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The necessary and sufficient criterion for system stability is that all of the poles of the closed-loop transfer function lie in the left half of the complex j-plane; i.e., for the above example the requirement is

where Re($) denotes the real part of s.

The general relationship between the location of the poles of the transfer function and the system stability may be determined by considering the linear differential equation

where the linear operator A with constant coefficients a, is

and N(t) is a disturbance torque. Substituting a trial solution of the form

into Eq. (18-26) with N(l) = 0, we obtain the characteristic equation anu>n + an_ 1 + • ■ • + a,to + ao=0 (18-29)

The complementary solution to Eq. (18-26), i.e., /49c = 0, is

i= i where the n values of to, are the roots or zeros of the characteristic equation. Thus, a necessary condition for system stability (i.e.. for the lim 9c(t) to exist) is that, for all i, Re(«,)<0.

The question of stability has thus been reduced to investigating the characteristics of the roots of Eq. (18-29). A pure imaginary root, i.e., Re(io,) = 0, results in an undamped oscillatory component of the solution, while a root with a positive real part results in an exponentially increasing component of the solution. All of the roots of Eq. (18-29) have negative real parts if and only if the Routh-Hunvitz criteria [Korn and Korn, 1968] are satisfied. These are

2. Either all the even or all the odd 7), i < n, defined below, are positive.

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