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.     

Estimation of uncertain models of activated sludge processes with interval observers

This paper deals with the designs of observers (software sensors) for uncertain models of wastewater treatment. We assume that the model (4 state variables) is such that the growth rate is unknown and other parameters (e.g. the organic concentration in the influent) are uncertain, i.e. we only know their upper and lower bounds. Given the measurements of the dissolved oxygen concentration, we build interval observers, giving dynamic bounds containing the variables to estimate. The width of these bounds are related to the width of the uncertainty bounds of the parameters. In some cases, we show that we can estimate exactly the unknown variables despite the uncertainties on the model.     

Robust fault and icing diagnosis in unmanned aerial vehicles using LPV interval observers

This paper proposes a linear parameter varying (LPV) interval unknown input observer for the robust fault diagnosis of actuator faults and ice accretion in unmanned aerial vehicles (UAVs) described by an uncertain model. The proposed interval observer evaluates the set of values for the state, which are compatible with the nominal fault‐free and icing‐free operation and can be designed in such a way that some information about the nature of the unknown inputs affecting the system can be obtained, thus allowing the diagnosis to be performed. The proposed strategy has several advantages. First, the LPV paradigm allows taking into account operating point variations. Second, the noise rejection properties are enhanced by the presence of the integral term. Third, the interval estimation property guarantees the absence of false alarms. Linear matrix inequality–based conditions for the analysis/design of these observers are provided in order to guarantee the interval estimation of the state and the boundedness of the estimation. The developed theory is supported by simulation results, obtained with the uncertain model of a Zagi Flying Wing UAV, which illustrate the strong appeal of the methodology for identifying correctly unexpected changes in the system dynamics due to actuator faults or icing.     

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.     

Design of functional interval observers for non‐linear fractional‐order systems

This paper considers the problem of designing functional interval observers for a class of non‐linear fractional‐order systems with bounded uncertainties. First, interval observers for linear functions of the state vector of the considered system are designed. Then, conditions for the existence of such interval observers are established and an effective algorithm for computing unknown observer matrices is provided in this paper. Finally, numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.     

基于区间观测器的执行器故障检测

针对同时具有未知非线性函数(包括系统不确定性、外部干扰等) 和执行器故障的非线性系统, 提出基于区间观测器的故障检测方法. 首先, 在假定执行器故障不出现的前提下, 基于未知非线性函数的上下界信息, 提出两种区间观测器设计方法; 然后, 利用这两种区间观测器的输出和系统的真实输出, 构造可以对执行器故障进行检测的残差, 以此实现基于区间观测器的执行器故障检测. 最后, 通过两个仿真例子验证了所提出方法的正确性和有效性.

     

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.     

Nonlinear predictive control of irregularly sampled multirate systems using blackbox observers

This work aims the development of an inferential nonlinear model predictive control (NMPC) scheme based on a nonlinear fast rate model that is identified from irregularly sampled multirate data, which is corrupted with unmeasured disturbances and measurement noise. The model identification is carried out in two steps. In the first step, a MISO fast rate nonlinear output error (NOE) model is identified from the irregularly sampled output data. In the second step, a time varying nonlinear auto-regressive (NAR) type model is developed using the residuals generated in the first step. The deterministic and stochastic components of the observer are parameterized using generalized ortho-normal basis filters (GOBF). The identified NOE and NAR models are combined to form MISO state observers. We then proceed to use these identified observers to formulate a nonlinear MPC strategy for controlling irregularly sampled multirate systems. The identified observers are used to generate inter-sample estimates of the irregularly sampled outputs and for performing future trajectory predictions. The efficacy of the proposed modeling and control scheme is demonstrated using simulations on a benchmark continuous fermentation process. This process exhibits input multiplicity and change in the sign of steady state gain in the operating region. The validity of the proposed modeling and control scheme is also established by conducting identification and control experiments on a laboratory scale heater-mixer setup. The proposed NMPC gives satisfactory regulatory as well as servo performance over a wide operating range in the irregularly sampled multirate scenario.     

