Root locus

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot). In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot). A graphical method that uses a special protractor called a 'Spirule' was once used to determine angles and draw the root loci. In addition to determining the stability of the system, the root locus can be used to design the damping ratio (ζ) and natural frequency (ωn) of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, a gain K can be calculated and implemented in the controller. More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead, PI, PD and PID controllers can be designed approximately with this technique. The definition of the damping ratio and natural frequency presumes that the overall feedback system is well approximated by a second order system; i.e. the system has a dominant pair of poles. This is often not the case, so it is good practice to simulate the final design to check if the project goals are satisfied. The root locus of a feedback system is the graphical representation in the complex s-plane of the possible locations of its closed-loop poles for varying values of a certain system parameter. The points that are part of the root locus satisfy the angle condition. The value of the parameter for a certain point of the root locus can be obtained using the magnitude condition. Suppose there is a feedback system with input signal X ( s ) {displaystyle X(s)} and output signal Y ( s ) {displaystyle Y(s)} . The forward path transfer function is G ( s ) {displaystyle G(s)} ; the feedback path transfer function is H ( s ) {displaystyle H(s)} . For this system, the closed-loop transfer function is given by Thus, the closed-loop poles of the closed-loop transfer function are the roots of the characteristic equation 1 + G ( s ) H ( s ) = 0 {displaystyle 1+G(s)H(s)=0} . The roots of this equation may be found wherever G ( s ) H ( s ) = − 1 {displaystyle G(s)H(s)=-1} . In systems without pure delay, the product G ( s ) H ( s ) {displaystyle G(s)H(s)} is a rational polynomial function and may be expressed as