Data
3.5.1
Model Order Determination
As shown in chapter 2, the sinusoidal input response of VVE has some special properties. The response contains only super-harmonics. The n-th kernel state produces n-th harmonic and lower order harmonics of same parity. Therefore, for a model with n kernel states, n-th harmonic is the highest order harmonic in the response. In the response to relatively small amplitude input, only x1(t) is significant. Hence, the response of VVM is approximately linear. These properties are in direct correlation with the forced oscillation wind tunnel data.
The number of kernel states to be considered in the model can be determined using Power Spectrum Density (PSD) of the harmonic input response data. As discussed in Section 3.2, the variation in aerodynamic forces and moments on the wing in response to large amplitude sinusoidal change in angle-of-attack contains only super-harmonics. For small amplitudes of input, the force and moment variations contain only first harmonic. Hence, the entire LAFO test data is scanned to determine the highest harmonic nmax present in any of the responses. Then, model structure for the aerodynamic load consisting of nmax kernel states is sufficient.
The experimental data can be marred by low-frequency noise, and hence the highest frequency peak in the PSD may not have sufficient power relative to noise. In such cases, it is left to the judgement of the modeler to consider a low order VVM. Such a model may not be the best-fit but is essentially devoid of the effects of noise.
3.5.2
Estimation of Time-scale using SAFO Data
The linear response model parameters, time-constant a(α0) and dynamic-gain K1(α0) are estimated in two steps at each angle-of-attack α0. The first step is identification of time-scale by linear regression. This is due to linear relation between in-phase versus out-of-phase derivatives. This is proved for VVM by linearizing the single-state model in Eq.(3.3) at α0, and then deriving an analytical representation of in-phase and out-phase derivatives of its response to sinusoidal input.
In a small amplitude forced oscillation test, the wind tunnel model is oscillated in pitch in a sinusoidal motion as α(t) = α0 + ∆αsin(ωt). The measured normal force coefficient CZ(t) is converted to in-phase derivative CZα,ω0(α0) and out-of-phase derivative CZα,ω0˙ (α0) by harmonic analysis of the time-series data. The pitching
frequency is non-dimensionalised as ¯ω = ω¯c
2V, where, ¯c is mean aerodynamic chord and V is the air speed. Therefore, the steady-state response of the normal force coefficient measured in a wind tunnel test is given by,
CZ(t) = CZ0(α0) + CZα,ω0(α0)∆α sin(ωt) + CZα˙,ω0(α0)¯ω∆α cos(ωt) (3.4)
Now consider the response of VVM given by Eq.(3.3). For small amplitude input, only first kernel state is significant and the model response is linear. So for small amplitude pitching motion Cdyn(t) = x1(t). Therefore, the output Cdyn(t) to the input
˙
α = ∆α¯ω cos(ωt) is given by, ˙
Cdyn(t) = ˙x1(t)
= a(α0)x1+ K1(α0) ˙α(t)
= a(α0)x1+ K1(α0)¯ω∆α cos(ωt) (3.5) Solving this differential equation, the steady-state solution Cdyn(t)ssis obtained as,
Cdyn(t)ss =
K1∆α¯ω2
a2+ ¯ω2 sin(ωt)−
K1∆α¯ωa
a2+ ¯ω2 cos(ωt) (3.6) Then the Eq.(3.1) is linearized at α0 and the above equation is substituted in it to get the total normal force coefficient as,
CZ(t)ss = CZ0(α0) + CZα,st(α0)∆αsin(ωt) + CZq(α0)¯ω∆α cos(ωt) + K1ω¯ 2 a2+ ¯ω2∆αsin(ωt)− K1a a2+ ¯ω2∆α cos(ωt) (3.7)
coefficient from the wind tunnel test and VVM response respectively. Hence, comparing them gives the relation between model parameters and experimental derivatives as,
CZα,ω0(α0) = CZα,st(α0) + K1ω¯2 a2+ ¯ω2 CZα,ω0˙ (α0) = CZq(α0)− K1a a2+ ¯ω2 (3.8)
Rearranging the terms in Eq.(3.8), a linear relation between CZα,ω0(α0) and CZα,ω0˙ (α0) is
evident as given in Eq.(3.9). CZα,ω0(α0) and CZα,ω0˙ (α0) are known from SAFO test data
at various α0 for at least three frequencies.
