Aircraft model block reference
8.2 The aircraft model block libraries
The blocks and subsystems from the aircraft model have been sorted six individual blocklibraries, which are directly accessible from the main FDC libraryFDCLIB:
1. Airdata, atmosphere, 2. Aerodynamics,
3. Engine forces and moments, 4. Gravity and wind forces, 5. Equations of motion, 6. Other (output-) equations.
Although these libraries are considered to be sublibraries ofFDCLIB, they can also be accessed directly by typing fdclibi in the command-line, where i represents the number of the blocklibrary according to the list above.
In addition,FDCLIBcontains a direct link to the system Beaver(the stand-alone version of the aircraft model), and a link to a sublibrary containing the subsystem equivalents ofBeaver. The stand-alone model and this sublibrary can also be opened directly from the command-line by typing beaver or fdclib10, respectively. In fig-ure 8.3, the main FDC library has been shown, with highighted links to the aircraft blocklibraries and the complete aircraft models. FDCLIB can be opened from the command-line by typing fdclib.
1The reason for using scalarInportandOutportblocks in the interface level of the aircraft model was that older versions of SIMULINKdid not allow the use of vector inputs or outputs in top-levels of systems. See the description ofBeaver Level 1for more information.
8.2. The aircraft model block libraries 121
Figure 8.3:Main FDC library with highlighted aircraft model sublibraries
The remainder of this chapter (pages 122 to 164) provides a detailed description of all blocks and subsystems from the nonlinear aircraft model, i.e. the system Beaveror one of its subsystem equivalents. The blocks have been sorted in al-phabetical order; their locations have been referenced with respect to the system Beaverand the main FDC library FDCLIB.
122 Chapter 8. Aircraft model block reference
12 ODEs
Beaver level 1 / Beaver level 2 / Aircraft Equations of Motion / 12 ODEs Main FDC library / Equations of motionType
Aircraft-independent subsystem, essential for solving the equations of motion. The sub-system is not masked.
Description
The subsystem 12 ODEs contains the twelve non-linear Ordinary Differential Equations that describe the aircraft dynamics. The equations are valid for all rigid bodies, assum-ing a flat, non-rotatassum-ing Earth; see chapter 2 for a detailed theoretical discussion. The time-derivatives of the twelve state variables are nonlinear functions of the state varia-bles themselves and of the external forces and moments. The latter depend upon the state variables, external control inputs, and external disturbances affecting the airplane.
Subsystems and/or blocks
The subsystem 12 ODEs contains four blocks:
Vabdot computes time-derivatives of true airspeed, angle of attack, and sideslip angle, pqrdot computes time-derivatives of the angular velocities along the body-axes of the
aircraft,
Eulerdot computes time-derivatives of the Euler angles,
xyHdot computes time-derivatives of the coordinates and the altitude above sea-level.
Inputs
x = [V α β p q r ψ θ ϕ xeye H]T state vector, x
Ftot = [FxFyFz]T total external forces, Ftot
Mtot = [L M N]T total external moments, Mtot
yhlp = [cos α sin α cos β sin β tan β sin ψ cos ψ sin θ cos θ sin ϕ cos ϕ]T
frequently used sines and cosines, yhlp ybvel∗ = [u+uw v+vw w+ww]T body-axes velocity components plus wind, ybvel∗ Outputs
˙x = [V˙ ˙α ˙β ˙p ˙q ˙r ˙ψ ˙θ ˙ϕ ˙xe ˙ye H˙ ]T time-derivative of state vector, xdot Note: the vector ˙x that leaves the subsystem 12 ODEs must be corrected for the direct contribution of ˙β to the aerodynamic side force Ya. This correction is taken into account by the block xdotcorr (Beaver), which has been described later.
Parameters
The block Vabdot reads the parameter vector GM1 from the MATLABworkspace; the block pqrdotrequires the matrix GM2. The definitions of GM1 and GM2 can be found in ap-pendix C. These variables can be loaded from the file AIRCRAFT.DAT, using the utility DATLOAD(see section 12.4.2). If this datafile has somehow inadvertently been deleted, it can be re-created by running the program MODBUILD (see section 12.6.1).
Connections
in: xcomes from the block Integrator; Ftotand Mtotcome from FMsort; yhlpcomes from Hlpfcn; ybvel∗is the sum of the output from uvw and the wind velocity components from the external input vector uwind.
out: ˙x (not corrected for the implicit nature of the ˙β-equation) is connected to the block xdotcorr (Beaver).
