Computing by observing: Simple systems and simple observers

We survey and extend the work on the paradigm called “computing by observing”. Its central feature is that one considers the behavior of an evolving system as the result of a computation. To this purpose an external observer records this behavior. In this way, several computational trade-offs between the observer and the observed system can be determined. It has turned out that the observed behavior of computationally simple systems can be very complex, when an appropriate observer is used. For example, a restricted version of context-free grammars with regular observers suffices to obtain computational completeness. As a second instantiation presented here, we apply an observer to sticker systems, an abstract model of DNA computing. Finally, we introduce and investigate the case where the observers can read only one measure of the observed system (e.g., mass or temperature), modeling in this way the limitations in the observation of real physical systems. Finally, a research perspective on the topic is presented.     

On uniform observation of nonuniformly observable systems

In [4,7,9,12], classes of nonlinear systems are considered for which observers can be designed. Although observability of nonlinear systems is known to be dependent on the input, the proposed observers have the property that the estimation error decays to zero irrespective of the input. In the first part of this paper, it is shown that this phenomenon follows from a common property of these systems: for all of them, the “unobservable states” with respect to some input, are in some sense “stable” (in the linear case, these systems are called detectable), and for this reason, a reduced order observer can be designed. In the second part is given a more general class of nonlinear systems for which such an observer can be designed.     

面向社交网络信息源定位的观察点部署方法

准确地定位社交网络上的信息扩散源点,对于网络信息扩散控制具有重要的现实意义。现有的一种可行方法是通过在网络中观察点搜集的过程信息对扩散源进行定位,定位准确率与观察点的选择紧密相关。针对网络中的信息扩散源定位问题,提出了一种网络观察点优化部署方法。考虑单信息源的信息扩散过程,首先分析了特定信息源定位准确率与观察点部署位置之间的关系,以此为基础,发现了与任意信息源定位准确率相关的关键因素。提出基于r覆盖率的观察点部署策略,以观察点集合的r覆盖率作为目标函数,实现了r覆盖率优先观察点选取算法。在模型网络与实际网络上进行了实验,验证了该方法的有效性。提出的观察点部署策略对于网络谣言、计算机病毒的控制具有重要意义。     

Robust observer design for nonlinear uncertain systems with LQ regulators

For a class of uncertain systems with linear nominal dynamics and nonlinear uncertainties, it has been shown (Katayama and Sasaki 1987) that linear quadratic (LQ) state feedback regulators can be used to provide robust asymptotic stability. In this paper, we study the combined observer-controller design problem, based on the linear state feedback regulator proposed by Katayama and Sasaki (1987), so that only output feedback is needed. Both full-order and reduced-order observers are considered. For the full-order observer, we propose an algorithm to synthesize the robust observer gain matrix. It is shown that with the observer it is still possible to achieve robust asymptotic stability. For the reduced-order observer, some conditions are derived to guarantee the robust asymptotic stabilizability of the uncertain systems. The trade-off between the magnitudes of controller and observer gains is clear in our approach. An example is used to illustrate the design process of the robust controller with full-order as well as reduced-order observers.     

Discrete-time observers for singularly perturbed continuous-timesystems

The use of discrete-time observers for nonlinear singularly perturbed continuous-time systems is explored. The observer design approach is based on inversion of state-to-measurement maps. The two-time-scale nature of the system is exploited by constructing separate reduced-order observers for the approximate slow and fast subsystems, using multirate measurements and computation. The reduced-order observers are compared to a full-order observer designed for the same system. The comparison shows that although the reduced-order observers exhibit some approximation error, they also result in reduced computational requirements if the inversion is performed on-line, reduced memory requirements if the inversion is implemented by look-up table and reduced stiffness. The trade-off between accuracy and implementation requirements always favors the reduced-order approach for systems with significant separation of time scales. Numerical examples are provided, including a case-study of an observer-based control system for permanent-magnet synchronous motors     

Moving horizon numerical observers of nonlinear control systems

In this note, we develop moving horizon numerical observers and analyze the error. In the error estimation, we take into consideration both the integration error and the optimization error. The design facilitates the use of a variety of numerical algorithms to form different observers. As a special case, an Euler-Newton observer is introduced. The numerical observer is independent of any optimization software or toolbox. Furthermore, the observer is formulated in a way that is especially efficient for systems with sampled measurement.     