CZα,ω0(α0) = aCZα,ω0˙ (α0) + [CZα,st(α0) + K1− CZq(α0)] (3.9)
The linear relation between in-phase versus out-of-phase derivative must be validated before estimation of the first kernel state parameters using SAFO data. In the parameter estimation process, in-phase and out-of-phase derivatives are estimated from the SAFO data. The effects of measurement noise and mild nonlinearities are removed. However, it is important to check the harmonics in small amplitude response and nonlinearity in static variation on the range of angle-of-attack corresponding to the amplitude used in the test.
Then, a(α0), K1(α0) at each α0 in the stall region can be estimated by the two-step regression method presented in [22]. This method is given in APPENDIX 1 and the estimation procedure is illustrated in the the next chapter.
3.5.3
Estimation of Nonlinear Model Parameters using LAFO Data
The parameters of the first kernel state represent linear-dynamical variations, and are identified from SAFO data at certain mean angle-of-attack intervals. However, the time- constant of vortex dynamics changes with angle-of-attack. Hence, the parameters of VVM are considered as functions of angle-of-attack. This in turn makes the unsteady aerodynamic model nonlinear for large-amplitude changes in angle-of-attack. It is important that the resulting nonlinear response of the model be identified to be consistent with the nonlinear variations captured in the LAFO data. Hence, defining a parameter function by simply using the linear interpolation of the values from the first step is inadequate. Therefore, the LAFO data should be first used to tune the parameter functions (a(α), K1(α)).
For large amplitude sinusoidal inputs, the nonlinear nature of variations is indicated by the presence of higher order harmonics. Hence, the higher order kernel states become important. So, the estimated parameter functions (a(α), K1(α)) are frozen, and then the
parameters of higher order kernel states are estimated by the output-error method. since VVM is linear in parameters, the estimates obtained from output-error method are also the maximum likelihood estimates [43].
Estimation of parameter functions becomes tricky due to the issues of, number of node-points and optimization algorithm to be used. As known from the system identification theory, a larger number of node-points than essential will cause the variance of the estimated values to be larger. This implies that the bounds on values of parameters at node-points will be bigger. Although larger number of node-points improve the accuracy of results, it reduces the fidelity of the model for simulation using random inputs. A proper choice of optimization algorithms is important because the parameters need to be estimated using a large data-set. A combination of both gradient and steepest- descent based methods are used in the current study. These issues are illustrated in the case studies presented in the Chapter 4.
3.5.4
Estimation of Input bounds for Convergence
Helie-Laroche algorithm for estimation of input bounds for convergence of Volterra series or Volterra variational equations was presented in Chapter 2. This method has been demonstrated to be effective using the example of Duffings oscillator system in Section 2.6. In case of VVM, the parameters are not constants but a function of angle-of-attack. Hence, the input bounds can also be computed as function of angle-of-attack. Its validity though is questionable as the parameters vary continuously with change in angle-of- attack. This is a mathematically complex problem which was not probed further in this work.
It was realized during its application to the unsteady aerodynamic models estimated for coefficients of GTA and F16XL that the estimated value is too small. This is because the Helie-Laroche algorithm gives a conservative estimate of bounds by considering the minima for any input type. The estimated bound corresponds to the case of constant input, i.e. zero frequency. Therefore, there is a need to extend the result to the case of input trajectories like step, ramp and harmonic input. An approach to compute input bound for the case of Duffings oscillator system considering harmonic input is due to Pang and Lang given in [87]. Several attempts to follow a combination of the two approaches to obtain an appropriate estimate considering harmonic inputs did not succeed.
Estimation of convergence bounds for the estimated unsteady models is apparently not necessary. With the estimated unsteady models for GTA and F16XL presented in the next chapter, the convergence was never found to be an issue. These models have been used in a variety of flight simulation studies, and the estimated VVMs have never been found to diverge to infinity or produce any unreasonable response. Therefore, for
the problem of modeling unsteady aerodynamic loads using VVM, convergence is not an important issue. This may also be a reason why it has been ignored in the applications to physiological systems presented in the book [48].