Observability Gramians Research Articles

Coupled Inertial Navigation and Flush Air Data Sensing Algorithm for Atmosphere Estimation

This paper describes an algorithm for atmospheric state estimation based on a coupling between inertial navigation and flush air data-sensing pressure measurements. The navigation state is used in the atmospheric estimation algorithm along with the pressure measurements and a model of the surface pressure distribution to estimate the atmosphere using a nonlinear weighted least-squares algorithm. The approach uses a high-fidelity model of atmosphere stored in table-lookup form, along with simplified models propagated along the trajectory within the algorithm to aid the solution. Thus, the method is a reduced-order Kalman filter in which the inertial states are taken from the navigation solution and atmospheric states are estimated in the filter. The algorithm is applied to data from the Mars Science Laboratory entry, descent, and landing from August 2012. Reasonable estimates of the atmosphere are produced by the algorithm. The observability of winds along the trajectory are examined using an index based on the observability Gramian and the pressure measurement sensitivity matrix. The results indicate that bank reversals are responsible for adding information content. The algorithm is applied to the design of the pressure measurement system for the Mars 2020 mission. A linear covariance analysis is performed to assess estimator performance. The results indicate that the new estimator produces more precise estimates of atmospheric states than existing algorithms.

Mean square state estimation for sensor networks

This paper deals with mean square state estimation over sensor networks with a fixed topology. Attention is focused on designing local stationary state estimators with a general structure while accounting for the network communication topology. Two estimator design approaches are proposed. One is based on the observability Gramian, and the other on the controllability Gramian. The computation of the estimator state-space matrices is recast as off-line convex optimization problems and requires the system asymptotic stability and global knowledge of the network topology. Convergence of the estimation error variance is ensured at each network node and a guaranteed performance in the mean square sense is achieved. The proposed approaches are also extended for designing robust filters to handle polytopic-type parameter uncertainty.

Frequency interval balanced truncation of discrete-time bilinear systems

This paper presents the development of a new model reduction method for discrete-time bilinear systems based on the balanced truncation framework. In many model reduction applications, it is advantageous to analyze the characteristics of the system with emphasis on particular frequency intervals of interest. In order to analyze the degree of controllability and observability of discrete-time bilinear systems with emphasis on particular frequency intervals of interest, new generalized frequency interval controllability and observability gramians are introduced in this paper. These gramians are the solution to a pair of new generalized Lyapunov equations. The conditions for solvability of these new generalized Lyapunov equations are derived and a numerical solution method for solving these generalized Lyapunov equations is presented. Numerical examples which illustrate the usage of the new generalized frequency interval controllability and observability gramians as part of the balanced truncation framework ...

On Submodularity and Controllability in Complex Dynamical Networks

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function. The results are illustrated on randomly generated systems and on a problem of power electronic actuator placement in a model of the European power grid.

A methodology for optimal placement of sensors in enclosed environments: A dynamical systems approach

Air quality has been an important issue in public health for many years. Sensing the level and distributions of impurities help in the control of building systems and mitigate long term health risks. Rapid detection of infectious diseases in large public areas like airports and train stations may help limit exposure and aid in reducing the spread of the disease. Complete coverage by sensors to account for any release scenario of chemical or biological warfare agents may provide the opportunity to develop isolation and evacuation plans that mitigate the impact of the attack. All these scenarios involve strategic placement of sensors to promptly detect and rapidly respond.This paper presents a data driven sensor placement algorithm based on a dynamical systems approach. The approach utilizes the finite dimensional Perron-Frobenius (PF) concept. The PF operator (or the Markov matrix) is used to construct an observability gramian that naturally incorporates sensor accuracy, location constraints, and sensing constraints. The algorithm determines the response times, sensor coverage maps, and the number of sensors needed. The utility of the procedure is illustrated using four examples: a literature example of the flow field inside an aircraft cabin and three air flow fields in different geometries. The effect of the constraints on the response times for different sensor placement scenarios is investigated. Knowledge of the response time and coverage of the multiple sensors aides in the design of mechanical systems and response mechanisms. The methodology provides a simple process for place sensors in a building, analyze the sensor coverage maps and response time necessary during extreme events, as well as evaluate indoor air quality. The theory established in this paper also allows for future work in topics related to construction of classical estimator problems for the sensors, real-time contaminant transport, and development of agent dispersion, contaminant isolation/removal, and evacuation strategies.

