Fractional-order optimization techniques for dynamic performance improvement in nonlinear control applications

Abstract

Operating under uncertain environmental conditions, nonlinear dynamic systems pose optimization challenges related to complicated interrelationships among conflicting objectives, disturbances, and nonlinear behavior. In this paper, a fractional-order minimax optimization approach combining fractional calculus operators and principles of robust optimization is proposed to enhance dynamic performance and reliability of solutions. A fractional-order differential operator is used within the proposed optimization formulation to include memory-dependent features. An optimization problem is set up using a minimax optimization criterion to optimize the worst-case value of the performance index. A fractional gradient-based iterative optimization operator is introduced to ensure rapid convergence. Mathematical properties of the proposed optimization framework are studied, including the existence of an optimum solution, conditions for boundedness, and guarantees of convergence of the optimization sequence. Efficiency of the proposed approach is illustrated on a nonlinear control system by means of comparison with traditional PID and integer-order optimization techniques. The obtained simulation results show improvement in terms of settling time, reduction in overshoot, and root mean square error values, proving the efficiency of fractional-order memory effects in optimizing dynamic performance. The proposed framework provides a solid mathematical basis for dealing with uncertain nonlinear optimization problems and can be extended to stochastic, distributed, and infinite-dimensional fractional optimization problems.