The absolute stability of a nonlinear control system

The recent introduction of the Popov theorem concerning the stability of a nonlinear control has made it possible to obtain results not available from other techniques. A single loop system with an isolated nonlinearity is considered; the linear portion of the system is described by the transfer function G(s). Of particular interest is the Aizerman conjecture, i.e. the conjecture that if the linear system, obtained by replacing the nonlinearity with a simple gain element, is stable for all gain K satisfying k� [less than or equal to] K [less than or equal to] k₂, then the nonlinear system is absolutely stable for all single-valued nonlinearities φ satisfying k� [less than or equal to] φ(σ)/σ [less than or equal to] k₂ and certain other constraints. k� is taken as 0 for a type 0 system and ε for a type 1 or higher system. By the Popov technique the Aizerman conjecture is shown to be true for the following types of third order systems: 1. Type 1 -- all pole 2. Type 1 -- one zero 3. Type 0 -- one zero in the LHP[.] The Aizerman conjecture is shown to be invalid for the general system of order higher than three and with arbitrary coefficients. However, it is valid for many specific systems. A graphical technique based on root locus methods is developed for determining whether or not the Aizerman conjecture can be established for any given specific system by the Popov criterion. The technique exposed consists of the following steps: 1. Plot the root locus of 1 + kG(s) = 0 to determine the maximum gain, k[subscript max], permitted by the linear system and the value of frequency at which the root locus crosses the jω axis. 2. Determine q as the slope of the Popov curve at the point at which it crosses the negative real axis. 3. Plot the root locus of 1 + k(sq + l)G(s) = 0 to determine its roots when k = k[subscript max]. 4. Plot the root locus of KH(s) = ± j where H(s) = 1 + k(sq + l)G(s). If the root locus of step 4 does not cross into the right half of the s-plane, then the Aizerman conjecture is applicable for the subject system. In the case of the all pole system it was found that an extrapolation of the tangent to the root locus of G(s) at the point of intersection with the imaginary axis, intercepts the negative real axis at the point -1/q. The same result was noticed for an example system containing a zero of transmission but its validity as a general result for systems containing zeros has not been established