The other parameters show that the tracking performance is acceptable until 15%
uncertainty level which coincides with the probability of instability and saturation bounds.
However, this 15% level was obtained considering the worst parameter combinations (the lower and upper bounds are the worst possible signal realizations). Simulating the MM tracker with the three real parameter sets from table 1.1 and 5.1 shows that really good tracking performance and also stability can be achieved with higher level of uncertainties also (see figure 5.5, for an enlarged transient see Appendix 8.22.3). To be honest it should be noted that neither noises and saw disturbance, nor estimator time delay are considered in this simulation. But the original 15% level meansTe= 0.036s ≈0.04s which is about two time step delay for the 50Hz estimator and tracking was good until this level with worst case parameters, noises and saw signal together.
15 20 25 30 35 40 45
−20
−10 0 10 20
φ [deg]
Time [s]
Tracking of roll angle and yaw rate Reference MM_mod1 MM_mod2 MM_mod3
15 20 25 30 35 40 45
−500 0 500
r [deg/s]
Time [s]
Reference MM_mod1 MM_mod2 MM_mod3
Figure 5.5: Good tracking results for the three different identified models (minimax case) As a summary, it can be stated that the minimax tracker has poorer system robustness and acceptable performance properties than the LQ tracker. This can be because of the interconnection with the open loop disturbance estimator. However, acceptable simulation performance was experienced until 15% uncertainty level for both LQ and MM tracker so, the overall performance of the two methods is the same, but the MM tracker is disturbance tolerant, which is a large advantage.
In the next section real flight test results and comparison with the PID control solution will be presented.
Chapter 6
Real flight test results and
comparison with the PID solution
Both the LQ and the minimax tracker are tested in real flight onboard the Ultrastick UAV and compared to the PID solution originally implemented onboard (see [70]). The attitude estimator developed in chapter 4 is used in all tests. The same roll doublet reference signal as in figure 2.3 was tracked several times meanwhile the yaw damper was also applied. The weighting for the LQ tracker was exactly the same as the best weighting in the Matlab simulations.
Q1 =<100, 0×7>, Q2 =<2000, 2>, R=<5·103, 5·104 >
For the minimax tracker on 13th September 2011. the weighting was exactly the same as the best in the Matlab simulations, however, this solution was too slow, that’s why the roll angle tracking error weight was increased to 600 and the rudder weight was increased to 100000 in the next test. The latter was increased to provide less oscillations in the yawing motion.
Q1 =<500, 0×7>, Q2 =<500, 2>, Ru =<21000, 30000>, Rd =I4·1012 For the minimax tracker on 19th July 2012. with improved performance it was:
Q1 =<500, 0×7>, Q2 =<600, 2>, Ru =<21000, 100000>, Rd=I4 ·1012 All the control codes were implemented on the same PhyCORE MPC555 32 bit, 40MHz floating point system on module which was used by [70]. Paw Yew Chai experi-enced that this module can work with a maximum of 10 to 15 dimensional controller state spaces. This shows the possible computational burden of microcontroller architecture.
The numerical test results are summarized in table 6.1 considering the two configura-tions of the minimax tracker together, because they gave similar results.
Ts1 andTs2 are the settling times to the 95% to 105% range. Eφand Er are the mean absolute values of the roll angle and yaw rate tracking errors. Ep, Eδa and Eδr are the normalized signal two norms of the roll rate, aileron and rudder deflections defined as:
ksk= 1 N
vu ut
XN i=1
s2i
Where i is the time index. The normalization by N (the number of samples) is required because now signals with different lengths should be compared (the length of each test case depends on decision of the RC pilot) . In the table, the minimum, maximum and mean values (from several cases) of each parameter are included.
Table 6.1: Real flight tracking parameters
Value Method Ts1 [s] Ts2 [s] Eφ[◦] Er[◦/s] Ep[◦/s] Eδa[◦] Eδr[◦]
MIN 0.68 0.34 1.78 5.03 0.87 0.06 0.01
MAX PID 2.66 1.16 4.56 7.92 1.62 0.13 0.14
MEAN 1.213 0.72 3.348 6.5 1.19 0.088 0.032
MIN 0.64 0.68 3.04 5.57 0.86 0.05 0.02
MAX LQ 4.5 2.08 4.78 9.06 1.53 0.2 0.03
MEAN 2.44 1.07 3.98 7.267 1.179 0.0855 0.027
MIN 0.72 2.6 5.52 3.92 0.43 0.05 0.06
MAX MM 4.98 4.92 7.6 9.25 4.77 0.23 0.13
MEAN 2.9 3.73 6.263 5.88 1.248 0.097 0.087
Considering the table the settling times are larger both with the LQ and minimax tracker. The mean value of the φ tracking error is about the same with the LQ, but much larger with the MM method. This clearly shows that the MM method is slower, as experienced also in the simulation tests. Regarding the error of yaw damping, the LQ mean is worse, but the MM mean is better than the PID mean. This can be closely related to the used rudder control energy. The LQ method uses the less, while the MM method uses the most rudder energy. So, the yaw damping with the MM method is better, but it uses more control energy. Considering the energy in the roll rate, the methods are about the same, the MM method uses a bit more energy. This also can be seen considering the aileron control energy.
