Form Of State Equation Research Articles

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Aerodynamic modeling and performance analysis of model predictive controller for fixed wing vertical takeoff and landing unmanned aerial vehicle

This research focuses mainly on the aerodynamic modelling and performance analysis of a model predictive controller for a hybrid fixed-wing vertical takeoff and landing of unmanned aerial vehicle. The aerodynamics, which comprises several aerodynamic characteristics including the lift, drag, and thrust coefficients, is modelled using Newton’s second law of motion. The force and moment equations were obtained, and they were then converted into matrix and state equation form, together with the transformation matrices. The essential equations with six degrees of freedom (6 DoF) and 12 state matrices including the control input and manipulating variables were obtained. The proposed model predictive controller (MPC) was designed using the optimal model predictive controller design parameters, such as a sampling time of 0.1, a prediction horizon of 15, a control horizon of 3 and additional controller settings. With a settling period of 3 s, an overshoot of 0.5865, and a steady state inaccuracy of 0.00785, the MATLAB simulation demonstrates that the system variables, including roll, pitch, and yaw, are stabilized. This MPC control is more effective in anticipating and optimizing the UAV than other control strategies. Eventually, the controller’s simulation on MATLAB Simulink demonstrates the controller’s ability to stabilize and control the system in a real-time application. Autonomous vertical takeoff and landing operations depend heavily on the mathematical model and architecture of the flight controller. Among the uses are inspection, monitoring, and rescue.

Analytical theory of composite beams with CSWs using the state space method

Based on the zig-zag theoretical model of corrugated steel web beam (CSW), the governing equations in the form of state equations are derived and their analytical solutions are obtained for different loads, variable cross-sections and boundary conditions. Through numerical examples, it is verified that the bending moment, deflection and stresses predicted by the present analytical method have a good agreement with the results of the three-dimensional finite element model. The applicability of the present method for non-prismatic continuous beams with CSWs under different loads is also demonstrated. Meanwhile, it is found that the quasi-plane assumption is still valid in non-prismatic continuous beams with CSWs. Through quantitative analysis of the flexural stresses in the top and bottom slabs, it is confirmed that the zig-zag theory in this study can well reflect the accordion effect of CSWs. Because of the inclined bottom slabs in the non-prismatic beams with CSWs, the Resal effect on the shear stresses in the CSWs and bottom slabs, which is caused by the vertical component of the axial force in the bottom slab, is discussed. A refined shear formulation is developed for more accurate shear stresses in the CSWs and bottom slabs, which provides fundamental theoretical support for the strength prediction of the non-prismatic beam with CSWs.

Machine-Learning Assisted Identification of Battery Life Models

Predictive battery life models are commonly utilized to extrapolate degradation trends observed during accelerated aging tests for simulation of degradation in real-world applications. Thus, fitting accelerated aging data as accurately as possible and with low uncertainty is crucial for making believable projections of battery lifetime, but it is challenging to identify algebraic expressions that accurately fit multivariate degradation trends. A review of models published in literature reveal some common expressions for fitting calendar aging data, which is only dependent on temperature and state-of-charge, but no consistency across many models for fitting cycle aging data, indicating the need for a statistically rigorous data driven approach for developing empirical models.This talk will describe a machine-learning assisted method for identification of predictive battery life models utilizing bilevel optimization and symbolic regression. Bilevel optimization with cross-validation is used to statistically determine cell- and stress-dependent model parameters, while symbolic regression identifies both linear and multiplicative candidate expressions to predict stress-dependent degradation rates by selecting low-order subsets of features from a generated feature library. Because model expressions are identified empirically, it is crucial to ensure resulting models behave according to physical expectations, so the stability of models for interpolation or extrapolation is interrogated qualitatively through simulation and quantitatively through cross-validation and uncertainty quantification via bootstrap resampling. This model identification approach substantially improves upon models identified purely using expert judgement in terms of both accuracy and uncertainty.Model simulation and validation is then conducted by deriving a state-equation form of the predictive model, enabling simulation of battery aging under dynamic stresses. This enables validation of the predictive battery model on lab-based tests with varying conditions or on drive-cycle or application-cycle testing protocols. Parameter uncertainty can be carried forward into model simulation, giving lifetime estimates and confidence windows for cell- or system-level lifetime. The financial impact of battery model uncertainty can be estimated by incorporating uncertainty into a technoeconomic model.

