Linear System SURVEY

Supplement for Flight Dynamics, Second Edition,
Robert F. Stengel

SURVEY.m is a MATLAB program that analyses linear, time-invariant (LTI) response for the stability and control-effect matrices computed inFLIGHTv2.m. A mathematical model based on the trim condition is stored in FmodelFile.mat and GmodelFile.mat. The SURVEY.m script provides the option of reducing these matrices into smaller models using the function LonLatDir.m. The function produces a single model at each call for either longitudinal or lateral-directional motions, in body or hybrid/stability axes, and with four or six elements in the state vector. The program is based on material found in Flight Dynamics, Second Edition, Princeton University Press, 2022.

SURVEY.m calculates initial-condition response, equilibrium response, control response, controllability and observability matrices, natural frequencies, damping ratios, real roots, eigenvalues and eigenvectors for the open-loop system. It forms single-input/single-output transfer functions for all possible combinations in the model selected by the user and makes their Bode, Nyquist, and Nichols plots; with a simple change, these plots can be generated for disturbance inputs as well. A transfer function model of human-pilot neuromuscular lags can be included in control responses. SURVEY.m is tutorial, and the USER controls program flow by making changes in the code. No explicit or implicit warranties are made regarding the accuracy or correctness of the computer code. The code is subject to continuing change, and user suggestions for improving the code are welcome.

Main Program for Analysis and Simulation (SURVEY)

TABLE OF CONTENTS

1. USER INPUT
   1. ANALYSIS FLAGS
   2. SELECTION FLAGS
2. NOMINAL FLIGHT CONDITION [Input from FmodelFile.mat]
3. Specification of SYSTEM MODEL DIMENSION AND AXES
4. NATURAL FREQUENCIES, DAMPING RATIOS, and REAL ROOTS using NatFreq.m
5. EIGENVALUES and EIGENVECTORS using StabMode.m
6. INITIAL CONDITION RESPONSE
7. SINGLE-INPUT/SINGLE-OUTPUT (SISO) ANALYSES
   1. CONTROLLABILITY and OBSERVABILITY with Single Input or Output
   2. Response to Control STEP INPUT
   3. Response to CONTROL IMPULSE
   4. Response to I.C. and CONTROL PROFILE
   5. EQUILIBRIUM RESPONSE to Control Perturbations
8. TRANSFER FUNCTION ANALYSES
   1. PILOT Transfer Function
   2. AIRCRAFT Transfer Functions
   3. BODE Plots
   4. NYQUIST Plots
   5. NICHOLS Charts
   6. ROOT LOCUS Plots
9. NOMENCLATURE

SURVEY.m makes calculations based on the (12 x 12) Fmodel, (12 x 7) Gmodel, and (12 x 3) Lmodel matrices that are stored in FmodelFile.mat and GmodelFile.mat by FLIGHTv2.m. The nominal trimmed flight condition also is stored in FmodelFile.mat. The chosen model is summarized in ControlData.mat. Examples of FmodelFile.mat, GmodelFile.mat, and ControlData.mat are included.

Supporting Functions

Reduced-Order Models (LonLatDir)

LonLatDir.m accepts stability and control-effect matrices for longitudinal and lateral-directional dynamics. It either passes the matrices unchanged, re-orders elements of the matrices, or generates a selected reduced-order model from them. It presents matrices for body or hybrid/stability axes, for longitudinal or lateral-directional partitions, and for fourth- or sixth-order computations. As shown, there are twelve original state elements, four longitudinal controls, three lateral-directional controls, and three disturbances.

Natural Frequency (NatFreq)

NatFreq.m presents complex-valued eigenvalues, natural frequencies, damping ratios, and time constants that identify oscillations, convergence, and divergence in the system's modes of motion.

Stability and Modes of Motion (StabMode)

StabMode.m presents a diagonal matrix of eigenvalues and a modal matrix of eigenvectors. Amplitudes of the eigenvector components for both original and velocity-weighted sets identify the relative participation of each state element in each mode of motion.

Flight Dynamics, Second Edition.