Nonlinear observer design via passivation of error dynamics

We present a new design scheme of nonlinear state observers (global, full order, asymptotic observers) through passivation of the error dynamics. In order to consider passivity of the error dynamics for the observer problem, we place a conceptual input and output on the generalized error dynamics which also includes the plant, and the strictness of passivity is extended with respect to a set in which the estimation error becomes zero. Then, output feedback passivation for the error dynamics will lead to the construction of a state observer. It is also shown that a nonlinear observer is generally vulnerable to measurement disturbance, in the sense that even an arbitrarily small measurement disturbance can lead to a blowup of the error state. However, due to the passivity of the error dynamics, the proposed nonlinear injection gain can be easily modified for the observer to be robust to measurement disturbances.     

State observers are unnecessary for induction motor control

Induction motors constitute a theoretically interesting and practically important class of nonlinear systems, which are evolving into a benchmark example for nonlinear control. They are described by a fifth-order nonlinear differential equation with two inputs, and only three state variables available for measurement. A lot of research in the field has been devoted to the design of observers which, combined with a suitable control strategy, would yield stable behaviour. Our contribution in this paper is to show that the control objective can be achieved without reconstruction of the motor state. Specifically, we present a globally stable nonlinear dynamic output feedback controller for torque tracking of induction motors which does not rely on state reconstruction ideas. Another important feature of our sheme is that the control law is globally defined, even in startup. This stems from the fact that we do not aim at linearizing the system dynamics, but instead exploit the energy dissipation properties of the motor model. For the sake of illustration we present the result for a model described in the stator frame (ab model), but the theory applies as well to models expressed in a rotating frame (dq model). We also show how, as a particular case of torque tracking, we can solve the rotor speed tracking problem.     

State and parameter estimation in chemical and biochemical processes: a tutorial

This paper aims at giving an overview of available results of state and parameter approaches for chemical and biochemical processes. It is largely organized as a tutorial and starts with a brief reminder concerning the design of extended Luenberger (ELO) and Kalman (EKO) observers, followed by an illustrative nonlinear observer algorithm. Evaluation of the performance of classical observers in presence of model uncertainties will serve as a basis for the motivation of designing asymptotic and interval observers, that do not require the knowledge of the process kinetics. The design of state observers with known kinetic models but uncertain kinetic parameters will then be considered via suggestions of improvements of the EKO and the introduction of two other types of observers (observers where the unknown parameters are used as design parameters; adaptive observers). Finally, the design of on-line parameter estimation schemes will be introduced. One of the objectives of the present survey is also to suggest new research directions.     

未知死区输入非线性系统的双观测器动态面控制

针对一类含死区输入的严格反馈非线性系统,提出基于双观测器的自适应鲁棒控制算法.动态面的每一步设计中,第1观测器即跟踪信号观测器对指令信号进行观测,并得到指令信号的差分信号,消除传统动态面控制中计算复杂问题.第二观测器即扰动观测器在线估计高阶动态面控制系统中每一步的不确定模型,与跟踪信号观测器实现双反馈控制,提高控制效果.通过李雅普诺夫方法分析了自适应鲁棒控制器的输入输出有界的特性.最后,实验验证基于双观测器的自适应鲁棒控制器的性能,结果表明本文方法控制效果较好.且采用反双曲正弦函数建立观测器,参数调节少,利于工业应用.     

Partial state and unknown input estimation for time-delay systems

This article considers the problem of estimating a partial set of the state vector and/or unknown input vector of linear systems driven by unknown inputs and time-varying delay in the state variables. Three types of reduced-order observers, namely, observers with delays, observers without internal delays and delay-free observers are proposed in this article. Existence conditions and design procedures are presented for the determination of parameters for each case of observers. Numerical examples are presented to illustrate the design procedures.     