Decentralized robust control of building structures based on sliding mode theory

SummaryIn this paper, a decentralized control algorithm is proposed for actively controlling the response of the flexible tall building structures under earthquake excitations. In the proposed approach, tall building structure was divided into some substructures in the form of state equation. The interaction of the subsystems and external excitations is conducted as bounded generalized force acting on the subsystems. A decentralized control algorithm of tall building structures is established based on the sliding model control theory. The control structure is described based on unit vector control. The control law consists of two parts: a linear control law uL and a nonlinear law uN. The linear control is merely a linear state feedback controller, whereas the nonlinear feedback controller incorporates the discontinuous or continuous nonlinear elements of the control law. Using the advantage of match conditions of sliding mode theory and the bounded feature of generalized force, the overall stability of decentralized control is also investigated. The actuator arrangement and matching conditions are discussed. The effectiveness of the proposed method is demonstrated by the numerical simulation of the decentralized control of a 20‐story benchmark structure under seismic excitations.

Application of state equations approach to solve the classical Kubelka-Munk differential equations in turbid environment

The Kubelka-Munk (KM) model is frequently used to describe the optical properties (absorption and scattering) of inhomogeneous media in terms of the measured reflection and transmittance. The classical KM is modeled by two fluxes which counterpropagate in the media and can be described by system of differential equations. This system, on the other hand, can also be rewritten in the form of state equations, widely used in many areas of engineering such as electrical circuits, control systems and more. Here, we describe the simple mathematical expression of light reflectance and transmittance in turbid media based on the KM light propagation model via a system of state equations. To this end, we demonstrate the use of the state equations to solve KM differential equations in a simple and straightforward manner. To validate the solution of our model against the KM solution, different turbid media including tissue phantoms, milk with varying concentrations of sugar, human hands, and mouse brain were tested. We used a spatial frequency domain imaging system to recover the absorption and scattering properties of each of the media. These measured properties were then introduced into both models to obtain diffuse light reflectance and transmittance information. Experimental results demonstrate significant concurrence between the two approaches within a minute difference in all tested cases. The results presented in this study demonstrate the validity of our alternative strategy to the traditional KM solution.

Validation of energy conditions in traversable wormhole for GBD modified theory

In this work, we explore the wormhole physics in a modified gravitational theory, called the generalized Brans–Dicke theory (GBD). In order to solve the equations in this theory, the parameterized forms for the shape function and the scalar-field function are taken. With the help of selecting a specific form of state equation, i.e. considering a relation between the transverse pressure (not the total pressure) and the energy density for matter that threads wormhole: [Formula: see text], we provide some special solutions of the traversable wormhole in GBD. Given that the violation of energy conditions (ECs) of matter often induces problems on the gravitational theory, it is important to inspect the ECs of matter in modified theories. In this GBD theory, we find that the weak energy condition, the null energy condition, the dominated energy condition and the strong energy condition could be satisfied for the matter near the throat of wormhole, depending on the parameterized models and the model-parameters values. Finally, using the classical reconstruction technique in the traversable wormhole physics, we derive a Lagrangian function of gravitational field for the GBD theory.

Machine-Learning Assisted Identification of Accurate Battery Lifetime Models with Uncertainty

Reduced-order battery lifetime models, which consist of algebraic expressions for various aging modes, are widely utilized for extrapolating degradation trends from accelerated aging tests to real-world aging scenarios. Identifying models with high accuracy and low uncertainty is crucial for ensuring that model extrapolations are believable, however, it is difficult to compose expressions that accurately predict multivariate data trends; a review of cycling degradation models from literature reveals a wide variety of functional relationships. Here, a machine-learning assisted model identification method is utilized to fit degradation in a stand-out LFP-Gr aging data set, with uncertainty quantified by bootstrap resampling. The model identified in this work results in approximately half the mean absolute error of a human expert model. Models are validated by converting to a state-equation form and comparing predictions against cells aging under varying loads. Parameter uncertainty is carried forward into an energy storage system simulation to estimate the impact of aging model uncertainty on system lifetime. The new model identification method used here reduces life-prediction uncertainty by more than a factor of three (86% ± 5% relative capacity at 10 years for human-expert model, 88.5% ± 1.5% for machine-learning assisted model), empowering more confident estimates of energy storage system lifetime.

