Fractional Optimal Control Within Caputo’s Derivative

Dynamic Constraint ◽

Caputo’S Derivative ◽

A general formulation and solution of fractional optimal control problems (FOCPs) in terms of Caputo fractional derivatives (CFDs) of arbitrary order have been considered in this paper. The performance index (PI) of a FOCP is considered as a function of both the state and control. The dynamic constraint is expressed by a fractional differential equation (FDE) of arbitrary order. A general pseudo-state-space representation of the FDE is presented and based on that, FOCP has been developed. A numerical technique based on Gru¨nwald-Letnikov (G-L) approximation of the FDs is used for solving the resulting equations. Numerical example is presented to show the effectiveness of the formulation and solution scheme.

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C ◽

Numerical Method for Solving Fractional Optimal Control Problems

10.1115/detc2009-87008 ◽

2009 ◽

Cited By ~ 9

Author(s):

Raj Kumar Biswas ◽

Siddhartha Sen

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Numerical Technique ◽

Control Problems ◽

Caputo Fractional Derivatives ◽

Liouville Derivative ◽

Fractional Differential ◽

The Right

A numerical technique for the solution of a class of fractional optimal control problems has been proposed in this paper. The technique can used for problems defined both in terms of Riemann-Liouville and Caputo fractional derivatives. In this technique a Reflection Operator is used to convert the right Riemann-Liouville derivative into an equivalent left Riemann-Liouville derivative, and then the two point boundary value problem is solved numerically. The proposed method is straightforward and it uses an available numerical technique to solve fractional differential equations resulting from the formulation. Examples considered here show that the numerical results obtained using this and other techniques agree very well.

Numerical solutions of fractional optimal control with Caputo–Katugampola derivative

Advances in Difference Equations ◽

10.1186/s13662-021-03580-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

N. H. Sweilam ◽

A. M. Nagy ◽

T. M. Al-Ajami

Keyword(s):

Optimal Control ◽

Fractional Derivative ◽

Optimal Control Problems ◽

Numerical Solutions ◽

Numerical Technique ◽

Ritz Method ◽

Derivative Operator ◽

AbstractIn this paper, we present a numerical technique for solving fractional optimal control problems with a fractional derivative called Caputo–Katugampola derivative. This derivative is a generalization of the Caputo fractional derivative. The proposed technique is based on a spectral method using shifted Chebyshev polynomials of the first kind. The Clenshaw and Curtis scheme for the numerical integration and the Rayleigh–Ritz method are used to estimate the state and control variables. Moreover, the error bound of the fractional derivative operator approximation of Caputo–Katugampola is derived. Illustrative examples are provided to show the validity and applicability of the presented technique.

Implementation of fractional optimal control problems in real-world applications

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0088 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1783-1796

Author(s):

Neelam Singha

Keyword(s):

Optimal Control ◽

Fractional Order ◽

Adomian Decomposition Method ◽

Necessary Optimality Conditions ◽

Order Model ◽

Adomian Decomposition ◽

Real World Applications ◽

Abstract In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order optimal control problem. The key motive is to perform numerical simulations that shall facilitate us in understanding the behavior of state and control variables. Further, the graphical interpretation of solutions effectively validates the applicability of the present analysis to investigate the growth of cancer cells in the presence of medical treatment.

Modeling, Actuation, and Control of an Active Fluid Vibration Isolator

Journal of Vibration and Acoustics ◽

10.1115/1.2889687 ◽

1997 ◽

Vol 119 (1) ◽

pp. 52-59 ◽

Cited By ~ 2

Author(s):

M. J. Panza ◽

D. P. McGuire ◽

P. J. Jones

Keyword(s):

Electrical Power ◽

Fluid Pressure ◽

Space Representation ◽

Vibration Isolator ◽

Mass System ◽

Integral Control ◽

Active Fluid ◽

In Series ◽

An integrated mathematical model for the dynamics, actuation, and control of an active fluid/elastomeric tuned vibration isolator in a two mass system is presented. The derivation is based on the application of physical principles for mechanics, fluid continuity, and electromagnetic circuits. Improvement of the passive isolator performance is obtained with a feedback scheme consisting of a frequency shaped notch compensator in series with integral control of output acceleration and combined with proportional control of the fluid pressure in the isolator. The control is applied via an electromagnetic actuator for excitation of the fluid in the track connecting the two pressure chambers of the isolator. Closed loop system equations are transformed to a nondimensional state space representation and a key dimensionless parameter for isolator-actuator interaction is defined. A numerical example is presented to show the effect of actuator parameter selection on system damping, the performance improvement of the active over the passive isolator, the robustness of the control scheme to parameter variation, and the electrical power requirements for the actuator.

