Differential Algebraic Approach Research Articles

Fault Estimation for a Quad-Rotor MAV Using a Polynomial Observer

This work addresses the problem of fault detection and diagnosis (FDD) for a quad-rotor mini air vehicle (MAV). Actuator faults are considered on this paper. The basic idea behind the proposed method is to estimate the faults signals using the extended state observers theory. To estimate the faults, a polynomial observer (Aguilar et al. 2011; Mata-Machuca et al., Commun Nonlinear Sci Numer Simul 15(12):4114---4130, 2010, BioSystems 100(1):65---69, 2010) is presented by using the available measurements and know inputs of the system. In order to investigate the diagnosability properties of the system, a differential algebra approach is proposed (Cruz-Victoria et al., J Frankl Inst 345(2):102---118, 2008; and Martinez-Guerra and Diop, IEE P-Contr Theor Ap 151(1):130---135, 2004). The effectiveness of the methodology is illustrated by means of numerical simulations.

The Differential-Algebraic Analysis of Symplectic and Lax Structures Related with New Riemann-Type Hydrodynamic Systems

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic hierarchy, proposed recently by O. D. Artemovych, M. V. Pavlov, Z. Popowicz and A. K. Prykarpatski, is developed. In addition to the Lax-type representation, found before by Z. Popowicz, a closely related representation is constructed in exact form by means of a new differential-functional technique. The bi-Hamiltonian integrability and compatible Poisson structures of the generalized Riemann type hierarchy are analyzed by means of the symplectic and gradient-holonomic methods. An application of the devised differential-algebraic approach to other Riemann and Vakhnenko type hydrodynamic systems is presented.

Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease

Cholera and many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. An important epidemiological question is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine whether parameters for a differential equation model of waterborne disease transmission dynamics can be identified, both in the ideal setting of noise-free data (structural identifiability) and in the more realistic setting in the presence of noise (practical identifiability). We used a differential algebra approach together with several numerical approaches, with a particular emphasis on identifiability of the transmission rates. To examine these issues in a practical public health context, we apply the model to a recent cholera outbreak in Angola (2006). Our results show that the model parameters—including both water and person-to-person transmission routes—are globally structurally identifiable, although they become unidentifiable when the environmental transmission timescale is fast. Even for water dynamics within the identifiable range, when noisy data are considered, only a combination of the water transmission parameters can practically be estimated. This makes the waterborne transmission parameters difficult to estimate, leading to inaccurate estimates of important epidemiological parameters such as the basic reproduction number (R0). However, measurements of pathogen persistence time in environmental water sources or measurements of pathogen concentration in the water can improve model identifiability and allow for more accurate estimation of waterborne transmission pathway parameters as well as R0. Parameter estimates for the Angola outbreak suggest that both transmission pathways are needed to explain the observed cholera dynamics. These results highlight the importance of incorporating environmental data when examining waterborne disease.

Hidden Symmetries of Lax Integrable Nonlinear Systems

Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.

Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited

A differential-algebraic approach to studying the Lax integrability of the generalized Riemann type hydrodynamic hierarchy is revisited and its new Lax representation is constructed in exact form. The bi-Hamiltonian integrability of the generalized Riemann type hierarchy is discussed by means of the gradient-holonomic and symplectic methods and the related compatible Poissonian structures for N = 3 and N = 4 are constructed.

Asymptotic synchronization of the Colpitts oscillator

In this paper we deal with the observer-based asymptotic synchronization problem for a class of chaotic oscillators. Some results based on a differential algebraic approach are used in order to determine the algebraic observability of unknown variables. The strategy consists of proposing a slave system (observer) which tends to follow asymptotically the master system. The methodology is tested in the real-time asymptotic synchronization of the Colpitts oscillator by means of a proportional reduced order observer (PROO) of free-model type.

Fault estimation using a polynomial observer: A real-time application

This work has been developed in the context of the faults diagnosis problem for nonlinear systems. The problem is viewed as the estimation of fault signals using extended state observers theory. A differential algebra approach is proposed to determine the observability and diagnosability of the system. A polynomial observer is used, to estimate the faults, for a multiple available outputs system. Other two schemes of nonlinear observers are used in the faults reconstruction process, only for comparison purposes. The first one is a reduced order observer while the second one is a sliding mode observer. The results of a real-time application are shown to illustrate these methodologies. The approaches were tested in the experimental setting Amira DTS200.

HIGH-ORDER OUT-OF-PLANE EXPANSION FOR 3D FIELDS

The precise determination of the dynamics in accelerators with complicated field arrangements such as Fixed Field Alternating Gradient accelerators (FFAG) depends critically on the ability to describe the appearing magnetic fields in full 3D. However, frequently measurements or models of FFAG fields postulate their behavior in the midplane only, and rely on the fact that this midplane field and its derivatives determine the field in all of space. The detailed knowledge of the resulting out-of-plane fields is critical for a careful assessment of the vertical dynamics. We describe a method based on the differential algebraic (DA) approach to obtain the resulting out-of-plane expansions to any order in an order-independent, straightforward fashion. In particular, the resulting fields satisfy Maxwell's equations to the order of the expansion up to machine precision errors, and without any inaccuracies that can arise from conventional divided difference or finite element schemes for the computation of out-of-plane fields. The method relies on re-writing the underlying PDE as a fixed point problem involving DA operations, and in particular the differential algebraic integration operator. We illustrate the performance of the method for a variety of practical examples, and obtain estimates for the orders necessary to describe the fields to a prescribed accuracy.