Optimization and Simulation of DMC Algorithm with Terminal Weight Based on LQR Controller

Based on the multivariable dynamic matrix control algorithm, there are many similarities between the dynamic matrix control algorithm and LQR in the form of state equation, state variable and objective function. We can use the objective function of LQR to improve the objective function of DMC and improve the control effect of the system. Using the given mathematical model, the terminal weighting matrix of LQR is applied to DMC algorithm, and the optimized multivariable dynamic matrix control algorithm is simulated. The results show that the algorithm can effectively reduce the response time of the system, make the system sink faster and reach the given value faster.

Математические модели магнитоэлектрических вентильных двигателей

The mathematical models of converter fed permanent magnet motors written in the coordinate axes are considered. Owing to pulse-width modulation, the currents and voltages of the motors are assumed to be sinusoidal. In the models with two state equations, the stator current is represented only by its quadrature-axis component, and the variable load angle appears in explicit or implicit form. These equations are most close to the similar equations of dc commutator motors. Third-order equations contain both stator current components; the load angle in them is fixed as constant one or varies to maintain the unity power factor for an arbitrary load. The models constructed in the form of state equations of the appropriate order, which are implemented in the MathCad software, illustrate possible functional regimes of the motor and indicate the ways of their possible implementations using means known from the practice.

Lumped‐parameter‐based electromagnetic transients simulation of non‐uniform single‐phase lines using state variable method

The characteristic impedance of a transmission line, a wire or a conductor changes in a non-uniform manner if the distance to the ground at all points longitudinally is not the same. Vertical conductors, transmission towers and sagging overhead lines are examples for the non-uniform lines. In this study, lumped-parameter-based state variable representation of the single-phase non-uniform line is described. From the lumped-parameter non-uniform line model a linear set of first-order differential equations is obtained in the form of state equations and this analytical expression is solved in closed form using MATLAB to obtain the transient response directly in the time domain. The closed-form solution has the advantage of obtaining the response of the system at an instant without the need for data in the previous states except for the initial conditions. The method also allows attaining the voltage and current profile of the system for any instant. In the illustrative cases presented, the systems with different surge impedance variations are considered and the surge response of a vertical conductor, an exponential line, and a horizontal cone and a vertical cone with constant and also varying propagation velocity are computed. The results are verified by those obtained using s-domain simulations of distributed-parameter transmission line and inverse Laplace transform.

Discrete-time chattering free exponentially stabilizing sliding mode scalar control via Lyapunov’s method

Linear time-invariant discrete-time plant, with no restrictions on the form of the state equation and with scalar control, is considered. The exponentially stabilizing state feedback control algorithm is developed by Lyapunov’s second method leading to the variable structure system with chattering free sliding modes. Essentially, the control algorithm drives the system from an arbitrary initial state to a prescribed so-called sliding subspace S, in finite time and with prescribed velocity estimate. Inside the sliding subspace S, the system is switched to the sliding mode regime and stays in it forever. The proposed algorithm is tested on the real system in practice, direct current (DC) servo motor, and simulation and experimental results are given. Also, it is compared with another already known algorithm from literature.

The Sensitivity of State Differential Game Vessel Traffic Model

Abstract The paper presents the application of the theory of deterministic sensitivity control systems for sensitivity analysis implemented to game control systems of moving objects, such as ships, airplanes and cars. The sensitivity of parametric model of game ship control process in collision situations have been presented. First-order and k-th order sensitivity functions of parametric model of process control are described. The structure of the game ship control system in collision situations and the mathematical model of game control process in the form of state equations, are given. Characteristics of sensitivity functions of the game ship control process model on the basis of computer simulation in Matlab/Simulink software have been presented. In the end, have been given proposals regarding the use of sensitivity analysis to practical synthesis of computer-aided system navigator in potential collision situations.

