Computation Of Jacobian Matrices Research Articles

Towards invariant extended Kalman filter-based resilient distributed state estimation for moving robots over mobile sensor networks under deception attacks

This paper studies the resilient distributed state estimation problem for moving robots over wireless sensor networks (WSNs) under deception attacks. The state of a moving robot consists of its position and orientation, evolving according to a nonlinear motion model, and the WSNs are mobile. These settings make the problem more challenging than most existing researches on the resilient distributed state estimation which recover stationary or linear states by static WSNs. To achieve the resilient distributed state estimation for the robot over a mobile sensor network (MSN), we propose a novel invariant extended Kalman filter-based resilient distributed state estimation approach. Specifically, based on the mappings between a Lie group and its Lie algebra, we first define a new type of estimation errors and linearize it. Utilizing the linearization results, the calculation of the current prior estimate based on the last posterior one is achieved by each sensor, which avoids the estimation biases induced by the computation of Jacobian matrices. Then, sensors in the MSN fuse the local prior estimates of their neighbors according to the covariance intersection fusion rule, obtaining more accurate estimates than their local prior ones. Finally, measurements collected by the sensors are used to correct the estimates. In the correction process, an adaptive gain is designed for each sensor to restrict the negative effects of the attacks on the estimates, which is crucial for the accomplishment of the resilient estimation task. We illustrate the performance of the approach by convergence analysis of the estimation errors based on the boundedness of the covariance matrix estimates computed in the state estimation process and verify it by simulation and experimental results.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

COVRECON: automated integration of genome- and metabolome-scale network reconstruction and data-driven inverse modeling of metabolic interaction networks.

One central goal of systems biology is to infer biochemical regulations from large-scale OMICS data. Many aspects of cellular physiology and organismal phenotypes can be understood as results of metabolic interaction network dynamics. Previously, we have proposed a convenient mathematical method, which addresses this problem using metabolomics data for the inverse calculation of biochemical Jacobian matrices revealing regulatory checkpoints of biochemical regulations. The proposed algorithms for this inference are limited by two issues: they rely on structural network information that needs to be assembled manually, and they are numerically unstable due to ill-conditioned regression problems for large-scale metabolic networks. To address these problems, we developed a novel regression loss-based inverse Jacobian algorithm, combining metabolomics COVariance and genome-scale metabolic RECONstruction, which allows for a fully automated, algorithmic implementation of the COVRECON workflow. It consists of two parts: (i) Sim-Network and (ii) inverse differential Jacobian evaluation. Sim-Network automatically generates an organism-specific enzyme and reaction dataset from Bigg and KEGG databases, which is then used to reconstruct the Jacobian's structure for a specific metabolomics dataset. Instead of directly solving a regression problem as in the previous workflow, the new inverse differential Jacobian is based on a substantially more robust approach and rates the biochemical interactions according to their relevance from large-scale metabolomics data. The approach is illustrated by in silico stochastic analysis with differently sized metabolic networks from the BioModels database and applied to a real-world example. The characteristics of the COVRECON implementation are that (i) it automatically reconstructs a data-driven superpathway model; (ii) more general network structures can be investigated, and (iii) the new inverse algorithm improves stability, decreases computation time, and extends to large-scale models. The code is available in the website https://bitbucket.org/mosys-univie/covrecon.

Efficient estimation of aquifer intrinsic permeability anisotropy using perturbation theory

