Observability Rank Condition Research Articles

The Unscented Kalman Filter for Nonlinear Parameter Identification of Adaptive Cruise Control Systems

This paper develops and investigates a dual unscented Kalman filter (DUKF) for the joint nonlinear state and parameter identification of commercial adaptive cruise control (ACC) systems. Although the core functionality of stock ACC systems, including their proprietary control logic and parameters, is not publicly available, this work considers a car-following scenario with a human-driven vehicle (leader) and an ACC engaged ego vehicle (follower) that employs a constant time-headway policy (CTHP). The objective of the DUKF is to determine the CTHP parameters of the ACC by using real-time observations of space-gap and relative velocity from the vehicle's onboard sensors. Real-time parameter identification of stock ACC systems is essential for assessing their string stability, large-scale deployment on motorways, and impact on traffic flow and throughput. In this regard, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_{2}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_\infty$</tex-math></inline-formula> string stability conditions are considered. The observability rank condition for nonlinear systems is adopted to evaluate the ability of the proposed estimation scheme to estimate stock ACC system parameters using empirical data. The proposed filter is evaluated using empirical data collected from the onboard sensors of two 2019 SUV vehicles, namely Hyundai Nexo and SsangYong Rexton, equipped with stock ACC systems; and is compared with batch and recursive least-squares optimization. The set of ACC model parameters obtained from the proposed filter revealed that the commercially implemented ACC system of the considered vehicle (Hyundai Nexo) is neither <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_{2}$</tex-math></inline-formula> nor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_\infty$</tex-math></inline-formula> string stable.

Distributed Fusion Estimation for Nonlinear Cyber-Physical Systems With Attacked Control Signals

This article studies the secure estimation problem for cyber-physical systems where the control input is falsified by false data injection attack. First, novel optimization objectives are constructed and attack estimation is performed from real and expected measurement outputs, where the estimator gains are jointly derived through solving quadratic problems recursively. Subsequently, the exponential stability of the proposed estimator is derived if the nonlinear observability rank condition is satisfied, and an introduced factor is given properly. Moreover, to enhance estimation performance, the distributed fusion criterion is designed and the weighting matrices are obtained without any statistical information. Finally, an intelligent localization system monitored by multiple sensors is given to illustrate the effectiveness of the proposed method.

Derivative-Free Observability Analysis for Sensor Placement Optimization of Bioinspired Flexible Flapping Wing System.

Observability analysis of a bioinspired flexible flapping wing system provides a measure of how well the states of flexible flapping wing micro-aerial vehicles can be estimated from real-time measurements during high-speed flight. However, the traditional observability analysis approaches have trouble in terms of lack of quantitative analysis index, high computational complexity, low accuracy, and unavailability in stochastic systems with memory, including bioinspired flexible flapping wing systems. Therefore, a novel derivative-free observability analysis method is proposed here based on the generalized polynomial chaos expansion. By formulating a surrogate model to represent the relationship between the cumulative measurement and the random initial state, the observability coefficient matrix is calculated and the observability rank condition is stated. Consequently, several observability indices are proposed to quantity the observability of the system. Altogether, the proposed method avoids the disadvantages of the traditional approaches, especially in assessing the observability degree of each state and the effect of stochastic noise on observability. The validation of the proposed method is first provided by demonstrating the equivalence between the traditional and proposed methods and subsequently by comparing the observability of the Lorenz system calculated via three different approaches. Finally, the proposed method is applied on a bioinspired flexible wing system to optimize the placement of sensors, which is consistent with the natural configuration of campaniform sensilla on the wing of the hawkmoth.

Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems

This article provides an extension of the observability rank condition to time-varying nonlinear systems. Previous conditions to check the state observability only hold for nonlinear systems that do not explicitly depend on time, or, for time-varying systems, they only account for the linear case. In this article, a general analytic condition is provided. The new condition reduces to the well-known observability rank condition in the two simpler cases of time-varying linear systems and time-invariant nonlinear systems. The proposed condition is applied to aerospace robotics to study the observability of a lunar-module-type system equipped with monocular vision and inertial sensors.