Understanding the breakdown of classic two-phase theory and spray atomization at engine-relevant conditions

A generalized framework for multi-component liquid injections is presented to understand and predict the breakdown of classic two-phase theory and spray atomization at engine-relevant conditions. The analysis focuses on the thermodynamic structure and the immiscibility state of representative gas-liquid interfaces. The most modern form of Helmholtz energy mixture state equation is utilized which exhibits a unique and physically consistent behavior over the entire two-phase regime of fluid densities. It is combined with generalized models for non-linear gradient theory and for liquid injections to quantify multi-component two-phase interface structures in global thermal equilibrium. Then, the Helmholtz free energy is minimized which determines the interfacial species distribution as a consequence. This minimal free energy state is demonstrated to validate the underlying assumptions of classic two-phase theory and spray atomization. However, under certain engine-relevant conditions for which corroborating experimental data are presented, this requirement for interfacial thermal equilibrium becomes unsustainable. A rigorously derived probability density function quantifies the ability of the interface to develop internal spatial temperature gradients in the presence of significant temperature differences between injected liquid and ambient gas. Then, the interface can no longer be viewed as an isolated system at minimal free energy. Instead, the interfacial dynamics become intimately connected to those of the separated homogeneous phases. Hence, the interface transitions toward a state in local equilibrium whereupon it becomes a dense-fluid mixing layer. A new conceptual view of a transitional liquid injection process emerges from a transition time scale analysis. Close to the nozzle exit, the two-phase interface still remains largely intact and more classic two-phase processes prevail as a consequence. Further downstream, however, the transition to dense-fluid mixing generally occurs before the liquid length is reached. The significance of the presented modeling expressions is established by a direct comparison to a reduced model, which utilizes widely applied approximations but fundamentally fails to capture the physical complexity discussed in this paper.

Sympathetic inrush current phenomenon with loaded transformers

The magnetization of a transformer is usually associated with the well-known phenomenon of inrush current. However, inrush current does not only affect the transformer being switched on. Instead, it has a significant impact also on all parallel connected transformers, which most certainly includes measurement transformers. This is known as a sympathetic inrush phenomenon. While the transformer being switched on might be a subject to a sudden high saturation level, the parallel transformers are gradually drawn to saturation as well, but of opposite polarity. The sympathetic inrush is expected in situations where the ohmic part represents a significant portion of the total system impedance. In contrast to other available literature on this subject, in this paper a modal approach to solving equivalent circuit by expressing differential equations in the state-equation form is selected. From the derivation of eigenvalues and eigenvectors, the phenomenon can be systematically investigated. A special attention will be given to circumstances, when the already operational transformer is fully loaded as within substations two or more transformers often operate in parallel, among which at least one of them is usually loaded. Finally, simulation results are compared to captured Wide Area Monitoring System (WAMS) measurements of the phenomenon and reasons for discrepancies are discussed.

Discriminant-based bistability analysis of a TMG-induced lac operon model supported with boundedness and local stability results

This paper presents the results of a theoretical and numerical study on the analysis of bistable behavior of the most studied gene regulatory network, the lac operon, in terms of the model parameters. The boundedness of the state variables for the considered model are demonstrated, the parameter values providing the existence of the multiple equilibria and thus the bistable behavior are determined, and a local stability analysis of the equilibria is performed. The parameter region yielding the existence of multiple equilibria is determined in an algebraic way based on discriminants. The model given in the state equation form is dened by the ordinary differential equations with the rational right-hand sides constituted within Hill and Michaelis{Menten approaches based on enzyme kinetics. The presented method can also be used in the parametric studies of other gene regulatory and metabolic networks given by state equations with rational right hand sides.