Inverse modeling of aquifer measurements aims at estimating the spatial distribution of rock properties that best fits the available measurements which, in turn, allows predicting water yields and pressures to an acceptable level of accuracy. Nonlinear automatic inverse methods require multiple solutions of the diffusion equation, also known as the groundwater flow equation. Although numerical solutions to the forward problems improve the understanding of complex groundwater flow systems, they are often computationally expensive, thereby rendering the inversion process inefficient. We introduce a gradient-based inversion algorithm that leverages perturbation theory to efficiently estimate the tensorial intrinsic permeability distribution of porous media from single-phase transient pressure measurements. With a maximum of two numerical simulations, the Joint Perturbation-Superposition (JPS) method allows one to compute flow-history-dependent Intrinsic permeability Sensitivity Functions (PSF) on the spatial-temporal domain. Regardless of the gradient-based iterative inversion technique, our method efficiently adapts the sensitivity functions yielded by perturbation theory to calculate the entries of the associated Jacobian matrix at every iteration. A two-dimensional synthetic model comprising multi-well conditions is examined for an anisotropic and spatially heterogeneous groundwater flow system. The perturbation method yields accurate and efficient parameter estimation by leveraging sensitivity functions for (1) the appropriate selection of input transient pressure measurements and (2) reducing the sequential calculation of Jacobian matrices. As a result, Monte Carlo uncertainties as low as <5% are obtained for the estimated tensorial intrinsic permeability distribution, with inversion times halved when compared to conventional techniques.

Efficient computation of Jacobian matrices for entropy stable summation-by-parts schemes

Entropy stable schemes replicate an entropy inequality at the semi-discrete level. These schemes rely on an algebraic summation-by-parts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semi-discrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, which are then used in the context of two-derivative explicit time-stepping schemes and implicit time-stepping.

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

Nonlinear optimal control for the dynamics of the underactuated slosh-container system

ABSTRACT Using Euler-Lagrange analysis the state-space model of the slosh-container system is obtained. This state-space description undergoes approximate linearization with the use of Taylor series expansion and the computation of Jacobian matrices around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The proposed control method stands for the solution of the nonlinear optimal control problem for the slosh-container system, under model uncertainties and external perturbations. To find the controller's feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking of setpoints for all state variables of the slosh-container system, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.

Compact sparse symbolic Jacobian computation in large systems of ODEs

This work introduces a novel algorithm that automatically produces computer code for the calculation of sparse symbolical Jacobian matrices. More precisely, given the code for computing a function f depending on a set of state (independent) variables x, where the code makes use of intermediate algebraic (auxiliary) variables a(x), the algorithm automatically produces the code for the symbolic computation of the matrix J=∂f/∂x in sparse representation.A remarkable feature of the algorithm developed is that it can deal with iterative definitions of the functions preserving the iterative representation during the whole process up to the final Jacobian computation code. That way, in presence of arrays of functions and variables, the computational cost of the code generation and the length of the generated code does not depend on the size of those arrays. This feature is achieved making use of Set–Based Graph representation.The main application of the algorithm is the simulation of large scale dynamical systems with implicit Ordinary Differential Equation (ODE) solvers like CVODE-BDF, whose performance are greatly improved when they are invoked using a sparse Jacobian matrix. However, the algorithm can be used in a more general context for solving large systems of nonlinear equations.The paper, besides introducing the algorithm, discusses some aspects of its implementation in a general purpose ODE solver front-end and analyzes some results obtained.

Sensitivity and depth of investigation from Monte Carlo ensemble statistics

ABSTRACTFor many geophysical measurements, such as direct current or electromagnetic induction methods, information fades away with depth. This has to be taken into account when interpreting models estimated from such measurements. For that reason, a measurement sensitivity analysis and determining the depth of investigation are standard steps during geophysical data processing. In deterministic gradient‐based inversion, the most used sensitivity measure, the differential sensitivity, is readily available since these inversions require the computation of Jacobian matrices. In contrast, differential sensitivity may not be readily available in Monte Carlo inversion methods, since these methods do not necessarily include a linearization of the forward problem. Instead, a prior ensemble is used to simulate an ensemble of forward responses. Then, the prior ensemble is updated according to Bayesian inference. We propose to use the covariance between the prior ensemble and the forward response ensemble for constructing sensitivity measures. In Monte Carlo approaches, the estimation of this covariance does not require additional computations of the forward model. Normalizing this covariance by the variance of the prior ensemble, one obtains a simplified regression coefficient. We investigate differences between this simplified regression coefficient and differential sensitivity using simple forward models. For linear forward models, the simplified regression coefficient is equal to differential sensitivity, except for the influences of the sampling error and of the correlation structure of the prior distribution. In the non‐linear case, the behaviour of the simplified regression coefficient as sensitivity measure is analysed for a simple non‐linear forward model and a frequency‐domain electromagnetic forward model. Differential sensitivity and the simplified regression coefficient are similar for prior intervals on which the forward model response is approximately linear. Differences between the two sensitivity measures increase with the degree of non‐linearity in the prior range. Additionally, we investigate the correlation between prior ensemble and forward response ensemble as sensitivity measure. Correlation yields a normalized version of the simplified regression coefficient. We propose to use this correlation and the simplified regression coefficient for determining depth of investigation in Monte Carlo inversions.