Volume 3C: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration Control, Analysis, and Identification ◽

Modeling, Actuation, and Control of an Active Fluid Vibration Isolator

10.1115/detc1995-0562 ◽

1995 ◽

Author(s):

Michael J. Panza ◽

Dennis P. McGuire ◽

Peter J. Jones

Keyword(s):

Electrical Power ◽

Fluid Pressure ◽

Space Representation ◽

Vibration Isolator ◽

Mass System ◽

Integral Control ◽

Active Fluid ◽

In Series ◽

Abstract An integrated mathematical model for the dynamics, actuation, and control of an active fluid/elastomeric tuned vibration isolator in a two mass system is presented. The derivation is based on the application of physical principles for mechanics, fluid continuity, and electromagnetic circuits. Improvement of the passive isolator performance is obtained with a feedback scheme consisting of a frequency shaped notch compensator in series with integral control of output acceleration and combined with proportional control of the fluid pressure in the isolator. The control is applied via an electromagnetic actuator for excitation of the fluid in the track connecting the two pressure chambers of the isolator. Closed loop system equations are transformed to a nondimensional state space representation and a key dimensionless parameter for isolator-actuator interaction is defined. A numerical example is presented to show the effect of actuator parameter selection on system damping, the performance improvement of the active over the passive isolator, the robustness of the control scheme to parameter variation, and the electrical power requirements for the actuator.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/dnaa037 ◽

2020 ◽

Author(s):

Ali Ketabdari ◽

Mohammad Hadi Farahi ◽

Sohrab Effati

Keyword(s):

Optimal Control ◽

Fractional Order ◽

Operational Matrix ◽

Two Dimensional ◽

Algebraic Equations ◽

Control Variables ◽

Bernstein Functions ◽

Abstract We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

LINEAR DYNAMIC MATHEMATICAL MODEL AND IDENTIFICATION OF MICRO TURBOJET ENGINE FOR TURBOFAN POWER RATIO CONTROL

Aviation ◽

10.3846/aviation.2019.11653 ◽

2019 ◽

Vol 23 (2) ◽

pp. 54-64 ◽

Cited By ~ 1

Author(s):

Khaoula Derbel ◽

Károly Beneda

Keyword(s):

Mathematical Model ◽

Power Plants ◽

Space Representation ◽

Nonlinear Behaviour ◽

Power Ratio ◽

Linear Quadratic ◽

Control Laws ◽

Turbofan Engines ◽

Micro turbojets can be used for propulsion of civilian and military aircraft, consequently their investigation and control is essential. Although these power plants exhibit nonlinear behaviour, their control can be based on linearized mathematical models in a narrow neighbourhood of a selected operating point and can be extended by using robust control laws like H∞ or Linear Quadratic Integrating (LQI). The primary aim of the present paper is to develop a novel parametric linear mathematical model based on state space representation for micro turbojet engines and the thrust parameter being Turbofan Power Ratio (TPR). This parameter is used by recent Rolls-Royce commercial turbofan engines but can be applied for single stream turbojet power plants as well, as it has been proven by the authors previously. An additional goal is to perform the identification for a particular type based on measurements of a real engine. This model has been found suitable for automatic control of the selected engine with respect of TPR, this has been validated by simulations conducted in MATLAB® Simulink® environment using acquired data from transient operational modes.

Research on a collocation approach and three metaheuristic techniques based on MVO, MFO, and WOA for optimal control of fractional differential equation

Journal of Vibration and Control ◽

10.1177/10775463211051447 ◽

2021 ◽

pp. 107754632110514

Author(s):

Asiyeh Ebrahimzadeh ◽

Raheleh Khanduzi ◽

Samaneh P A Beik ◽

Dumitru Baleanu

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Focal Point ◽

Control Problems ◽

Operational Matrices ◽

State Variables ◽

Caputo Fractional Derivatives ◽

Fractional Optimal Control Problems

Whale Optimization ◽

Exploiting a comprehensive mathematical model for a class of systems governed by fractional optimal control problems is the significant focal point of the current paper. The efficiency index is a function of both control and state variables and the dynamic control system relies on Caputo fractional derivatives. The attributes of Bernoulli polynomials and their operational matrices of fractional Riemann–Liouville integrations are applied to convert the optimization problem to the nonlinear programing problem. Executing multi-verse optimizer, moth-flame optimization, and whale optimization algorithm terminate to the most excellent solution of fractional optimal control problems. A study on the advantage and performance between these approaches is analyzed by some examples. Comprehensive analysis ascertains that moth-flame optimization significantly solves the example. Furthermore, the privilege and advantage of preference with its accuracy are numerically indicated. Finally, results demonstrate that the objective function value gained by moth-flame optimization in comparison with other algorithms effectively decreased.

State Space Representation of the Open-Loop Dynamics of the Space Shuttle Main Engine

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2896475 ◽

1991 ◽

Vol 113 (4) ◽

pp. 684-690 ◽

Cited By ~ 10

Author(s):

Ahmet Duyar ◽

Vasfi Eldem ◽

Walter C. Merrill ◽

Ten-Huei Guo

Keyword(s):

State Space ◽

Linear Models ◽

Space Shuttle ◽

Open Loop ◽

Space Representation ◽

Structure Estimation ◽

Loop Dynamics ◽

Main Engine ◽

A parameter and structure estimation technique for multivariable systems is used to obtain state space representation of open loop dynamics of the space shuttle main engine (SSME). The parametrization being used is both minimal and unique. The simplified linear models may be used for fault detection studies and control system design and development.