4.5 Simulation Study
4.5.2 Nonlinear Model
86 4 Altitude Control of Helicopters with Unknown Dynamics
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
−10
−8
−6
−4
−2 0 2
time [s]
observer error [ft/s]
0 0.2 0.4 0.6 0.8 1
−0.2
−0.1 0 0.1
time [s]
tracking error [ft]
= 0.001 = 0.01 = 0.02
= 0.02
= 0.01
= 0.001
Fig. 4.3 Effect ofon tracking errors and observer errors 2
4Pd Rd
«d 3 5D
2
4 0 1 0
0 0 1
2; 00042036 3 5
2 4d
d
d 3 5C
2 4 0
0 2; 000
3 5ref;
ref.t /D 8ˆ ˆˆ ˆˆ
<
ˆˆ ˆˆ ˆ:
0 if 0t < 0:5 0:1 if 0:5t < 2 0 if 2t < 3 0:1if 3t < 4:5
0 if t 4:5
: (4.108)
It can be seen in Fig.4.4 that the tracking performance of our controller is comparable with that of [38] for the pitch tracking task. From Fig.4.5, we see that the control input and neural weights are bounded. Similar to the results in Sect.4.5.1.1, the control of [38] exhibits more fluctuations, and the neural weights are larger, illustrating the different mechanisms at work in the two control schemes.
0 1 2 3 4 5
−0.1
−0.05 0 0.05 0.1
time [s]
pitch [rad]
by [52]
by the proposed control
desired
Fig. 4.4 Comparison of pitch tracking performance between the proposed control and that of [38]
0 1 2 3 4 5
−0.2
−0.1 0 0.1 0.2 0.3
control position [rad]
1 2 3 4
0 0.005 0.01 0.015 0.02
norm of neural weights 0 50
0.5 1 1.5 2
time [s]
by [52]
by the proposed control
by [52]
by the proposed control
Fig. 4.5 Comparison of control input and norm of neural weights, given by
qk OWTk2C k OVk2F, between the proposed control and that of [38]. For the bottom graph, the left scale corresponds to the proposed control while the right scale corresponds to that of [38]
of the operating point, for which the linearized model is valid. Nevertheless, it was demonstrated that the proposed adaptive NN control is effective for the linear models considered, which constitute a subclass of general non-affine systems considered in the control design. In this section, we extend the investigations further
88 4 Altitude Control of Helicopters with Unknown Dynamics to the case of a nonlinear helicopter model. Both full-state and output feedback cases are considered.
Consider the nonlinear model of the X-cell 50 model helicopter in vertical flight [48, 93]:
P x1D x2
P
x2D a0Ca1x2Ca2x22C.a3Ca4x4p
a5Ca6x4/x32 P
x3D a7Ca8x3C.a9sinx4Ca10/x32Cath P
x4D x5
P
x5D a11Ca12x4Ca13x23sinx4Ca14x5K2u
yD x1; (4.109)
where x1 denotes altitude; x2 denotes altitude rate; x3 denotes rotor speed; x4
denotes the collective pitch angle; x5 denotes the collective pitch rate; ath D 111:69s2 is a constant input to the throttle; and u is the input to the collective servomechanisms. The parameters are
K1D 0:1088s2 K2D0:25397s2a0D 17:67ms2 a1D 0:1s2 a2D 0:1s2 a3D5:31104 a4D1:5364102 a5D2:82107a6D1:632105 a7D 13:92s2 a8D 0:7s2 a9D 0:0028 a10D 0:0028 a11D434:88s2 a12 D800s2 a13D 0:1 a14D 65s2:
(4.110)
Let outputybe the altitudex1. By restricting the throttle input to be constant, we obtain a SISO system in which u is the only input variable forcing the outputy to track a desired trajectoryyd.t /, which we define as
yd.t /D5:50:5sint: (4.111)
It can be shown that the system has strong relative degree 4, with the subsystem given by
P1D2Dx2
P2D3Da0Ca1x2Ca2x22C.a3Ca4x4p
a5Ca6x4/x32 P3D4D.a1C2a22/3Ca3x23C
a4 a6
2p
a5Ca6x4
x23x5
C2.a3Ca4x4p
a5Ca6x4/x3.a8x3C.a9sinx4Ca10/x32/
P4Db.x/Cg.x/u2; (4.112)
where
g.x/D K2x32
a4 a6
2p
a5Ca6x4
: (4.113)
The derivation ofb.x/in (4.112) is omitted, and we proceed to verify that the system indeed satisfies the assumptions supposed in the control design. Assump- tions4.1and4.2are obviously satisfied from (4.108) and (4.112) respectively.
