz condition. By constructing the novel Lyapunov–Krasovskii functional while utilizing the free‐weighting matrices approach, the one‐sided Lipschitz condition and the quadratic inner‐bounded condition, novel sufficient conditions, which guarantee the observer error converge asymptotically to zero, are established for a class of nonlinear dynamical systems with interval time‐varying delay in terms of the linear matrix inequalities. The computing method for observer gain matrix is given. Finally, two examples illustrate the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Interval estimation for uncertain systems with time-varying delays

The estimation problem for uncertain time-delay systems is addressed. A design method of reduced-order interval observers is proposed. The observer estimates the set of admissible values (the interval) for the state at each instant of time. The cases of known fixed delays and uncertain time-varying delays are analysed. The proposed approach can be applied to linear delay systems and nonlinear time-delay systems in the output canonical form. It involves the properties of quasi-monotone/Metzler/cooperative systems. In this framework, it is shown that if under a suitable coordinate transformation the delay-free subsystem is cooperative, then the delayed estimation error dynamics inherits this property. The conditions to find the observer gains are formulated in the form of LMI. The framework efficiency is demonstrated on examples of nonlinear systems.     

An LMI approach to robust fault estimation for a class of nonlinear systems

The paper presents a robust fault estimation approach for a class of nonlinear discrete‐time systems. In particular, two sources of uncertainty are present in the considered class of systems, that is, an unknown input and an exogenous external disturbance. Thus, apart from simultaneous state and fault estimation, the objective is to decouple the effect of an unknown input while minimizing the influence of the exogenous external disturbance within the framework. The resulting design procedure guarantees that a prescribed disturbance attenuation level is achieved with respect to the state and fault estimation error while assuring the convergence of the observer. The core advantage of the proposed approach is its simplicity by reducing the fault estimation problem to matrix inequalities formulation. In addition, the design conditions ensure the convergence of the observer with guaranteed performance. The effectiveness of the proposed approach is demonstrated by its application to a twin rotor multiple‐input multiple‐output system. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust adaptive observer for nonlinear systems with unmodeled dynamics

Unmodeled dynamics exist in almost all applications of observers due to the impossibility of using exact and detailed models. It is highly desired that the observers can dominate the effects of unmodeled dynamics independently to prevent the state estimations from diverging and to get the precise estimations. Based on adaptive nonlinear damping, this paper presents a robust adaptive observer for multiple-input multiple-output nonlinear systems with unknown parameters, uncertain nonlinearities, disturbances and unmodeled dynamics. The observer only has one adaptive parameter no matter how high the order of the system is and how many unknown parameters there are. With the proposed observer, neither estimating the unknown parameters or solving linear matrix inequalities is needed. It is shown that the state estimation error is uniformly bounded and can be made arbitrarily small.     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Circle-criterion approach to discrete-time nonlinear observer design

This paper addresses the design of discrete-time nonlinear observers through the circle criterion. The new design method is mainly devoted to either globally Lipschitz systems or bounded-state systems whose nonlinearities can be decomposed into a linear combination of positive-slope nonlinearities. The observer design is not restricted to systems with positive-slope nonlinearities, but it encompasses systems with non-positive-slope nonlinearities too. Stability conditions of the observation error are given in terms of numerically tractable linear matrix inequalities. Illustrative examples are presented in order to highlight the main features and advantages of the new proposed technique over some classical designs.     

Fault tolerant control of uncertain dynamical systems using interval virtual actuators

In this paper, a model reference fault tolerant control strategy based on a reconfiguration of the reference model, with the addition of a virtual actuator block, is presented for uncertain systems affected by disturbances and sensor noise. In particular, this paper (1) extends the reference model approach to the use of interval state observers, by considering an error feedback controller, which uses the estimated bounds for the error between the real state and the reference state, and (2) extends the virtual actuator approach to the use of interval observers, which means that the virtual actuator is added to the control loop to preserve the nonnegativity of the interval estimation errors and the boundedness of the involved signals, in spite of the fault occurrence. In both cases, the conditions to assure the desired operation of the control loop are provided in terms of linear matrix inequalities. An illustrative example is used to show the main characteristics of the proposed approach.     

Lipschitz非线性系统观测器设计新方法

考虑了非线性项满足Lipschitz条件的非线性系统观测器设计问题, 利用Lyapunov方法给出了新的判断观测误差稳定性的条件, 并由所给的条件通过求解线性矩阵不等式来设计观测器. 通过算例与其它方法进行的比较, 说明了所提方法的有效性.     

Cooperative observers and regulators for discrete‐time multiagent systems

This paper investigates the design of distributed observers for agents with identical linear discrete‐time state‐space dynamics networked on a directed graph interaction topology. The digraph is assumed to have fixed topology and contain a spanning tree. Cooperative observer design guaranteeing convergence of the estimates of all agents to their actual states is proposed. The notion of convergence region for distributed observers on graphs is introduced. It is shown that the proposed cooperative observer design has a robustness property. Application of cooperative observers is made to the synchronization problem. A command trajectory generator and pinning control are employed for synchronizing all the agents to a desired trajectory. Complete knowledge about the agent's state is not assumed. A duality principle is shown for observers and state feedback for distributed discrete‐time systems on graph topologies. Three different observer/controller architectures are proposed for dynamic output feedback regulator design, and they are shown to guarantee convergence of the estimate to the true state and synchronization of all the agents' states to the command state trajectory. This provides design methods for cooperative regulators based on a separation principle. It is shown that the observer convergence region and feedback control synchronizing region for discrete‐time systems are inherently bounded, so that the conditions for observer convergence and state synchronization are stricter than the results for the continuous‐time counterparts. This is in part remedied by using weighting of different feedback coupling gains for every agent. Copyright © 2012 John Wiley & Sons, Ltd.     

Design of LPV observers for LPV time-delay systems: an algebraic approach

The design of reduced order observer for linear parameter varying (LPV) time-delay systems is addressed. Necessary conditions guaranteeing critical structural properties for the observation error dynamics are first provided through nonlinear algebraic matrix equalities. An explicit parametrisation of the family of observers fulfilling these necessary conditions is then derived. Finally, an approach based on linear matrix inequalities is provided and used to select a suitable observer within this family, according to some criterion; e.g. maximisation of the delay margin or guaranteed suboptimal ?₂-gain. Examples from the literature illustrate the efficiency of the approach.     

Closed loop observers bundle for uncertain biotechnological models

We propose nonlinear observers for a class of biotechnological processes. These observers are an extension of the open loop asymptotic observers (observers with unknown inputs) devoted to biotechnological systems for which some parts of the model are unknown. We take benefit of the additional outputs which are (nonlinear) functions of the state to design a closed loop observer. The global convergence of these nonlinear observers is proven. We use these observers to design interval based observers which predict guaranteed intervals in which the state is lying. We run simultaneously a broad set of interval observers and we select the best ones. The method is illustrated with a model describing the bioconversion of a substrate using micro-organisms in a bioreactor.     

非线性离散系统基于观测器的反馈控制

针对一类非线性离散系统,首先提出了一种新的容易实现的状态观测器设计了方案,并证明了观测器的收敛性,其次设计了系统基于观测器的输出反馈稳定化控制器,最后给出了数值算例,仿真结果表明,本文设计方法的有效性。     

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.     

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.     

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.