The HHC Algorithm for Helicopter Vibration Reduction Revisited

The Higher-Harmonic Control (HHC) algorithm is re-examined from a rigorous control theory-oriented perspective. A brief review of the history and developments of HHC is given, followed by a careful development of the algorithm. The paper proceeds to perform an analytic convergence and robustness analysis. On-line identification with the adaptive variant of the algorithm is also addressed. A new version of the algorithm, relaxed HHC, is also introduced and shown to have beneficial robustness properties. Some numerical results comparing these variants of the HHC algorithm applied to helicopter vibration reduction are also presented. The results presented in this paper unify and extend previous work on the higher-harmonic control algorithm. Nomenclature 0n Zero matrix of size n× n A Matrix relating plant input and output between updates A Adjoint matrix of A, (Ā) D Matrix defined to be TQT + R Ff Exponential window In Identity matrix of size n× n J Quadratic-form cost function JT̂,∞ Converged cost function for adaptive control JT,α,∞ Converged cost function for relaxed control Kk+1 Matrix defined to be ∆uk+1Pk+1 k Integer index between control inputs m Number of harmonics in control input M Matrix defined to be D−1(TTQ + S) Nb Number of rotor blades Pk Matrix defined as (∆Uk∆Uk ) −1 ∗Ph.D. Candidate, Student Member AIAA. †Professor, Member AIAA. ‡Francois-Xavier Bagnoud Professor, Fellow AIAA. 1 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference 19 22 April 2004, Palm Springs, California AIAA 2004-1948 Copyright © 2004 by D. Patt, L. Liu, J. Chandrasekar, D. S. Bernstein and P. P. Friedmann. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. p Number of harmonics in measured output Q Weighting matrix for plant output in cost function of size 2p× 2p R Positive definite weighting matrix for control input in cost function of size 2m× 2m S Cross-weighting term in objective function J T Sensitivity matrix relating control input to plant output T̂ Estimate of the sensitivity matrix T T̂LS Least-squares estimate of the sensitivity matrix T tk Time of controller update uk Control input at time tk, vector of length 2m uk,opt The optimal control law W Matrix relating plant response to disturbance w Disturbance to plant zk Plant output at time tk, vector of length 2p z0 Initial output condition of plant output α Relaxation factor γf Exponential window factor εk+1 Vector defined to be ∆zk+1 − T̂LSk∆uk+1 (λA)i The i eigenvalue, λi, of matrix A λαi αλi + 1− α Λmax Maximum real part of eigenvalues Λmin Minimum real part of eigenvalues σi The real part of the eigenvalue λi ιi The imaginary part of the eigenvalue λi ∆Uk Matrix composed of relative parameters ∆uk ∆uk Relative parameter uk − uk−1 ∆Zk Matrix composed of relative parameters ∆zk ∆zk Relative parameter zk − zk−1 Γs Matrix defined to be (I− M̂∆T)−1 τ Time interval between control updates μ Helicopter advance ratio /rev Frequency of rotor revolution rank Matrix rank spec(A) Spectrum of A, spec(A) = {λA1 , . . . , λAn} ρs(A) Spectral radius of A, max 1≤i≤n |λAi | = sprad(A) = ρs(A)