A Kalman Filter for Nonlinear Attitude Estimation Using Time Variable Matrices and Quaternions.

The nonlinear problem of sensing the attitude of a solid body is solved by a novel implementation of the Kalman Filter. This implementation combines the use of quaternions to represent attitudes, time-varying matrices to model the dynamic behavior of the process and a particular state vector. This vector was explicitly created from measurable physical quantities, which can be estimated from the filter input and output. The specifically designed arrangement of these three elements and the way they are combined allow the proposed attitude estimator to be formulated following a classical Kalman Filter approach. The result is a novel estimator that preserves the simplicity of the original Kalman formulation and avoids the explicit calculation of Jacobian matrices in each iteration or the evaluation of augmented state vectors.

A Nonlinear H-infinity Control Approach for Autonomous Truck and Trailer Systems

A nonlinear optimal control method is developed for autonomous truck and trailer systems. Actually, two cases are distinguished: (a) a truck and trailer system that is steered by the front wheels of its truck, (b) an autonomous fire-truck robot that is steered by both the front wheels of its truck and by the rear wheels of its trailer. The kinematic model of the autonomous vehicles undergoes linearization through Taylor series expansion. The linearization is computed at a temporary operating point that is defined at each time instant by the present value of the state vector and the last value of the control inputs vector. The linearization is based on the computation of Jacobian matrices. The modeling error due to approximate linearization is considered to be a perturbation that is compensated by the robustness of the control scheme. For the approximately linearized model of the autonomous vehicles an H-infinity feedback controller is designed. This requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The stability of the control loop is confirmed through Lyapunov analysis. It is shown that the control loop exhibits the H-infinity tracking performance which implies elevated robustness against modeling errors and external disturbances. Moreover, under moderate conditions the global asymptotic stability of the control loop is proven. Finally, to implement state estimation-based control for the autonomous vehicles, through the processing of a small number of sensor measurements, the H-infinity Kalman Filter is proposed.

Artificial Bee Colony based Static State Estimation for Power Systems

This paper applies an artificial bee colony (ABC) based optimization, inspired from the foraging behavior of honey bees in searching foods, in solving the power system state estimation problem. The method does not require computation of Jacobian matrices but involves simply calculation of the objective function. In this approach each probable solution is represented by the position of food source. The proposed algorithm (PA) has been applied on IEEE-6, -14, -30 and -57 bus test systems and results are presented to exhibit its superiority in applying it in energy management systems.

Non‐linear optimal control for the hot‐steel rolling mill system

Control of hot-steel rolling mills aims at raising the levels of quality of the related industrial production and at minimising the cost of the electric energy consumed by such industrial units. This paper proposes a non-linear optimal control approach for the hot-steel rolling mill system. The non-linear dynamic model of the hot-steel rolling mill undergoes approximate linearisation around a temporary operating point which is recomputed at each iteration of the control method. The linearisation relies on Taylor series expansion and on the calculation of the system's Jacobian matrices. For the approximately linearised model of the hot-steel rolling process, an H-infinity feedback controller is designed. This controller provides the solution of the non-linear optimal control problem for the system under model uncertainty and external perturbations. For the computation of the controller's feedback gain, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a robust state estimator.

