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Fig. 8. Root locus plots (see Franklin et al., 1994) of the transfer function of G(s) given by Eqn 7 for five characteristic values of the dimensionless constant, . Each plot depicts the locus of poles (roots of the denominator) of the closed-loop system (Fig. 2) under P-control. The three open-loop poles (roots of the denominator of Eqn 7) are indicated by crosses, and therefore there are three branches of the root locus (magenta, green, blue). There is an open-loop zero (root of the numerator of Eqn 7) at –1, indicated by a circle. The small inset plot (d vs t) for each root locus depicts a typical response of the hypothetical closed-loop system. For stability, all of the poles of the closed-loop system must be in the open left-half-plane, that is, they must have negative real parts. (A) For <1/9, all of the poles are in the left-half-plane; the inset shows an over-damped response of dvst. (B) For =1/9, the system would be critically damped with K_P=3. (C) For 1/9<<1, the system would be underdamped under P-control. (D) For =1, the system would be oscillatory for all choices in gain, K_P. (E) For >1, the system would be unstable. Since approaches or exceeds 1 for behaviorally relevant running speeds (Eqn 22), these graphs preclude the possibility of P-control. Stability can be greatly improved by adding a derivative feedback term, as in Eqn 11, enabling larger values of . Imag., imaginary.

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