Parametric Sensitivity Functions Research Articles

Parameter-varying partial differential equation to model the global change impacts on wildlife populations

Accurate models of wildlife population dynamics are needed to assess the causes of global biodiversity decline. In this paper, new parameter-varying PDE models are proposed with an original methodology for parametric estimation from ecological data. The model structure makes it possible to introduce the environmental heterogeneity which can characterize global changes. Particular attention was paid to the implementation of the identification procedure: true parametric sensitivity functions in the optimization algorithm, a Galerkin method with a proper orthogonal decomposition for the PDE solution and a pre-estimate to initialize the iterative procedure. The tools are validated on simulation data and then applied to a real application modelling the impact of climate change and agricultural intensification on the population of a passerine bird.

Elliposoidal estimates of trajectory sensitivity of multi-dimensional processes based on generalized singular values problems

The problem of studying the sensitivity of control processes to parameter variations is considered. To solve the problem, the trajectory sensitivity apparatus was used, the use of which, together with the state-space method, made it possible to construct sensitivity models. Based on the models, ellipsoidal estimates of the trajectory sensitivity functions in terms of the state, output, and error of linear multidimensional continuous systems in the form of majorants and minorants are determined. Calculations are performed using the generalized singular value decomposition of matrices composed of trajectory parametric sensitivity functions. The resulting ellipsoidal estimates, due to the meaningful possibilities of the generalized singular value decomposition of matrices, have the property of minimal sufficiency. The estimation method makes it possible, using the left singular basis corresponding to the extremal generalized singular values, to determine subspaces in the state, output, and error spaces that are characterized at each moment of time by the largest and smallest additional motion in terms of the norm. The right singular basis allows us to determine subspaces in the parameter space that generate the largest and smallest additional motion in the norm. The proposed approach solves the problem of “optimal nominal”, that is, the problem of choosing the nominal value of the parameter vector of the plant units that provide the multidimensional controlled process with the smallest value of the ellipsoidal estimates of the trajectory sensitivity functions, as well as to compare the flow of multidimensional controlled processes according to the ellipsoidal estimates of the trajectory parametric sensitivity.

REGARDING THE QUALITY OF ASSEMBLY OF THE SCREW CONNECTIONS OF THE MOUNTING ASSEMBLY OF BALL BEARINGS

Application of method of vibrodiagnostics is based on the presence of cross-correlation dependences of dynamic descriptions of all knots from contact pressure in connection through the change of inflexibility of joints. The change of inflexibility of separate elements causes the change of the resulted inflexibility of knot, and at the same time and characteristic it to dynamic descriptions of the artificially created oscillation field: frequencies and amplitudes of resonance vibrations, indexes of damping, phase correlations. There is a task of determination of method of control of the most responsible technological frame-clamping operations at making of such objects: quality of contact connections, ball-bearing knots, fixing of previous pull of ball-bearings and control of isotropicness, inflexibility and dissipative descriptions of the system of pendant of rotor at all stages of drafting, including in the functional state. The primary objective of this work is research of question of influence of non-perfect resiliency and different inflexibility of the supporting bearings on the dynamic reaction of object at external oscillation excitation, determination of estimation of the different inflexibility bearings, determination of accordance of axial inflexibility of ball-bearings supports, them by a basic value. The dynamic model of knot and certain functions of influence of parameters of the system is considered on high-quality indexes. The dynamic model of precision rotor system (PRS) is investigated. Vibration tests at the stage of assembling the structure of the ORS showed that in the process of assembling the elements change the natural frequencies of the system and the quality factor of the oscillating circuits. The resulted dependences allow giving an estimation degree of non-linearity of resilient description and, as a result, different inflexibility of bearings supports as a result of oscillation researches. The sensitivity functions of the design parameters of the complex press-threaded connection in relation to the vibration dynamic parameters – frequencies and amplitudes of resonant oscillations – do not differ qualitatively from those obtained previously.

Lexicographic derivatives of nonsmooth glucose-insulin kinetics under normal and artificial pancreatic responses