Nonlinear Observability for Relative Orbit Determination with Angles-Only Measurements

This paper presents nonlinear observability criteria for the relative orbital dynamics represented by the solutions of the two-body problem. It is assumed that a chief is on a circular orbit with a prescribed orbital radius, and it measures lines-of-sight toward a deputy only. A differential geometric method, based on the Lie derivatives, is used to derive sufficient conditions for observability of the orbital properties of the deputy. It is shown that under certain geometric conditions on the relative configuration between the chief and the deputy, the nonlinear relative motion is observable from angles-only measurements. The second part of this paper presents a quantitative measure of observability for the relative orbits, and it is formulated by generalizing the observability Gramian of linear dynamic systems. An extended Kalman filter is also developed to numerically illustrate the observability of nonlinear relative orbits with angles-only measurements and to show correspondence between the proposed observability measure and filtered solution accuracy.

Sensor Configuration Problem: Application to a Membrane Separation Unit

For high performance model based control applications the state and parameter estimation algorithms are essential. Furthermore the accuracy of the resulting estimates highly depends on what is being measured. In this work we address the output channel selection problem by making use of the degree of observability measures defined by the observability gramians of the associated system and possible different sensor configurations. The observability gramians for large scale and nonlinear first principle models are difficult to compute. Instead the data based approximations, the empirical observability gramians, are constructed. The degree of observability measures are used for comparing the spectral properties of the observability gramians to decide the sensor configuration. The output channel selection procedure is applied to an industrial membrane filtration system.

Evaluation of a Power System Stability Based on Usage of Virtual Digital Analyzer

For early detection of the threat of a cascading failure we need to use the immune information system, the main subsystem of which is a virtual stability analyzer. We propose to implement the analyzer by usage of discrete linearized models of energy systems and apparatus of Lyapunov direct method, based on the study of discrete Lyapunov equation solutions, called Gramians. The functional of stability loss risk is equal to the square of H2-norm of the matrix transfer function, which is proportional to energy of the system. We propose to use a functional risk assessment through spectral decomposition of controllability and observability Gramians for evaluating of the energy accumulated in combinational dominant eigenvalues of the system. In comparison with the method of modal analysis the proposed approach allows to evaluate the impact of synergies between weak-stable eigenvalues and their combinations to the system stability loss risk and to detect the potential source of stability loss.

Adaptive Optimal PMU Placement Based on Empirical Observability Gramian

Abstract: In this paper, we compare four measures of the empirical observability gramian, including the determinant, the trace, the minimum eigenvalue, and the condition number, which can be used to quantify the observability of system states and to obtain the optimal PMU placement for power system dynamic state estimation. An adaptive optimal PMU placement method is proposed by automatically choosing proper measures as the objective function. It is shown that when the number of PMUs is small and thus the observability is very weak, the minimum eigenvalue and the condition number are better measures of the observability and are preferred to be chosen as the objective function. The effectiveness of the proposed method is validated by performing dynamic state estimation on an Northeast Power Coordinating Council (NPCC) 48-machine 140-bus system with the square-root unscented Kalman filter.

Input and Output Normal Forms for Model Order Reduction of Linear Systems

Abstract This paper considers the problem of model order reduction by transforming the system into input and output normal forms. The reachability gramian in the input normal form is the identity matrix and the observability gramian is a diagonal matrix. Conversely, the observability gramian in the output normal form is the identity matrix and the reachability gramian is a diagonal matrix. The elements of the non-identity diagonal gramians in both normal forms are the squares of the system Hankel singular values. This fact determines the equivalent role which both normal forms play in model order reduction. The implemented projection is nearly orthogonal up to a scaling with the elements of a diagonal matrix. In the paper are shown the relations between the transformed system descriptions and the reachability and observability operators. Major influence for the output energy distribution in the input normal form has the observability operator, while the input energy is uniformly distributed. Alternatively, the input energy distribution in the output normal form is due to the reachability operator action, while the output energy is uniformly distributed. Several experiments are performed confirming the equivalent role of the input and output normal forms in the procedures of system approximation.