A New Algorithm to Design Minimal Multi‐Functional Observers for Linear Systems

Designing minimum possible order (minimal) observers for multi‐input multi‐output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi‐functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.     

一类非线性系统的变结构鲁棒观测器

研究了系统中不确定因素不满足匹配条件且具有外部干扰的变结构鲁棒观测器的设计问题,构造的观测器适用于标称系统是线性定常系统或可化正则型的仿射非线性系统。采用Ｌｙａｐｕｎｏｖ函数为稳定观测器的判别条件,使观测器在外有部干扰时具有一致最终有界的构造误差。     

Active vision

We investigate several basic problems in vision under the assumption that the observer is active. An observer is called active when engaged in some kind of activity whose purpose is to control the geometric parameters of the sensory apparatus. The purpose of the activity is to manipulate the constraints underlying the observed phenomena in order to improve the quality of the perceptual results. For example a monocular observer that moves with a known or unknown motion or a binocular observer that can rotate his eyes and track environmental objects are just two examples of an observer that we call active. We prove that an active observer can solve basic vision problems in a much more efficient way than a passive one. Problems that are ill-posed and nonlinear for a passive observer become well-posed and linear for an active observer. In particular, the problems of shape from shading and depth computation, shape from contour, shape from texture, and structure from motion are shown to be much easier for an active observer than for a passive one. It has to be emphasized that correspondence is not used in our approach, i.e., active vision is not correspondence of features from multiple viewpoints. Finally, active vision here does not mean active sensing, and this paper introduces a general methodology, a general framework in which we believe low-level vision problems should be addressed.The author is Yiannis     

Observer Design—A Survey

 This paper surveys the results of observer design for linear time-invariant (L-T-I) deterministic irreducible open-loop systems (OLS), the most basic type of OLS. An observer estimates Kx(t) signal where K is a constant and x(t) is the state vector of the OLS. Thus, an observer can be used as a feedback controller that implements state feedback control (SFC) or Kx(t)-control, and observer design is therefore utterly important in all feedback control designs of state space theory. In this survey, the observer design results are divided into three categories and for three respective main purposes. The first category of observers estimate signal Kx(t) only with a given K, and this survey has four conclusions: 1) Function observer that estimates Kx(t) directly is more general than state observer that estimates x(t), and may be designed with order lower than that of state observer, and the additional design objective is to minimize observer order; 2) The function observer design problem has already been simplified to the solving of a single set of linear equations only while seeking the lowest possible number of rows of the solution matrix, and an apparently most effective and general algorithm of solving such a problem can guarantee unified upper and lower bounds of the observer order; 3) Because such a single set of linear equations is the simplest possible theoretical formulation of the design problem and such theoretical observer order bounds are the lowest possible, and because the general, simple, and explicit theoretical formula for the function observer order itself do not exist, the theoretical part of this design problem is solved; 4) Because the function observer order is generically near its upper bound, further improvement on the computational design algorithm so that the corresponding observer order can be further reduced, is generically not worthwhile. The second category of observers further realize the loop transfer function and robustness properties of the direct SFC, and the conclusion of this survey is also fourfold: 1) To fully realize the loop transfer function of a practically designed Kx(t)-control, the observer must be an output feedback controller (OFC) which has zero gain to OLS input; 2) If parameter K is separately designed before the observer design, as in the separation principle which has been followed by almost all people for over half of a century, then OFC that estimates Kx(t) does not exist for almost all OLS s; 3) As a result, a synthesized design principle that designs an OFC first and is valid for almost all OLS s, is proposed and fully developed, the corresponding K will be designed afterwards and will be constrained by the OFC order as well as the OFC parameters; 4) Although the Kx(t)-control is constrained in this new design principle and is therefore called the "generalized SFC" (as compared to the existing SFC in which K is unconstrained), it is still strong enough for most OLS''s and this new design principle overcomes many fundamental drawbacks of the existing separation principle. The third category of observers estimate Kx(t) signal at special applications such as fault detection and identification and systems with time delay effects. Using directly the result of OFC that estimates Kx(t) of the second category, these observers can be generally and satisfactorily designed.     