Parametric sensitivities of nonsmooth glucose-insulin kinetics models are investigated in this article. In particular, we study several variations on a model of the so-called Intravenous Glucose Tolerance Test (IVGTT), which measures a subject’s insulin response to glucose over time and yields data useful for characterizing and diagnosing diabetes. Nonsmoothness refers here to continuous behavior that is punctuated by “switches” (physiological events) arising due to biochemical and control thresholds being crossed. Insulin secretion from a typical pancreatic response and insulin infusion from an external device (for diabetic patients) are both considered here. The presence of nonsmoothness means that classical sensitivity theory and associated numerical methods are ill-equipped to handle these models. Motivated by this, lexicographic directional differentiation and lexicographic “sensitivity functions” are used, which are recently developed tools in nonsmooth analysis. Using this new approach, we derive nonsmooth sensitivity systems associated with the considered glucose kinetics models, whose unique solutions provide local sensitivity information analogous to classical forward parametric sensitivity functions. Moreover, the theory is practically implementable and the resulting sensitivity functions are computationally relevant, as they can be supplied to dedicated nonsmooth numerical methods (e.g., optimization). The nonsmooth sensitivity systems are solved numerically, giving new insights into the model, and thus into the dynamics of glucose kinetics under the simulated conditions. It is hoped that the results here may be used to improve the design of assessments for diabetes (e.g., the IVGTT) and its treatment.

Nonsmooth Hessenberg differential-algebraic equations

Hessenberg differential-algebraic equations (DAEs) of size ν with nonsmooth right-hand side functions are studied. Well-posedness theory established in this article includes consistent initialization robust to parametric perturbations, and local existence and uniqueness of solutions in the classical senses. The aforementioned results follow from “regularity” of initial data, which is stated in terms of participating functions determining a Lipschitz homeomorphism. Said regularity implies the nonsmooth Hessenberg DAEs have (local) generalized differentiation index ν, and can be verified with matrix-theoretic conditions involving, for example, Clarke Jacobian projections. Regular solutions on an interval of interest are shown to have maximal continuations and to be lexicographically smooth with respect to problem parameters. Consequently, lexicographic directional differentiation can be applied to yield an auxiliary, nonsmooth “high-index” sensitivity system whose solution characterizes generalized derivative information of the reference solution of interest with respect to parameters. This theory is computationally relevant as the parametric sensitivity functions can be evaluated in a tractable way due to recent progress in nonsmooth automatic/algorithmic differentiation, and can be supplied to dedicated nonsmooth numerical equation-solving or optimization methods. This allows for numerically implementable dynamic optimization methods; open-loop optimal control theory is given which provides subdifferential elements of objective functions via solving a nonsmooth equation system involving the parametric sensitivity functions. As a corollary of the results in this article, foundational theory for smooth Hessenberg DAEs having classical differentiation index ν is established in a rigorous manner.

Nonlinear system identification using fractional Hammerstein–Wiener models

The paper deals with identification of fractional order nonlinear systems based on Hammerstein–Wiener models. An output error approach is developed using the robust Levenberg–Marquardt algorithm. It presents the difficulty of the parametric sensitivity functions calculation which requires a heavy computational load at each iteration. To overcome this drawback, the fractional nonlinear system is reformulated under a regression form, and the gradient and the Hessian can be obtained in a closed form without using the parametric sensitivity functions. The method’s efficiency is confirmed on numerical simulations, and its feasibility is illustrated with its application to the modeling of an experimental arm robot.

Identification of fractional Hammerstein system with application to a heating process

In this paper, fractional Hammerstein system identification is considered, where the linear block is of fractional order. The original discrete Hammerstein system is first converted to a fractional polynomial nonlinear state-space model (PNLSS), which allows a better parameterization of the model. An output-error identification approach is developed based on the robust Levenberg–Marquardt algorithm, whose nevralgic point is the calculation of parametric sensitivity functions. These last are developed as a multivariable fractional PNLSS model which effectively reduces the computational effort. Various simulations are used to test the method’s efficiency and its statistical performance is analyzed using Monte Carlo simulation. Finally, the method is evaluated through a heating experimental benchmark. The obtained results show good agreement with the real system outputs.

Modelling and parameter identification for a hybrid dynamical system in microbial fed-batch culture

In this paper, we consider the modelling and identification in the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open-loop glycerol input and pH logic control. Taking into account the hybrid characteristic of the fed-batch operation, we present a nonlinear hybrid dynamical system, which is developed by embedding the discrete process of adding glycerol and alkali into the dynamical system of batch culture, to formulate this fermentation process. Some important properties of the solution to the proposed system are then discussed, including the existence, uniqueness, boundedness and regularity. To estimate the unknown parameters in the system, a parameter identification problem subject to continuous state inequality constraints is proposed, and its identifiability is also proved. Subsequently, the parametric sensitivity functions of the system are given and utilized to obtain the requisite gradient information for further numerical computation. Finally, to solve the identification problem, a gradient-based algorithm is constructed in conjunction with constraint transcription and the smoothing approximation technique. Numerical simulations show that the validated hybrid system is fit for describing the fed-batch processes as observed from the experimental results.