The plots of the flight test results can be seen in figures from 6.1 to 6.12. Every plot includes time histories from several flights. Enlarged figures can be found in Appendix 8.23. Notice that in the LQ tracker cases the aircraft φ angle is a bit below 20◦ while around −20◦. So, it tracks the negative angle easier. This is because of the negative torque disturbance resulting from electrical motor and propeller reaction torque which is negative with a propeller rotating to the right. So, this shows the worse disturbance rejection properties of the LQ tracker. As a summary it can be stated that all two methods are applicable on a real system with acceptable performance. Comparing them to the µ synthesis based roll angle tracking controller in [70] which was tested also on Ultrastick aircraft with MPC555 microcontroller they give better results. They track the non smoothed doublet signal better than that controller tracks the smoothed one. The real flight performance of that controller is shown in figure 6.13.
Finally, a few words about the possible further application of the algorithms. The developed roll angle tracking and yaw damping algorithms can be used in aircraft waypoint
and trajectory tracking tasks. In such problems the aircraft turn can be initiated by the banking of the aircraft through its roll angle. Waypoint and trajectory tracking algorithms were also developed and published by the author in [Bau11] and [BB12]. Two representative figures about waypoint and spatial (sinusoidal) trajectory tracking can be seen in 6.14 and 6.15.
6.1 Summary
This chapter compares real flight test results obtained with the LQ and MM trackers onboard the Ultrastick test aircraft with results obtained from a PID tracking solution (the same solution as is used in the Matlab simulation tests).
0 5 10 15
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with PID control
Figure 6.1: Tracking of the φ doublet ref-erence signal on 10th December 2009. PID control
0 5 10 15 20
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with PID control
Figure 6.2: Tracking of the φ doublet ref-erence signal on 17th June 2010. PID con-trol
0 5 10 15
−30
−20
−10 0 10 20 30
r [deg/s]
Time [s]
Tracking of yaw rate with PID control
Figure 6.3: yaw damping on 10th Decem-ber 2009. PID control
0 5 10 15 20
−150
−100
−50 0 50 100 150
r [deg/s]
Time [s]
Tracking of yaw rate with PID control
Figure 6.4: yaw damping on 17th June 2010. PID control
0 5 10 15 20
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with LQ tracker control
Figure 6.5: Tracking of the φ doublet ref-erence signal on 13th September 2011. LQ tracker control
0 5 10 15 20
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with LQ tracker control
Figure 6.6: Tracking of the φ doublet ref-erence signal on 29th September 2011. LQ tracker control
0 5 10 15 20
−40
−30
−20
−10 0 10 20 30 40
r [deg/s]
Time [s]
Tracking of yawrate with LQ tracker control
Figure 6.7: yaw damping on 13th Septem-ber 2011. LQ tracker control
0 5 10 15 20
−40
−30
−20
−10 0 10 20 30
r [deg/s]
Time [s]
Tracking of yaw rate with LQ tracker control
Figure 6.8: yaw damping on 29th Septem-ber 2011. LQ tracker control
0 5 10 15 20 25
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with MM tracker control
Figure 6.9: Tracking of the φ doublet refer-ence signal on 13th September 2011. MM tracker control
0 5 10 15 20
−30
−20
−10 0 10 20 30
φ [deg]
Time [s]
Tracking of roll angle with MM tracker control
Figure 6.10: Tracking of the φdoublet refer-ence signal on 19th July 2012. MM tracker control
0 5 10 15 20 25
−30
−20
−10 0 10 20 30 40
r [deg/s]
Time [s]
Tracking of yaw rate with MM tracker control
Figure 6.11: yaw damping on 13th September 2011. MM tracker control
0 5 10 15 20
−60
−40
−20 0 20 40 60
r [deg/s]
Time [s]
Tracking of yaw rate with MM tracker control
Figure 6.12: yaw damping on 19th July 2012. MM tracker control
Figure 6.13: Tracking performance of a µ synthesis based controller from [70] page 138.
figure 6.18 (published with the permission of the author)
Figure 6.14: Waypoint tracking Figure 6.15: Sinusoidal trajectory tracking
Chapter 7
New scientific results
Thesis 1I formulated and solved the problem of finite horizon, LQ optimal output track-ing for DT, LTI systems with Lagrange multiplier method in a more detailed way then published in literature. This detailed solution gives deeper insight into the effect of refer-ence signal on the system which makes it possible to derive a nice infinite horizon solution in Thesis 2. This solution could not be derived without knowing the obtained detailed structure of the costate variable.
I extended the finite horizon output tracking problem to the minimax case, when system disturbances are also considered in the functional to be minimized. A structurally similar, but otherwise different solution is derived.
