Nonzero Initial Conditions Research Articles

Barrier Adaptive Iterative Learning Control for Tank Gun Control Systems Under Nonzero Initial Error Condition

In this paper, a barrier adaptive iterative learning control scheme is proposed to solve the trajectory-tracking problem for tank gun control systems under nonzero initial error condition. A novel construction method of rectified reference trajectory is presented for dealing with the initial position problem of iterative learning control for tank gun control systems. With a quadratic form barrier Lyapunov function adopted to controller design, the quadratic form of system error is constrained within the preset range during each iteration. Adaptive iterative learning control technique and robust control technique are jointly used to compensate for the parametric/nonparametric uncertainties and nonsymmetric deadzone nonlinearity. As the iteration number increases, the system state of tank gun control systems may accurately track the rectified reference trajectory, which leads to a excellent tracking performance during the part operation interval of tank gun control systems. Simulation results are presented to verify the effectiveness of the proposed barrier adaptive iterative control scheme.

High-Order Internal Model Based Barrier Iterative Learning Control for Time-Iteration-Varying Parametric Uncertain Systems With Arbitrary Initial Errors

In this paper, a high-order internal model based adaptive iterative learning control scheme is proposed to solve the trajectory tracking problem for a class of nonlinear systems with time-iteration-varying parametric uncertainties which are generated from a high-order internal model. A time-varying boundary layer is constructed to remove the nonzero initial error condition in ILC design. An adaptive iterative learning law is designed to deal with the time-iteration-varying parametric uncertainties. For improving the robustness and safety, a barrier Lyapunov function is adopted to controller design, thus making the filtering error constrained during each iteration. Even there exist nonzero initial state errors, the norm of tracking error vector will asymptotically converge to a tunable residual set as the iteration number increases. Simulation results show the effectiveness of the propose high-order internal model based filtering-error constraint adaptive learning scheme.

VIBRATIONS OF AN INFINITE PIECE-HOMOGENEOUS TWO-LAYER PLATE UNDER THE INFLUENCE OF NORMAL LOAD

This article examines the effect of normal load on an infinite piecewise homogeneous two-layer plate when the materials of the upper and lower layers of the plate are elastic. The transverse displacement of the points of the contact plane of a two-layer plate is determined, satisfying the approximate equation obtained in [1], in the case of replacing viscoelastic operators with elastic Lyame coefficients, respectively. For a rectangular infinite two-layer piecewise homogeneous plate under non-zero initial conditions, the frequencies of natural oscillations are calculated and an analytical solution to this problem is constructed. The obtained theoretical results for solving dynamic problems of transverse oscillation of piecewise homogeneous two-layer plates of constant thickness, taking into account the elastic properties of their material, allow us to more accurately calculate the transverse displacement of the points of the contact plane of the plates under normal external loads.

A non-linear disturbance observer-based cooperative adaptive cruise control for real traffic scenarios

Cooperative adaptive cruise control (CACC) has a strong potential to improvise highway traffic capacity and ease traffic disturbances. Extensive exploration is not carried out in the area of CACC for a cut-in maneuver. Contemporary control strategies proposed for CACC cannot regulate the peaking of control input and thus the acceleration/deceleration of following vehicles when applied for various real traffic scenarios. This paper aims to develop a non-linear disturbance observer-based sliding mode control to control a CACC system for various traffic scenarios. The proposed observer estimates the uncertainty present in the actuator dynamics and the preceding vehicle’s acceleration as the lumped disturbance at the same time, it adjusts the observer gain to alleviate the peaking of control input. The stability of individual vehicles and the string stability of vehicle platoon are derived The performance of the proposed scheme is validated with various traffic scenarios, that is, cut-in maneuver, cut-out maneuver, and non-zero initial conditions. The effectiveness of the proposed scheme is demonstrated by comparing it with a linear disturbance observer-based control.

A probabilistic point of view on peak effects in linear difference equations

It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that “solutions of stable difference equations probably experience peak”. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.

Comparison between two generalized Nash models with a same non-zero initial condition

Abstract Initial condition can impact the forecast precision especially in a real-time forecasting stage. The discrete linear cascade model (DLCM) and the generalized Nash model (GNM), though expressed in different ways, are both the generalization of the Nash cascade model considering the initial condition. This paper investigates the relationship and difference between DLCM and GNM both mathematically and experimentally. Mathematically, the main difference lies in the way to estimate the initial storage state. In the DLCM, the initial state is estimated and not unique, while that in the GNM is observed and unique. Hence, the GNM is the exact solution of the Nash cascade model, while the DLCM is an approximate solution and it can be transformed to the GNM when the initial storage state is calculated by the approach suggested in the GNM. As a discrete solution, the DLCM can be directly applied to the practical discrete streamflow data system. However, the numerical calculation approach such as the finite difference method is often used to make the GNM practically applicable. Finally, a test example obtained by the solution of the Saint-Venant equations is used to illustrate this difference. The results show that the GNM provides a unique solution while the DLCM has multiple solutions, whose forecast precision depends upon the estimate accuracy of the current state.

