Fault-tolerant Flight Control

Control Configuration and Performance Consideration

For each trimming condition case, the optimal fault-tolerant controller is computed. The generic closed-loop structure of the system G(s) with a controller K(s), including the weighting functions We, Wd, and Wu, is shown in Figure 5.14.

Fig. 5.14 Closed-loop system, including weighting functions.

To shape the output z =

"

z1

z2

#

according to the robustness and performance requirements, the system includes the following weighting functions:

First, We, for minimising the tracking error was designed to track the heading and

captured by the upper bound:

|S(jω)| < 1

We

, ∀ω

We was chosen to be a low-pass filter, as shown in Figure 5.15, and tuned for the

required manoeuvres in each trimming case. Table 5.3 lists parameters of We for

heading angle ψ at different flying conditions.

Table 5.3 We parameters for tracking requirements of the heading angle (ψ).

We parameters Ascending Cruise Descending

Low-frequency gain, dB 6 13 14.9 Crossover frequency, rad/s 0.04 0.11 0.085 High-frequency gain, dB −40 −40 −20

To reduce the tracking efforts for a more robust stabilising solution, the heading (ψ) and bank (ϕ) angles tracking requirements in We are identical in shape, however,

their magnitudes are different. Figure 5.15 shows the illustration of Bode diagram for We for bank angle.

Fig. 5.15 Bode diagram of tracking weighting functions We for bank angle (at cruise).

Second, we consider Wd for shaping the variation of the nominal value of the fault

(∆) over different frequencies. This weighting function could be frequency-dependent or just a constant. In this work, a 2 × 2 constant matrix was assumed with values of ±1.

Finally, Wu is needed for shaping the control efforts and capturing the actuator

5.4 Non-linear Testing and Validation 97

Table 5.4 Wu parameters for control efforts.

Wu Ascending Cruise Descending

Aileron 0.1226 0.09 0.075 Rudder 0.16 0.135 0.0975

this weight is decreased, γ (the robustness condition) decreases, but the deflection increases, and vice versa. In addition, to use the aileron actuator more than the rudder for a rolling motion, Wu for the rudder exceeds the weight of the aileron.

Table 5.4 lists the Wu parameters.

Robust Control Computation

The closed-loop system in Figure 5.14 is given in the generalised plant P form (Figure 5.3) with the following relationship:

     z1 z2 r − y      =      We −WeG 0 Wu I −G        w u  

The H∞ controller is computed assuming that uncertainty is neglected. It is

necessary to find the nominal stabilising solution that fulfils the condition:

∥Tzr∥∞ = We(I + GK)−1 WuK(I + GK)−1 _∞ <1

The value of γ = ∥Tzr∥∞ that fulfils the robust controller requirements was found

by adjusting the weighting functions and using the MATLAB function hinfsyn for computation. For robust control design against uncertainty, that represents an actuator loss-of-effectiveness fault, the faults are modelled as follows:

∆ =   30% 0 0 1%  

for 30% fault in the aileron and 1% fault in the rudder. The generalised plant P , depicted in Figure ?? in Appendix B.0.1, has the relationship

        Wd z1 z2 r − y         =         0 0 I 0 We −WeG Wu 0 Wu −G I −G              ud w u     

The µ-synthesis controller based on D − K iteration was computed in terms of the upper bound of (µ), whereas H∞ was analysed in terms of the (γ) bounds,

as mentioned before. Computational optimisation and iterations yielded the best possible results for the robustness conditions of the controllers, as listed in Table 5.5.

Table 5.5 Robustness condition results for nonlinear model.

Controller The condition Ascending Cruise Descending

H∞ γ < 1 1.153 1.151 1.154

µ-synthesis µ < 1 0.994 0.998 0.999

The results indicate that the conditions of stability and performance robustness were guaranteed when the µ-synthesis control technique was used, but could not be achieved when the H∞ controller was used for the aircraft under different operating

conditions. Aircraft motion was examined for those two controllers with the 30% loss of effectiveness aileron fault. Aircraft response, when using the µ-synthesis, is damped with shorter settling time compared to the H∞ based response as shown

in Figure 5.16. Moreover, the response when using the µ-synthesis had neither an overshoot nor a steady state error.

Aileron deflections when using the H∞ controller fluctuate very rapidly as shown

in Figure 5.17. It also saturated swiftly reaching the maximum deflection limit. On the other hand, the aileron deflections when using the µ-synthesis controller had a very acceptable rate of changes and position limits, which assures the actuator dynamics. These results emphasise the robustness and viability of the µ-synthesis controller against loss of effectiveness faults, which eventually enhance the aircraft’s stability and performance.

5.4 Non-linear Testing and Validation 99

Fig. 5.16 Bank angle response using the two controllers with 30% faulty actuators.

Fig. 5.17 Aileron deflections with 30% loss of effectiveness faults.