5.3 Application of Aerodynamic Mechanical Power to Thrust Computa-
5.3.2 Thrust control
The proposed hierarchical controller for regulation of thrust is shown in Figure 5.6. The feedforward terms are included to limit the offsets associated with the simple proportional gains in the control architecture. In the motor model, M1 is a combina-
tion of (5.3a) and (5.3c) and models the non-linear dynamics relatingva andia. M2
§5.3 Application of Aerodynamic Mechanical Power to Thrust Computation and Control115
Algorithm 1Thrust Computation
1: Datac0,c1, c2,c3,c4,d0,d1,N,∆,e.
2: Local stateold[λsk].
3: For each measurementCPam(t) =
ˆ
Pam
v3 at timet. 4: Setk=1; Set λs1=old[λsk]−∆
5: fork=1 . . .N do
6: ifk = 2then; Setλs2 =old[λsk];
7: Use (5.11) to compute λi;
8: Use (5.8) to computeCT;
9: Use (4.64) to computeκ;
10: Use (5.11) and (5.9) to computeCPam;
11: Compute f(λsk) =CPam(t)−CPam;
12: ifk >2and |f(λsk))− f(λsk−1)|<ethen break
13: Computeλsk+1= λsk− f(λsk) λ s k−λsk−1 f(λs k)−f(λsk−1);return 14: Setold[λsk] =λsk; 15: OutputT =CTv2;
given by (5.2b) and M3 is the non-linear model that maps Pam to T as outlined in
Section 5.3.1.
The low-level current controller is crucial in generating fast v response using high-gain inner-loop control. This avoids the need for exceptionally high gains in the outer-loop where noise in the estimated feedback signals for thrust become a prob- lem. Similar to the aerodynamic mechanical power controller, the low-level current controller is defined by va =vf f(Td)−K1p ia−ida , (5.12)
where vf f(Td)is a feedforward voltage defined by f1(Td)in Figure 5.8. In a similar
manner to the aerodynamic mechanical power controller, a negative (unstable) gain −K1pin the control design is proposed. This is a key aspect of the control design and is important in generating the desired rise time of the full system. To understand the role of the gain, consider the linearisation of the inner-loop current control around some fixed constant thrust condition in static free air derived in Appendix A with closed-loop transfer function given by
H1(s) = ia va = K 1 p KeKq+Ra(Irs+δ) Irs+δ−K1pKeKq+Ra(Irs+δ),
where δ is a damping factor associated with the linearisation of the aerodynamic torque and Kq = Kq0−Kq1iˆa
(3.9). From the stability condition derived in Ap- pendix A given by (A.2),K1
pis such thatK1p ≤ KeKqδ+Raδ, then the poles of this transfer function can be placed close to the imaginary axis ensuring fast rise time with very
Table 5.2: Feedforward terms.
Term Polynomial R2
f1(Td) −0.2245Td2+2.4835Td+0.6416 0.9939
f2(Td) 0.8949Td−0.5660 0.9930
high overshoot of the current response. A large overshoot in the current response provides the surge of power necessary to spin up (or spin down) the rotor. In order to prevent over-current, the computed voltage is saturated based on the maximum current through the ESC and the instantaneous v measurement using (2.6). Since the actual control design is a pure proportional control and the inherent current dy- namics are stable, the saturation will not destabilise the system. Of course, if K1p is chosen too large, then the current response could actually become unstable. Based on the stability bound on K1
p in (A.2),K1p is chosen (K1p =0.01) such that the current
dynamics have a high stable overshoot with damping factor less than 0.1.
In the outer-loop, governing actual thrust control, a feedforward and proportional integral (PI) feedback controller is proposed for robust control Åström and Hagglund [1988]. The feedback controller design of this system is a straightforward linear de- sign once the inner current control loop is stabilised. The overall control architecture includes only a single integrator at the outer-level to avoid dynamic complexity. The stability analysis using linearisation about static hovering condition of the outer-loop controller is also presented in Appendix A.
The feedforward terms in the control architecture are obtained by considering steady state hover and static free air conditions (|~vs|=0) and letting∆=kai2a in (3.7)
to be dealt with by the integral term of the robust outer-loop proportional integral thrust controller. The feedforward term f1(Td) is derived from (2.6), which gives
1 K2 e v d a−idaRa2=v2d, hence, Td= CT K2 e vda−idaRa 2 .
Thus a quadratic model for f1(Td)is derived. To obtain f2(Td), consider the electrical
torque (τ = Kqia) and aerodynamic torque (τa = CQv2) at steady state (i.e. ˙v = 0) where CQ is the torque coefficient. Hence ia = CKQqv2. This current to rotor speed relationship is consistent with the model derived in Section 3.4 given by (3.9). From this, it is easily seen that Td = CTCKQqia, hence, a linear function relatingTd andia is
obtained for f2(Td).
Hence to effectively and robustly control thrust, the polynomial coefficients of
f1(Td),f2(Td)need to be identified using. These polynomial feedforward terms along
with their R2 values are summarised in Table 5.2. Furthermore, experimental and computed estimates are shown on Figure 5.9 for the motor-rotor system described in Section 3.4.
§5.3 Application of Aerodynamic Mechanical Power to Thrust Computation and Control117
Figure 5.8: The proposed thrust controller architecture.