Observability Analysis of Continuous‐Time LTI Systems with Limited Derivative Data

We consider continuous‐time LTI systems with either unknown‐input or with lack of information about the input and output derivatives. We compute the unknown‐input observability subspace and the observability subspace with unknown derivatives of input and output. We first formulate the unknown‐input observability subspace via projection matrices, then show that through having the unknown‐input observability subspace, one can easily evaluate the effect of known input and output signals but unknown derivatives on the observability subspace. Our method is demonstrated on the dynamics of a longitudinal aircraft in steady‐state flight.     

On observability of nonlinear steady-state generators

An output regulation problem can be converted into a stabilization problem in a systematic framework, thanks to an appropriately designed internal model. However, the successful construction of an internal model (IM) relies on the observability condition of a steady-state generator (SSG). In the literature, the observability condition is restricted to linear or linearized SSGs. This paper introduces a novel observability condition for nonlinear SSGs. Nonlinear observability leads to an effective combination of the internal model principle and the certainty equivalence principle for a more general class of systems.     

UIO sequence based checking sequences for distributed test architectures

This study addresses the construction of a preset checking sequence that will not pose controllability (synchronization) and observability (undetectable output shift) problems when applied in distributed test architectures that utilize remote testers. The controllability problem manifests itself when a tester is required to send the current input and because it did not send the previous input nor did it receive the previous output it cannot determine when to send the input. The observability problem manifests itself when a tester is expecting an output in response to either the previous input or the current input and because it is not the one to send the current input, it cannot determine when to start and stop waiting for the output. Based on UIO sequences, a checking sequence construction method is proposed to yield a sequence that is free from controllability and observability problems.     

网络调度中的能控性与能观性研究

为了研究网络调度对能控性与能观性的影响,对输入、输出通道都存在通信约束的网络控制系统的调度进行建模;利用通信序列表示通道的调度情况,最终将模型转化为离散切换系统;在静态调度下,利用循环不变子空间理论及其相关的性质,给出了网络控制系统能控性、能观性成立的充要条件,并给出了原有控制系统与经网络静态调度后的系统之间能控性、能观性的关系;最后用仿真实例说明了所提结论的正确性。     

Output analysis for terminating simulations with partial observability

In this paper we propose methods of output analysis for terminating simulations when the researcher is unable to fully specify the initial state of the simulation using observations of the real system. We call this situation “partial observability” and argue that it is common in practice, especially in the case of complex agent-based simulations. We provide classification of situations where not all input parameters or simulation state variables are observable and for each case we propose a method of terminating simulation output analysis. In particular, we focus on the situation where a rare event needs to be analyzed, since it requires careful design of a simulation experiment in order to minimize the computational budget and avoid bias in the output predictions.     

Output integral sliding mode control based on algebraic hierarchical observer

The problem of the realization of integral sliding mode controllers based only on output information is discussed. The implementation of an output integral sliding mode controller ensures insensitivity of the state trajectory with respect to the matched uncertainties from the initial time moment. In the case when the number of inputs is more than or equal to the number of outputs, the closed loop system, describing the output integral sliding mode dynamics, is shown to lose observability. For the case when the number of inputs is less than the number of outputs, a hierarchical sliding mode observer is proposed. The realization of the proposed observer requires a filtration to obtain the equivalent output injections. Assigning the first order low-pass filter parameter small enough (during this filter realization), the convergence time and the observation error can be made arbitrarily small. The results obtained are illustrated by simulations.     

Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria

This paper presents some new criteria for uniform and nonuniform asymptotic stability of equilibria for time-variant differential equations and this within a Lyapunov approach. The stability criteria are formulated in terms of certain observability conditions with the output derived from the Lyapunov function. For some classes of systems, this system theoretic interpretation proves to be fruitful since—after establishing the invariance of observability under output injection—this enables us to check the stability criteria on a simpler system. This procedure is illustrated for some classical examples.     

Stable‐protocol output consensualization for high‐order swarm systems with switching topologies

Stable‐protocol (SP) output consensus analysis and design for high‐order linear time‐invariant swarm systems with switching topologies are dealt with. Firstly, on the basis of observability decomposition, a dynamic output feedback consensus protocol with switching topologies is given. Then, a necessary and sufficient condition for SP output consensus is shown, and an explicit expression of the output consensus function is presented, which is independent of switching topologies. Furthermore, necessary and/or sufficient conditions for SP output consensualization, which are independent of the number of agents, are shown respectively. Finally, a numerical example is shown to demonstrate theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

From classic observability to a simple fuzzy observability for fuzzy discrete-event systems

In order to determine uncertainties from restricted available information, fuzzy discrete-event systems (FDESs), or fuzzy discrete-event dynamic systems (FDEDSs), were recently proposed. These frameworks include fuzzy states and events occurring simultaneously with different membership degrees. Fuzzy states and events have been used to describe uncertainties that occur often in practical problems, such as treatment planning for HIV/AIDS patients, sensory information processing for robotic control, and fault diagnosis problems. In order to measure information associated with FDESs or FDEDSs, the classical discrete event system (DES) observability has been turned into fuzzy observability for FDESs or FDEDSs. The newly proposed method allows ease of defining degrees of observability so that uncertainties in FDESs or FDEDSs can be dealt with effectively. This gives an opportunity to design better decision-making systems. To calculate the observability degree, a simple fuzzy observability checking method is introduced, and two examples are elaborated upon to illustrate the presented method. Finally, the newly proposed method is tested on a heating, ventilating, and air-conditioning (HVAC) system.     