Modeling and parameter estimation of a nonlinear switching system in fed-batch culture with pH feedback

In this paper, we propose a nonlinear switching system to formulate the coupled fed-batch culture with pH feedback of glycerol to 1,3-propanediol by Klebsiella pneumoniae and discuss its some basic properties. The parametric sensitivity functions are also given to obtain the derivatives of solutions with respect to the unknown kinetic parameters. To estimate those parameters from the experimental data, we present a parameter estimation problem with inequality path constraints. By constraint transcription and local smoothing technique, the estimation problem is transformed and solved by an optimization algorithm, which is constructed via sensitivity functions approach. Finally, an illustrative numerical example shows the appropriateness of the nonlinear switching system and the validity of optimization algorithm.

Spatial heterogeneity of passive electrical transfer properties of neuronal dendrites due to their metrical asymmetry

The complex and diverse geometry of neuronal dendrites determines the different morphological types of neurons and influences the generation of complex and diverse discharge patterns at the cell output. The recent finding that each temporal pattern has its spatial signature in the form of a combination of high- and low-depolarization states of asymmetrical dendritic branches with active membrane properties raises the question of the nature of such characteristic spatial heterogeneity of electrical states. To answer this, we consider passive dendrites as a conventional reference case using the known current transfer functions, which we complete by corresponding parametric sensitivity functions. These functions for metrically asymmetrical bifurcations of different sizes, as the simplest elements constituting arborizations of arbitrary geometry, are analyzed under different membrane conductivity conditions related to the intensity of activation of ion channels. Characteristic relationships are obtained on the one hand among the size (branch lengths), metrical asymmetry (difference between sister branches in length and/or diameter), and membrane conductivity, and on the other hand, for the difference between the branches in their current transfer effectiveness as an indicator of their electrical asymmetry (heterogeneity). These relationships (i) allow the introduction of a biophysically based criterion for the electrical distinction between metrically asymmetrical branches, (ii) show how the difference first increases and then decreases with increasing membrane conductivity, and (iii) show that the greatest electrical heterogeneity occurs in a lower or higher range of conductivity, corresponding to larger or smaller bifurcation size. As a consequence, the characteristic low-, medium-, and high-conductance states are derived such that metrically asymmetrical parts of simple and complex trees are electrically distinct when the membrane conductivity lies in the size-related medium range, and indistinct otherwise.

A signal convolution method for estimation of controller parameter sensitivity functions for tuning of feedback control systems by an iterative process

Optimisation of controller parameters can be achieved through the use of parameter sensitivity functions in an iterative tuning process. Unlike some other methods for the generation of sensitivity functions, the approach described in this paper avoids the need for explicit a priori knowledge of the plant model structure or parameters, systematic adjustments of controller parameters being made entirely through the processing of signals obtained from tests on the closed-loop system. The methods, based on convolution, may be applied to the estimation of controller parameter sensitivity functions in linear closed-loop systems using simple test inputs, such as step signals. This approach differs from the Iterative Feedback Tuning algorithm in that it does not involve repeated experiments involving system outputs from one test being used as reference inputs in a second test. The signal convolution approach is illustrated through an example involving a simple two-input two-output liquid-level control system involving two coupled tanks.

Optimization of hybrid discrete/continuous dynamic systems

Many engineering applications call for the open loop optimization of a hybrid (discrete/continuous) dynamic system. Examples include the design of operating procedures for process start-up, shut-down and changeovers, the design of emergency shutdown systems, or the optimal design of inherently dynamic processes such as those operated in a batch, semi-continuous and/or periodic manner. The most intriguing class of problems are those in which the optimal trajectories are characterized by a sequence of switches and/or jumps at events, some of which are dependent on the state of the system satisfying certain conditions (state or implicit events), and it is necessary to search over several alternative sequences of events to find the optimal one. The potential for numerical optimization procedures to make optimal sequencing decisions in hybrid dynamic systems is explored. A general formulation of the hybrid optimal control problem is presented. Novel existence and uniqueness results for the parametric sensitivity functions of a hybrid system show that parameter optimization of hybrid dynamic systems (including sequencing decisions) is in general nonsmooth, but also smooth in many important cases. For illustration, the design of a minimum time changeover operation for a pressure vessel avoiding the formation of explosive mixtures is considered. In closing, progress on more systematic approaches to the solution of the resulting nonsmooth optimization problems are discussed.

An approach to the selection of optimal sensor locations in distributed parameter systems

This paper presents an approach to the selection of optimal sensor locations in distributed parameter systems, which distinguishes the purposes of state estimation from the purposes of parameter estimation. In the first case, the optimality criterion is based on a measure of independence between the sensor responses, while in the second case, it is based on a measure of independence between the parameter sensitivity functions. The procedure, which is general and can be applied to models with any degree of complexity, is illustrated with the optimal placement of temperature sensors in a catalytic fixed-bed reactor. Some numerical results for the on-line estimation of temperature and concentration profiles as well as for the estimation of unknown model parameters are discussed.

