MATHEMATICAL MODEL STRUCTURES

In one way, model building is closely connected to experiments and observations. This is called empirical model building. Formulation of a theory can be called ‘‘model building’’ because it gives an analytical basis to study the system. Theory is a proposed concept of the system studied, e. g., the theory of Newton’s laws of motion lends to equations of motion (EOM) of an aerospace vehicle (Chapter 3). These EOM form the basis of mathematical models of such a vehicle and are the backbone of flight control, flight simulation, parameter estimation from flight data, and handling qualities analysis (HQA). This is the process of application of basic laws of physics: force/moment balance for mechanical systems, Kirchoff’s laws/Maxwell’s electromagnetic laws for electrical/electronic systems, and energy balance for thermal and propulsion systems. Thus, model building consists of (1) selection of a mathematical structure based on knowledge of the physics of the problem, (2) fitting of parameters to the experimental data (estimation), (3) verifica – tion/testing of the model (diagnostic checks), and finally its use for a given purpose.

In most aircraft identification/estimation applications [4], the model can be represented by a set of rigid body equations in the body axis system [5]. The basic EOM, derived from Newtonian mechanics, define the aircraft characteristic motion. They are based on the fundamental assumption that the forces and moments acting on the aircraft can be synthesized. In many practical cases, the forces and moments, which include the aerodynamic, inertial, gravitational, and propulsive forces, are approximated by terms in the Taylor’s series expansion. This invariably leads to a model that is linear in parameters. Aircraft mathematical models are treated in Chapters 3 through 5. A more complete model can be justified for the correct description of aircraft dynamics. The degree of relationship between the complexity of the model and measurement information needs to be ascertained. To arrive at a complex and complete model, one would need a series of repeated experiments under various environmental conditions and a large amount of data. This will ensure the statistical consistency of the modeling results in the face of the uncertainties of measurement of noise/errors. However, how much of this additional effort/data will be useful in building the complete model will depend on the methods of reduction of the data and engineering judgment.

There are two types of models: parametric and nonparametric. In a parametric model the behavior of a system is captured by certain coefficients or parameters: state-space, time-series, and transfer-function (TF) models. The model structure is either assumed known or determined by processing the experimental data. A non­parametric model captures the behavior in terms of an impulse response or spectral density curve, and no model structure is assumed. Near equivalence is present between these two types of models, especially for linear systems (or linearized systems). Given a set of data, an approach to construct a function, y = f (x, t), is an integral part of mathematical model building. This function should be parameterized in terms of a finite number of parameters. Thus, the function ‘‘f ’’ is mapping from the set of x values to the set of y values. This parameterization could be a ‘‘tailor – made’’ [6], orthogonal function expansion, or based on artificial neural networks (ANNs) [7]. If a true system cannot be parameterized by a finite number of parameters, then one can use an expanded set of parameters as more and more data become available for analysis. This is a problem of parsimony. A set of minimum number of parameters is always preferable for good predictability by the identified model. A complex model does not necessarily mean that it is an excellent (predict­ive) model of the system. Interestingly, one can say that the exact model of a system is the system itself! The most important aspect of mathematical model building is to arrive at an adequate model that serves the purpose for which it is designed. One can look for a model that is probably approximately correct.

For tailor-made model structures, i. e., physical parameterization, it is necessary to bring physical modeling closer to system identification. This aspect is difficult to realize, since the parameters should be identifiable from the input/output data. These models are often in the form of state-space structures. These models are based on the basic principles of the system’s behavior and involve the physics of the process and are often termed as phenomenological models. The simulation of these models is complex, but they have a large validity range [6], e. g., complete simulation of the flight dynamics of an aerospace vehicle. In so-called black box models, like finite impulse response (FIR), autoregressive-exogenous (ARX) inputs, and Box-Jenkin’s models for linear systems [8], the a priori information about the system is either very little or nil. The canonical state-space models and difference equation models have parameters that have necessarily no physical meaning. These models, often called readymade models, represent the approximate observed behavior of the process. For nonlinear systems, one can use Volterra models and multilayer perceptrons (MLPNs), i. e., neural networks (NWs) [9]. Another black box nonlinear structure is formed by fuzzy models that are based on FL [7]. Since always some a priori information about a system will be available, these black box models can be considered as gray box models. Also, it is possible to preprocess the data to get some preliminary information on the statistics of the noise affecting the system, thereby leading to gray box models.