Trajectory Tracking and Stabilization of Nonholonomic Wheeled Mobile Robot Using Recursive Integral Backstepping Control

In this paper, a generalized nontriangular normal form is presented to facilitate designing a recursive integral backstepping control for the class of underactuated nonholonomic systems, i.e., wheeled mobile robots (WMRs) that perform posture stabilization and trajectory tracking in environments without obstacles. Based on the differential geometry theory, we develop a multiple input multiple output (MINO) generalization of normal form using the input-output feedback linearization technique. Then, the change of variables (diffeomorphism) transform the state-space model of WMR, incorporating both kinematic and dynamic models into nontriangular normal form. As a result, the system dynamics can be represented as internal and external dynamics. The nonlinear internal dynamics of WMR pose serious challenges to design a suitable controller due to its internal dynamics being not minimum phase and non-strict feedback form structure. The proposed backstepping controller is designed in two steps. First, a standard integral backstepping controller is designed to stabilize the robot’s orientation angle. Then, a recursive integral backstepping control technique is applied to achieve asymptotic convergence of position error to zero. Hence, both asymptotic posture stabilization and trajectory tracking are achieved in semi-global regions, except the nonzero initial condition of the orientation angle. The asymptotic stability of the entire closed-loop system is shown using the Lyapunov criteria.

A New Method for Laplace Transforms of Multiterm Fractional Differential Equations of the Caputo Type

AbstractThe Laplace transform method is one of the powerful tools in studying the fractional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multiterm Caputo FDEs of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multiterm Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of nonzero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the solution to the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multiterm Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag–Leffler functions.

Cryptanalysis of Some Self-Synchronous Chaotic Stream Ciphers and Their Improved Schemes

In this paper, a cryptanalysis method that combines a chosen-ciphertext attack with a divide-and-conquer attack by traversing multiple nonzero component initial conditions (DCA-TMNCIC) is proposed. The method is used for security analysis of [Formula: see text]-D ([Formula: see text]) self-synchronous chaotic stream ciphers that employ a product of two chaotic variables and three chaotic variables ([Formula: see text]-D SCSC-2 and [Formula: see text]-D SCSC-3), taking 3-D SCSC-2 as a typical example for cryptanalysis. For resisting the combinational effect of the chosen-ciphertext attack and DCA-TMNCIC, several improved chaotic cipher schemes are designed, including 3-D SCSC based on a nonlinear nominal system (3-D SCSC-NNS) and [Formula: see text]-D SCSC based on sinusoidal modulation ([Formula: see text]-D SCSC-SM ([Formula: see text])). Theoretical security analysis validates the improved schemes.

The Explicit Formula for Solution of Anomalous Diffusion Equation in the Multi-Dimensional Space

This paper intends on obtaining the explicit solution of $$n$$ -dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $$t^{-\alpha}E_{1-\alpha,1-\alpha}(-t^{1-\alpha})$$ , $$\alpha\in(0,1),$$ where $$E_{\alpha,\beta}$$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

Splitting model reduction for bilinear control systems

AbstractIn this paper, based on low‐rank balanced truncation (BT), we present a splitting model reduction method for bilinear control systems with nonzero initial conditions. By analyzing Volterra series, the original bilinear control system is split up into two systems: one is a bilinear control system with zero initial condition, and the other is our constructed control system. Specifically, we prove that the Volterra series containing initial condition of the original system converges to the output of our constructed control system. Thus, we develop a new splitting model reduction method that involves reducing two systems independently by low‐rank BT method and then combining the reduced outputs into an aggregate output. A nonlinear RC circuit and a heat transfer model are taken as numerical examples to illustrate the effectiveness of our proposed method.

Commutativity of Sixth-Order Time-Varying Linear Systems

This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system $$A$$ , we find its commutative pair, that is a new differential system $$B$$ of order $$m \le 6$$ ; the obtained result will be used to show the input–output equivalency between the cascaded-connected systems $$AB$$ and $$BA$$ . Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11.

Design and Smartphone Implementation of Chaotic Duplex H.264-Codec Video Communications

In this paper, a chaotic duplex H.264-codec-based secure video communication scheme is designed and its smartphone implementation is also carried out. First, an improved self-synchronous chaotic stream cipher algorithm equipped with a sinusoidal modulation, a multiplication, a modulo operation and a round down operation (SCSCA-SMMR) is developed. Using the sinusoidal modulation and multiplication, the improved algorithm can resist the divide-and-conquer attack by traversing multiple nonzero component initial conditions (DCA-TMNCIC). Meanwhile, also by means of the round down operation and modulo operation, on the premise that the DCA-TMNCIC does not work, the original keys cannot be further deciphered only by the known-plaintext attack, the chosen-plaintext attack and the chosen-ciphertext attack, respectively. Then, the Android low-level multimedia support infrastructure MediaCodec class is used to access low-level media encoder/decoder components and the H.264 hardware encoding/decoding is performed on real-time videos, so the chaotic video encryption and decryption can be realized in real-time by smartphones. Security analysis and smartphone experimental results verify the effectiveness of the proposed method.

