Fuzzy Optimal Control Problem Research Articles

Refinements the solutions of fuzzy optimal control problems

Refinements the solutions of fuzzy optimal control problems

Fuzzy approximation-based optimal consensus control for nonlinear multiagent systems via adaptive dynamic programming

This paper investigates the fuzzy approximation-based optimal consensus control problem for nonlinear multiagent systems with unknown perturbations. By constructing local error dynamics, the considered optimal consensus problem is reformulated as finding Nash-equilibrium solutions to zero-sum games. Then, by using sliding mode control technology and the concept of hierarchical design, a series of control signals are sequentially designed to regulate the consensus error and minimize the local value function. In addition, an identifier-critic architecture is developed by using generalized fuzzy hyperbolic models, where the identifier is employed to relax the requirement for complete system dynamics information, and the hierarchical sliding mode surface-based critic network is applied to approximate optimal control inputs. Finally, A simulation example is presented to illustrate the validity of the proposed approach.

Extremal Solutions for Surface Energy Minimization: Bicubically Blended Coons Patches

A Coons patch is characterized by a finite set of boundary curves, which are dependent on the choice of blending functions. For a bicubically blended Coons patch (BBCP), the Hermite cubic polynomials (interpolants) are used as blending functions. A BBCP comprises information about its four corner points, including the curvature represented by eight tangent vectors, as well as the twisting behavior determined by the four twist vectors at these corner points. The interior shape of the BBCP depends not only on the tangent vectors at the corner points but on the twist vectors as well. The alteration in the twist vectors at the corner points can change the interior shape of the BBCP even for the same arrangement of tangent vectors at these corner points. In this study, we aim to determine the optimal twist vectors that would make the surface an extremal of the minimal energy functional. To achieve this, we obtain the constraints on the optimal twist vectors (MPDs) of the BBCP for the specified corner points by computing the extremal of the Dirichlet and quasi-harmonic functionals over the entire surface with respect to the twist vectors. These twist vectors can then be used to construct various quasi-minimal and quasi-harmonic BBCPs by varying corner points and tangent vectors. The optimization techniques involve minimizing a functional subject to certain constraints. The methods used to optimize twist vectors of BBCPs can have potential applications in various fields. They can be applied to fuzzy optimal control problems, allowing us to find the solution of complex and uncertain systems with fuzzy constraints. They provide us an opportunity to incorporate symmetry considerations for the partial differential equations associated with minimal surface equations, an outcome of zero-mean curvature for such surfaces. By exploring and utilizing the underlying symmetries, the optimization strategies can be further enhanced in terms of robustness and adaptability.

Dynamic Analysis and Fuzzy Fixed-Time Optimal Synchronization Control of Unidirectionally Coupled FO Permanent Magnet Synchronous Generator System

This paper focuses on dynamic analysis and the fuzzy fixed-time optimal synchronization control problem of unidirectionally coupled fractional-order (FO) permanent magnet synchronous generator (PMSG) system. The synchronization model between FO master and slave PMSGs with capacitive and resistive couplings is built. The dynamic analysis fully reveals its abundant dynamical behaviors including chaotic oscillations and gives stability/instability boundaries with the designed numerical method. In controller design, the hierarchical type-2 fuzzy neural network (HT2FNN) with a transformation is designed to approximate unknown functions, the fixed-time command filter matched up with the compensating signal is proposed to achieve precise estimate and fast convergence, and a fixed-time preconfigured performance function integrated with a smooth and invertible function is built to realize fixed-time convergence and performance constraint. Then a fuzzy fixed-time optimal synchronization control scheme fusing with the HT2FNN, filter, performance function and optimal control is developed under the FO backstepping theory. The stability analysis proves that all signals of the closed-loop system are bounded along with the cost function being minimized. Finally, numerical simulation results verify the feasibility and advantages of our scheme.

Distributed Fuzzy Optimal Consensus Control of State-Constrained Nonlinear Strict-Feedback Systems.