Type browse 12odes at the command-line for on-line help.
8.2. The aircraft model block libraries 123
Accel
Beaver level 1 / Beaver level 2 / Additional Outputs / Accel Main FDC library / Other (output-) equationsType
Aircraft-independent masked subsystem block, not essential for solving the equations of motion.
Description
The block Accel computes some accelerations and specific forces (outputs of accelerome-ters) in the aircraft’s center of gravity. These output variables are useful for several FCS design tasks, e.g. turn-coordination by means of feedback of the specific force along the YB-axis, or manoeuvre load limiting. However, these variables are not required to actually solve the equations of motion themselves! Computing accelerations outside the center of gravity will require additional terms, which are not taken into account in this block; refer to ref.[14] for more details.
Equations
The equations used by the block Accel have been discussed in some more detail in sec-tion 3.6.
• Kinematic accelerations ax,k, ay,k, and az,k along the body-axes, measured in the vehi-cle’s c.g.:
These accelerations are measured in units of g which explains why the terms have been divided by g0. W =m g is the total weight of the aircraft, measured in [N]. Note: the kinematic accelerations differ from the body-axes velocity rates ˙u, ˙v, and ˙w, which are computed in the block uvwdot, because the latter do not contain the angular and trans-lational velocity cross-product terms.
• Outputs of body-axis accelerometers (specific forces) Ax, Ay, and Az, measured at the vehicle’s c.g.:
Ax = ax,k+sin θ = (Fx−Xgr)/ W Ay = ay,k−cos θ sin ϕ = (Fy−Ygr)/ W Az = az,k+cos θ cos ϕ = (Fz−Zgr)/ W These specific forces are also measured in units of g.
Inputs
Accelneeds the parameter vector GM1 to be present in the MATLABworkspace, in order to extract the mass m of the aircraft (the mass has been implemented as a parameter, i.e.
it is assumed to remain constant during the relative short time intervals considered in simulations). The definition of GM1 can be found in appendix C. GM1 can be loaded from the fileAIRCRAFT.DAT, using the utility DATLOAD (see section 12.4.2). If this datafile
124 Chapter 8. Aircraft model block reference
has somehow inadvertently been deleted, it can be re-created by running the program MODBUILD(see section 12.6.1).
Connections
in: Ftotcomes from the block FMsort; Fgravcomes from Gravity.
out: yaccis not used by any other block from the aircraft model.
Type browse accel at the command-line for on-line help.
8.2. The aircraft model block libraries 125
Additional Outputs
Beaver level 1 / Beaver level 2 / Additional Outputs Main FDC library / Other (output-) equationsType
Aircraft-independent subsystem, essential for solving the equations of motion. The sub-system is not masked.
Description
The subsystem Additional Outputs contains some useful output equations which are not required to solve the state equations, and which do not logically fit in other subsystems from the nonlinear aircraft model. The user is free to add other ‘additional output blocks’
to this subsystem, or delete available blocks from this subsystem.
However, it should be noted that the output signals from this subsystem are not meant to be used as inputs to the forces and moments blocks or the equations of motion themselves.
The ‘additional outputs’ may be functions of the state variables, their time-derivatives, external inputs and/or disturbances, or outputs from other subsystems from the aircraft model.1
Subsystems and/or blocks
The subsystem Additional Outputs contains three blocks:
Accel computes specific forces (outputs of accelerometers) and kinematic accelerations, Flpath computes some flight-path related variables,
uvwdot computes the time-derivatives of the body-axes velocity components.
Inputs
x = [V α β p q r ψ θ ϕ xeye H]T state vector, x
˙x = [V˙ ˙α ˙β ˙p ˙q ˙r ˙ψ ˙θ ˙ϕ ˙xe ˙ye H˙ ]T time-derivative of state vector, xdot yhlp = [cos α sin α cos β sin β tan β sin ψ cos ψ sin θ cos θ sin ϕ cos ϕ]T
frequently used sines and cosines, yhlp
Ftot = [FxFyFz]T total external forces, Ftot
Fgrav = [XgrYgrZgr]T gravity forces, Fgrav
Outputs
yfp = [γ fpa χΦ]T flight-path variables, yfp
yuvw = [ ˙u ˙v ˙w]T time-derivatives of body-axes velocities, yuvw yacc = [AxAyAzax,kay,kaz,k]T specific forces and accelerations, yacc Parameters
The block Accel reads the parameter vector GM1 from the MATLABworkspace; the block Flpathrequires the initial value of the state vector, xinco. GM1 can be loaded from the fileAIRCRAFT.DAT, using the utility DATLOAD (see section 12.4.2); its definition can be found in appendix C. If this datafile has somehow inadvertently been deleted, it can be re-created by running the program MODBUILD (see section 12.6.1). The initial value of the state vector can be computed using the aircraft trim program ACTRIM (see section 11.1), or loaded from file using the utility TRILOAD (see section 12.4.2).
1Since the outputs fromAdditional Outputsare not involved in the solution of the state equations themselves, the use of time-derivatives of state variables within this subsystem does not yield algebraic loops (see section section 6.4.1). However, if outputs from this subsystem are fed back to the input-side of the aircraft model, e.g. via an autopilot control loop, an algebraic loop may inadvertently be created. This can cause simulations to slow down considerably, and such an algebraic loop may be too complicated for SIMULINKto solve altogether. For this reason, feeding back ‘additional outputs’ into the aircraft model should be prevented.
126 Chapter 8. Aircraft model block reference
Connections
in: x comes from the block Integrator; ˙x comes from xfix; yhlp comes from Hlpfcn; Ftot
comes from FMsort; Fgravcomes from Gravity.
out: yfpfrom Flightpath, yuvwfrom uvwdot, and yaccfrom Accel are not connected to any other block in the aircraft model.
Type browse moreouts at the command-line for on-line help.
8.2. The aircraft model block libraries 127
Aerodynamics Group (Beaver)
Beaver level 1 / Level 2 / Aerodynamics Group (Beaver) Main FDC library / AerodynamicsType
Aircraft-dependent subsystem, essential for solving the equations of motion. The subsys-tem is not masked.
Description
The subsystem Aerodynamics Group (Beaver) contains blocks for computing the aerody-namic forces and moments that act on the aircraft under consideration. In this case the aerodynamic model is valid for the DHC-2 Beaver aircraft. It uses model data from ref.[34] (see also section 3.3.1). Notice that the subsystem Aerodynamics Group does not compute propeller slipstream related contributions to the aerodynamic forces and mo-ments; that task is performed by the subsystem Engine Group instead.
Subsystems and/or blocks
The subsystem Aerodynamics Group (Beaver) contains three blocks:
Aeromod computes dimensionless aerodynamic force and moment coefficients for the aircraft under consideration (currently Aeromod contains the aerodynamic mo-del of the Beaver from ref.[34], which has been described in section 3.3.1), Dimless computes dimensionless angular velocities,
FMdims converts the dimensionless force and moment coefficients to dimensional for-ces and moments.
Since the block Aeromod (Beaver) is aircraft-dependent, it needs to be replaced if the user wants to implement a model of a different aircraft in the FDC framework. However, if the new aircraft model also uses different definitions of the non-dimensional angular ve-locities, it may be more convenient to replace the complete subsystem Aerodynamic Group instead. Either way, this modular construction allows the user to build a comprehensive SIMULINKlibrary of aerodynamic models.
Inputs
x = [V α β p q r ψ θ ϕ xeye H]T state vector, x
uaero = [δeδaδrδf ]T aerodynamic control inputs, uaero
yad1 = [a M qdyn]T basic airdata variables, yad1
Outputs
ydl = [ 2Vpb qcV 2Vrb ]T dimensionless angular velocities, ydl Caero = [CXaCYaCZaClaCma Cna ]T aerodynamic force and moment coefficients, Caero FMaero = [XaYaZa La Ma Na]T dimensional aerodynamic forces and moments, FMaero Parameters
The block Dimless and the block FMdims read the parameter-vector GM1 from the MAT
-LABworkspace; the block Aeromod (Beaver) needs the parameter-matrix AM. The defin-itions of these variables can be found in appendix C. The variables can be loaded from the fileAIRCRAFT.DAT, using the utility DATLOAD (see section 12.4.2). If this datafile has somehow inadvertently been deleted, it can be re-created by running the program MOD-BUILD(see section 12.6.1).