Applications of the Poincaré–Hopf Theorem: Epidemic Models and Lotka–Volterra Systems

This article focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré–Hopf theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria and ensures that the system is in fact locally exponentially stable. We apply this result to the susceptible–infected–susceptible (SIS) networked epidemic model, and a generalized Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralized feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincaré–Hopf approach, we identify a necessary and sufficient condition, under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady state), and monotone systems theory is used to show that this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.

Analytic Approach for Solving System of Fractional Differential Equations

In this paper, Sumudu transformation (ST) of Caputo fractional derivative formulae are derived for linear fractional differential systems. This formula is applied with Mittage-Leffler function for certain homogenous and nonhomogenous fractional differential systems with nonzero initial conditions. Stability is discussed by means of the system's distinctive equation.

Realization of a Fourth-Order Linear Time-Varying Differential System with Nonzero Initial Conditions by Cascaded two Second-Order Commutative Pairs

Decomposition is an important tool that is used in many differential systems for solving real engineering problems and improving the stability of a system. It involves breaking down of high-order linear systems into lower-order commutative pairs. Commutativity plays an essential role in mathematics, and its applications are extended in physical science and engineering. This paper explicitly expresses all form of necessary and sufficient conditions for decomposition of any kind of fourth-order linear time-varying system as commutative pairs of two second-order systems. Regarding the nonzero initial conditions, additional requirements are derived in order to satisfy the decomposition process. In this paper, explicit method for reducing fourth-order linear time-varying systems (LTVS) into two second-order commutative pairs is derived and solved. The method points out the effect of disturbance and sensitivity on the systems and also highlights the necessary and sufficient conditions for commutativity of the decomposed systems. The results are illustrated by solving some examples.

Stability of a Certain Class of Second Order Switched Systems

Abstract This paper present study of stability of a special two-dimensional system with switching. Its subsystems have second order and are Hurwitz. The first subsystem is linear system presented in Frobenius form and the second one has the same eigenvalues as first one but has swapped states and corresponding realization. The considered problem is to find all permissible eigenvalues that provide asymptotic stability to the switched system under any switching signal and non-zero initial condition. The study is based on common Lyapunov functions which existence is checked using known approach that consists in studying the stability of a convex combination of the subsystems matrices.

Robust System Identification for Anemia Management

Abstract Anemia is the condition in which patients suffer from the deficiency of the red blood cells to carry the oxygen in the body. One of the many causes of anemia is chronic kidney disease (CKD). CKD is a disease in which kidneys are partially or completely damaged, which results in a deficiency of the oxygen-carrying red blood cells. CKD is common in older people, and external human recombinant erythropoietin (EPO) is required to maintain healthy levels of hemoglobin (Hb). In order to effectively address the impact of inter and intra-individual variability in dose-response characteristics in CKD patients, individualized patient-specific models are required instead of traditional population-based models. In this research, individualized patient models are developed by using patient-specific time-domain data with robust system identification techniques. For control-oriented system identification, two robust identification techniques are investigated: (1) l1 robust identification considering zero initial conditions and (2) Semi-blind robust system identification considering non-zero initial conditions. The performance of these two techniques is compared and it is shown that the Semi-blind robust identification technique gives better results as compared to l1 robust identification.

THE COMPELLED FLUCTUATIONS OF A RECTANGULAR TWO-LAYER PIECEWISE-HOMOGENEOUS PLATE OF A CONSTANT THICKNESS

This article discusses forced vibrations of a rectangular two-layer piecewise homogeneous plate of constant thickness, when the material of the upper layer of the plate is elastic and the other satisfies Maxwell’s model, that is, viscoelastic. The transverse displacement of points of the contact plane of a two-layer plate is determined, which satisfies the approximate equation obtained in [1], replacing only the viscoelastic operators of the upper layer of the plate with the elastic Lames coefficients, respectively. Fluctuation rectangular is free a piecewise-homogeneous plate at nonzero initial conditions, frequencies of own fluctuations are calculated, and the analytical decision of this problem is under construction. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely cross-section displacement of points of a plane of contact of plates at non-stationary external loadings.

Short Transient IIR Multinotch Filter With Time-Varying Parameters and Nonzero Initial Conditions

In this article, an analytical concept and hardware verification of the time-varying infinite impulse response narrow multiband-stop filter with nonzero initial conditions are presented. The time variability of the filter parameters and nonzero initial conditions are introduced to the filter structure in order to reduce its transient response. Such a filter can be very useful in various measurement systems. In particular, short transient filters are very desirable for processing signals whose initial part is of great importance. In this article, the algorithm for finding the length of the initial conditions vector is presented. As an example, the proposed IIR multinotch filter is used for removing power-line interferences from the ECG signal. Computer simulations, hardware verification, and performance indices demonstrate that the proposed time-varying IIR multinotch filter with nonzero initial conditions possesses much better properties than the traditional time-invariant filter with zero initial conditions.