Adaptive neurofuzzy H-infinity control of DC–DC voltage converters

A novel adaptive neurofuzzy H-infinity control approach to feedback control of DC–DC converters, being used in applications of renewable energy generation, is developed. The form and the parameters of the differential equations that constitute the dynamic model of the controlled system are considered to be unknown, while there is only knowledge about the order of the system. The model of the controlled system undergoes approximate linearization around a temporary operating point which is re-computed at each iteration of the control algorithm. The linearization procedure makes use of Taylor series expansion and the computation of Jacobian matrices. For the approximately linearized model of the DC–DC converter, it is possible to design a stabilizing H-infinity feedback controller, provided that knowledge about the matrices of the linearized state-space description is available. In such a case, the computation of the feedback controller’s gain comes from the solution of an algebraic Riccati equation taking also place at each iteration of the control method. However, since the elements of these matrices are unknown, it is proposed to estimate them with the use of neurofuzzy approximators. Actually, neurofuzzy networks are employed for learning the constituent functions of the system’s dynamic model. Based on these function estimates, the system’s Jacobian matrices are also obtained and this allows the implementation of the H-infinity feedback controller. To assure the stability of the control loop, the learning rate of the neurofuzzy approximators is chosen from the requirement that the first derivative of the system’s Lyapunov function to be always a negative one. The global asymptotic stability and the robustness properties of the control method are proven through Lyapunov stability analysis.

A nonlinear optimal control approach to stabilization of a macroeconomic development model

A nonlinear optimal (H-infinity) control approach is proposed for the problem ofstabilization of the dynamics of a macroeconomic development model that is known as the Grossman-Helpman model of endogenous product cycles. The dynamics of the macroeconomic developmentmodel is divided in two parts. The first one describes economic activities in a developed country andthe second part describes variation of economic activities in a country under development which tries tomodify its production so as to serve the needs of the developed country. The article shows that throughcontrol of the macroeconomic model of the developed country, one can finally control the dynamicsof the economy in the country under development. The control method through which this is achievedis the nonlinear H-infinity control. The macroeconomic model for the country under developmentundergoes approximate linearization round a temporary operating point. This is defined at each timeinstant by the present value of the system’s state vector and the last value of the control input vectorthat was exerted on it. The linearization is based on Taylor series expansion and the computation of theassociated Jacobian matrices. For the linearized model an H-infinity feedback controller is computed.The controller’s gain is calculated by solving an algebraic Riccati equation at each iteration of thecontrol method. The asymptotic stability of the control approach is proven through Lyapunov analysis.This assures that the state variables of the macroeconomic model of the country under developmentwill finally converge to the designated reference values.

A Nonlinear Optimal Control Approach to Stabilization of Business Cycles of Finance Agents

The article proposes a new nonlinear optimal control method for the stabilization of the business cycles of interconnected finance agents. First, the dynamics of the interacting finance agents and of the associated business cycles is described by a model of coupled nonlinear oscillators. Next, this dynamic model undergoes approximate linearization round a temporary operating point which is defined by the present value of the system’s state vector and the last value of the control inputs vector that was exerted on it. The linearization procedure is based on Taylor series expansion of the dynamic model and on the computation of Jacobian matrices. Next, for the linearized model of the interacting finance agents, an H-infinity feedback controller is designed. The computation of the feedback control gain requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. Through Lyapunov stability analysis it is proven that the control scheme is globally asymptotically stable.

A Nonlinear H-Infinity Approach to Optimal Control of PEM Fuel Cells

A new nonlinear H-infinity control approach is applied to PEM fuel cells. First, the dynamic model of the PEM fuel cells undergoes approximate linearisation, through Taylor series expansion, round local operating points which are defined at each time instant by the present value of the system's state vector and the last value of the control input that was exerted on it. The linearisation procedure requires the computation of Jacobian matrices at the aforementioned operating points. The modelling error, which is due to the truncation of higher order terms in the Taylor series expansion is perceived as a perturbation that should be compensated by the robustness of the control loop. Next, for the linearized equivalent model of the PEM fuel cells, an H-infinity feedback control loop is designed. This approach, is based on the concept of a differential game that takes place between the control input (which tries to minimize the deviation of the state vector from the reference setpoints) and the disturbance input (that tries to maximize it). In such a case, the computation of the optimal control input requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The known robustness properties of H-infinity control enable compensation of model uncertainty and rejection of the perturbation terms that affect the PEM fuel cells system. The stability of the control loop is proven through Lyapunov analysis. Actually, it is shown that H-infinity tracking performance is succeeded, while conditionally the asymptotic stability of the control loop is also demonstrated. The efficiency of the proposed control scheme for the PEM fuel cells model is further confirmed through simulation experiments.