An efficient algorithm to test the observability of rational nonlinear systems with unmeasured inputs

This work proposes an efficient algorithm to examine the observability and identifiability of rational nonlinear systems in the presence of unmeasured and unknown inputs. The proposed algorithm allows for determining whether the dynamic states, unknown parameters and unmeasured inputs of a dynamical system can be, in theory, successfully identified from a given set of input–output measurements. The underlying theory of the algorithm is based on a further extension of the recently suggested extended Observability Rank Condition while focusing on rational instead of analytic nonlinearities. For the robust development of the algorithm, a power series based framework is established for computing the observability matrix efficiently. The occurring framework substantially alleviates the computational burden of the standard implementations of the extended Observability Rank Condition approaches, which allows for applications to real-world engineering systems that are often large and complex. Several examples of large-scale and high-complexity engineering structures are used to demonstrate the performance and capability of the algorithm. Furthermore, the proposed algorithm is used to investigate the feasibility of monitoring a sub-system that is independently separated from a full system under the introduction of additional unmeasured inputs.

A robust algorithm to test the observability of large linear systems with unknown parameters

This work proposes a robust algorithm to examine the observability of linear systems whose dynamic states and parameters are to be identified. The observability of a dynamical system plays a fundamental role in predicting whether it would be successful to use system identification methods to estimate the dynamic states and parameters of the system from a given set of input–output measurements. The motivation of the development of the suggested algorithm arises from the need to address the significant physical memory requirements of the standard implementation of the Observability Rank Condition (ORC). The high computational cost of the ORC substantially limits its applicability to real-world engineering systems, even when the underlying dynamics can be reasonably approximated as linear. The framework of the algorithm is obtained through the derivation of a recursive formula for the computation of the observability matrix of linear systems with unknown parameters. To further improve the efficiency, robust numerical implementations of the algorithm are achieved through random realizations of the dynamic states and parameters and the use of modular operations. The superior performance of the algorithm is demonstrated using several examples of large linear dynamical models of engineering systems containing up to thousands of dynamic states and parameters.

Response based track profile estimation using observable train models with numerical and experimental validations

Condition monitoring of railway tracks is essential in guaranteeing the running safety of railways. Track profiles are the primary source of external excitation for a train system. While Track Recording Vehicle is often utilized for maintenance purposes, this particular vehicle is expensive and difficult to use for small railway operators. Therefore, track profile estimation through in-service vehicle response measurements, which potentially provides efficient and frequent measurement, has been studied. However, the quantitative evaluation of the vertical and lateral track profile irregularities is still challenging as the inverse analysis solutions are sometimes inaccurate and even unstable. In this paper, numerical analyses are first carried out to evaluate track profiles from acceleration and angular velocity responses measured on a train car body. For the inverse analysis, an Augmented State Kalman Filter is utilized to solve the problem using 4 degrees of freedom observable train models. The sensor installation locations are investigated through observability rank condition analysis with different measurement layout. Secondly, a field experiment is carried out in a local Japanese in-service railway network to estimate track profile from car body motions. Smartphones are utilized for the field test measurements as prevalent sensing devices. The effectiveness of the proposed approach is demonstrated with the observable train model. Numerical analyses and field experiments clarify the proposed track profile estimation

Observability of modally reduced order models with unknown parameters

Modally reduced order models are commonly adopted in system inversion. Their observability requires specific attention, since these models only accurately describe the dynamic behavior of the underlying system in a limited frequency range. This paper elaborates a methodology to investigate the observability of modally reduced order models with unknown parameters. The focus is on a particular type of model where the quasi-static contribution of the out-of-band modes is accounted for using so-called dummy modes. The observability test is performed by means of the commonly used Observability Rank Condition (ORC). The proposed methodology is illustrated by multiple examples from structural engineering. It is found that modally reduced order models serve as a valuable alternative for full order models when applied in system inversion. Not only are they computationally much less demanding, but due to their strong link with the underlying full order model, they also allow for the identification of physical parameters, such as mass or stiffness.

