6.3 Dynamic Control Allocation using the Quadratic Programming approach
6.4.3 Control Allocation
Once the virtual control command v(t) is computed, it is now a task of allocating the total control efforts on the available aircraft control surfaces. At trimmed flying conditions, the desired steady state control distribution matrix that resembles the relationships between total required rolling, pitching, and yawing efforts and the corresponding control surfaces is chosen to be:
S = −5.0 −0.05 −0.12 5.0 −0.05 0.12 −10 −0.02 −1.0 10 −0.02 1.0 0.02 0 −8.0
Where the weights are W1 = diag ([2 2 2 2 2]) and W2 = diag([8 8 10 10 10]).
Control Allocation without saturation
The results of the computation of the control allocation filter using Matlab for the unsaturated formula yields sets of transfer functions that relate the inputs (v) to the outputs u. Hence, the transfer function of input vroll to inner right aileron, neglecting
the very small terms (i.e. < 1x10−7), is:
ξir vroll
= −18.03 s5+ 51.7 s4−49.4 s3+ 15.8 s2
s5−2.9 s4+ 2.7 s3 −0.86 s2
The transfer function of input vroll to inner left aileron is:
ξil vroll
= 18.03 s5−51.7 s4+ 49.4 s3−15.8 s2
s5−2.9 s4+ 2.7 s3 −0.86 s2
ξor vroll
= −15.5 s5+ 44.1 s4−41.7 s3+ 13.17 s2
s5−2.9 s4+ 2.7 s3 −0.86 s2
transfer function of input vroll to outer left aileron is:
ξol vroll
= 15.5 s5−44.1 s4 + 41.7 s3 −13.17 s2
s5−2.9 s4+ 2.7 s3 −0.86 s2
transfer function of input vroll to rudder is:
ζ vroll =
0.85 s5−2.4 s4+ 2.3 s3−0.73 s2
s5−2.9 s4+ 2.7 s3 −0.86 s2
transfer function of input 2 (vpitch) to inboard right and left ailerons:
ξi vpitch =
−0.003 s5−0.006 s4−0.003 s3
s5−2.9 s4+ 2.7 s3−0.86 s2
transfer function of vpitch to outboard right and left ailerons:
ξo vpitch
= −0.0008 s5−0.0015 s4−0.0007 s3
s5−2.9 s4+ 2.7 s3−0.86 s2
transfer function of input vpitch to rudder:
ζ vpitch =
−7x10−22 s2−7x10−22 s
s5 −2.9 s4+ 2.7 s3−0.86 s2 = 0
transfer functions of input 3 (vyaw) to inboard right aileron
ξir vyaw
= −0.69 s5+ 2 s4−1.9 s3+ 0.62 s2
s5−2.9 s4+ 2.7 s3−0.86 s2
transfer function of input vyaw to inboard left aileron
ξil vyaw
= 0.69 s5 −2 s4+ 1.9 s3−0.62 s2
s5−2.9 s4+ 2.7 s3−0.86 s2
transfer function of input vyaw to outboard right aileron
ξor vyaw
= −0.56 s5+ 1.6 s4−1.5 s3+ 0.46 s2
s5−2.9 s4+ 2.7 s3−0.86 s2
transfer function of input vyaw to outboard left aileron
ξol vyaw
= 0.56 s5−1.6 s4+ 1.5 s3 −0.46 s2
s5−2.9 s4+ 2.7 s3−0.86 s2
6.4 Flight Control Allocation System for multi-aileron aircraft 127
ζ vyaw
= −9.23 s5+ 26.3 s4−25.04 s3+ 7.94 s2
s5−2.9 s4+ 2.7 s3−0.86 s2
For each actuator, the related transfer functions were added together giving the total required deflections sent to the control surface. Figure 6.9 shows the transfer functions between components of the virtual control command and each actuator. To test the control laws, the following were assumed:
ϕ= 30 3 ≤ t ≤ 30 0 otherwise
Fig. 6.9 Summation of transfer functions for each effector.
The result of using a no-saturation DCA are analysed in terms of the control deflections and the response of the aircraft, as shown in Fig. 6.10 and Fig. 6.11. Outer ailerons have identical deflections with different signs. Similarly for the inner ailerons. The rate of change of deflection was acceptable, but the outer ailerons exceeded the maximum deflection limit (±30 deg), as expected, due to the non-constraint computations.
For aircraft response, although the rise time was satisfactory, there was a 16% steady state error for the requested rolling angle, even with high actuator deflections.
Fig. 6.10 Control surfaces’ deflections using non-saturated DCA.
Fig. 6.11 Aircraft response using non-constrained DCA.
Constrained Dynamic Control Allocation
In this computation method, the control input u(t) is calculated based on the virtual control command optimisation function, using the weighted least-squares solver [144]. Thus, in order to give more emphasis on the virtual control command part that generates the rolling motion, the weights corresponding to the pitch and yaw motion are kept small, W v = diag([10 1 1]). Results of using constrained DCA technique are shown in Fig. 6.12 for controls deflections and Fig. 6.13 for aircraft response. The deflection plot shows a higher rate of change (reaching 20 deg/s) in the controls compared to the unsaturated computation, yielded in faster aircraft
6.4 Flight Control Allocation System for multi-aileron aircraft 129
response. Moreover, the deflections were constrained by the maximum deflection position. It is shown that when one actuator saturates, the other actuators margin of deflection is exploited.
Fig. 6.12 Control surface deflections using constrained DCA.
The aircraft response is fast (rise time is 2.8 second) and well damped, resulting in a relatively small steady state error (<3%).
Fig. 6.13 Aircraft response using dynamic control allocation.
The constrained dynamic control allocation has the following merits over the non-saturation case. First, it has very small steady state tracking error in the aircraft response. Moreover, it takes position limits into account and utilises the optimised computation for rate limits, which produces a quicker and more accurate response.
Taking these limits into account in the design process is advisable because they will comply with actuator dynamics in real-life applications. In the next section, a fault-tolerant control allocation scheme will be developed based on the constrained control allocation approach.