Parametric sensitivity functions for hybrid discrete/continuous systems

The general equations for the parametric sensitivity functions of a broad class of hybrid discrete/continuous dynamic systems where the continuous part is described by differential–algebraic equations (DAEs) are presented. For the cases where this continuous part is an ordinary differential equation system (ODEs) or a linear time invariant DAE, sufficient conditions for the existence and uniqueness of the sensitivity functions are derived. Numerical computation of these sensitivity functions has been implemented as a generic functionality in a mathematical modeling environment. Special cases and application examples are used for illustration.

Adaptive Chandrasekhar filter for linear discrete-time stationary stochastic systems

This paper considers the design problem of adaptive filters based on the state-space models for linear discrete-time stationary stochastic signal processes. The adaptive state estimator consists of both the predictor and the sequential prediction error estimator. The discrete Chandrasekhar filter developed by author is employed as the predictor and the nonlinear least-squares estimator is used as the sequential prediction error estimator. Two models are presented for calculating the parameter sensitivity functions in the adaptive filter. One is the exact model called the linear innovations model and the other is the simplified model obtained by neglecting the sensitivities of the Chandrasekhar X and Y functions with respect to the unknown parameters in the exact model.

Accuracy of manipulation robots dynamics

A computer oriented method for deriving sensitivity functions of dynamic parameters of the mechanism and actuators for an open chain manipulator system is presented. The manipulator system is observed under two modes: the nominal mode (feedforward), where the manipulator is driven by nominal input to the actuators, and the deviation mode, where the manipulator motion is controlled by local feedback loops. These sensitivity functions can be used for evaluating the quality of the chosen robot parameters with respect to sensitivity of their variation, and in evaluating the effect of local feedback loops in the case of parameter variation. The algorithms are illustrated by an example of a six degree-of-freedom robotic manipulator.

Optimal design of superconducting turbogenerators via polyparametric sensitivity analysis

In this paper the dynamic behaviour of a superconducting turbogenerator is simulated by means of the Park's non-linear mathematical model in the state-space form so as to analyze the influence of the machine parameters simultaneous variations upon its dynamic behaviour via polyparametric sensitivity analysis approach. The computer program worked out by the authors is used to simulate a 2,000 MVA, three-phase, two-pole superconducting turbogenerator as well as to generate its parametric sensitivity functions. The machine model is subjected to a step input simulating a sudden three-phase short circuit occurring at the unit terminals. Its response is observed in the time domain simultaneously with the sensitivity functions, so as to determine the influence of each parameter variations on the dynamic behaviour of the simulated system. Investigation is here confined to the machine resistive parameters and only the ones which largely affect the machine dynamic response will be considered. The values of these ones are selected which achieve the most suitable machine dynamic response. Moreover a polyparametric sensitivity vector is derived from the parametric sensitivity functions in order to investigate the mutual influence of the parameter variations in their chosen ranges.

Parameter and Structural Sensitivity of Econometric Models

Sensitivity analysis which can be used as an evaluation technique for mathematical models, is discussed in relation to an econometric model. Parameter sensitivity functions are used to define a performance criterion by which a comparison may be made of the simulation properties of the econometric model for different estimation methods. Structural parameter sensitivity functions are introduced to evaluate the way in which various variables are used in the structure of an econometric model. The simple KLEIN 1 model is used as an example for the analysis, but the sensitivity methods discussed in the paper can easily be extended to accommodate more complicated macroeconometric models containing intricate lag patterns and non-1inearities.

Sensitivity method for online optimisation of a synchronous-generator excitation controller

The paper describes a technique for the optimisation of the parameter settings of the excitation controller of a synchronous machine which is suitable for on-site use. The optimisation method is based on an integral squarederror criterion, and involves the use of parameter sensitivity functions to estimate the parameter changes needed to alter system responses to approach some predetermined or desirable form. The computation of the parameter sensitivity functions is achieved by a direct method, which does not demand explicit identification of the controlled system and which yields the sensitivity functions of all output variables to all adjustable parameters from the results of a single-step test. The technique may be implemented using a small digital computer. The paper includes the results of simulation studies on unsynchronised and on loaded generators, together with results of micromachine experiments.

Technique for direct evaluation of parameter sensitivity functions

A method is presented for the determination of parameter sensitivity functions of a simple feedback system from measured response data obtained during a step test. Results of simulation studies and a practical implementation involving a synchronous-generator-excitation system are presented.