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Conclusion and outlooks
Published in Dževad Belkić, Karen Belkić, Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications, 2010
We significantly expand and generalize the concept of Froissart doublets as the main part of the comprehensive strategy of signal-noise separation, which can be used in the whole field of signal processing as the most reliable method to date for disentangling genuine from spurious information. In system theory as well as in control theory from engineering, stability of the system’s parameters is the prerequisite for the system’s overall performance. Both zeros and poles of response functions play their crucial roles in the sought stability of systems. Not unexpectedly, the Padé approximant has been firmly established over the years as the optimal response function in both system and control theory. This fully coheres with the like experience from quantum physics as emphasized throughout this book and mentioned in the outlined example (i) and (ii). Moreover, Proissart doublets are the proof that zeros of the Padé response function are critical to finding stability of the examined system. Genuine zeros remain in the final output of the performed data analysis as stable structures of the studied system, while spurious ones as unstable Froissart zeros are washed out by being canceled by the corresponding spurious poles at the coincident positions in the complex-valued frequency spectra.
Digital Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
As it has been seen, the transfer function is a rational function in either q or z. In any case, the denominator of this transfer function when equated to zero forms the system characteristic equation. This characteristic function plays the same role as was discussed for continuous-time systems, where it is used in the stability analysis of the respective systems. Stability analysis for discrete-time systems is given in the next sections. The roots of the characteristic equation are also known as the poles of the system, while the roots of the numerator of the transfer function are known as the zeros of the system. Both the zeros and poles play an important role in the determination of the system response.
Laplace Transform
Published in Nassir H. Sabah, Electric Circuits and Signals, 2017
where K = am /bn. The values –z1, –z2, ..., –zm are zeros of F(s), because F(s) = 0 when s assumes any of these values. The values –p1, –p2, ..., –pn are poles of F(s), because F(s) → ∞ when s assumes any of these values. In general, zeros and poles may be complex, in which case they must occur in complex conjugate pairs, because the a′s and b′s are real in the LTs of physical systems.
Advanced operation and monitoring the economic performance of ammonia production based on natural gas steam reforming by using programmed feedforward–Ratio–Cascade controllers
Published in Chemical Engineering Communications, 2022
Determination procedure of three single-input and single-output (SISO) decoupled models in the form of plant model transfer functions used the list of CVs and MVs in Table 2. Utilization of data-driven modeling served for development of the feedforward, ratio, cascade, and gain-scheduled APC structure. The MATLAB System Identification Toolbox uses the transient response for transfer functions determination. According to the transient response of the ammonia production rate on a step change of the typical process disturbance variables given in Table 9 (included in Supplementary Material file) the first-order transfer functions (control, process, and disturbance transfer functions) were determined from the model to describe their dynamic response. Using closed-loop simulation, the parameters of the SISO model were identified. Then, the SISO model is converted to a discrete-time state space model for use in the APC controller design. The principal formulation of the SISO model and determination of the transfer functions was the same as in the previous work of Zecevic and Bolf (2020). Table 5 shows controller transfer function expressions with the minimum fit to estimation data of 95% and the same are included in Supplementary Material file. ATV technique by Honeywell was used due to possibility to suppress disturbances by adjustment of the auto tuner parameters. ATV technique was used for preliminary determination of the ultimate period and the ultimate gain from which the starting proportional, integral, and derivative gains were obtained. A small limit-cycle disturbance was set up between the controller output and the controlled variables, such that whenever the process variable crosses the set point, the controller output is changed. The ATV tuning method is as follows: determine a reasonable value for the valve change (h represent this value) then move the valve for +h%, wait until the process variable starts moving, then move valve for −2h%, and when the PV crosses the set point, move the OP for +2h%. The following auto tuner parameters were used: α (ratio Ti/Td) = 4.50, β (gain ratio) = 0.25, φ (phase angle) = 60.0, h (relay hysteresis) = 0.1%, and d (relay amplitude) = 5.00%. After preliminary definition of the controller’s parameters by the ATV technique, the MATLAB Control System Designer served for additionally tuning to get optimal performance and response of APC structure for all three PID controllers. This tuning procedure delivers results according to the desired system performance requirements by zeros and poles location adjustment on the real and imaginary axes in a root locus editor. The system performance requirements for primary/master PID controller are: