Lazoriv N. Empirical modeling and synthesis of the automatic temperature control system for a muffle furnace

Muffle furnaces are small thermal objects that are used both for laboratory research and in industrial production of small parts. The technological mode of the muffle furnace includes three phases: heating a furnace to a given temperature, maintaining a temperature at a given level, and cooling a furnace. The furnace is powered by an electrical energy source to which heating elements made of materials with high ohmic resistance are connected. From a modeling perspective, a muffle furnace is an object with distributed parameters, which complicates obtaining mathematical models in "input‒output" terms. To simplify modeling, a muffle furnace is divided into zones, each of which is considered as an object with concentrated parameters. As a result, a nonlinear equations system is obtained, including a number of thermophysical parameters, for the determination of which additional experiments are required. After linearization of the obtained equations system and transition to the complex domain, an object’s structural diagram is obtained, based on which an automatic control system of the muffle furnace temperature regime is synthesized. Synthesizing an automatic control system becomes more complicated if a muffle furnace has two independent sources of electrical energy. In such furnaces, there are cross-connections between inputs and outputs, that complicates the creation of an adequate mathematical model and worsens the quality indicators of the control process. An urgent scientific task is to conduct experimental research aiming to build adequate mathematical models of the muffle furnace as an automatic control object and, on this basis, creating an automatic control system with improved dynamic properties, which will allow improving the quality of industrial products. Experimental studies aimed at obtaining acceleration characteristics for two channels of influence transmission from the input to the output of the muffle furnace. Obtained experimental data enabled construction of transient characteristics of a muffle furnace on four "input-output" signal transmission channels. Analysis showed that they have an aperiodic character and were approximated by transfer functions as the ratio of two polynomials of m and n degrees. Using the improved method of areas, algorithmic and software support for the synthesis of empirical models of a muffle furnace with two independent furnaces was developed. The criterion for models selection was the sum of ordinates deviation squares of the empirical model from the experimental data at the observation points. As a result, four transfer functions for each signal transmission channel with the same structures are obtained, with m=2 and n=3. The constructed structural diagram of the muffle furnace showed the presence of cross connections. To compensate for them, a cross-connection compensator is included in the control circuit. Based on the autonomy conditions, the compensator’s transfer function is obtained. The presence of the compensator in the direct control channel leads to the appearance of two single-loop control systems. Each of two independent circuits contains a controller with a PI or PID control algorithm. For PID and PI control algorithms, the settings parameters of the regulators were calculated based on the minimum value of the generalized quadratic criterion. The parameters of the muffle furnace transfer functions have errors caused by the inaccuracy of the experimental data and the chosen approximation method. To study the impact of inaccuracies of model parameters on the stability of control systems, model parameters were considered as fuzzy values with a triangular membership function, which was approximated by a Gaussian function. The transition from triangular to Gaussian membership functions made it possible to apply the rules of fuzzy arithmetic to the polynomials of the numerator and denominator of the transfer functions in order to obtain the expressions of the transfer functions taking into account the vagueness of their parameters. It is shown that with unclear parameters of empirical models, the amplitude stability margins remained unchanged, while the phase ones decreased slightly, which does not lead to a loss of stability of automatic control systems. Muffle furnace crosslink compensators with implemented on digital computing devices. The transition from continuous to discrete models was implemented by the Hankel method, which made it possible to simplify the process of modeling and calculating the transfer functions of discrete compensators. The results of simulation modeling showed a satisfactory compensation of the action of cross control channels, and the quality indicators of automatic control systems satisfy the technical requirements for automatic control systems of muffle furnaces