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Verification of Aizerman's conjecture for a class of third-order systems

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2 Author(s)
Bergen, A.R. ; University of California, Berkeley, CA, USA ; Williams, I.

The second method of Lyapunov is used to validate Aizerman's conjecture for the class of third-order nonlinear control systems described by the following differential equation:tdot{e} + a_{2}ddot{e} + a_{1}dot{e} + a_{0}e + f(e)=0In this case, the stability of the nonlinear system may be inferred by considering an associated linear system in which the nonlinear functionf(e)is replaced byke. If the linear system is asymptotically stable fork_{1} < k < k_{2}, then the nonlinear system will be asymptotically stable in-the-large for anyf(e)for whichk_{1} < frac{f(e)}{e} < k_{2}.The Lyapunov function used to prove this result is determined in a straightforward manner by considering the physical behavior of the system at the extreme points of the allowable range ofk.

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Automatic Control, IRE Transactions on  (Volume:7 ,  Issue: 3 )