Research

15 pages, 388 KiB

Open AccessArticle

Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games

by Vasile Drăgan, Ivan Ganchev Ivanov, Ioan-Lucian Popa and Ovidiu Bagdasar

Mathematics 2021, 9(21), 2713; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212713 - 26 Oct 2021

Cited by 1 | Viewed by 1169

Abstract

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear [...] Read more.

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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13 pages, 1694 KiB

Open AccessFeature PaperArticle

Feedforward of Measurable Disturbances to Improve Multi-Input Feedback Control

by Javier Rico-Azagra and Montserrat Gil-Martínez

Mathematics 2021, 9(17), 2114; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172114 - 01 Sep 2021

Cited by 1 | Viewed by 1502

Abstract

The availability of multiple inputs (plants) can improve output performance by conveniently allocating the control bandwidth among them. Beyond that, the intervention of only the useful plants at each frequency implies the minimum control action at each input. Secondly, in single input control, [...] Read more.

The availability of multiple inputs (plants) can improve output performance by conveniently allocating the control bandwidth among them. Beyond that, the intervention of only the useful plants at each frequency implies the minimum control action at each input. Secondly, in single input control, the addition of feedforward loops from measurable external inputs has been demonstrated to reduce the amount of feedback and, subsequently, palliate its sideband effects of noise amplification. Thus, one part of the action calculated by feedback is now provided by feedforward. This paper takes advantage of both facts for the problem of robust rejection of measurable disturbances by employing a set of control inputs; a previous work did the same for the case of robust reference tracking. Then, a control architecture is provided that includes feedforward elements from the measurable disturbance to each control input and feedback control elements that link the output error to each control input. A methodology is developed for the robust design of the named control elements that distribute the control bandwidth among the cheapest inputs and simultaneously assures the prescribed output performance to correct the disturbed output for a set of possible plant cases (model uncertainty). The minimum necessary feedback gains are used to fight plant uncertainties at the control bandwidth, while feedforward gains achieve the nominal output response. Quantitative feedback theory (QFT) principles are employed. An example illustrates the method and its benefits versus a control architecture with only feedback control elements, which have much more gain beyond the control bandwidth than when feedforward is employed. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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25 pages, 8514 KiB

Open AccessArticle

Leveraging Elasticity to Uncover the Role of Rabinowitsch Suspension through a Wavelike Conduit: Consolidated Blood Suspension Application

by Sara I. Abdelsalam and Abdullah Z. Zaher

Mathematics 2021, 9(16), 2008; https://0-doi-org.brum.beds.ac.uk/10.3390/math9162008 - 22 Aug 2021

Cited by 59 | Viewed by 2436

Abstract

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along [...] Read more.

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along with heat transfer. It is of interest in this work to study the effects of EOFs, which are rigid spherical particles that are suspended in the Rabinowitsch fluid, the Grashof parameter, heat source, and elasticity on the shear stress of the Rabinowitsch fluid model and flow quantities. The solutions are achieved by taking long wavelength approximation with the creeping flow system. A comparison is set between the effect of pseudoplasticity and dilatation on the behaviour of shear stress, axial velocity, and pressure rise. Physical behaviours have been graphically discussed. It was found that the Rabinowitsch and electroosmotic parameters enhance the shear stress while they reduce the pressure gradient. A biomedical application to the problem is presented. The present analysis is particularly important in biomedicine and physiology. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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16 pages, 9213 KiB

Open AccessArticle

On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates

by Odysseas Kosmas, Pieter Boom and Andrey P. Jivkov

Mathematics 2021, 9(14), 1689; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141689 - 19 Jul 2021

Viewed by 1746

Abstract

The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of [...] Read more.

The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The standard barycentric coordinate system in this work is restricted to three-dimensional (3-D) convex polytopes. We show that for an explicit energy expression, applicable also to non-convex polytopes, the general barycentric coordinates constitute fundamental tools. We define, in addition, the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We, finally, prove that the resulting expressions can consistently describe the deformation of solids. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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18 pages, 321 KiB

Open AccessFeature PaperArticle

Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

by Fátima Cruz, Ricardo Almeida and Natália Martins

Mathematics 2021, 9(14), 1665; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141665 - 15 Jul 2021

Cited by 1 | Viewed by 1575

Abstract

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper [...] Read more.

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

38 pages, 515 KiB

Open AccessArticle

Quadratic First Integrals of Time-Dependent Dynamical Systems of the Form

{\ddot{q}}^{a} = - Γ_{b c}^{a} {\dot{q}}^{b} {\dot{q}}^{c} - ω (t) Q^{a} (q)

by Antonios Mitsopoulos and Michael Tsamparlis

Mathematics 2021, 9(13), 1503; https://0-doi-org.brum.beds.ac.uk/10.3390/math9131503 - 27 Jun 2021

Cited by 6 | Viewed by 1659

Abstract

We consider the time-dependent dynamical system

{\ddot{q}}^{a} = - Γ_{b c}^{a} {\dot{q}}^{b} {\dot{q}}^{c} - ω (t) Q^{a} (q)

where

ω (t)

is a non-zero arbitrary function and the [...] Read more.

