Various methods and techniques for the TDM spectrum analysis and its pole loci estimation exist see a survey by Pekař and Gao 15. The dominant (i.e., usually the rightmost) subset of poles can match the pole loci of a high order system however, one must be careful about other (uncontrolled) TDM poles, especially of high-frequency ones. The infinite nature of the TDM spectrum yields its advantages and disadvantages when estimating the actual system dynamics. The pole loci most significantly determine the dynamic and stability properties of the model 14. All infinitely many solutions of the CE constitute the TDM spectrum of characteristic values (or poles). However, these models are infinite-dimensional because of the transcendental form of the characteristic equation (CE) 13. Even a simple TDM can express the dynamics of a high-order non-delay model 11, 12 with sufficient accuracy for control design. TDMs have the form of functional differential equations, or more specifically, delay differential equations (DDEs), instead of PDEs. On the other hand, time-delay models (TDMs) may be very good estimators of some systems and processes dynamics, even if any significant physical delay is not supposed to appear in the process. However, such an approach can be unreasoning as the solution of partial differential equations (PDEs)-representing the reign of many industrial process models-often results in functions with lumped and distributed delays 8, 9, 10. As complex systems include internal feedback loops, internal delays must be considered along with the input–output ones nevertheless, internal delays are often ignored when process modeling. In modern discrete-time control systems, delays also arise from the human–machine interaction and signal sampling and processing 7. Delays appear mainly due to mass, energy, and data transportation in the process and network interconnections, and their existence is closely related to distributed parameter systems. It is a well-known fact that dozens of industrial processes, including chemical ones, as well as social, economic, and other everyday systems are affected by latencies and delays 1, 2, 3, 4, 5, 6. Although the obtained unordinary time and frequency domain responses may yield satisfactory results for control tasks, the identified model parameters may not reflect the actual values of process physical quantities. The second step attempts to estimate the remaining model parameters by various numerical optimization techniques and also to enhance all model parameters via the Autotune Variation Plus relay experiment for comparison. The benefit of this technique is that multiple model parameters can be estimated under a single relay test. Then, this result is transformed to the estimation of the initial characteristic equation parameters of the complex infinite-dimensional heat-exchanger model using the exact dominant-pole-loci assignment. The first one adopts the simple model with a low computational effort using the saturated relay that provides a more accurate estimation than the standard on/off test. The identification procedure has two substantial steps. Processes of this type and construction are widely used in industry. The second part intends to apply the simple infinite-dimensional model for relay-based parameter identification of a more complex model of a heating–cooling process with heat exchangers. The derived spectrum constitutes an infinite set, making it a suitable and simple-enough representative of even high-order process dynamics. The first part aims at the rigorous and complete analysis of pole loci of a simple delayed model, the characteristic function of which is represented by a quasi-polynomial with a non-delay and a delay parameter. The focus of this contribution is twofold.
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