Connections
in: xcomes from the block Integrator; uaerois an external input vector containing aero-dynamic control inputs; yad1comes from the block Airdata1.
out: ydl from Dimless is used by the block Aeromod; Caero from Aeromod is used by FMdims; FMaerofrom FMdims is used by the block FMsort.
Type browse aerogrp at the command-line for on-line help.
128 Chapter 8. Aircraft model block reference
Aeromod (Beaver)
Beaver level 1 / Beaver level 2 / Aerodynamics Group / Aeromod (Beaver) Main FDC library / AerodynamicsType
Aircraft-dependent masked subsystem block, essential for solving the equations of mo-tion.
Description
The block Aeromod (Beaver) contains the aerodynamic model of the DHC-2 Beaver air-craft. The model is based on ref.[34], which expresses the aerodynamic forces and mo-ments in terms of nonlinear polynomial functions of state variables and external aerody-namic control inputs. This block needs to be updated if one wishes to implement a model of a different aircraft within the FDC framework. Thanks to the black-box structure of Aeromod, it is relatively easy to change the structure of the equations if necessary, e.g. to incorporate look-up table functions, which often define the aerodynamic equations for different airplane models.
Note: Aeromod (Beaver)does not take into account contributions to the forces and mo-ments due to the propeller slipstream. This is done in the masked subsystem block Eng-modwithin the subsystem Engine Group.
Equations
The aerodynamic force and moment coefficients of the Beaver are expressed in terms of nonlinear polynomial functions of the state variables and aerodynamic control inputs.
The model includes longitudinal-lateral cross-coupling effects, as well as unsteady aero-dynamics, but it does not take into account the influence of compressibility, as airspeed is assumed to be low [34]. The equations have been described in more detail in section 3.3.1.
• Coefficients of the aerodynamic forces and moments, measured in the body-fixed refer-ence frame:
The coefficients of the polynomials represent the stability and control derivatives of the Beaver; they have been listed in table B.3 of appendix B. A closer look at the internals of the block Aeromod (Beaver) reveils that the polynomial evaluation has been implemented in practice by means of a multiplication of the vector:
with the constant parameter matrix AM in which the stability and control derivatives of the Beaver are contained.
Notice that the direct contribution of the time-derivative of the sideslip angle ˙β to the aerodynamic side-force coefficient CYa has not been taken into account in this block: the
8.2. The aircraft model block libraries 129
last element of the multiplication vector equals zero instead of ˙βb/2V, i.e. the bracketed term in the CYaequation is neglected by Aeromod (Beaver)! This was necessary in order to prevent the ˙β equation from becoming implicit, which would have yielded an algebraic loop in the simulation model (see section 6.4.1). The error which results from neglecting this term is corrected in the separate block xdotcorr.
Inputs
x = [V α β p q r ψ θ ϕ xeye H]T state vector, x
uaero = [δeδaδrδf ]T aerodynamic control inputs, uaero ydl = [ 2Vpb qcV 2Vrb ]T dimensionless angular velocities, ydl Outputs
Caero = [CXaCYaCZaClaCma Cna ]T aerodynamic force and moment coefficients, Caero Parameters
The block Aeromod reads the parameter matrix AM from the MATLABworkspace. AM contains the stability and control coefficients of the Beaver; see appendix C for its exact definition. This matrix can be loaded from the fileAIRCRAFT.DAT, using the utility DAT-LOAD(see section 12.4.2). If this datafile has somehow inadvertently been deleted, it can be re-created by running the program MODBUILD (see section 12.6.1).
Connections
in: x comes from the block Integrator; uaero is an external input vector, containing the aerodynamic control inputs; ydlcomes from Dimless.
out: Caerois connected to FMdims.
Type browse aeromod at the command-line for on-line help.
130 Chapter 8. Aircraft model block reference
Aircraft Equations of Motion (Beaver)
Level 1 / Level 2 / Aircraft Equations of Motion Main FDC library / Equations of MotionType
Aircraft-independent subsystem, except for the block xdotcorr (Beaver), essential for solv-ing the equations of motion. The subsystem is not masked.