Nonlinear observers for autonomous Lipschitz continuous systems

This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct observers of a remarkably simple structure: A linear state variable filter followed by a nonlinearity. The observer is established by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist, and by giving an explicit expression for its nonlinearity. It is demonstrated that the existence conditions are satisfied, and hence our results include a new observer which is not high-gain, for the wide class of smooth systems. It is shown that the observer can as well be designed to realize an arbitrary, finite accuracy rather than ultimate exactness. On a compact region of the state space, this requires only observability of the system. A corresponding numerical design procedure is described, which is easy to implement and computationally feasible for low order systems.     

Multi-modelling as new estimation schema for high-gain observers

The presented paper proposes a novel method of observer design. A new two-layer observer structure is introduced. The first layer consists of multiple high-gain observers. The latter is built to connect the first layer observers into single one. As the new contribution, the new mapping is defined between an unknown state and measurable outputs allowing to explore new estimation schema. Hence, the proposed method enhances the estimation process for linear and nonlinear systems. Furthermore, it is shown that the introduced observation scheme improves the transients. Illustrative examples are calculated to show the properties of the new observation method.     

Design and sensitivity analysis of a reduced-order rotor flux optimal observer for induction motor control

This paper aims to give simple and effective design criteria of rotor flux reduced-order observers for motion control systems with induction motors. While the observer is optimized for rotor and stator resistance variations, a sensitivity analysis is carried out in the presence of variations of all the motor parameters by means of either transfer function from true to observed rotor flux or simulation in a MATLAB-SIMULINK environment, assuming the voltages supplying the motor to be different from those supplying the observer. The sensitivity analysis makes it possible to establish design criteria for the observer in question. The behaviour of the proposed reduced order observer is compared with current model and voltage model observers. Experimental results are also given, with the dual aim of confirming the validity of this design method and showing that the observer can be implemented on microprocessor-based devices.     

Nonlinear robust observer design using an invariant manifold approach

This paper presents a method to design a reduced order observer using an invariant manifold approach. The main advantages of this method are that it enables a systematic design approach, and (unlike most nonlinear observer design methods), it can be generalized over a larger class of nonlinear systems. The method uses specific mapping functions in a way that minimizes the error dynamics close to zero. Another important aspect is the robustness property which is due to the manifold attractivity: an important feature when an observer is used in a closed loop control system. A two degree-of-freedom system is used as an example. The observer design is validated using numerical simulation. Then experimental validation is carried out using hardware-in-the-loop testing. The proposed observer is then compared with a very well known nonlinear observer based on the off-line solution of the Riccati equation for systems with Lipschitz type nonlinearity. In all cases, the performance of the proposed observer is shown to be very high.     

State estimation for linear hybrid systems with periodic jumps and unknown inputs

In order to solve the state estimation problem for linear hybrid systems with periodic jumps and unknown inputs, some hybrid observers are proposed. The proposed observers admit a Luenberger‐like structure and the synthesis is given in terms of linear matrix inequalities (LMIs). Therefore, the proposed observer designs are completely constructive and provide some input‐to‐state stability properties with respect to unknown inputs. It is worth mentioning that the structure of the hybrid observers, as well as the structure of the LMIs, depends on some observability properties of the flow and jump dynamics, respectively. Then, in order to compensate the effect of the unknown inputs, a hybrid sliding‐mode observer is added to the Luenberger‐like observer structure, providing exponential convergence to zero of the state estimation error despite certain class of unknown inputs. The existence of the hybrid observers and the unknown input hybrid observer is guaranteed if and only if the hybrid system is observable and strongly observable, respectively. Some numerical examples illustrate the feasibility of the proposed estimation approach.     

