Robust Stability Analysis of a Linear Time-Periodic Active Helicopter Rotor

Abstract

Robust stability of a helicopter hingeless rotor model used for air resonance alleviation is investigated in this paper. This aeroelastic phenomenon can be described naturally considering the rotor as a whole through a time-periodic change of coordinates called multiblade coordinates transformation. To assess robust stability of the resulting continuous linear time-periodic system, this paper defines a set of uncertainties specific to the helicopter rotor problem. The uncertain system is written in the form of a linear fractional representation and transformed using a frequency-lifting technique. This leads to a time-invariant representation of the uncertain system, the so-called harmonic transfer function. The extension of the standard -analysis to such lifted systems is proposed in the paper. Compared with a similar approach from literature based on an elementary example, the technique requires a moderate increase of the number of inputs and outputs, thus reducing the numerical effort involved in the computation of the -bounds. The methodology, applied to the uncertain closed-loop helicopter rotor, shows robust stability for the defined set of uncertainty. The results are compared to an analysis performed on the base of a linear time-invariant system to quantify how strongly robust stability bounds are underestimated when the periodicity is not accounted for. Nomenclature A, B, C, D = system, control, measurement, and feedthrough matrices of a linear time-periodic system A, B, C, D = Toeplitz form of time-periodic matrices A, B, C, and D G = harmonic transfer function Y GU G ibc = nominal single input/single output individual blade control model used in the weight definitioñ G i max ibc = admissible single input/single output system allowing the maximal variation from nominal model G ibc G k o ;k i = truncation of the harmonic transfer function G to the k i th input's and k o th output's harmonics G nom = nominal plant with additional inputs/outputs for the parametric uncertainty fG n g n2Z = elements of the harmonic transfer function G H mix = uncertain closed-loop augmented plant in -structure M = blade lagging moments defined in multiblade coordinates, Nm s = complex frequency T = fundamental period of a linear time-periodic system, s Tt = multiblade coordinates transformation from multiblade coordinates to individual blade control u, x, y = input, state, and output signal vectors of a state-space representation U, X , Y = frequency-lifted form of vectors u, x, and y u ibc = trailing-edge flaps command transformed in individual blade control u ibc Tt u unc = multiplicative input of the lumped uncertainty, defined in multiblade coordinates u 1 , u 2 = augmented plant inputs, used for the parametric uncertainty u 3 = multiplicative input of the lumped uncertainty, defined in individual blade control V, W = frequency-lifted form of the uncertainty output and input vectors v and w v, w = uncertainty output and input vectors of a linear fractional representation W = single input/single output individual blade con...

Efficient Calculation of Stabilization Parameters in RF Power Amplifiers

“…The HTF formalism, introduced by Wereley and Hall in [16], is a general formulation that serves to analytically relate the sidebands of an input/output representation of a PLTV system. The HTF(s) has the form of a conversion matrix and has widely proven its applicability for studying periodic systems of all types, from helicopter rotors to wind turbines [17]- [19].…”