Appendix C. Linear System Survey

Supplement for Flight Dynamics, a book by Robert F. Stengel

The SURVEY program accepts the linear, time-invariant stability and control-effect matrices, Fmodel and Gmodel, from FLIGHT as inputs and performs a variety of analyses on them. It first provides the option of reducing these matrices into smaller models using the function LonLatDir. The function produces a single model at each call for either longitudinal or lateral-directional motions, in body or stability axes, and with four or six elements in the state vector. The remaining SURVEY functions operate on either the original twelve-dimensional model or on the reduced model produced by LonLatDir. The program calculates transient response, static response, controllability and observability matrices, natural frequencies, damping ratios, and real roots. It forms single-input/single-output transfer functions selected by the user and makes their Bode, Nyquist, and Nichols plots. SURVEY is tutorial, and the user must control program flow by making changes in the code. The interested reader could consider adding root locus plots and Monte Carlo evaluation to the main program. No explicit or implicit warranties are made regarding the accuracy or correctness of the computer code.

C.1 Main Program for Analysis and Simulation (SURVEY)

As shown, SURVEY is set up to make calculations based on the 12 x 12 Fmodel and 12 x 7 Gmodel produced by FLIGHT, with the generic business jet model trimmed for flight at V = 102 m/s and h = 3,050 m. The flags are set to generate a fourth-order, body-axis, lateral-directional model from the data set; to calculate eigenvalues, eigenvectors, and the transfer function from aileron to roll rate; and to show the Bode plot, Nyquist plot, and Nichols chart for the transfer function.

C.2 Supporting Functions

C.2.1 Reduced-Order Models (LonLatDir)

LonLatDir accepts stability and control-effect matrices for combined longitudinal and lateral-directional dynamics as input and 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, stability, or hybrid 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, and three lateral-directional controls

C.2.2 Transient Response (Trans)

Trans calculates and plots transient response to an arbitrary array of initial conditions and step control inputs, calling the MATLAB function lsim to perform the simulation.

C.2.3 Static Response (Static)

Static computes static control and command response for fourth-order systems with no disturbances and two control inputs (elevator and throttle in the longitudinal case, aileron and rudder for the lateral-directional model). Static control equilibrium does not exist for the higher-order models presented here because the stability matrix is singular.

C.2.4 Controllability and Observability (ConObs)

The matrix rank tests described in Chapter 4 are performed by calling the MATLAB functions ctrb and obsv in ConObs.

C.2.5 Natural Frequency (NatFreq)

NatFreq presents eigenvalues sorted in descending order and natural frequencies, damping ratios, and real roots using the MATLAB functions esort and damp.

C.2.6 Stability and Modes of Motion (StabMode)

StabMode presents eigenvalues and eigenvectors, as well as amplitudes of the eigenvector components for both original and velocity-weighted sets.

Flight Dynamics.