In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a linear time-invariant (LTI) 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). Evans also invented in 1948 an analog computer to compute root loci, called a "Spirule" (after "spiral" and "slide rule"); it found wide use before the advent of digital computers.
Uses
If a pole sits at a location s = σ + j ω {\displaystyle s=\sigma +j\omega } , then the system contains a mode y m o d e ( t ) ∝ e σ t cos ( ω t + ϕ ) {\displaystyle y_{mode}(t)\propto e^{\sigma t}\cos {(\omega t+\phi )}} with growth rate σ and phase offset φ at the oscillation frequency ω. All modes must have negative growth rates for the system to be asymptotically stable. This stability condition is often phrased as all poles needing to lie in the left hand plane, i.e. σ < 0 {\displaystyle \sigma <0} . For stable modes, the (positive) decay rate is then -σ. When single complex pairs of poles lie on the imaginary axis σ = 0 {\displaystyle \sigma =0} (excluding the origin), the system is considered to be marginally stable. 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 (with angle cos − 1 ( ζ ) {\displaystyle \cos ^{-1}(\zeta )} from the negative real axis), and lines of constant natural frequency can be drawn as circles centered at the origin with radius ωn. 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, lead, lag, PI, PD and PID controllers can be designed approximately with this technique. The definition of the damping ratio and natural frequency in the paragraph above presumes that the overall feedback system is stable and well approximated by a second-order system. This happens when the system satisfies the "dominant poles" approximation. A complex pair of poles dominates when every other pole lies sufficiently farther left, e.g. | σ | > 5 | σ d o m | {\displaystyle |\sigma |>5|\sigma _{dom}|} . Equivalently, the dominant poles are the one with the smallest decay rate − σ d o m {\displaystyle -\sigma _{dom}} . A single pole on the real axis might also dominate, in which case the system can be approximated as a first-order system. Note that the factor of 5 is a common heuristic rather than a rule, derived from e − 5 < 1 % {\displaystyle e^{-5}<1\%} . Additionally, nearby zeros may weaken the effect of poles. So all controllers designed with this approximation should be simulated with the full transfer function to verify that the design goals are satisfied.
Definition The root locus of a LTI 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 LTI 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
T ( s ) = Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle T(s)={\frac {Y(s)}{X(s)}}={\frac {G(s)}{1+G(s)H(s)}}}
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