Absolute stability analysis for negative-imaginary systems

Tracking control for systems with slope‐restricted hysteresis nonlinearities

“…The nonlinearity (·) = diag(Φ 1 , … , Φ m ) belongs to the class as discussed in Section 3.1. The slope function is in diagonal form s I, we follow the similar exposition of Section 4, which leads to the closed-loop state Equation (14) to have the following form:…”

“…Alternatively, recent research efforts are focused on to develop control schemes directly based on the mathematical formulations of input‐output characteristics of the hysteresis . Interesting results using specific system properties such as dissipativity and negative imaginary can be found in the literature . Most of these results either only comment on boundedness of the states or do not provide any systematic design guideline.…”

“…[9][10][11][12] Interesting results using specific system properties such as dissipativity and negative imaginary can be found in the literature. [13][14][15][16] Most of these results either only comment on boundedness of the states or do not provide any systematic design guideline. Both clockwise and counterclockwise circulation of hysteresis characteristics are separately considered in the work of Ouyang and Jayawardhana 13 ; whereas Valadkhan et al 17 can only tackle counterclockwise hysteresis.…”

Comment on “Absolute stability analysis for negative-imaginary systems” [Automatica 67 (2016) 107–113]

“…The main result (Theorem 9) in (Dey et al, 2016) is a sufficient stability condition for strictly proper and strongly strict negative-imaginary (SSNI) systems in positive feedback with a diagonal, memoryless, slope-restricted nonlinearity. In classical language the result for single-input single-output (SISO) systems may be stated succinctly as "SSNI systems satisfy the Kalman conjecture with positive feedback"; Theorem 9 in (Dey et al, 2016) also provides the natural generalization of this statement to multivariable systems. Dey et al (2016) prove their result via a Lur'e-Postnikov type Lyapunov function, Popov multipliers and loop transformation.…”

“…In classical language the result for single-input single-output (SISO) systems may be stated succinctly as "SSNI systems satisfy the Kalman conjecture with positive feedback"; Theorem 9 in (Dey et al, 2016) also provides the natural generalization of this statement to multivariable systems. Dey et al (2016) prove their result via a Lur'e-Postnikov type Lyapunov function, Popov multipliers and loop transformation. In fact it can be shown via simple application of the Popov criterion.…”