Compartmental modelling of the pharmacokinetics of a breast cancer resistance protein

A mathematical model for the pharmacokinetics of Hoechst 33342 following administration into a culture medium containing a population of transfected cells (HEK293 hBCRP) with a potent breast cancer resistance protein inhibitor, Fumitremorgin C (FTC), present is described. FTC is reported to almost completely annul resistance mediated by BCRP in vitro. This non-linear compartmental model has seven macroscopic sub-units, with 14 rate parameters. It describes the relationship between the concentration of Hoechst 33342 and FTC, initially spiked in the medium, and the observed change in fluorescence due to Hoechst 33342 binding to DNA. Structural identifiability analysis has been performed using two methods, one based on the similarity transformation/exhaustive modelling approach and the other based on the differential algebra approach. The analyses demonstrated that all models derived are uniquely identifiable for the experiments/observations available. A kinetic modelling software package, namely FACSIMILE (MPCA Software, UK), was used for parameter fitting and to obtain numerical solutions for the system equations. Model fits gave very good agreement with in vitro data provided by AstraZeneca across a variety of experimental scenarios.

A Differential Algebra Approach to Identifiability of HIV/AIDS Models

In this paper we check a priori global identifiability of a nonlinear HIV model taken from the recent literature by using a recently proposed differential algebra approach. The HIV model is rather complex and has not yet been successfully analyzed by traditional techniques. Here, we demonstrate that identifiability of this complex model can be tested by a differential algebra algorithm based on previous work of the author. The algorithm actually provides a general and reliable tool to test global identifiability of several HIV models proposed in the literature. We have recently developed a computer algebra software tool, called DAISY (Differential Algebra for Identifiability of SYstems), based on differential algebra, which can be used to automatically check global identifiability of (linear and) nonlinear models described by polynomial or rational differential equations. The software can be used by researchers with a minimum of mathematical background and is available on the web site http://www.dei.unipd.it/~pia/.

Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg–de Vries hydrodynamical equations

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg–de Vries dynamical system.

Evaluating the structural identifiability of the parameters of the EBPR sub-model in ASM2d by the differential algebra method

The calibration of ASMs is a prerequisite for their application to simulation of a wastewater treatment plant. This work should be made based on the evaluation of structural identifiability of model parameters. An EBPR sub-model including denitrification phosphorus removal has been incorporated in ASM2d. Yet no report is presented on the structural identifiability of the parameters in the EBPR sub-model. In this paper, the differential algebra approach was used to address this issue. The results showed that the structural identifiability of parameters in the EBPR sub-model could be improved by increasing the measured variables. The reduction factor η NO 3 was identifiable when combined data of aerobic process and anoxic process were assumed. For K PP , X PAO and q PHA of the anaerobic process to be uniquely identifiable, one of them is needed to be determined by other ways. Likewise, if prior information on one of the parameters, K PHA , X PAO and q PP of the aerobic process, is known, all the parameters are identifiable. The above results could be of interest to the parameter estimation of the EBPR sub-model. The algorithm proposed in the paper is also suitable for other sub-models of ASMs.

Identifiability of a Hodgkin-Huxley type ion channel under voltage step measurement conditions

In this paper, we analyze the identifiability properties of a Hodgkin-Huxley (HH) type voltage dependent ion channel model under voltage clamp circumstances. The elimination of the differential variables is performed, and the identifiability of various parameters is analyzed using the differential algebra approach and the algorithm based on the Taylor series expansion of the output. It is shown that the model is structurally non-identifiable using certain commonly applied parametrizations, and hidden dependences between physical parameters are unravelled. Using the results of the identifiability analysis, physically meaningful examples are shown when the model parameters are different but the system outputs are identical.

Fault detection and isolation in nonlinear systems

This paper deals with fault detection and identification in dynamic systems when the system dynamics can be modeled by smooth nonlinear differential equations including affine, bilinear or linear parameter varying (LPV) systems. Two basic approaches will be considered, these apply differential algebraic and differential geometric tools. In the differential algebraic approach the state elimination methods will be used to derive nonlinear parity relations. In the specific case when a reconstruction of the fault signal is needed the dynamic inversion based approach will be investigated. This approach will also be studied from geometric point of view. The geometric approach, as proposed by Isidori and De Persis, is suitable to extend the detection filter and unknown input observer design approaches (well elaborated for LTI systems) to affine nonlinear systems. Beyond the development of the theory of fault detection and identification it is equally important to offer computable methods and to analyze the robustness properties against uncertainties. Both the observer based and the inversion based approaches will be elaborated for LPV systems that may offer computational tools inherited from linear systems and also allow to design for robustness utilizing results from H ∞ robust filtering and disturbance attenuation.