This article investigates the distributed fuzzy optimal consensus control problem for state-constrained nonlinear strict-feedback systems under an identifier-actor-critic architecture. First, a fuzzy identifier is designed to approximate each agent's unknown nonlinear dynamics. Then, by defining multiple barrier-type local optimal performance indexes for each agent, the optimal virtual and actual control laws are obtained, where two fuzzy-logic systems working as the actor network and critic network are used to execute control behavior and evaluate control performance, respectively. It is proved that the proposed control protocol can drive all agents to reach consensus without violating state constraints, and make the local performance indexes reach the Nash equilibrium simultaneously. Simulation studies are given to verify the effectiveness of the developed fuzzy optimal consensus control approach.

On Atangana-Baleanu fuzzy-fractional optimal control problems

Optimal control is a very important field of study, not only in theory but in applications, and fractional optimal control is also a significant branch of research in theory and applications. Based on the concept of fuzzy process, a fuzzy fractional optimal control problem is presented. In this article, we derived the necessary and sufficient optimality conditions for a class of fuzzy-fractional optimal control problems (FFOCPs) with gH-Atangana-Baleanu fuzzy-fractional derivative expressed in Caputo sense. The main aim is to find the best possible control that minimizes the fuzzy performance index and satisfies the related ABC fuzzy-fractional dynamical systems. We also presented some examples for more illustration of the subject.

A numerical method for approximating the solution of fuzzy fractional optimal control problems in caputo sense using legendre functions

Fractional order differential equations accurately model dynamic systems and processes. In some of the fractional optimal control problems (FOCPs), due to the ambiguity in the initial conditions and the transfer of ambiguity to the solution, it is necessary to use fuzzy mathematics. In this paper, a numerical method is presented to approximate the solution for a class of Fuzzy Fractional Optimal Control Problems (FFOCPs) using the Legendre basis functions. The fuzzy fractional derivative is described in the Caputo sense. The performance index of an FFOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fuzzy Fractional Differential Equations (FFDEs). After obtaining Euler–Lagrange equations for FFOCPs and the necessary and sufficient conditions for the existence of solutions, using the definition of generalized Hukuhara differentiability (types I, II), the problem is considered in two cases. Then the distance function and an approach similar to the variational type along with the Lagrange multiplier method are used to formulate and solve the equations in a system. Time-invariant and time-varying examples are provided to assess the presented method. Numerical results show a similar trend for the state and control variables for various numbers of Legendre polynomials. Also, the convergence of state and control variables for the time-invariant system can be seen, and the same is true for control variables for the time-varying system.

An Application of Variational Minimization: Quasi-Harmonic Coons Patches

For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation ( p d e ), whose solution in general is the desired surface as the unknown function of surface parameters. The solution of this partial differential equation is known only for very few cases. Instead of solving the corresponding partial differential equation, we exploit an ansatz method (Ahmad et al. (2013), Ahmad et al. (2014) Ahmad et al. (2015)), used for Coons patches spanned by finite number of boundary curves for a quasi-minimal surface as the extremal of r m s of mean curvature. The ansatz method targets a slightly perturbed surface (the rational blending interpolants-based Coons patch, Hermite cubic polynomials interpolants-based Coons patch, and Ferguson surface (vanishing twist vectors) in our case) that comprises initially a nonminimal surface plus the product of a real parameter with a variational function of surface parameters (vanishes at the boundary curves along the unit normal to the slightly perturbed surface). The variational function can be deliberately chosen as the product of linear functions and the mean curvature of the initial nonminimal surface such that it is zero at the boundary curves and then replace this mean curvature by the mean curvature of the resulting surface to find a surface of reduced area, and the process can be repeated for further improvement. In this article, we extend the ansatz method (1) for the rational blending functions interpolants-based Coons patch with a parameter in their form for its different values and (2) for the blending functions comprising Hermite cubic polynomial interpolants-based Coons patch (bicubically blended Coons patch (BBCP) and the Ferguson surface). The ansatz method can be extended for the variational extremal of the surfaces for fuzzy optimal control problems (Filev et al. (1992), Emamizadeh (2005), Farhadinia (2011), Mustafa et al. (2021)).

