Uncertain Fractional-Order Multi-Objective Optimization Based on Reliability Analysis and Application to Fractional-Order Circuit with Caputo Type

Reliability analysis is critical to the study of uncertain dynamic systems under noise excitation. This paper focuses on the uncertain fractional-order multi-objective optimization model and its application. Firstly, based on the infinite time-based expected utility criterion, we propose two innovative first-hitting criteria, namely the reliability index criterion and the reliability lifetime criterion. Secondly, by replacing the ordinary derivative with the Caputo fractional-order derivative and introducing uncertain variables to the system, we develop an uncertain fractional-order differential equations to better model the actual electronic system. Subsequently, combining two proposed first-hitting criteria, we present a novel uncertain fractional-order multi-objective optimization problem, which is then transformed into the corresponding single-objective crisp optimization model. Lastly, the proposed model is applied to a fractional-order resistor–capacitance circuit model, where the analytic expression of the uncertainty distribution for the first-hitting time and the sufficient condition for the optimal solution are derived for the order of 1. Also, we give numerical examples to illustrate how our optimal solution fluctuates under various constraints.