On global and local observability of nonlinear polynomial systems: a decidable criterion

Abstract It is very difficult to check the observability of nonlinear systems. Even for local observability, the observability rank condition provides only a sufficient condition. Much more difficult is the verification of global observability. This paper deals with the local and global observability analysis of polynomial systems based on algebraic geometry. In particular, we derive a decidable criterion for the verification of global observability of polynomial systems. Our framework can also be employed for local observability analysis.

Observability of nonlinear systems with unmeasured inputs

Abstract This paper presents a geometric algorithm to investigate the theoretical observability of nonlinear systems with partially measured inputs and outputs. The algorithm is based on Lie algebra and applies to systems whose state and measurement equations are analytical and affine in all inputs. It investigates whether the system satisfies a necessary observability condition that is named the Observability Rank Condition for systems with Direct Feedthrough (ORC-DF). The presented algorithm allows to assess the observability of the dynamic system states, the identifiability of constant-in-time parameters, and the ability to track unmeasured inputs, which is referred to as system invertibility. It is also shown how the developed methodology can be extended to investigate the observability of non-smooth systems that can be broken into different smooth branches, often encountered in mechanical applications related to sliding and damage. Possible applications are illustrated with several examples from structural engineering.

Realization Theory for LPV State-Space Representations With Affine Dependence

We present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA). We show that minimality of LPV-SSAs is equivalent to observability and span-reachability rank conditions, and that minimal LPV-SSAs of the same input-output map are isomorphic. We present necessary and sufficient conditions for existence of an LPV-SSA in terms of the rank of a Hankel-matrix and a Ho-Kalman-like realization algorithm.

Existence of an observation window of finite width for continuous-time autonomous nonlinear systems

In this note, the relationship between notions of observability for continuous-time nonlinear system related to distinguishability, observability rank condition and K-function has been investigated. It is proved that an autonomous nonlinear system that is observable in both distinguishability and rank condition sense permits an observation window of finite width, and it is possible to construct a K-function related to observability for such system.

Observability of autonomous navigation of distributed satellite system using relative position measurements

The purpose of this paper is to investigate the observability of autonomous navigation system of two satellite system using relative position measurements only. The concept of local $k$-order observability is proposed and introduced into analysis of nonlinear system. Necessary and sufficient conditions of local 4-order, 5-order and 6-order observability of the autonomous navigation system are presented and mathematically proved. The relationships between these three conditions as well as the sufficient and necessary condition of satisfying observability rank condition for the autonomous navigation system are given and proved.

On the observability and identifiability of nonlinear structural and mechanical systems

SUMMARY The question of a priori observability of a dynamic system, that is, whether the states of a system can be identified given a particular set of measured quantities is of utmost importance in multiple disciplines including biology, economics, and engineering. More often than not, some of the parameters of the system need to be identified, and thus the issue of identifiability, that is, whether the measurements result in unique or finite solutions for the values of the parameters, is of interest. Identifiability arises in conjunction with the question of observability, when the notion of states may be augmented to include both the actual state variables of the dynamic system and its parameters. This results in the formulation of a nonlinear augmented system even though the dynamic equations of motion of the original system might be linear. In this work, three methods for the observability and identifiability of nonlinear dynamic systems are considered. More specifically, for a system whose state and measurement equations are analytic, the geometric Observability Rank Condition, which is based on Lie derivatives may be used. If the equations are rational, algebraic methods are also available. These include the algebraic observability methods and the algebraic identifiability algorithms which determine the finiteness or uniqueness of the solutions for the parameters. The aforementioned methods are used to study the observability and identifiability of suitable problems in civil engineering and highlight the connections between them and the corresponding concepts. Copyright © 2014 John Wiley & Sons, Ltd.

