Lyapunov-based switching control of nonlinear systems using high-gain observers

We consider output feedback stabilization of uniformly observable uncertain nonlinear systems when the uncertain parameters belong to a known but comparably large compact set. In a previous paper, we proposed a new logic-based switching control to improve the performance of continuous high-gain-observer-based sliding mode controllers. Our main goal here is to show that similar techniques can be exploited for solving challenging control problems for a more general class of uncertain nonlinear systems. We require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if, after a dwell-time period, the derivative of the Lyapunov function does not satisfy a certain inequality. A high-gain observer is used to estimate the derivatives of the output as well as the derivative of the Lyapunov function. Another goal of this paper is to introduce a switching strategy that uses on-line information to decide on the controller to switch to, instead of using a pre-sorted list as in our previous work. The new strategy can improve the transient performance of the system.     

On the performance of high-gain observers with gain adaptation under measurement noise

We address the problem of state observation for a system whose dynamics may involve poorly known, perhaps even nonlocally Lipschitz functions and whose output measurement may be corrupted by noise. It is known that one way to cope with all these uncertainties and noise is to use a high-gain observer with a gain adapted on-line. The proposed method, while presented for a particular case, relies on a “generic” analysis tool based on the study of differential inequalities involving quadratic functions of the error system in two coordinate frames plus the gain adaptation law. We establish that, for bounded system solutions, the estimated state and the gain are bounded. Moreover, we provide an upper bound for the mean value of the error signals as a function of the observer parameters. Since due to perturbations the gain adaptation law may drive the observer/plant interconnection to nearby boundary of its stability region, oscillatory behavior may emerge. To overcome this issue, we suggest an adaptive procedure based on a space averaging technique involving several copies of the observer.     

Sonar-based robot navigation using nonlinear robust observers

This paper addresses the sonar-based navigation of mobile robots. The extended Kalman filtering (EKF) technique is considered, but from a deterministic, nonstochastic, point of view. For this problem, new results are presented on the robustness of the nonlinear observation scheme. The original feature is that the region-of-convergence question is posed in its complete nonlinear framework, that is, considering the dynamics not only of the estimation error ζ(t), but also of the covariance matrix P(t). In this way the approach followed makes less conservative the treatment and improves the convergence analysis. The proposed ideas were tested successfully on simulation experiments of a mobile platform.     

Continuous-time norm-constrained Kalman filtering

This paper considers continuous-time state estimation when part of the state estimate or the entire state estimate is norm-constrained. In the former case continuous-time state estimation is considered by posing a constrained optimization problem. The optimization problem can be broken up into two separate optimization problems, one which solves for the optimal observer gain associated with the unconstrained state estimates, while the other solves for the optimal observer gain associated with the constrained state estimates. The optimal constrained state estimate is found by projecting the time derivative of an unconstrained estimate onto the tangent space associated with the norm constraint. The special case where the entire state estimate is norm-constrained is briefly discussed. The utility of the filtering results developed are highlighted through a spacecraft attitude estimation example. Numerical simulation results are included.     

Nonlinear approximate observers for feedback control

A class of approximate nonlinear observers is presented and, when used to implement stabilizing state feedback control laws, shown to yield asymptotically stable output feedback regulation.     

From flatness, GPI observers, GPI control and flat filters to observer-based ADRC

In this article, we establish the route taken by the author, and his research group, to bring differential flatness to the realm of active disturbance rejection control (ADRC). This avenue entitled: 1) generalized proportional integral observers (GPIO), as natural state and disturbance observers for flat systems, 2) generalized proportional integral (GPI) control, provided with extra integrations, to produce a modular controller known as flat filters (FF’s) and, finally, 3) the establishing of an equivalence of observer based ADRC with FF’s. The context is that of pure integration systems. The obtained controllers depend only on the order of the flat system and they are to be directly used on the basis of the available flat output signal in a universal, modular, fashion. The map is complemented with the relevant references where the intermediate techniques were illustrated and developed, over the years, in connection with laboratory experimental implementations.     