We consider the time-dependent dynamical system

{\ddot{q}}^{a} = - Γ_{b c}^{a} {\dot{q}}^{b} {\dot{q}}^{c} - ω (t) Q^{a} (q)

where

ω (t)

is a non-zero arbitrary function and the connection coefficients

Γ_{b c}^{a}

are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that

I = K_{a b} {\dot{q}}^{a} {\dot{q}}^{b} + K_{a} {\dot{q}}^{a} + K

where the unknown coefficients

K_{a b}, K_{a}, K

are tensors depending on

t, q^{a}

and impose the condition

\frac{d I}{d t} = 0

. This condition leads to a system of partial differential equations (PDEs) involving the quantities

K_{a b}, K_{a}, K,

ω (t)

and

Q^{a} (q)

. From these PDEs, it follows that

K_{a b}

is a Killing tensor (KT) of the kinetic metric. We use the KT

K_{a b}

in two ways: a. We assume a general polynomial form in t both for

K_{a b}

and

K_{a}

; b. We express

K_{a b}

in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either

ω (t)

or

Q^{a} (q)

. We consider first that

ω (t)

is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities

Q^{a} (q)

to be the generalized time-dependent Kepler potential

V = - \frac{ω (t)}{r^{ν}}

and determine the functions

ω (t)

for which QFIs are admitted. We extend the discussion to the non-linear differential equation

\ddot{x} = - ω (t) x^{μ} + ϕ (t) \dot{x}

(μ \neq - 1)

and compute the relation between the coefficients

ω (t), ϕ (t)

so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

24 pages, 5938 KiB

Open AccessArticle

High-Order Filtered PID Controller Tuning Based on Magnitude Optimum

by Damir Vrančić and Mikuláš Huba

Mathematics 2021, 9(12), 1340; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121340 - 09 Jun 2021

Cited by 16 | Viewed by 2365

Abstract

The paper presents a tuning method for PID controllers with higher-order derivatives and higher-order controller filters (HO-PID), where the controller and filter orders can be arbitrarily chosen by the user. The controller and filter parameters are tuned according to the magnitude optimum criteria [...] Read more.

The paper presents a tuning method for PID controllers with higher-order derivatives and higher-order controller filters (HO-PID), where the controller and filter orders can be arbitrarily chosen by the user. The controller and filter parameters are tuned according to the magnitude optimum criteria and the specified noise gain of the controller. The advantages of the proposed approach are twofold. First, all parameters can be obtained from the process transfer function or from the measured input and output time responses of the process as the steady-state changes. Second, the a priori defined controller noise gain limits the amount of HO-PID output noise. Therefore, the method can be successfully applied in practice. The work shows that the HO-PID controllers can significantly improve the control performance of various process models compared to the standard PID controllers. Of course, the increased efficiency is limited by the selected noise gain. The proposed tuning method is illustrated on several process models and compared with two other tuning methods for higher-order controllers. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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15 pages, 1790 KiB

Open AccessArticle

A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

by Elsayed Badr, Sultan Almotairi and Abdallah El Ghamry

Mathematics 2021, 9(11), 1306; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111306 - 07 Jun 2021

Cited by 12 | Viewed by 4270

Abstract

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, [...] Read more.

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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22 pages, 337 KiB

Open AccessArticle

On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions

by Usman Riaz, Akbar Zada, Zeeshan Ali, Ioan-Lucian Popa, Shahram Rezapour and Sina Etemad

Mathematics 2021, 9(11), 1205; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111205 - 26 May 2021

Cited by 11 | Viewed by 1724

Abstract

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are [...] Read more.

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

34 pages, 1420 KiB

Open AccessArticle

Stability Analysis and Optimal Control of a Fractional Order Synthetic Drugs Transmission Model

by Meghadri Das, Guruprasad Samanta and Manuel De la Sen

Mathematics 2021, 9(7), 703; https://0-doi-org.brum.beds.ac.uk/10.3390/math9070703 - 24 Mar 2021

Cited by 9 | Viewed by 2003

Abstract

In this work, a fractional-order synthetic drugs transmission model with psychological addicts has been proposed along with psychological treatment. The effects of synthetic drugs are deadly and sometimes even violent. We have studied the local and global stability of the model with different [...] Read more.

In this work, a fractional-order synthetic drugs transmission model with psychological addicts has been proposed along with psychological treatment. The effects of synthetic drugs are deadly and sometimes even violent. We have studied the local and global stability of the model with different criterion. The existence and uniqueness criterion along with positivity and boundedness of the solutions have also been established. The local and global stabilities are decided by the basic reproduction number

R_{0}

. We have also analyzed the sensitivity of parameters. An optimal control problem has been formulated by controlling psychological addiction and analyzed by the help of Pontryagin maximum principle. These results are verified by numerical simulations. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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14 pages, 427 KiB

Open AccessArticle

The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method

by Nehad Ali Shah, Ioannis Dassios, Essam R. El-Zahar, Jae Dong Chung and Somaye Taherifar

Mathematics 2021, 9(2), 141; https://0-doi-org.brum.beds.ac.uk/10.3390/math9020141 - 11 Jan 2021

Cited by 10 | Viewed by 2128

Abstract

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu [...] Read more.

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu decomposition method. The solution procedure reveals that the variational iteration transform method and Shehu decomposition method is effective, reliable and straightforward. The variational iteration transform methods solve non-linear problems without using Adomian’s polynomials and He’s polynomials, which is a clear advantage over the decomposition technique. The solutions achieved are compared with the corresponding exact result to show the efficiency and accuracy of the existing methods in solving a wide variety of linear and non-linear problems arising in various science areas. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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