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.

Discussion on Optimality Conditions of Fuzzy Optimal Control Problems

In this study, a novel methodology for detecting fuzzy optimal solutions of fuzzy optimal control problems governed by fuzzy differential equations is presented. For this purpose, first the concept of optimal solution is proposed by considering a partial ordering on fuzzy numbers space which is based on a new parametric representation for α-level set of fuzzy numbers. Using this partial ordering and parametric representation, the discussed fuzzy optimal control problems are reduced to general optimal control problems in parametric form. Then by Pontryagin’s principle, a candidate for the solution of the original problem is derived. The procedure of extracting candidate solution is illustrated for two experimental problems, which improve the recent results of Refs. 10 and 17.

Approximate solutions of fuzzy optimal control problems using sigmoid-weighted neural networks

Optimal control problem is one of the most challenging subjects in control theory. It has numerous applications in science and engineering. In this study, we are motivated to obtain the solution of fuzzy optimal control problems via universal approximation capability of a single-layer feedforward artificial neural network. First, we transform the fuzzy optimal control problems into systems of first-order ordinary differential equations via fuzzy Pontryagin’s minimum principle and fuzzy Hamiltonian function. Then, we solve these systems by using a single-layer feedforward sigmoid-weighted neural network. The numerical examples are presented to determine the simplicity and efficiency of the proposed method.

The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concept

In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler–Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM).

Adaptive fuzzy output feedback inverse optimal control for vehicle active suspension systems

In this paper, the adaptive fuzzy inverse optimal output feedback control problem for vehicular active suspension systems (ASSs) is addressed. The considered vehicular active suspension system (ASS) is a quarter-car model with electromagnetic actuators and contains unknown nonlinear characteristics. The fuzzy logic systems (FLSs) are used to approximate unknown nonlinearities, and an auxiliary system model is constructed. Based on auxiliary system model, a fuzzy state observer is designed to estimate immeasurable states. By using adaptive backstepping design technique and inverse optimal principle, an adaptive fuzzy output feedback inverse optimal strategy is proposed. The developed adaptive fuzzy inverse optimal output feedback control method not only guarantees that the attitude of the vehicle is stabilized, but also achieves inverse optimization in relation to the cost functional. Finally, the simulation studies are given in the cases of both bump road and sinusoid road displacement input, and the simulation results show the validity of the proposed control method.

Fuzzy optimal control for nonlinear systems with time-varying delay via sampled-data controller

This work is devoted to a fuzzy sampled-data optimal control problem for nonlinear systems with time-varying delay in Takagi–Sugeno fuzzy form. Based on the Lyapunov–Krasovskii functional theory, some novel stability criteria are established. A fuzzy sampled-data controller is developed to ensure that the fuzzy closed-loop sampled-data control system is asymptotically stable and the guaranteed cost performance is also minimized. It is shown that the obtained results are less conservative in the stability analysis. Two examples of the computer-simulated truck-trailer system and the continuous stirred tank reactor system are provided to show the effectiveness and the merits of the proposed design.

Observer-Based Fuzzy Adaptive Inverse Optimal Output Feedback Control for Uncertain Nonlinear Systems

In this article, an observer-based fuzzy adaptive inverse optimal output feedback control problem is studied for a class of nonlinear systems in strict-feedback form. The considered nonlinear systems contain unknown nonlinear dynamics and their states are not measured directly. Fuzzy logic systems are applied to identify the unknown nonlinear dynamics and an auxiliary nonlinear system is constructed. Based on this auxiliary system, a fuzzy state observer is first designed to estimate the immeasurable states. By using the inverse optimal principle and adaptive backstepping design theory, an observer-based fuzzy adaptive inverse optimal output feedback control scheme is then developed. The proposed inverse optimal control scheme need not assume that the states are measurable. It also guarantees that the closed-loop system is semiglobally uniformly ultimately bounded, and achieves the optimal control objective as well. Finally, two simulation examples are provided to check the validity of the presented control method.

