2015
DOI: 10.1016/j.automatica.2015.07.005
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Abstract: a b s t r a c tThe Kalman conjecture is known to be true for third-order continuous-time systems. We show that it is false in general for second-order discrete-time systems by construction of counterexamples with stable periodic solutions. We discuss a class of second-order discrete-time systems for which it is true provided the nonlinearity is odd, but false in general. This has strong implications for the analysis of saturated systems.

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Cited by 69 publications
(46 citation statements)
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“…These difficulties in part are related to well-known Aizerman's and Kalman's conjectures on the global stability of nonlinear control systems, which are valid from the standpoint of simplified analysis by the linearization, harmonic balance, and describing function methods (note that all these methods are also widely used to the analysis of nonlinear oscillators used in VCO [12], [13]). However the counterexamples (multistable high-order nonlinear systems where the only equilibrium, which is stable, coexists with a hidden periodic oscillation) can be constructed to these conjectures [48], [53].…”
Section: G(t)mentioning
confidence: 99%
“…These difficulties in part are related to well-known Aizerman's and Kalman's conjectures on the global stability of nonlinear control systems, which are valid from the standpoint of simplified analysis by the linearization, harmonic balance, and describing function methods (note that all these methods are also widely used to the analysis of nonlinear oscillators used in VCO [12], [13]). However the counterexamples (multistable high-order nonlinear systems where the only equilibrium, which is stable, coexists with a hidden periodic oscillation) can be constructed to these conjectures [48], [53].…”
Section: G(t)mentioning
confidence: 99%
“…Now, the use of (44) in Lemma 4,(14), implies that Lemma 9(ii), (36), holds implying the existence of 0 , ( >…”
Section: Lemma 10 If and Are Sufficiently Small Thenmentioning
confidence: 99%
“…The property of asymptotic hyperstability generalizes that of absolute stability [13][14][15] which generalizes the most basic concept of stability of dynamic systems. See, for instance [2,3,11,[13][14][15][16][17][18][19][20][21][22][23][24][25][26], and references therein. It is well known that closed-loop hyperstability is, by nature, a powerful version of closed-loop stability since it refers to the stability of a hyperstable linear feed-forward plant (in the sense of the positive realness of the associated transfer matrix) under a wide class of feedback controllers applied.…”
Section: Introductionmentioning
confidence: 99%
“…A related property is that time-invariant dynamic linear systems which are externally positive, while they have positive real or strictly positive real transfer matrices, are, in addition, hyperstable or asymptotically hyperstable, that is, globally Lyapunov stable for any nonlinear and/or time-varying feedback device satisfying a Popov's-type inequality for all time [11,12]. Such a property of asymptotic hyperstability generalizes that of absolute stability [13][14][15], which generalizes the most basic concept of stability of dynamic systems. See, for instance, [13,14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29] and references therein.…”
Section: Introductionmentioning
confidence: 99%