Order Linear Systems Research Articles

Commutativity of Cascaded Connected Fractional Order Linear Time-Varying Systems

Commutativity of Cascaded Connected Fractional Order Linear Time-Varying Systems

Moth Flame Optimization for Model Order Reduction of Complex High Order Linear Time-Invariant Systems

Moth Flame Optimization for Model Order Reduction of Complex High Order Linear Time-Invariant Systems

Linearized Models of the Coupled Rotorcraft Flight Dynamics and Acoustics for Real-Time Noise Prediction

This article demonstrates the linearization of the coupled rotorcraft flight dynamics and aeroacoustics to provide realtime acoustic predictions in generalized maneuvering flight. To demonstrate the methodology, the study makes use of a nonlinear simulation model of a generic utility helicopter (PSUHeloSim) that is coupled with an aeroacoustic solver based on a marching cubes algorithm. A periodic equilibrium of the coupled rotorcraft flight dynamics and acoustics is first found at a desired flight condition using a modified harmonic balance trim solution method. Next, the nonlinear time-periodic dynamics are linearized about that periodic equilibrium and transformed into an equivalent higher order linear time-invariant system in harmonic decomposition form. Composite aeroacoustic measures are included as an output of this system. To speed up runtime and make control design tractable, the order of these harmonic decomposition models is reduced via residualization to an 8-state model where the states are representative of the rigid-body dynamics of the aircraft. This 8-state model is shown to provide accurate acoustic response predictions for small-amplitude pilot inputs and to abate runtime by a factor of approximately 104, thus enabling acoustic predictions in generalized maneuvering flight that are significantly faster than real time. The 8-state model is subsequently used to demonstrate the use of linear system tools for the dynamic analysis of the coupled rotorcraft flight dynamics and acoustics.

Simplifying the electronic wedge brake system model through model order reduction techniques

The electronic wedge brake (EWB) uses self-reinforcement principles to optimise stopping power, but its mathematical model has various actuation angles and system dynamics making controller design complex and computationally burdensome. Therefore, the model order reduction (MOR) is made based on three factors that may have a negligible influence on the EWB system: the motor inductance, lead screw axial damping, and wedge mass. Six reduced order model (ROM) types were proposed when one, two, or all factors were ignored. The ROM accuracy was analysed using the frequency and time domain. The percentage of root means square error (RMSE) response value between the EWB benchmark model, and the predicted response based on the ROM was found to be less than 2%, with ROM size reduced from 5 to 2 orders. It guarantees that the new ROM series will be useful for simpler EWB controller design. The proposed ROM simplifies the original model drastically while retaining accuracy at an adequate level. Even though the simplest EWB model is a 2nd order linear system, the best ROM vary depending on EWB design parameters.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Examining reachability of fractional dynamical systems with delays in control utilizing ψ-Hilfer pseudo-fractional derivative

This research focuses on the reachability criteria of fractional dynamical systems with delays in control of order ϑ ∈ (0, 1) and type ϱ ∈ [0, 1] in the context of control theory. The study introduces the utilization of the ψ-Hilfer pseudo fractional derivative to describe the dynamics of systems with delays in control. We obtained the necessary and sufficient conditions for the reachability criteria of fractional order linear dynamical systems by employing the positive definiteness of Gramian matrices. We arrived the sufficient conditions for the reachability criteria of fractional order nonlinear dynamical systems using Banach’s fixed point theorem. A few numerical examples are made available for a better understanding of the theoretical conclusions.