Modeling and control of fuzzy discrete event systems

In order to make it possible to effectively represent deterministic uncertainties and vagueness as well as the human subjective observation and judgement inherent to many real-world problems, especially those in biomedicine, we introduce, in this paper, fuzzy states and fuzzy events and generalize (crisp) discrete event systems (DES) to fuzzy DES. The largely graph-based current framework of the crisp DES is unsuitable for the expansion, and we have thus reformulated it using state vectors and event transition matrices which can be extended to fuzzy vectors and matrices by allowing their elements to take values between 0 and 1. To measure information related to fuzzy DES, we generalize the crisp DES observability. The new observability allows one to determine whether or not the system output observed is sufficient for decision making. Finally, we extend the optimal control of DES to fuzzy DES. The new fuzzy DES theory is consistent with the existing theory, both at the conceptual and the computation levels, in that the former contains the latter as a special case when the memberships must be either 0 or 1. Numerical examples are provided to illustrate the theoretical development.     

High‐order sliding‐mode observer–based input‐output linearization

The problem of output control in multiple‐input–multiple‐output nonlinear systems is addressed. A high‐order sliding‐mode observer is used to estimate the states of the system and identify the discrepancy between the nominal model and the real plant. The exact and finite‐time estimation may be tackled as long as the system presents the algebraic strong observability property. Thus, a continuous robust input‐output linearization strategy can be obtained with respect to a prescribed output. As a consequence, the closed‐loop dynamics performs robustly to uncertainties/perturbations. To illustrate the advantages of the proposed method, we introduce a study case that demands a robust linear system behavior: the self‐oscillations induced in an underactuated mechanical system through a two‐relay controller. Experiments with an inertial wheel pendulum illustrate the feasibility of the proposed approach.     

Non-linear output feedback stabilization on a bounded region of attraction

In this paper it is shown that, for multi-input single-output non-affine non-linear systems, when a state feedback control stabilizes an equilibrium point of a plant with a certain bounded region of attraction, it is also stabilized by an output feedback controller with arbitrarily small loss of the region. Moreover, the proposed output feedback controller has the dynamic order n which is the same as the order of the plant. From any given state feedback, an explicit form of the overall controller is provided. A sufficient condition presented for the result is shown to be necessary and sufficient for regional uniform observability when the system is input affine. Thus, the result can be regarded as a regional separation principle for affine non-linear systems.     

一种泛函级数建模的新方法及其应用

针对非线性动态系统较难做任意精度逼近的这一问题,提出了使用Volterra级数高阶核估算的全新估计方法。该方法在核函数理论基础上,构造特殊线性空间,将求解Volterra级数的各阶核的问题转化为求用输出观测向量在希尔伯特空间中某一子空间上的投影的问题,使原本复杂、难于计算的非线性系统的Volterra级数的逼近问题在所构建的线性空间中巧妙地以向量内积的方式解决。给出了具体计算方法。相比于其他时域或频域估计Volterra核的方法,该算法的优点在于理论体系严密、计算量不会随着阶数增高而成几何级数增加,辨识精度高,理论上能够辨识出任意阶的核,改善了现有的估计Volterra核的方法难以估计超过4阶或更高阶核的缺点,特别能够应用在对动态系统和强非线性系统的建模上。通过对电厂汽轮机轴系统的辨识和仿真,证明了该方法的有效性。     

On "Over-Sized" High-Gain Practical Observers for Nonlinear Systems

In this paper, it is shown that the performances of a class of high-gain practical observers can be improved by estimating the time derivatives of the output up to an order that is greater than the dimension of the system, which is assumed to be in observability form and, possibly, time-varying. Such an improvement is achieved without increasing the gain of the observers, thus allowing their use in a wide variety of control and identification applications.      

切换系统不变集的输出反馈镇定

利用一个非负函数V (x), 推广了小时间状态模可观测和大时间状态模可观测的定义, 提出了小时间V (x)可观测和大时间V (x)可观测的概念. 利用V (x)可观测的概念, 讨论了子系统无源的切换系统不变集的有界输出反馈镇定及动态输出反馈镇定. 利用多Lyapunov函数证明了闭环系统渐近稳定. 仿真例子验证了文中结论正确性.     

Stability of the Kalman filter for continuous time output error systems

The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of continuous time output error systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability alone is sufficient to ensure the asymptotic stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered systems themselves. The exponential or polynomial convergence of the Kalman filter is then further analyzed for particular cases of stable or unstable output error systems.     

Dissipativity preserving balancing for nonlinear systems — A Hankel operator approach

In this paper we present a version of balancing for nonlinear systems which is dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and the required supply of the original system are the controllability and the observability functions of a modified, asymptotically stable, system. Then Hankel singular value theory can be applied and the axis singular value functions of the modified system equal the nonlinear extensions of “similarity invariants” obtained from the required supply and available storage of the original system. Furthermore, we also consider an extension of normalized comprime factorizations and relate the available storage and required supply with the controllability and observability functions of the factorizations. The obtained relations are used to perform model order reduction based on balanced truncation, yielding dissipative reduced order models for the original systems. A second order electrical circuit example is included to illustrate the results.     

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.     

Observer design for time-varying systems

The state estimation problem is considered for linear time–varying continuous systems with non–lexicographic–fixed observability basis. A two–stage design strategy is presented for constructing observers for this class of systems. At the first stage, by augmenting the given system with an auxiliary system, the original system is transformed to a new augmented model whose observability indices are lexicographic–fixed. A specific augmentation scheme is proposed in order to find this auxiliary system. At the second stage, a design procedure is given to construct minimum–order observers for the augmented system by utilizing the new input and output information. The canonical transformation technique is also employed to simplify the design procedure. Finally, with the help of a simple reverse transformation, solutions to the original problem arc obtained. An example is also given to illustrate the design techniques.