Iterative learning control of a class of discrete-time singular system

Singular systems are also called descriptor systems. Compared with normal systems, singular systems have become one of effective tools which can describe and characterize varieties of real systems, because they can better describe physical properties of the systems. Up to now, the analyses and syntheses of singular systems have been widespread applied to linear matrix inequality (LMI) method. The method requires known systems to have accurate mathematical model information, however, real systems have difficulties in obtaining their accurate mathematical models, owing to the fact that real systems are frequently subjected to all kinds of interferences, uncertainties, and nonlinear factors. Specially, in the case of singular systems, obtained LMI conditions often have a constraint equation or LMI is semi-definite, which makes it more difficult to solve LMI. Therefore, in order to avoid the above two problems occurring in settling state tracking issue for singular systems, in the meantime, for convenience of computating control algorithm and information storage, in this paper we propose a discrete-time iterative learning control algorithm for a class of discrete-time singular system with repetitive running characteristics in finite time interval. The specific process is divided into two steps. First, the class of discrete-time singular system is decomposed into normal discrete-time state equation and algebraic equation form by nonsingular transformation. Accordingly, the singular system state is also decomposed into two parts. Among them, the dimension of the first part state is equal to singular matrix rank and another is equal to system dimension minus singular matrix rank. In addition, the control law of last iterative learning is modified by using two tracking errors at two different times: one error is real-time tracking error generated from the comparison between the first part state and its desired state and another is tracking error at a previous time generated from comparison between the second part state and its desired state. And thus a new control law of next iterative learning is obtained, such that, as for any given real singular system, its state may completely track the desired state as long as selected learning gain can satisfy the convergence condition of the algorithm. Further, the convergence of the control algorithm is theoretically proved by compression mapping method, and thus its sufficient convergence condition is given in the sense of -norm. The results indicate that the proposed iterative learning control algorithm can make system state realize the perfect tracking of desired state as iteration number gradually increases in finite time interval, and the convergence of the algorithm only depends on system parameters and learning gain rather than initial value of control variable. The simulation example finally verifies the effectiveness of the proposed algorithm.

Sensitivity of the Game Control of Ship in Collision Situations

Abstract The paper introduces the application of the theory of deterministic sensitivity control systems for sensitivity analysis taking place in game control systems of moving objects, such as ships. The sensitivity of parametric model of game ship control process and game control in collision situations - sensitivity to changes in its parameters have been presented. First-order and k-th order sensitivity functions of parametric model of the process and game control are described. The structure of the game ship control system in collision situations and the mathematical model of game control process in the form of state equations are given. Characteristics of sensitivity functions of the model and game ship control process on the base of computer simulation in Matlab/Simulink software have been presented. At the end are given proposals regarding the use of sensitivity analysis to practical synthesis of computer-aided system navigator in potential collision situations.

Determining the bistability parameter ranges of artificially induced lac operon using the root locus method

This paper employs the root locus method to conduct a detailed investigation of the parameter regions that ensure bistability in a well-studied gene regulatory network namely, lac operon of Escherichia coli (E. coli). In contrast to previous works, the parametric bistability conditions observed in this study constitute a complete set of necessary and sufficient conditions. These conditions were derived by applying the root locus method to the polynomial equilibrium equation of the lac operon model to determine the parameter values yielding the multiple real roots necessary for bistability. The lac operon model used was defined as an ordinary differential equation system in a state equation form with a rational right hand side, and it was compatible with the Hill and Michaelis–Menten approaches of enzyme kinetics used to describe biochemical reactions that govern lactose metabolism. The developed root locus method can be used to study the steady-state behavior of any type of convergent biological system model based on mass action kinetics. This method provides a solution to the problem of analyzing gene regulatory networks under parameter uncertainties because the root locus method considers the model parameters as variable, rather than fixed. The obtained bistability ranges for the lac operon model parameters have the potential to elucidate the appearance of bistability for E. coli cells in in vivo experiments, and they could also be used to design robust hysteretic switches in synthetic biology.

English

The proposed research work on the topic Infinite Impulse Response Structure and Its Application is basically state space representation for the digital filter of different form obtained from analog filter and transformation to digital one. In the normal course of analysis the digital filter transfer function are realized in different forms using time delay elements and multipliers. This realization can be eased if digital filter are represented in state space techniques opening a new field for computation analysis. This aspect has been incorporated in the proposed research work things outlining the introduction of IIR filter and different structure including ladder and wave structure realization, state space description of IIR filter is presented in detail including the normal form which has complex structure but simple form of state equation. IIR filter design is based on state space technique using Lattice structure. The generalized technique of state space equation realized from general form of digital filter transfer function has been developed then the technique is demonstrated using differential form of various filter like LPF, HPF and BPF for the required order of analog filters. Statistical analysis of digital filter like round off error and dynamic scaling has been incorporated, where the two direct computes have been lumped together in the common state space representation. Autocorrelation and uncorrelation between noise and input samples gives round off noise and dynamic scaling give expression for the factor forms, round off error and dynamic range. This aspect on statistical analysis of digital filter has opened a new field for research.