Admissibility and Observability of Jeffreys Type of Overdamped Second Order Linear Systems

Admissibility and Observability of Jeffreys Type of Overdamped Second Order Linear Systems

Controllability of networked multiagent systems based on linearized Turing’s model

Turing’s model has been widely used to explain how simple, uniform structures can give rise to complex, patterned structures during the development of organisms. However, it is very hard to establish rigorous theoretical results for the dynamic evolution behavior of Turing’s model since it is described by nonlinear partial differential equations. We focus on controllability of Turing’s model by linearization and spatial discretization. This linearized model is a networked system whose agents are second order linear systems and these agents interact with each other by Laplacian dynamics on a graph. A control signal can be added to agents of choice. Under mild conditions on the parameters of the linearized Turing’s model, we prove the equivalence between controllability of the linearized Turing’s model and controllability of a Laplace dynamic system with agents of first order dynamics. When the graph is a grid graph or a cylinder grid graph, we then give precisely the minimal number of control nodes and a corresponding control node set such that the Laplace dynamic systems on these graphs with agents of first order dynamics are controllable.

Exact internal controllability and exact internal synchronization for a kind of first order hyperbolic system

In this paper we investigate the exact controllability and exact synchronization in L2 space for a 1-D first order linear hyperbolic system with internal controls located on some part of the domain. Based on the exact boundary controllability theory, we first use the constructive method to establish the exact controllability by internal controls for a time reversible system with inhomogeneous boundary conditions. The method is then adapted to prove the exact internal null controllability for a general system with homogeneous boundary conditions. These results can be then used to establish the exact internal synchronization for the system, and related subjects are further studied.

An Algorithm for the Solution of Second Order Linear Fuzzy System With Mechanical Applications

In this paper, we consider homogeneous and non-homogeneous second order linear fuzzy systems under granular differentiability. The concept of continuous n-dimensional fuzzy functions on the space of n-dimensional fuzzy numbers are introduced. Developed an algorithm for the solution of a non-homogeneous second order linear fuzzy system under granular differentiability. The proposed algorithm is applied to solve some well-known mechanical problems with fuzzy uncertainty.

Suboptimal control law for a multi fractional high order linear quadratic regulator system in the presence of disturbance

A new method for finding a suboptimal control law for a multi fractional spring–mass system as a fractional high order linear quadratic regulator system in the presence of known disturbance is introduced. The method is presented in a general form for solving practical multi fractional linear quadratic regulator systems with disturbances. After defining a closed-loop controller, formulations for finding the suboptimal controller are derived. In the proposed method, there is no need to solve the fractional vector differential equation. Simulation studies demonstrate the applicability of the method for practical purposes. Some new types of a spring–mass–damper system are considered.

USE OF ORDINARY DIFFERENTIAL EQUATIONS APPLIED TO MECHANICAL AND ELECTRICAL PROBLEMS WITH FIRST ORDER LINEAR SYSTEMS

In recent years there has been an increased interest in the study of mathematics applied to engineering to study aspects of its teaching. In the present research work, a compilation of analytical concepts of differential equations applied to the solution of problems of electrical circuits, a branch of electrical engineering and mechanical engineering problems, specifical problems of spring-mass mechanical systems, through the description of the analytical solutions of direct current electrical circuits and spring-mass mechanical systems, applying differential equations. This method stands out among the diverse mathematical treatments used to solve this type of engineering problem, both for its demonstrative elegance and capacity to offer a total answer. Likewise, practical engineering examples related to the studied subject were presented and the results were obtained and analyzed by using the mathematical software, Wolfram Mathematica.

Robust H[formula omitted]-PID control Stability of fractional-order linear systems with Polytopic and two-norm bounded uncertainties subject to input saturation

This paper deals a type of H∞ proportional–integral–derivative (PID) control mechanism for a type of structural uncertain fractional order linear systems by convex Polytopic and two-norm bounded uncertainties subject to input saturation which mainly focuses on the case of a fractional order α such that 0<α<1. The Gronwall–Bellman lemma and the sector condition of the saturation function are investigated for system stability analysis and stabilization. The main strategy of the presented strategy is to restore fractional order PID controller design under input saturation problem from static output feedback controller design. Unlike existing strategies, non-iterative strategy is used to get optimal output feedback based on the LMI. On the premise of a linear matrix inequality algorithm, the SOF control laws can be obtained. After that, the fractional-order PID controller is recovered from the SOF controller. A numerical example is provided in order to show the validity and superiority of the proposed method.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

On the transient behavior and gain design for a class of second order homogeneous systems

In this note, we consider a class of homogeneous second order systems which contains two relevant sub-classes: the first one in a controller form and the second one in an observer form. For both of them, we provide necessary and sufficient conditions on the gain parameters to guarantee oscillatory or non-oscillatory behavior of the solutions. This is in the same spirit of tuning of second order linear systems, where the gains can be designed to obtain an overdamped, critically damped or underdamped behavior. The analysis of all the cases is performed in a unified framework based on a search for invariant sets which are analogous to the invariant eigenspaces of linear systems.

