Determining configuration parameters for proportionally integrated differentiating controllers by arranging the poles of the transfer function on the complex plane
DOI:
https://doi.org/10.15587/1729-4061.2021.242869Keywords:
automatic control, PID controller, system of equations, configuration parameters, process quality indicatorsAbstract
This paper reports a solution to the problem of determining the configuration parameters of PID controllers when arranging the poles of the transfer function of a linear single-circuit automated control system for a predefined set of control objects.
Unlike known methods in which the task to find the optimal settings of a PID controller is formed as a problem of nonlinear programming, in this work a similar problem is reduced to solving a system of linear algebraic equations.
The method devised is based on the generalized Viète theorem, which establishes the relationship between the parameters and roots of the characteristic equation of the automatic control system. It is shown that for control objects with transfer functions of the first and second orders, the problem of determining the configuration parameters of PID controllers has an unambiguous solution. For control objects with transfer functions of the third and higher orders, the generated problem is reduced to solving the redefined system of linear algebraic equations that has an unambiguous solution when the Rouché–Capelli theorem condition is met.
Such a condition can be met by arranging one of the roots of the characteristic equation of the system on a complex plane. At the same time, the requirements for the qualitative indicators of the system would not always be met. Therefore, alternative techniques have been proposed for determining the configuration parameters of PID controllers. The first of these defines configuration parameters as a pseudo solution to the redefined system of linear algebraic equations while the second produces a solution for which the value of the maximum residual for the system of equations is minimal.
For each case, which was used to determine the settings of PID controllers, such indicators of the control process as overshooting and control time have been determined
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