Two-stage approach to state and force estimation in rigid-link multibody systems

A novel two-stage approach is presented for improving the estimates of both the kinematic state and the unknown external forces in rigid-link multibody systems with negligible joint clearance. The approach is said to be a two-stage one because the estimation process is carried out by two observers running simultaneously and only partially coupled in order to reduce model uncertainties. Nonlinear Kalman filters are employed at both stages. In the first stage, a kinematic observer estimates an augmented system state (i.e., positions, velocities and accelerations) by employing the kinematic constraint equations and some measurements of kinematic quantities as inputs and outputs. Therefore, it is unbiased by external forces and uncertainties on any dynamic parameters. In the second stage, a force observer estimates the external forces by employing dynamic models. The input of the force observer is the kinematic state, while the correction is performed through some direct or indirect measurements of the known forces. Numerical assessment of the theory developed is provided through a slider–crank mechanism. The results achieved through the proposed approach are compared with those yielded by traditional unknown input observers based on a single-stage dynamic estimation. An extensive statistical analysis is carried out at varying levels of measurement noise. Two different strategies are followed in the synthesis of the non-linear Kalman filters. The comparison clearly shows the advantages and the effectiveness of the new two-stage approach.     

Application of the least squares algorithm to the observer designfor linear time-varying systems

In this paper, it is shown that the least squares algorithm with covariance reset, which is originally developed for the purpose of constant parameter identification, can be effectively applied to the observer design for a general linear time-varying system. The new observer successfully avoids many of the disadvantages of other time-varying observers, such as slow convergence rate, heavy computation load, high amplification of measurement noise, and the inapplicability to systems with time-varying observability indexes or discontinuous parameter variations     

Study on Four Disturbance Observers for FO-LTI Systems

This paper addresses the problem of designing disturbance observer for fractional order linear time invariant (FO-LTI) systems, where the disturbance includes time series expansion disturbance and sinusoidal disturbance. On one hand, the reduced order extended state observer (ROESO) and reduced order cascade extended state observer (ROCESO) are proposed for the case that the system state can be measured directly. On the other hand, the extended state observer (ESO) and the cascade extended state observer (CESO) are presented for another case when the system state cannot be measured directly. It is shown that combination of ROCESO and CESO can achieve a highly effective observation result. In addition, the way how to tune observer parameters to ensure the stability of the observers and reduce the observation error is presented in this paper. Finally, numerical simulations are given to illustrate the effectiveness of the proposed methods.      

Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach

In this paper the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance‐observer‐based control (DOBC) framework. The unknown external disturbances are supposed to be generated by an exogenous system, where some classic assumptions on disturbances can be removed. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. Design schemes are presented for both the full‐order and reduced‐order disturbance observers via LMI‐based algorithms. For the plants with known nonlinearity, it is shown that the full‐order observer can be constructed by augmenting the estimation of disturbances into the full‐state estimation, and the reduced‐order ones can be designed by using of the separation principle. For the uncertain nonlinearity, the problem can be reduced to a robust observer design problem. By integrating the disturbance observers with conventional control laws, the disturbances can be rejected and the desired dynamic performances can be guaranteed. If the disturbance also has perturbations, it is shown that the proposed approaches are infeasible and further research is required in the future. Finally, simulations for a flight control system is given to demonstrate the effectiveness of the results. Copyright © 2005 John Wiley & Sons, Ltd.     

Positive state‐bounding observer for positive interval continuous‐time systems with time delay

This paper is concerned with the problem of positive observer synthesis for positive systems with both interval parameter uncertainties and time delay. Conventional observers may no longer be applicable for such kind of systems due to the positivity constraint on the observers, and they only provide an estimate of the system state in an asymptotic way. A pair of positive observers with state‐bounding feature is proposed to estimate the state of positive systems at all times in this paper. A necessary and sufficient condition for the existence of desired observers is first established, and the observer matrices can be obtained easily through the solutions of a set of linear matrix inequalities (LMIs). Then, to reduce the error signal between the system state and its estimates, an iterative LMI algorithm is developed to compute the optimized state‐bounding observer matrices. Finally, a numerical example is presented to show the effectiveness and applicability of the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.