Separation results for the stabilization of nonlinear systems using different high-gain observer designs

High-gain observers have been used in the design of output feedback controllers due to their ability to robustly estimate the unmeasured states while asymptotically attenuating disturbances. The available techniques for the design of high-gain observers can be classified into three groups: pole-placement algorithms, Riccati equation-based algorithms, and Lyapunov equation-based algorithms. In [1], we presented separation results for globally bounded stabilizing state feedback controllers when the high-gain observer is designed using pole-placement so as to create a closed-loop system with two-time-scale structure. In this paper, we show that the separation results of [1] hold for the other observer designs.     

On state observers for nonlinear systems

For linear systems, it is known that there exists a state observer if and only if the system is detectable, or in other words, asymptotically stabilizable by output injection. In this paper, it is shown that asymptotic stabilizability by output injections is neither necessary nor sufficient for the construction of a nonlinear observer which may not have an exponential error decay rate. The concept of uniform observers is introduced and necessary conditions and sufficient conditions are given for the construction of uniform observers.     

High gain observers with updated gain and homogeneous correction terms

Exploiting dynamic scaling and homogeneity in the bi-limit, we develop a new class of high gain observers which incorporate a gain update law and nonlinear output error injection terms. A broader class of systems can be addressed and the observer gain is better fitted to the incremental rate of the nonlinearities. The expected improved performance is illustrated.     

GNSS signals processing via linear and extended Kalman filters

In this paper, a new linear estimator that enables solution of the user position estimation problem has been designed for the case of global navigation satellite system (GNSS) utilization. The proposed recursive linear Kalman filter (LKF) does not need any linearization for nonlinear GNSS measurements and enables accurate estimation of the user coordinates and the distance measuring errors resulting from the time differences between the user and the satellite (clock bias). Comparison of the presented linear and conventional extended Kalman filters for the stationary and moving users' position estimation via GNSS measurements is performed. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Sliding-mode observers for systems with unknown inputs: A high-gain approach

Sliding-mode observers can be constructed for systems with unknown inputs if the so-called observer matching condition is satisfied. However, most systems do not satisfy this condition. To construct sliding-mode observers for systems that do not satisfy the observer matching condition, auxiliary outputs are generated using high-gain approximate differentiators and then employed in the design of sliding-mode observers. The state estimation error of the proposed high-gain approximate differentiator based sliding-mode observer is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. Finally, the unknown input reconstruction using the proposed observer is analyzed and then illustrated with a numerical example.     

Nonlinear tension observers for web machines

Precise regulation of span tension in web-handling machines is critical to ensure product quality. Elevated tension causes web fracture during operation; depressed or highly variable tension leads to irregular and unacceptable spool. Web-handling machines typically employ costly load cell sensors to measure span tension. This work investigates the use of nonlinear estimators or observers to provide cost-effective span tension estimates from system measurements. After presenting the dynamic equations of a web-handling machine, their structure is exploited to design a reduced-order, nonlinear observer with linear time-varying error dynamics. Exponential stability of the error dynamics is demonstrated. Performance of the estimator is experimentally compared with two competing designs on a real web-handling machine.     

Observer design for continuous-time dynamical systems

We review the main design techniques of state observer design for continuous-time dynamical systems, namely algorithms which reconstruct online the full information of a dynamical process on the basis of partially measured data. Starting from necessary conditions for the existence of such asymptotic observers, we classify the available methods depending on the detectability/observability assumptions they require. We show how each class of observer relies on transforming the system dynamics in a particular normal form which allows the design of an observer, and how each observability condition guarantees the invertibility of its associated transformation and the convergence of the observer. Finally, some implementation aspects and open problems are briefly discussed.     