Условие оптимальности типа принципа максимума Понтрягина в задаче управления линейными разностными уравнениями дробного порядка

The optimal control problem of fractional order linear two-dimensional difference equations sys-tems is considered. It is assumed that the control function is included in the boundary condition, and the functional is linear. A necessary and sufficient optimality condition is proved in the discrete maximum principal form. A sufficient optimality condition is proved in the nonlinear but convex cost functional case.

把持タスクにおける仮想力と力学系を用いた軌道修正による遠隔操作支援

This paper presents a dynamical system based approach to assistive teleoperation for object grasping. According to the target position and the current user input, a virtual desired goal is generated for a stable second order linear dynamical system via linear interpolation. In order to achieve collision avoidance between the target object and the robot finger, a virtual force is generated as an input to the dynamical system for online trajectory modification. Experimental results of an object grasping task are presented to demonstrate the effectiveness of the proposed assistive approach in a simulated robot teleoperation system.

Non-asymptotic and robust estimation for fractional order pseudo-state space model using an algebraic parametric method

This paper mainly investigates the derivative estimation problem for a class of fractional order linear systems. After applying the Laplace transform to the considered pseudo-state space representation model, a series of multiplications and derivations are applied to eliminate unknown initial conditions, which define an annihilator. Then, through strict mathematical derivation, an input-output integral equation is deduced when returning into the time domain. Based on the obtained equation and thanks to the fractional Leibniz formulas, algebraic integral formulas are provided, which can non-asymptotically and robustly estimate the fractional derivatives of the system output. Moreover, error analysis in discrete noisy cases is given to study the performance of the proposed differentiator. Finally, numerical simulations are provided to show the effectiveness of the proposed method.

On a two-dimensional model of generalized thermoelasticity with application

A 2D first order linear system of partial differential equations of plane strain thermoelasticity within the frame of extended thermodynamics is presented and analyzed. The system is composed of the equations of classical thermoelasticity in which displacements are replaced with velocities, complemented with Cattaneo evolution equation for heat flux. For a particular choice of the characteristic quantities and for positive thermal conductivity, it is shown that this system may be cast in a form that is symmetric t-hyperbolic without further recurrence to entropy principle. While hyperbolicity means a finite speed of propagation of heat waves, it is known that symmetric hyperbolic systems have the desirable property of well-posedness of Cauchy problems. A study of the characteristics of this system is carried out, and an energy integral is derived, that can be used to prove uniqueness of solution under some boundary conditions. A numerical application for a finite slab is considered and the numerical results are plotted and discussed. In particular, the wave propagation nature of the solution is put in evidence.

Suboptimal control of linear fuzzy systems

In this paper, we study first order linear fuzzy systems under generalized differentiability and present the general form of their solutions. Next, the fuzzy optimal control problem of these systems is considered to optimize the expected values of the appropriate objective fuzzy functions and the Pontryagin Maximum Principle is used to obtain a necessary optimality condition in the form of a fuzzy boundary value problem. Using the necessary optimality condition, the constant formulas for fuzzy optimal control function and the corresponding fuzzy state function are proposed. Due to the different formulations of a unique dynamical phenomenon in the fuzzy setting and different cases of the strongly generalized differentiability, for each fuzzy optimal control problem, we obtain a suboptimal solution. Finally, some examples including the optimal control of a cruise system, a RLC circuit system and a Mass-Spring-Damper system are given to illustrate our results.