Modal structural acoustic sensing with minimum number of optimally placed piezoelectric sensors

Structural acoustic sensing is a method of obtaining radiated sound pressure from a vibrating structure using vibration information. Structural acoustic sensing is used in active structural acoustic control for attenuating the sound radiated from a structure. In this paper, a new approach called Modal Structural Acoustic Sensing (MSAS) is proposed for estimating the pressure radiated from a vibrating cylindrical shell using piezoelectric sensors. The motion equations of a cylindrical shell in conjunction with piezoelectric patches are derived based on the Donnel-Mushtari shell theory. The locations of the piezoelectric sensors are optimized by the Genetic Algorithm based on maximizing the observability gramian matrix. The Kirchhoff-Helmholtz integral is used for estimating the sound pressure radiated from the cylindrical shell. Numerical simulations are performed to demonstrate the advantages of the proposed approach in comparison with previous methods such as discrete structural acoustic sensing and distributed modal sensors. Results show that the MSAS can increase the estimation accuracy and decrease the controller dimensionality and the number of required sensors.

Observability analysis of DVL/PS aided INS for a maneuvering AUV.

Recently, ocean exploration has increased considerably through the use of autonomous underwater vehicles (AUV). A key enabling technology is the precision of the AUV navigation capability. In this paper, we focus on understanding the limitation of the AUV navigation system. That is, what are the observable error-states for different maneuvering types of the AUV? Since analyzing the performance of an underwater navigation system is highly complex, to answer the above question, current approaches use simulations. This, of course, limits the conclusions to the emulated type of vehicle used and to the simulation setup. For this reason, we take a different approach and analyze the system observability for different types of vehicle dynamics by finding the set of observable and unobservable states. To that end, we apply the observability Gramian approach, previously used only for terrestrial applications. We demonstrate our analysis for an underwater inertial navigation system aided by a Doppler velocity logger or by a pressure sensor. The result is a first prediction of the performance of an AUV standing, rotating at a position and turning at a constant speed. Our conclusions of the observable and unobservable navigation error states for different dynamics are supported by extensive numerical simulation.

Gyroscopic sensing in the wings of the hawkmoth Manduca sexta: the role of sensor location and directional sensitivity

The wings of the hawkmoth Manduca sexta are lined with mechanoreceptors called campaniform sensilla that encode wing deformations. During flight, the wings deform in response to a variety of stimuli, including inertial-elastic loads due to the wing flapping motion, aerodynamic loads, and exogenous inertial loads transmitted by disturbances. Because the wings are actuated, flexible structures, the strain-sensitive campaniform sensilla are capable of detecting inertial rotations and accelerations, allowing the wings to serve not only as a primary actuator, but also as a gyroscopic sensor for flight control. We study the gyroscopic sensing of the hawkmoth wings from a control theoretic perspective. Through the development of a low-order model of flexible wing flapping dynamics, and the use of nonlinear observability analysis, we show that the rotational acceleration inherent in wing flapping enables the wings to serve as gyroscopic sensors. We compute a measure of sensor fitness as a function of sensor location and directional sensitivity by using the simulation-based empirical observability Gramian. Our results indicate that gyroscopic information is encoded primarily through shear strain due to wing twisting, where inertial rotations cause detectable changes in pronation and supination timing and magnitude. We solve an observability-based optimal sensor placement problem to find the optimal configuration of strain sensor locations and directional sensitivities for detecting inertial rotations. The optimal sensor configuration shows parallels to the campaniform sensilla found on hawkmoth wings, with clusters of sensors near the wing root and wing tip. The optimal spatial distribution of strain directional sensitivity provides a hypothesis for how heterogeneity of campaniform sensilla may be distributed.