Homogeneous approximations and local observer design

This paper is concerned with the construction of local observers for nonlinear systems without inputs, satisfying an observability rank condition. The aim of this study is, first, to define an homogeneous approximation that keeps the observability property unchanged at the origin. This approximation is further used in the synthesis of a local observer which is proven to be locally convergent for Lyapunov-stable systems. We compare the performance of the homogeneous approximation observer with the classical linear approximation observer on an example. Mathematics Subject Classification. 93B07, 93B29, 16W25.

Observability at an initial state for polynomial systems

We consider observability at an initial state for polynomial systems. When testing for local observability for nonlinear systems, the observability rank condition based on the inverse function theorem is commonly used. However, the rank condition is a sufficient condition, and we cannot test for global observability using the rank condition. In this paper, first we derive necessary and sufficient conditions for global observability at an initial state for continuous-time polynomial systems. Then, necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring. Using the same procedure, we derive both global and local observability conditions for discrete-time polynomial systems. Each condition is characterized by a finite set of equations, since polynomial rings are Noetherian. Finally, examples demonstrate the proposed criteria for testing for observability.

Observability Analysis of Nonlinear Systems Using Pseudo-Linear Transformation

In the linear control theory, the observability Popov-Belevitch-Hautus (PBH) test plays an important role in studying observability along with the observability rank condition and observability Gramian. The observability rank condition and observability Gramian have been extended to nonlinear systems and have found applications in the analysis of nonlinear systems. On the other hand, there is no observability criterion for nonlinear systems corresponding to the PBH test. In this study, we generalize the observability PBH test for nonlinear systems using pseudo-linear transformation.

Observability of nonlinear control systems on time scales

The problem of observability of the family of nonlinear control systems defined on any time scale and parameterised by time instant t 0 is developed. Necessary and sufficient conditions for observability are given. Because time scale is a rich model of time that includes continuous- and discrete-time models as standard cases, the obtained results extend classical observability rank conditions given for continuous-time systems and discrete-time systems.

Structural Conditions on Observability of Nonlinear Systems

In this paper parameter space and Lebesgue measurement are introduced into analysis of nonlinear systems. Structural observability rank condition is defined and together with the distinguishabililty the structural observability criterions of nonlinear systems are obtained. It proves that when the parameters are not identifiable the solutions with the same time but different parameters are also indistinguishable. Differential geometry and algebraic methods are used to investigate the observability problem, and it is proved that there are some relations between these two methods. Finally, examples are used to illustrate applications of the structural observability criterions.

Observability of viscoelastic fluids

We apply the observability rank condition to study the observability of various viscoelastic fluids under imposed shear or extensional flows. In this paper the observability means the ability of determining the viscoelastic stress from the time history of the observations of the first normal stress difference. We consider four viscoelastic models: the upper convected Maxwell (UCM) model, the Phan–Thien–Tanner (PTT) model, the Johnson–Segalman (JS) model and the Giesekus model. Our study reveals that all of the four models have observability for all stress components almost everywhere under shear flow whereas under extensional flow most of the models have no observability for the shear stress component. More specifically, for UCM and JS models under imposed shear flow, the observations of the first normal stress difference allow the reconstruction of all components of viscoelastic stress. For UCM and JS models under extensional flow, the two normal stress components can be determined from the measurements of the first normal stress difference; the shear stress component does not affect the evolution of the normal stress components and consequently it cannot be extracted from the observations. Under shear flow, the PTT and Giesekus models have observability almost everywhere. That is, all components of the viscoelastic stress can be determined from the observations when the vector formed by the components of viscoelastic stress does not lie on a certain surface. Under extensional flow, the PTT model has observability almost everywhere for normal stress components whereas the Giesekus model has observability almost everywhere for all stress components. We also run simulations using the unscented Kalman filter (UKF) to reconstruct the viscoelastic stress from observations without and with noises. The UKF yields accurate and robust estimates for the viscoelastic stress both in the absence and in the presence of observation noises.