Bayesian Framework

Note that effect of the case breach fault on the dynamics of the internal gas flow in SRBs is reduced to the effective modification of the nozzle throat area as explained above. It is, therefore, possible to infer SRB parameters using a Bayesian framework for an analysis of the overpressure faults due to the changes of the nozzle throat area. In particular, we showed that this algorithm can accommodate sudden changes of the model parameters and, therefore, is suitable for developing of the hybrid probabilistic IVHM of SRBs.

Here we briefly reproduce the results related to the analysis of the abrupt changes of the model parameters. The dynamics of the LDPM (20) can be in general presented in the form of Euler approximation of the set of ODEs on a discrete time lattice {t_{k} = *hk, k* = 0, 1, ...*K*} with time constant h.

where , , L-dimensional state of the system (20), is a diagonal noise matrix with two first non-zero elements a_{1} and a_{2}, is a vector field representing the rhs of this system, and c are parameters of the model. Given prior distribution for the unknown model parameters in a form of Gaussian distribution we can apply our theory of Bayesian inference of dynamical systems to

Here the vector field is parameterized in special form 1, , where U(x) is a block-matrix with N blocks of the form , is LxL unit matrix, and .

To verify the performance of this algorithm for the diagnostics of the case breach fault we first assume the nominal regime of the SRB operation and check the accuracy and the time resolution with which parameters of the internal ballistics can be learned from the pressure signal only. To do so we notice that equations for the nozzle throat radius r_{t}, burn distance *R*, and combustion chamber volume can be integrated analytically for a given by measurements time traces of pressure and substituted into the equations for pressure dynamics. We notice further that for small noise intensities the ration of dimensionless pressure and density then we obtain the following equation for the pressure dynamics:

where s_{t}, S_{b}, and V are known functions of time for a given pressure time trace and parameters

, and *D* are to be inferred in the nominal regime.

Table 1 The results of the parameter estimation of the model (20) in the nominal regime. The total time of the measurements in this test was T = 1 sec, the sampling rate was 1 kHz, and the number of measured points was N=1000.

We, therefore, conclude that the parameters of the nominal regime can be learned with good accuracy during the first second of the flight and can be assumed known in the further numerical tests of the Bayesian algorithm.

Our next step is to assume that the fault was initiated 0.2 sec after the flight in nominal regime and the time dependence of the case breach fault is general a nonlinear function of time (see results of the Section 0.0). We introduce the following general form of the nonlinear fault dynamics

*Click to enlarge image above*

Figure 10. (left) Actual fault dynamics is shown by the blue solid line. The time interval elapsed from the case breach fault is shown by the solid black line the predicted trajectories of the fault are shown by thing green solid lines. The time moment of the prediction is indicated by a vertical black line. (right) The distribution of the predicted values of the fault is shown by solid blue line in comparison with the true value of the fault indicated by the vertical black line.

We now investigate convergence of the distribution of the predicted values of the whole radius as a function of time elapsed from the onset of the fault. We assume known nominal parameters of the model and infer coefficients a

_{i}in the equation of the pressure dynamics, which in the presence of the case breach fault takes the form

Click to enlarge image aboveThe results of the numerical calculations are shown in Figure 10.

We illustrated our approach for example of the case breach faults. A detailed model has been developed that takes into account

- Real propellant geometry by using a key assumption that the burn distance defines uniquely the burning area of the propellant;
- The ablation of the nozzle throat and the nozzle exit that includes the time evolution of roughness at the nozzle walls;
- The fault dynamics that includes fault geometry and the heating of the hole walls in the metal case;
- The time evolution of the volume of the combustion chamber.
Under given assumptions the derived model can reproduce very accurately the time-traces of the internal SRB ballistics observed in the ground test firing.

The derived LDPM is validated using a FLUENT high-fidelity model of the case breach fault. Using results of the numerical solution of the high-fidelity model in FLUENT we show that obtained LDPM can be used to find analytically a time evolution of the axial distributions of the gas flow parameters in the combustion chamber and in the nozzle.

The obtained LDPM is then incorporated into a Bayesian inferential framework as a part of the on-board FD&P system for the next generation of NASA Heavy-Lift Launch Vehicles. It is shown that the obtained LDPM allows one to track in real time parameters of the SRB during the flight, to diagnose case breach fault, and to predict its future values.

We note that a number of other SRB fault modes (including e.g. combustion instabilities, bore choking, and nozzle throat faults) can be reduced to the analysis of the time variation of the parameters of the SRB. In particular, the LDPM allows one to track in time changes in the burning area, volume of the combustion chamber, area of the effective nozzle throat, and thrust. Therefore, the method developed in the present work method can be used to build FD&P system for a variety of the fault modes in the SRB.

We also note that the derived LDPM can be used to analyze various fault modes, including the nozzle blocking fault.