My publications related to Thesis 1: [BKB08a], [P´et08], [P´et09], [BKB09b], [BKB09a].
Thesis 2I solved the problem of infinite horizon, LQ optimal, causal reference tracking in a LQ optimal way for constant, and in a LQ sub-optimal way for time-varying references, considering DT, LTI causal systems without deterministic disturbances (2.1). The solution is based on the finite horizon result of Thesis 1and considers reference signal extrapolation to guarantee causality and make the stated problem solvable. This extrapolation causes the sub-optimality in case of time-varying references.
My publications related to Thesis 2: [BRSB08], [P´et08], [Bau08], [P´et09], [BKB08a], [BB11b].
Thesis 3 I derived solvability conditions for the Discrete Algebraic Riccati Equation (DARE) resulting in the infinite horizon LQ tracking problem from the functional (2.11).
My publications related to Thesis 3: [BKB08b].
Thesis 4I solved the problem of infinite horizon, LQ optimal, causal minimax reference tracking in a LQ optimal way for constant, and in a LQ sub-optimal way for time-varying references and disturbances, considering DT, LTI causal systems with deterministic dis-turbances (3.1). The solution is based on the finite horizon minimax result of Thesis 1 and considers reference signal extrapolation to guarantee causality and make the stated problem solvable. This extrapolation causes the sub-optimality in case of time-varying references.
My publications related to Thesis 4: [BKB09a], [BKB09b], [P´et10b], [BB11c].
Thesis 5 I developed a multi-mode attitude estimation Extended Kalman Filter. It switches between the measurements and uses wind estimation and correction to obtain as accurate estimates as possible. It calculates the integral of quaternion dynamic equa-tion with a closed form soluequa-tion of the Heun formula which can further improve accuracy.
My publications related to Thesis 5: [P´et10a], [BB10], [BB11a].
Chapter 8 Appendix
This appendix contains the detailed derivation of the results published in the thesis, the detailed description of some variables and detailed test results.
8.1 The applied basic methods
The thesis and the underlying work is built upon the following methods and tools from systems and control theory.
LQ optimal (output tracking) control [3],[53]. considered in case of LTI discrete time systems with the following system model:
xk+1 =Axk+Bu˜k
ykr =Crxk
yk =Cxk
(8.1)
The goal of design is to follow a prescribed reference signal r with the reference output yr by minimizing the following functional:
J(x,x,˜ u) =˜ 1 2
N−1X
k=0
(xk−x˜k)T Q(xk−x˜k) + ˜uTkRu˜k
+ + (xN −x˜N)T Q(xN −x˜N)
where:
Q=CTQ1C+CrTQ2Cr C =
I −CrT CrCrT−1
Cr
˜
xk =CrT CrCrT−1
rk=Hrk
(8.2)
The standard Lagrange multiplier method can be used to minimize this functional by deriving the costate variable at first for finite, then for infinite horizons (if exists).
LQ optimal minimax (output tracking) control [13],[49]. considered in case of LTI discrete time systems with the following system model:
xk+1 =Axk+Bu˜k+Gdk
ykr =Crxk
yk =Cxk
(8.3) The goal of design is to follow a prescribed reference signalrwith the reference outputyr together with attenuating the disturbanced and by minimizing the following functional:
J(x,x,˜ u, d˜ k) =
= 1 2
N−1
X
k=0
((xk−x˜k)T Q(xk−x˜k) + ˜uTkRuu˜k−
−γ2dTkRddk) + (xN −x˜N)T Q(xN −x˜N) where:
Q=CTQ1C+CrTQ2Cr
C =
I−CrT CrCrT−1
Cr
˜
xk =CrT CrCrT−1
rk =Hrk
(8.4)
Here also the standard Lagrange multiplier method can be used to minimize the functional by deriving the costate variable at first for finite, then for infinite horizons (if exists).
Extended Kalman Filter for nonlinear systems [42]. Considering the following nonlinear system (linearized around the actual state):
˙
x=f(x) → Ak = ∂f(x)
∂x xk
y=h(x) → Ck= ∂h(x)
∂x xk
xk+1 =Akxk+Vkvk yk=Ckxk+wk
E{vk}= 0, E{wk}= 0, Nk =E{vkvkT}, Wk=E{wkwTk}
(8.5)
The goal is to obtain the unbiased, minimum variance estimation of the system states from the measured outputy. This goal can be achieved by the following algorithm:
xk+1 =f(ˆxk)
Pk+1=AkPkATk +VkNkVkT
Kk+1 =Pk+1Ck+1T Ck+1Pk+1Ck+1T +Wk+1
−1
ˆ
xk+1 =xk+1+Kk+1(yk+1−h(xk+1)) P = (I−K C )P
(8.6)
Here ˆxis the estimated state of the system andP is the state estimation error covari-ance matrix (Pk=E{(xk−xˆk)(xk−xˆk)T}).