To verify Assumption4.4, we first note, from a practical standpoint, that the collective pitch angle,x4, is restricted within a range, typically from 0 to 0.44 rad [83]. It can be verified that the bracketed terms in (4.113) are virtually constant:
they take values in the rangeŒ1:4; 1:5103. Thus, the control coefficientg.x/
in (4.113) is always negative. Together with the fact that rotor speedx3is nonzero during flight, it can be concluded that there does not exist any control singularities or zero crossings ofg.x/. Therefore, the first part of Assumption4.4is satisfied.
Remark 4.22. Although the second part of the assumption, thatg.x/ > 0, does not correspond to this example, there is no loss of applicability of the theoretical results, as explained in Remark4.14. The control is still valid under a simple change of sign, i.e., uD unnub.
Lastly, it is not difficult to verify the existence of a function
g0.x/D2 0
@ja8j C ja9sinx4Ca10jjx3j C
a6 4.a5Ca6x4/1:5
2
a42paa6
5Ca6x4
1 A> 0;
8x32RC; x42Œ0; 0:44;
(4.114) which fulfils Assumption4.5for the case ofg.x/ < 0. Note that this function need not be known; we only need to show its existence.
The control parameters are chosen ask1 D 2:0,k2 D 3:0,k2 D4:5,k4 D5:5, kb D 0:6, while the NN parameters are D 0:01, W D 50I, V D 20:4I, V D0:055andW D0:05. For the high gain observer, we chooseD5104, 1 D 4,2 D 6,3 D 4,N2 D 5104,N3 D 5108, andN4 D 11011. The saturation limits of the control are˙400mrad. The initial conditions arex.0/D Œ5:2; 0; 95:36; 0; 0T,WO D0, andVO D0.
From Fig.4.6, it can be seen that good tracking performance is achieved by the proposed adaptive NN control. The tracking performance for the full-state and output feedback cases are similar for the choice of made. The initial error is efficiently reduced and the altitude trajectory lies in close proximity of the desired sinusoidal trajectory. We compare the performance of the NN controller with a linear PD controller
upd DKp.yyd/CKd.yP Pyd/; (4.115)
90 4 Altitude Control of Helicopters with Unknown Dynamics
0 5 10 15 20
5 5.2 5.4 5.6 5.8 6 6.2
time [s]
altitude [m]
PD (dashed)
desired(dash-dotted)
output feedback(dotted) full−state feedback(solid)
Fig. 4.6 Comparison of tracking performance between adaptive NN and PD control for nonlinear helicopter model
whereKp D5;000rad andKd D 500rad s are chosen so that the tracking errors are reasonably small and the control magnitude is constrained tojupdj 400mrad.
Although steady state errors are comparable between PD and NN control, the PD control gives poorer transient performance as it attempts to compensate for the initial error, due to the inability of the linear PD control to adequately compensate for the effects of nonlinearity and coupling. Clearly, a dynamic model compensator is essential to achieve better performance.
The boundedness of the control input and the neural weights, for full-state and output feedback NN control, as well as the PD control, are shown in Fig.4.7.
The size of input signal under PD control is much larger than that under NN control, as seen by the fact that the PD control signal initially fluctuates between the saturation limits. This can be explained by the fact that a large PD control gain is required to compensate for nonlinearities, thus amplifying the control effort greatly when the initial error is large.
In Fig.4.8, it is shown that the rotor speed and collective pitch angle, for both full- state and output feedback NN control, are bounded. In particular, it is confirmed that the collective pitch angle remains in the regionŒ0; 0:44rad as restricted in practical operations.
0 2 4 6 8 10
−500 0 500
control input [mrad]
0 2 4 6 8 10
0 20 40 60
time [s]
norm of neural weights
PD
output feedback
full-state feedback
Fig. 4.7 Top: Control signals under adaptive NN and PD control. Bottom: Norm of neural weights, given by
qk OWTk2C k OVk2F, under full-state and output feedback NN control
0 2 4 6 8 10 12 14
0.21 0.215 0.22 0.225 0.23
time [s]
collective pitch [rad]
0 2 4 6 8 10 12 14
95.3 95.4 95.5
rotor speed [rad/s]
output feedback full-state feedback
Fig. 4.8 Top: Speed of rotor. Bottom: Collective pitch angle
92 4 Altitude Control of Helicopters with Unknown Dynamics