Nonlinear Luenberger‐like observers for nonlinear MIMO systems

This paper deals with the design of observers for a class of continuous time nonlinear multi‐input multi‐output systems with nonlinear outputs. Geometric tools are used to transform the original system into an appropriate observer canonical form. Furthermore, a pole placement technique is used to obtain a desired transient response of resulting error dynamics. The observer design is presented for two cases. In both cases, it is shown that the observer gain can be obtained from the solution of a Riccati equation. An illustrative example of state estimation in induction motors is presented to explain the proposed observer design. The performance of the method is also verified by numerical simulations. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Time-varying increasing-gain observers for nonlinear systems

A full-order state observer for a class of nonlinear continuous-time systems is presented as generalization of the high-gain observer for having a time-varying gain that is let to be small in the first time instants, increases over time up to its maximum, and then is kept constant. The global stability of the resulting estimation error is proved by means of a Lyapunov functional via a change of coordinate. The design of such an observer is obtained by solving a nonlinear programming problem and using series expansions to set the time-varying gain.     

基于卡尔曼滤波器组的多重故障诊断方法研究

针对缺乏有效的用于处理多重(两重及以上)加性故障隔离问题的诊断方法的现状,本文提出了一种新的基于卡尔曼滤波器组的控制系统多重故障的检测与隔离算法.通过构造多个结构不同的卡尔曼滤波器并设计相应的残差,使得每个残差仅对执行机构或传感器某个故障敏感而对其余故障不敏感,最终实现多重故障检测与隔离.除此之外,通过理论推导以及仿真分析,证明了所提出的故障检测与隔离算法的优越性.     

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.     

A new perspective on Gaussian sigma-point Kalman filters

There are a variety of Gaussian sigma-point Kalman filters (GSPKF) existing in the literature which are based on different quadrature rules. Their performances are always compared with each other on accuracy and robustness from the numerical-integration perspective, where the number of the sigma points and their corresponding weights are the main reasons resulting in the different accuracy. A new perspective on the GSPKF is proposed in this paper, which is the Mahalanobis distance ellipsoid (MDE). From the MDE perspective, GSPKFs differ from each other on accuracy and robustness mainly because they enclose different probability concentrations. This characteristic is evident when using the high-degree GSPKFs to filter the low-dimensional nonlinear systems. Two classical nonlinear system examples are used to demonstrate the proposed point of this paper. Moreover, some suggestions are given on how to select an appropriate GSPKF for a given nonlinear system.     

广义容积卡尔曼滤波

传统CKF采用三阶球面径向容积定律来计算非线性积分,该定律将球面数值积分与径向积分相结合,难以构造高阶CKF算法。此外,CKF在许多非线性问题上表现出估计精度低等问题。为了解决以上问题,提出了一种广义CKF族,所提算法彻底抛弃了球面径向积分定律。进一步指出,传统CKF是这种滤波算法的特殊形式。实验结果表明,高阶CKF比传统的非线性滤波器准确性更高。     

Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter

The performance of Bayesian state estimators, such as the extended Kalman filter (EKF), is dependent on the accurate characterisation of the uncertainties in the state dynamics and in the measurements. The parameters of the noise densities associated with these uncertainties are, however, often treated as ‘tuning parameters’ and adjusted in an ad hoc manner while carrying out state and parameter estimation. In this work, two approaches are developed for constructing the maximum likelihood estimates (MLE) of the state and measurement noise covariance matrices from operating input-output data when the states and/or parameters are estimated using the EKF. The unmeasured disturbances affecting the process are either modelled as unstructured noise affecting all the states or as structured noise entering the process predominantly through known, but unmeasured inputs. The first approach is based on direct optimisation of the ML objective function constructed by using the innovation sequence generated from the EKF. The second approach - the extended EM algorithm - is a derivative-free method, that uses the joint likelihood function of the complete data, i.e. states and measurements, to compute the next iterate of the decision variables for the optimisation problem. The efficacy of the proposed approaches is demonstrated on a benchmark continuous fermenter system. The simulation results reveal that both the proposed approaches generate fairly accurate estimates of the noise covariances. Experimental studies on a benchmark laboratory scale heater-mixer setup demonstrate a marked improvement in the predictions of the EKF that uses the covariance estimates obtained from the proposed approaches.