rootlocusmenettelyä

Root locus menettelyä, often translated as root locus method, is a graphical technique used in control systems engineering to analyze the stability and transient response of a closed-loop system. It plots the locations of the closed-loop poles as a system parameter, typically the controller gain, is varied from zero to infinity. The path traced by these poles is called the root locus.

The method is based on the characteristic equation of the closed-loop system, which is generally of the form 1 + KG(s)H(s) = 0, where K is the adjustable parameter (gain), G(s) is the forward path transfer function, and H(s) is the feedback path transfer function. The roots of this equation are the closed-loop poles. By plotting these roots for all possible values of K, the root locus diagram reveals how the system's stability and response change with the gain.

Key features of a root locus diagram include asymptotes, breakaway and break-in points, and intersections with the imaginary axis. The number of asymptotes is determined by the difference between the number of open-loop poles and zeros. Breakaway points indicate where the locus leaves a real axis segment, and break-in points show where it enters a real axis segment. Intersections with the imaginary axis are critical as they often correspond to the onset of sustained oscillations and instability.

The root locus method provides valuable insights into system design. It allows engineers to select an appropriate gain K to achieve desired performance characteristics, such as stability margins and settling time, without needing to recalculate the closed-loop poles for every gain value. It is a fundamental tool in classical control theory.