Model-based control of the flying helicopter simulator: evaluating and optimizing the feedback controller

The numerical evaluation and optimization of the feedback controller parameters of the model-based control implemented in the flying helicopter simulator is subject of this paper. The German Aerospace Center operates this helicopter as a flying testbed for numerous applications, e.g., pilot assistance and in-flight simulation. Initially, the elements of the model-based control are presented. A genetic algorithm and the Nelder–Mead simplex method used for optimization are described. Two simple objective functions to rate parameter sets in the time domain are presented, and a Simulink^® model of the helicopter dynamics and the controller structure are used to find optimized sets. The first function, called “Delta Rating”, consists of a normalized integral of the absolute error between commanded and measured states. The second function incorporates the Delta Rating, but is enhanced by a penalty on overshoots. The controllers found are further evaluated using a frequency domain approach consisting of a weighted sum of the differences in amplitude and phase, also considering the coherence at the corresponding frequency. Apart from the Simulink^® model, a ground-based simulator is used to evaluate the standard and the optimized controllers.

AC, RC, TC:

Attitude, rate, translational rate command

DLR:

Deutsches Zentrum für Luft- und Raumfahrt/German Aerospace Center

DR:

Delta Rating

FB, FF:

Feedback, feedforward controller

FHS:

Flying helicopter simulator

IAE:

Integrated absolute error

MBCS:

Model-based control system

MIMO:

Multiple input multiple output

SISO:

Single input single output

A, B, C, D :

State-space representation matrices (standard form)

β :

Sideslip angle [°]

C(s):

Transfer function matrix of feedback controller

d :

Disturbance vector

δ _p :

Pilot inputs (% deflection)

δ _lon, δ _lat :

Longitudinal, lateral cyclic pilot control (% deflection)

δ _ped, δ _col :

Pedal and collective pilot control (% deflection)

G(s):

Transfer function matrix

I :

Identity matrix

J :

Objective function

J _ave :

Objective function in frequency domain, weighting amplitude and phase difference

J _DR, J _DROv :

Delta Rating objective function in time domain, without and with additional overshoot criterion

K :

Gain

k :

Vector of feedback parameters

L :

Derivatives of the identified model

M _c(s):

Command model transfer function matrix

P :

Pole of a transfer function

P(s), P _M(s):

Real, identified helicopter transfer function matrices

p, q, r :

Roll, pitch, and yaw rate (rad/s)

u, v, w :

Longitudinal, lateral, and vertical airspeed component, aircraft-fixed system (m/s)

u _FF, u _FB :

Vectors of actuator inputs to helicopter from feedforward and feedback controller (rad)

v _d :

Vertical airspeed component, earth-fixed system (m/s)

V _TAS :

True airspeed (m/s)

W :

Weighting factor

ΔX _IAE :

Vector of absolute integrated errors

x _c, x _m :

Vectors of commanded, measured states

Δx :

Vector of differences between commanded and measured states

\( \gamma_{xy}^{2} \) :

Coherence

Φ, Θ, Ψ:

Euler roll, pitch, and yaw angles (rad)

φ :

Transfer function phase (°)

z :

Height, earth-fixed system (m)

ω _n, ζ :

Natural frequency and damping

Authors and Affiliations

1. German Aerospace Center, DLR, Institute of Flight Systems, Lilienthalplatz 7, 38108, Braunschweig, Germany

   Johannes Hofmann & Antje Dittmer

Corresponding author

Correspondence to Johannes Hofmann.

This paper is based on a presentation at the German Aerospace Congress, September 27–29, 2011, Bremen, Germany.

Reprints and permissions