Nonlinear Optimal Control of Oxygen and Carbon Dioxide Levels in Blood

Nonlinear H-infinity control is proposed for the regulation of the levels of oxygen and carbon dioxide in the blood of patients undergoing heart surgery and extracorporeal blood circulation. The levels of blood gases are administered through a membrane oxygenator and the control inputs are the externally supplied oxygen and the aggregate gas supply (oxygen plus nitrogen). The proposed control method is based on linearization of the oxygenator’s dynamical model through Taylor series expansion and the computation of Jacobian matrices. The local linearization points are defined by the present value of the oxygenator’s state vector and the last value of the control input that was exerted on this system. The modelling errors due to linearization are considered as disturbances which are compensated by the robustness of the control loop. Next, for the linearized model of the oxygenator an H-infinity control input is computed at each iteration of the control algorithm through the solution of an algebraic Riccati equation. With the use of Lyapunov stability analysis it is demonstrated that the control scheme satisfies the H-infinity tracking performance criterion, which signifies improved robustness against modelling uncertainty and external disturbances. Moreover, under moderate conditions the asymptotic stability of the control loop is also proven. To estimate non-measurable state vector elements of the oxygenator the H-infinity Kalman Filter is proposed and thus state estimation-based control is also applied. The efficient performance of the control scheme is confirmed through simulation experiments.

An H-infinity approach to optimal control of doubly-fed reluctance machines

Abstract: An H-infinity approach is developed for the problem of nonlinear optimal control of doubly-fed reluctance machines. The dynamic model of the machines is subjected to linearization round local operating points, through Taylor series expansion and the computation of Jacobian matrices. For the linearized model an H-infinity feedback controller is designed, capable of compensating for the modelling error of the approximate linearization as well as for external perturbations affecting the machine. The computation of the feedback control gain is based on the solution of an algebraic Riccati equation that is performed at each iteration of the control algorithm. Lyapunov stability analysis for the control loop of the reluctance machine arrives at an H-infinity tracking performance criterion, and finally the asymptotic stability of the control loop is demonstrated. The excellent tracking performance of the H-infinity control method is confirmed through simulation experiments.

Nonlinear H-infinity control of multi-phase electric machines

Abstract: The use of multi-phase electric machines exhibits specific advantages such as increased power comparing to the three-phase machines and robustness to failures. In this paper, the dynamic model of the 6-phase synchronous electric machine undergoes first an approximate linearisation, through Taylor series expansion. The linearization is performed round local operating points which are defined at each time instant by the present value of the system’s state vector and the last value of the control input that was exerted on it. The linearisation procedure requires the computation of Jacobian matrices at the aforementioned operating points. The modelling error, which is due to the truncation of higher order terms in the Taylor series expansion is perceived as a perturbation that should be compensated by the robustness of the control loop. Next, for the linearized equivalent model of the 6-phase synchronous electric machine, an H-infinity feedback control loop is designed. This approach, is based on the concept of a differential game that takes place between the control input (which tries to minimize the deviation of the state vector from the reference setpoints) and the disturbance input (that tries to maximize it). In such a case, the computation of the optimal control input requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The known robustness properties of H-infinity control enable compensation of model uncertainty and rejection of the perturbation terms that affect the 6-phase synchronous machine. The stability of the control loop is proven through Lyapunov analysis. Actually, it is shown that H-infinity tracking performance is succeeded, while conditionally the asymptotic stability of the control loop is also assured. The efficiency of the proposed control scheme for the 6-phase synchronous machine is further confirmed through simulation experiments.