Identification
The Abramov-Goman model of unsteady aerodynamic loads (which is based on PoDE), was applied to model the longitudinal and lateral directional coefficients of GTA, F16XL, Delta-60 wing and X31, with satisfactory results [20, 22, 23]. This model is the most intuitive extension to the Goman-Khrabrov model in [11], to account for nonlinearity in the unsteady variation of aerodynamic coefficients. As shown in previous sections, VVM is mathematically equivalent to PoDE with two conditions, that the PoDE model has single stable steady state and it is subject to certain input bounds. However, there are differences between the two in their consideration for system identification. Hence, a comparative study of the two structures is presented in this section.
The topology of the PoDE model can exhibit nonlinear phenomena like critical states crossing, hysteresis and limit cycles oscillations, period doubling leading to chaos etc. This is because PoDE model can have multiple equilibrium and it can move from the region of attraction of one to another in response to inputs. Such dynamics cannot be represented by Volterra series or VVM, as these models always converge to a particular equilibrium state when the input is within certain bounds, as proved in [68]. In this sense, the topology of VVM is similar to linear dynamical systems but stretched to higher number of parameters to account for nonlinearity of the response in time. Therefore,
although the two models are equivalent when their dynamics is restricted to the region of attraction of a particular stable equilibrium, their global dynamics for a wide range of inputs can be significantly different. There is currently no mathematical solution to either bridge this gap or an alternative structure which exhibits the properties of both the models. This difference in topology of the two model structures offers some advantages and disadvantages. The capability of PoDE model to capture unsteady aerodynamic loads in presence of static hysteresis in CZst(α) and Cmst(α), as well as critical state crossing in
Cnst(β) and Clst(φ) is an important advantage. The identification using such a model can
be performed using ”Bifurcational Model of Static Hysteresis” presented by Abramov in his Thesis [20]. These phenomena and application of BMSH are discussed in detail in Chapter 5. Such aerodynamic phenomena cannot be modeled using VVM. The complementary nature of PoDE model properties to VVM can be useful for modeling many aerodynamic systems.
However, the number of parameters in PoDE for tuning the static hysteresis and unsteady variation of an aerodynamic coefficient is sometimes insufficient. In Chapter 6, the nonlinear and unsteady modeling of rolling moment due to the Abrupt Wing Stall phenomena is presented. In this case, a cubic PoDE model with just four parameters is not good enough to capture the unsteady variations and static hysteresis in Cl versus β. Hence, a novel model called ”Bifurcational Model of Aerodynamic Asymmetry” is proposed and is demonstrated to produce satisfactory results.
A practical problem in using PoDE model for system identification is that the topology of the identified model can be significantly different from that of the physical system in some cases. For a particular set of values of estimated parameters of PoDE, the identified model may exhibit bifurcation dynamics which do not exist for the physical system under consideration. Hence, during parameter estimation of a PoDE model, it is important to include additional constraints on parameters in order to have a single real equilibrium. This problem does not arise for VVM as the response always converges to zero, which in turn can be translated to the stable steady-state value of the system.
Nonlinearity of many physical systems are typically excited for high energy inputs, while for low energy inputs dynamics is approximately linear. Amplitude dependence of nonlinearity is an inherent feature of VVM. If the input amplitude to VVM is scaled by a factor of γ, then xn(t) gets scaled by a factor of γn. Therefore, for a sinusoidal input of particular frequency, if amplitude is decreased, relative contribution of higher order kernels to total response decreases in magnitude. For sufficiently small amplitude input the system response is approximately linear. This is true for any input shape in general. Thus, VVM is the most intuitive model to represent physical systems that show approximately linear behavior for small amplitude and nonlinear for larger amplitude
inputs. In case of PoDE model, the correlation between the responses to small or large amplitude inputs is not so distinct. The linear or nonlinear nature of the response is case specific. The model fit obtained by parameter estimation is good if the system characteristics actually correspond to the order of polynomial nonlinearity used in the model structure.
For the PoDE model, parameters can be estimated by least-squares technique and validated using the criteria like goodness-of-fit or parameter co-variances from System identification theory. However, the order of polynomial nonlinearities to be included in the model structure cannot be determined apriori. In case of VVM, the kernel states to be included in the model structure can be determined directly using harmonic input response data over systems operational conditions as presented in the previous section.
A higher degree polynomial term in PoDE can produce any number of harmonics in response to sinusoidal input, which may not be the case as seen in the experimental data of the system. On the other hand, the kernel states of a VVM model produce a definite set of harmonics in response to sinusoidal input. Thus, the VVM model structure can have exact correlation with the harmonics of available system response, while PoDE model may not.
As nonlinearity of the output of PoDE model increases, higher order kernel states are essential to produce an accurate match between the responses of the two models. This is illustrated in the next section, and more examples are available in [66]. However, the estimated model response using same experimental data is significantly different. In case of PoDE model, the relative magnitudes of parameters get tuned automatically to match nonlinearity in the experimental response data. In case of VVM, higher order kernel states need to be included in the model structure explicitly.
A truncated VVM is convergent for certain bounds on input, and we presented a criteria for convergence as a lower bound on the maximum magnitude of input. Although Helie-Laroche algorithm gives a conservative estimate, it specifies the likely order of magnitude of the input bound. Currently there are no algorithms in literature to get an accurate estimate of the input bounds for convergence.
In case of VVM it is straight forward to consider a second order differential operator in the model, to get a model similar to the ”ONERA Dynamic Stall model”. This is presented in detail in chapter (5). In case of PoDE, it becomes a Duffings oscillator and gives rise to all the complex nonlinear dynamics exhibited by it. Thus, VVM structure provides flexibility to included higher order differential formulation while a PoDE model does not.
To conclude, VVM structure is suitable for modeling systems which have only super- harmonics in PSD of response, and an approximately linear response for small amplitude
input; while PoDE structure is better suited for those systems which have sub-harmonics or large number of super-harmonics in PSD of the response, and exhibit bifurcations for certain input conditions.