Description
The subsystem Aircraft Equations of Motion (Beaver) contains the generic nonlinear six-degree-of-freedom rigid-body equations of motion. It determines the time-derivatives of the state variables from the aircraft model, and the time-trajectories of the state variables themselves. Aircraft Equations of Motion (Beaver) contains one aircraft-dependent element:
the block xdotcorr (Beaver) is needed to correct the time-derivative of the sideslip angle for the implicit nature of the sideforce equation in the aerodynamic model. Because of this, this subsystem is not entirely aircraft-independent; the current implementation has been tailored to the DeHavilland DHC-2 Beaver.
Subsystems and/or blocks
The subsystem Aircraft Equations of Motion (Beaver) contains two blocks:
uvw computes the body-axes velocity components as a function of the true air-speed, angle of attack, and sideslip angle,
xdotcorr applies corrections to the time-derivatives of the state variables in order to take into account the implicit nature of the aerodynamic forces and moments model (the current implementation is valid for the Beaver model, where the time-derivative of the sideslip angle is corrected to account for the contribution of
˙β to the aerodynamic sideforce Ya).
In addition, Aircraft Equations of Motion (Beaver) contains the subsystem:
12 ODEs contains the state equations, which express the time-derivatives of the state variables in terms of the external forces and moments, and the state variables themselves.
An Integrator-block is used determine the time-trajectories of the state variables as a func-tion of their time-derivatives and initial condifunc-tion.
Inputs
Ftot = [FxFyFz]T total external forces, Ftot
Mtot = [L M N]T total external moments, Mtot
uwind = [uwvwww ˙uw ˙vww˙w]T wind velocity components along body-axes and their time-derivatives, uwind yatm = [ρ ps T µ g]T basic atmospheric properties, yatm yhlp = [cos α sin α cos β sin β tan β sin ψ cos ψ sin θ cos θ sin ϕ cos ϕ]T
frequently used sines and cosines, yhlp Outputs
x = [V α β p q r ψ θ ϕ xeye H]T state vector, x
˙x = [V˙ ˙α ˙β ˙p ˙q ˙r ˙ψ ˙θ ˙ϕ ˙xe ˙ye H˙ ]T time-derivative of state vector, xdot ybvel = [u v w]T body-axes velocity components, ybvel Parameters
The block xdotcorr (Beaver) reads the parameter matrix AM and the vector GM1 from the MATLABworkspace. The subsystem 12 ODEs needs the parameter vector GM1 and the matrix GM2. The definitions of these variables can be found in appendix C. The variables
8.2. The aircraft model block libraries 131
can be loaded from the fileAIRCRAFT.DAT, using the utility DATLOAD (see section 12.4.2).
If this datafile has somehow inadvertently been deleted, it can be re-created by running the program MODBUILD (see section 12.6.1). The block Integrator requires the initial value of the state vector, xinco, which can be computed using the aircraft trim program ACTRIM (see section 11.1), or loaded from file using the utility TRILOAD (see section 12.4.2).
Connections
in: Ftotand Mtotcome from FMsort; uwindis an external input containing wind and tur-bulence velocities and their time-derivatives; yatmcomes from Atmosph; yhlpcomes from Hlpfcn.
out: x, which leaves the block Integrator, is connected to many other blocks in the air-craft model; ˙x from xfix is connected to Accel, Flpath, and uvwdot; ybvelfrom uvw is connected to xyHdot.
Type browse eqmotion at the command-line for on-line help.
132 Chapter 8. Aircraft model block reference
Airdata1
Beaver level 1 / Beaver level 2 / Airdata Group / Airdata1 Main FDC library / Airdata, atmosphereType
Aircraft-independent masked subsystem block, essential for solving the equations of mo-tion.
Description
The block Airdata1 is used to compute some essential airdata variables. This information is usually required to be able to compute the aerodynamic and propulsive forces and moments, and hence, to solve the equations of motion. For the Beaver model, only the dynamic pressure qdynis actually needed to solve the state equations, but models of faster flying aircraft usually also require the Mach number M. Other airdata (-related) equations have been implemented in the blocks Atmosph, Airdata2, and Airdata3.
The block Airdata1 is used to compute some essential airdata variables. This information is usually required to be able to compute the aerodynamic and propulsive forces and moments, and hence, to solve the equations of motion. For the Beaver model, only the dynamic pressure qdynis actually needed to solve the state equations, but models of faster flying aircraft usually also require the Mach number M. Other airdata (-related) equations have been implemented in the blocks Atmosph, Airdata2, and Airdata3.