Fault Detection and Isolation in Nonlinear Systems

This paper deals with fault detection and identification in dynamic systems when the system dynamics can be modeled by smooth nonlinear differential equations including affine, bilinear or linear parameter varying (LPV) systems. Two basic approaches will be considered, these apply differential algebraic and differential geometric tools.In the differential algebraic approach the state elimination methods will be used to derive nonlinear parity relations. In the specific case when a reconstruction of the fault signal is needed the dynamic inversion based approach will be investigated. This approach will also be studied from geometric point of view. The geometric approach, as proposed by Isidori and De Persis, is suitable to extend the detection filter and unknown input observer design approaches (well elaborated for LTI systems) to affine nonlinear systems.Beyond the development of the theory of fault detection and identification it is equally important to offer computable methods and to analyze the robustness properties against uncertainties. Both the observer based and the inversion based approaches will be elaborated for LPV systems that may offer computational tools inherited from linear systems and also allows to design for robustness utilizing results from H∞ robust filtering and disturbance attenuation.

Differential-Algebraic Approach to Dynamic Simulations of Flash Drums with Rigorous Evaluation of Physical Properties

Differential-Algebraic Approach to Dynamic Simulations of Flash Drums with Rigorous Evaluation of Physical Properties — The dynamics of flash drums is simulated with rigorous physical properties calculations using an equation of state to model each phase present. The formulation results in a set of differential-algebraic equations (DAE), whose differential equations describe the material and energy balances and the algebraic equations result from the conditions for thermodynamic equilibrium inside the drum. PSIDE (Parallel Software for Implicit Differential Equations) (Lioen et al., 1998) is used to solve the set of DAE with an efficient differential-algebraic approach. In this approach, the equations are solved simultaneously, with direct iterations in temperature, phase volumes and mole number of each component in each phase. The results show the efficiency of this methodology for the simulation of flash drums.

Differential-algebraic approach to large deformation analysis of frame structures subjected to dynamic loads

A nonlinear mathematical model for the analysis of large deformation of frame structures with discontinuity conditions and initial displacements, subject to dynamic loads is formulated with arc-coordinates. The differential quadrature element method (DQEM) is then applied to discretize the nonlinear mathematical model in the spatial domain, An effective method is presented to deal with discontinuity conditions of multivariables in the application of DQEM. A set of DQEM discretization equations are obtained, which are a set of nonlinear differential-algebraic equations with singularity in the time domain. This paper also presents a method to solve nonlinear differential-algebra equations. As application, static and dynamical analyses of large deformation of frames and combined frame structures, subjected to concentrated and distributed forces, are presented. The obtained results are compared with those in the literatures. Numerical results show that the proposed method is general, and effective in dealing with discontinuity conditions of multi-variables and solving differential-algebraic equations. It requires only a small number of nodes and has low computation complexity with high precision and a good convergence property.

A novel hybrid optimization algorithm for diferential-algebraic control problems

Dynamic optimization problems can be numerically solved by direct, indirect and Hamilton-Jacobi-Bellman methods. In this paper, the differential-algebraic approach is incorporated into a hybrid method, extending the concepts of structural and differential indexes, consistent initialization analysis, index reduction and dynamic degrees of freedom to the optimal control problem. The resultant differential-algebraic optimal control problem is solved in the following steps: transformation of the original problem into a standard nonlinear programming problem that provides control and state variables, switching time estimates and costate variables profiles with the DIRCOL code; definition of the switching function and the automatically generated sequence of index-1 differential-algebraic boundary value problems from Pontryagin’s minimum principle by using the developed Otima code; and finally, application of the COLDAE code with the results of the direct method as an initial guess. The proposed hybrid method is illustrated with a pressure-constrained batch reactor optimization problem associated with the slack variable method.

Sliding-mode Observer for Uncertainty Estimation in a Class of Chemical Reactor: A Differential-Algebraic Approach

The problem of the on-line estimation of the reaction heat in a continuous stirred tank reactor from temperature measurements is addressed in this paper. The proposed uncertainty observer is based on differential algebraic techniques, such that the algebraic observability condition of the uncertainty from noisy temperature measurements is easily verified and the observer structure is very simple, which lead to feasible implementation. The observer proposed is robust against noisy measurements and sustained disturbances. The good performance of the observer is shown by means of numerical simulations and is compared with a nonlinear Luenberger-type observer.

EChem++ – An object-oriented problem solving environment for electrochemistry. Part 5. A differential-algebraic approach to the error control of adaptive algorithms

Adaptive algorithms promise to provide general tools for the solution of dynamic electrochemical simulation models. We describe a modeling strategy which enables an adaptive step size selection in time based on the control of errors in the concentrations both of dissolved and adsorbed molecules and additionally in the electrochemical response, e.g. the current. Combined with automatic finite element grid adaption in space, a robust algorithm is obtained and included in the EChem++ problem solving environment. Simulations of chronoamperometric and cyclic voltammetric experiments in a one-dimensional cell geometry under linear diffusion conditions demonstrate the applicability of the approach. Provisions to integrate potential jumps in the time step control enable a flexible handling of multi-pulse chronoamperometry. Further applications include fast and slow kinetic adsorption processes and those including surface reactions.