Fuzzy Optimal Control Problem of Several Variables

The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

Adaptive fuzzy optimal control for a class of active suspension systems with full‐state constraints

In this study, an adaptive fuzzy inverse optimal control problem is investigated for a class of vehicle active suspension systems. Since active suspension systems have dynamic characteristics of complexities and spring non-linearities, the fuzzy logic systems are utilised to learn the unknown non-linear dynamics. In addition, there exist the constraints of the displacements of the sprung and unsprung masses, vertical vibration speeds, and current intensity in the considered suspension system, therefore, the Barrier Lyapunov functions are introduced into the control design to ensure that the full-state constraints are not overstepped. The inverse optimal control method is adopted by constructing an auxiliary system, which circumvents the assignment of solving a Hamilton–Jacobi–Bellman equation and brings about an inverse optimal controller associated with a meaningful objective functional. Based on Lyapunov stability theory and backstepping recursive design algorithm, a fuzzy adaptive optimal control scheme is developed. It is proved that the proposed control scheme not only guarantees that the vertical vibration of the vehicle is stabilised by the electromagnetic actuator but also achieves the goal of inverse optimality with regard to the cost functional. Finally, the simulation studies check the validity of the presented control strategy.

Fuzzy interval optimal control problem

This paper proposes the discrete-time fuzzy interval optimal control problem via single-level constrained fuzzy arithmetic. Fuzzy interval functions are established and allow us to evaluate the dynamics of the fuzzy interval difference equation. Moreover, fuzzy interval dynamic programming is developed to obtain the optimal fuzzy interval solution helping us to make a realistic decision. A numerical example illustrates the theory and provides the practical solution for an user.

Fuzzy Bang-Bang control problem under granular differentiability

In this paper, a class of fuzzy optimal control problem called fuzzy Bang-Bang (FBB) control problem is revisited. The FBB control problem aims at finding control inputs which transfer the states of an uncertain dynamical system to the origin in a minimum time. Since the conditions and/or parameters of the dynamical system are uncertain, the FBB control problem is a challenge task. To address the problem, we first introduce the concepts of the granular integral and derivative of fuzzy functions whose domain is uncertain. In addition, the notion of granular partial derivative of a multivariable fuzzy function whose variables are fuzzy functions with uncertain domains is presented. Then, we propose a theorem which is proved to be applicable to the FBB control problem. Moreover, taking the relative-distance-measure fuzzy interval arithmetic and horizontal membership functions into consideration, we further give complementary theorems to ensure that if the problem has a solution, then the controller assumes its boundary values. The simulated results confirm this theoretical conclusion. These findings may enrich our insight of the behavior of the uncertain dynamical system subjected to the FBB control problem, and guide to predict the uncertain trajectories.

Observer-Based Adaptive Fuzzy Fault-Tolerant Optimal Control for SISO Nonlinear Systems.

This paper investigates adaptive fuzzy output feedback fault-tolerant optimal control problem for a class of single-input and single-output nonlinear systems in strict feedback form. The considered nonlinear systems contain unknown nonaffine nonlinear faults and unmeasured states. Fuzzy logic systems are used to approximate cost function and unknown nonlinear functions, respectively. It is assumed that the states of the systems to be controlled are unmeasurable, thus an adaptive state observer is developed. To solve the nonaffine nonlinear fault control design problem, filtered signals are introduced into the adaptive backstepping control design procedures, and in the framework of adaptive critic technique and fault-tolerant control technique, a novel adaptive fuzzy fault-tolerant optimal control scheme is developed. The stability of the closed-loop system is proved by using Lyapunov stability theory. The simulation results verify the effectiveness of the proposed control strategy.