Synthesis of a Fuzzy controller by a second-order object with delay

The paper proposes a method for using Fuzzy controller tools to synthesize optimal control of a second-order dynamic object with a delay. The idea is to construct a phase surface that combines optimal relay control away from the equilibrium state region and linear control in the region itself. This approach made it possible to avoid self-oscillations in a steady state while maintaining the properties of optimal control in terms of speed. The switching trajectory in phase space, corresponding to the solution of the optimal control problem according to the Maximum principle, is determined by the method of inverse time calculation of the second-order difference equation of the plant. The region near the equilibrium state, where the linear controller is used, is determined from the results of modeling the motion of a point in phase space with optimal control for an object with a delay. This region is represented by an ellipse that describes motion in phase space in a self-oscillatory mode. To further eliminate self-oscillations in this area, a linear controller is used, tuned by means of solving the variational optimal control problem. It is proposed to use a tool for the synthesis of Fuzzy controllers, where the switching surface and calculation of the control value can be set arbitrarily. As a result, a variable controller structure is proposed to combine these two approaches. The resulting Fuzzy controller model is represented by a standard FLS structure, which was implemented in Python on the Orange Pi embedded computer. To connect to the current control object, an industrial controller FX3U-24MR is used, connected to a computer via a Modbus network. Full-scale tests are presented on a hot water supply temperature control object, which corresponds as closely as possible to the object model under study. The method, idea and results obtained in the work can be applied and investigated in the synthesis of control of dynamic objects in sliding mode to solve current problems related to eliminating the unwanted Chattering effect.