Simultaneous state and unknown input reconstruction using cascaded high-gain observers

This paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs (UIs) for a large class of nonlinear MIMO systems. After an appropriate change of state coordinates, a set of cascade high-gain observers (CHGO) are designed in such a way that each of them provides an estimation of only one component of the UI vector, except the last one which gives a final adjustment of the whole state variables. Such design achieves a boundary of the state estimation error which can be arbitrarily small by properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme.     

A note on stability, robustness and performance of output feedback nonlinear model predictive control

In recent years, nonlinear model predictive control (NMPC) schemes have been derived that guarantee stability of the closed loop under the assumption of full state information. However, only limited advances have been made with respect to output feedback in the framework of nonlinear predictive control. This paper combines stabilizing instantaneous state feedback NMPC schemes with high-gain observers to achieve output feedback stabilization. For a uniformly observable MIMO system class it is shown that the resulting closed loop is asymptotically stable. Furthermore, the output feedback NMPC scheme recovers the performance of the state feedback in the sense that the region of attraction and the trajectories of the state feedback scheme can be recovered to any degree of accuracy for large enough observer gains, thus leading to semi-regional results. Additionally, it is shown that the output feedback controller is robust with respect to static sector bounded nonlinear input uncertainties.     

Partial state observers for linear systems with unknown inputs

We consider the problem of constructing partial state observers for discrete-time linear systems with unknown inputs. Specifically, for any given system, we develop a design procedure that characterizes the set of all linear functionals of the system state that can be reconstructed through a linear observer with a given delay. By treating the delay as a design parameter, we allow greater flexibility in estimating state functionals, and are able to obtain a procedure that directly produces the corresponding observer parameters. Our technique is also applicable to continuous-time systems by replacing delayed outputs with differentiated outputs.     

An Active Disturbance Rejection Approach to Leader‐Follower Controlled Formation

This article describes the design of a linearizing, observer‐based, robust dynamic feedback control scheme for output reference trajectory tracking tasks in a leader‐follower non‐holonomic car formation problem. The approach is based on the cars' kinematic models. A radical simplification in the form of a global ultra‐model is proposed on the follower's exact open loop position tracking error dynamics obtained via flatness considerations. This results in a system described by an additively disturbed set of two, second order integrators with non‐linear velocity dependent control input gain matrix. The unknown additive disturbances are modeled as absolutely uniformly bounded time signals which may be locally approximated by arbitrary elements of a sufficiently high degree family of Taylor polynomials. Linear high‐gain Luenberger observers of the generalized proportional integral (GPI) type may be readily designed. These observers include the self updating internal model of the unknown disturbance input vector components in the form of generic, instantaneous, time‐polynomial models. The proposed (GPI) observers, which are the dual counterpart of GPI controllers [17], achieve a simultaneous disturbance estimation and tracking error phase variables estimation. This on‐line gathered information is used to advantage on the follower's feedback controller thus allowing for a simple, yet efficient, disturbance and control input gain cancelation effort. The results are applied to have the follower track a time‐delayed version of the actual leader's trajectory. Experimental results are presented which illustrate the robustness and viability of the proposed approach.     

含未知输入观测器的广义系统故障诊断

基于未知输入观测器设计和故障诊断的概念,讨论含未知输入的Lipschitz条件下非线性广义系统传感器故障诊断问题。在非线性广义系统中,通过引入传感器的故障信号,重新构造非线性广义系统,设计基于未知输入观测器,在满足Lipschitz条件下,实现了传感器故障的检测与分离。给出数值仿真算例验证该算法的有效性。     

Design of reduced-order observers for nonlinear sampled-data strict-feedback systems with actuator dynamics and disturbances

We design reduced-order observers for nonlinear sampled-data strict-feedback systems with actuator dynamics and disturbances. First, we use the property of the Euler model of sampled-data systems and the structure of discrete-time observers to design reduced-order observers of the Euler model. Then, we show that the designed observers are semiglobal and practical in T for the exact model. We also give both numerical and practical examples to illustrate the proposed design of reduced-order observers.