Thesis - Beaver simulation

Chapter 4. Implementation of the nonlinear

'Beaver' model in SIMULINK.

4.1 Introduction,

The models described in chapter 2 and appendices A to C can now be transferred into a SIMULINKsystem. There are different possible ways to do this. Ref.[25] described how the models can be programmed as a set of MATLABfunctions. This yielded a structure which was quite flexible, but only for experienced users. In this chapter, the systems will be implemented as SIMULINK block-diagrams, which are much faster, more flexible, and more user-friendly than the models from ref.[25]. Adding new output equations, deletingunwanted relations, or changing submodels to implement other aircraft is now really straightforward. In this chapter, the general structure of the models will be outlined. Appendix F gives a detailed description of all blocks from the SIMULINKimplementation. See chapter 6 for some examples which illustrate how to use the SIMULINKmodels in practice.

4.2 Modular structure of the aircraft models.

The dynamics of a n aircraft can be written as a set of nonlinear state equations, which can be derived from basic Newton mechanics and some kinematic relations. In appendix A, this derivation is described in detail; chapter 2 gives a short summary of the resulting equations. It will now be shown how the dynamic model of the aircraft dynamics can be divided into a number of submodels, which, in turn, can be implemented as different blocks in SIMULINK.

There are twelve state equations: three equations for the velocities along the aircraft's body axes (in this report, these expressions have been replaced by equations for the angle of attack, sideslip angle, and true airspeed), three equations for the rotational velocities along the aircraft's body axes, three equations which define the attitude of the aircraft relatively to the earth, and finally, three equations which define the position of the aircraft relatively to the earth. Consequently, there are twelve state variables.

The time-derivatives of the state variables are nonlinear functions of the states themselves and of external forces and moments acting upon the aircraft. The most important contributions to the total forces and moments are caused by aerodynamic influences, the operation of the powerplant (includingslipstream effects), the influence of gravity, and additional terms which must be taken into account in non-steady atmosphere. These contributions can be modelled separately and implemented as separate SIMULINKblocks.

The forces and moments are functions of the state variables and external inputs and/or disturbances, such as deflections of control surfaces, engine power settings, and components of wind and atmospheric turbulence. They also depend upon a number of airdata variables, which, in turn, are functions of some atmospheric properties, such as temperature, air density, and pressure. Besides the equations for the forces and moments themselves, all other equations can be implemented as separate SIMULINKfunction blocks, and the functions which are closely related can be grouped together.

Before examining the implementation of these models in SIMULINK,it is useful to consider the model structure used in the DUT-flightsimulator first. Figure 4-1 shows the sequence of calculations used in this simulator ", see ref.[l9]. A number of subroutines (aircraft module, engine module, ...)have to be called to calculate the time derivatives, i.e., to evaluate the state equations. The time derivatives of the state variables are integrated numerically with integration routines which have been included centrally within the simulation program. Before starting the actual simulations, a numerical trim routine is used to calculate steady-state values of the state and input variables, valid for a flightcondition, which must be specified by the user (airspeed, altitude, flap setting, engine speed). Refs.[G] and [la] describe the structure of this simulation software in more detail.

Figure 4 1 . Modular structure of the DUT-flightsimulator software (sequence of computations).

The figure shows the calculation sequence, used for off-line simulation. I t is necessary to include additional modules todetermine the deflections of the actuators of the motion system, the indications of the cockpit instruments, and the visual system display for on-line piloted flightsimulation.

If a part of the aircraft model needs to be changed, most other blocks can remain the same. For instance, removing all 'additional outputs' from this block-diagram can be done within only a few seconds. Adding other outputvariables takes some more time, since a new output-function block must be constructed, but implementing such blocks within this structure shouldn't be too difficult if the definitions of all input and outputvectors are known. Replacing the engine forces and moments model and the aerodynamic model of the 'Beaver' for similar models of other aircraft is also quite simple.

The parameters of the different submodels, such as coefficients for the aerodynamic and engine forces and moments models, have to be defined within the MATLABworkspace, using the M-files MODBUILDand LOADERSee. appendix F for more details.

It is necessary to define the initial values of the state and input variables in the vectors xinco (=x,),uaO (=u,(O)), and utO (=u,(O))in the MATLABworkspace. Normally, states and inputs which yield a certain user-specified steadystate flight condition are used as initial conditions. They need to be calculated before the actual simulation is started, using a numerical aircraft trim routine, similarly to the DUT-flightsimulator calculating sequence shown in figure 4-1.

In appendix E, it will be shown how the steady-state values of the input and state variables can be found numerically. The resulting aircraft trim routine is a n M-file called ACTRIM, which will be treated in section F.2.7 of appendix F. The user can save trimmed flight conditions to MAT-files, so it will not be necessary to recompute the steady-state condition every time a simulation is started.

In order to further integrate linear and nonlinear analysis, it is useful to linearize the aircraft models in several working points. Linearized aircraft models play a very important role in the AACS design and analysis, because the whole linear system theory can be applied to such models. In this report, it will be illustrated that the standard SIMULINKlinearization routine LINMOD can be used for this purpose. The working point is again defined by the values of x and u which yield a steady-state flight condition.

The combination of the nonlinear aircraft model in SIMULINK,the aircraft trim routine, and the linearization tools from SIMULINKmakes it possible to do the whole linear and nonlinear off-line (control-) system analysis in the same working environment. In this way, it is much easier to make the step from linear to nonlinear system analysis, which will encourage the users to do more experiments to analyze the systems, and a t the same time reduce the risk of making errors.

4.3 Adding external disturbances,navigationand approach models, and control loops.

If some additional input and output ports are included in the system shown in figure 4-2, it is possible to implement the nonlinear aircraft model as a subsystem within larger simulation models. In this way, other SIMULINKmodels can be connected to the 'Beaver' model. Examples of such models are: blocks to generate external disturbances and navigation signals (which may be needed for some control loops),control structures, and models of actuators and sensors. To keep the file size within limits, and to ensure a clear, modular structure, it is recommended to implement each model as a separate S-function. Different