Controllability and observability analysis of continuous-time multi-order fractional systems

This paper proposes some analytical criteria for controllability and observability analysis of fractional order systems describing by multi-order state space equations. The controllability and observability gramians are presented in which their non-singularity is equivalent to controllability and observability of the mentioned systems. Moreover, to overcome the gramian calculation complexities, the controllability and observability matrices are introduced. It is proved that the sufficient condition for controllability or observability of a multi-order fractional system is that its controllability or observability matrix is nonsingular. Some illustrative examples are given to show the efficiency of the proposed method.

Observability conservation by output feedback and observability Gramian bounds

Though it is a trivial fact that the observability of a linear state space system is conserved by output feedback, it requires a rigorous proof to generalize this result to uniform complete observability, which is defined with the observability Gramian. The purpose of this paper is to present such a proof. Some issues in existing results are also discussed. The uniform complete observability of closed loop systems is useful for the analysis of some adaptive systems and of the Kalman filter.

A Novel Method for Seeking Optimal Placement of TCSC to Damp Oscillations in Power Systems

Thyristor controlled series capacitors (TCSC) damping controller is considered as a useful device to improve the stability of transmission power systems. Seeking the optimal placement of TCSC controller has become a critical issue for the power system stability. From the working point of view, the TCSC controller is used to improve the voltage magnitude, reactive power and oscillation damping of power systems. The effectiveness of the location of TCSC controller, particularly in the damping of power oscillations depends on its location and input signals in the power system during time of grid fault. In this paper, an energetic method based on controllability and observability Gramians in continuous-time is proposed to determine the optimal location of TCSC. The effectiveness of the proposed method has been verified on the 39-bus New England power system.

Parametric solutions to the discrete periodic regul ator equations

Forward-time discrete periodic regulator matrix equations (FDPRE) and reverse-time discrete periodic regulator matrix equations (RDPRE) are firstly studied. The controllability and observability Gramians related with the lifting time variant system are utilized to constitute solutions. Parametric and explicit solutions to these two equations are presented in the form of iteration. The proposed method can provide sufficient design degree of freedom, since there are infinite solutions to be obtained. A detailed algorithm is also presented and its correctness is checked by a numerical example.

Analysis of multiplicity of Hankel singular values of control systems

Hankel singular values are important input-output invariants of linear control systems, and in this paper we are interested in determining the amount of different values. It is shown that for SISO systems the problem reduces to the analysis of controllability of a certain auxiliary system. For MIMO systems, a method is proposed for constructing the minimal order polynomial such that its roots are equal to the squared Hankel singular values. In the case of maximal multiplicity of the Hankel singular values, the controllability and observability Gramians of the system are proved to be mutually inverse up to a certain scalar multiplier.

Robust PID Tuning of Autonomous Underwater Vehicle Using Harmonic Search Algorithm Based on Model Order Reduction

The objective of this paper is to present Genetic Algorithm (GA) and Harmonic Search Algorithm (HSA) based on PID tuning and Model Order Reduction with Balance Truncation (MORBT). This method is used for constructing a second order balanced truncation model order reduction for symmetric linear time invariant third order control systems. These methods are closely related to the proper and improper controllability and observability Gramians and Hankel singular values of descriptor systems. The HS (Harmonic Search) algorithm mimics behaviours of music players in an improvisation process, in order to find a better state of harmony which can be translated into a solution vector in the optimization process. PID (proportional-integral-derivative) controller provides sufficient stability margins and good time responses. It is now possible to design an optimal PID controller with properties of a GA and HSA. In control strategies, like PID controller are successfully designed to control the autonomous underwater vehicle. The elementary focus is to simulate the controller response.

On a Hankel-based Measure of Decentralized Controllability and Observability

This paper discusses the controllability of linear time-invariant (LTI) systems with decentralized controllers. Whether an LTI system is controllable (by LTI controllers) with respect to a given information structure can be determined by testing for fixed modes. Measures have been developed to further determine how far a system is from having a fixed mode, but these measures cannot actually be computed in most cases. We thus seek an easily computable, non-binary measure of controllability for LTI systems with decentralized controllers of arbitrary information structure. We develop a measure based on the Hankel operator, that combines the controllability gramian, the observability gramian, and a version of the cross-gramian which incorporates the information structure.