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8/18/2019 Design of Robust Pitch Controller for an Aircraft Autopilot
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Design of Robust Pitch Controller foran Aircraft Autopilot
Zeashan H. Khan, Faisal Saud,Iftikhar Makhdoom & Naveed Ur Rehman
N E S C O M
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Motivation
Dynamic model of an airplane (derived from the flightmechanics equations) does not perfectly represent
the behavior of the real aircraft.
It is necessary to deal with the associated modeluncertainties.
Model perturbations (inside the control BW)
High frequency unmodeled/neglected dynamics(beyond the control BW)
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Plant Model Longitudinal model of a medium category UAV
Flight condition is 60 Kts airspeed & 1000 ft altitude Longitudinal states are Forward speed (u), Downward speed (w), Pitch
rate (q), Pitch angle (θ) and Height (h).
Control inputs are Elevator deflection (δe) and throttle (δt)
Output is Pitch angle (θ) In state space is as follows:
[ ] [ ]0 0&0 1 0 0 0
0 0
0 0
9.6551- 28.3890-
0 5.0120-
193.1017 0.2593
0 30.8680 0 0.9988- 0.0496
0 0 1.0 0 0
0 0.0482 9.7514- 1.1940- 0.0563-
0 0.4766- 28.5681 2.9936- 0.4871-
0 9.7951- 1.4781- 0.4761 0.1131-
==
=
=
DC
B A
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Poles of the plant Two dynamics are represented in PZ map
Phugoid, the lower frequency dynamics has poor damping while Short period, the higher frequency dynamics has good damping
P1 = -6.3598 ±4.751i (SP)
P2 = -0.0693 ±0.2403i (PH)
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Singular Values of OL Plant
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Robust StabilizationMcFarlane Glover LSDP
Modern H∞ optimization approach
Incorporate simple performance/robustness tradeoff
Based on concepts from classical Bode plot methods
Multivariable
Robust-stability guaranteed in face of plant perturbationsand uncertainties
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Robust Stabilization Classical gain and phase margins are unreliable indicators
of robust stability when defined for each channel (or loop),
taken one at a time, because simultaneous perturbations inmore than one loop are not then catered for.
It is common to model uncertainty by norm boundeddynamic matrix perturbations.
Robustness levels can then be quantified in terms of themaximum singular values of various closed loop transferfunctions.
Consider the stabilization of a plant G, which has a
normalized left coprime factorization:G = M-1N-------- (1)
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H∞∞∞∞ robust stabilization problem
M ∆
1− N
K
N ∆
u y
+
+
+ -
φ
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H∞∞∞∞ robust stabilization problem
)()( 1 N M p N M G ∆+∆+= −
A perturbed plant model can then be written as [1].
where ∆M and ∆N are stable unknown transfer functions, whichrepresent the uncertainty in the nominal plant model G. The
objective of robust stabilization is to stabilize not only the
nominal model G, but also a family of perturbed plants defined
by
{ }ε
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where ε >0 is the stability margin. To maximize this stabilitymargin is the problem of robust stabilization of normalized
coprime factor plant description.The stability probability is robust if and only if the nominal
feedback system is stable and
H∞∞∞∞ robust stabilization problem
ε γ
1)( 11 ≤−
=∞
−−∆
M GK I I
K
where γ is the H∞ norm from ϕ to [u y]’ and is the sensitivityfunction for this feedback arrangement.
------- (4)
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The lowest achievable value of γ and the corresponding maxstability margin ‘ε ‘ are given by Glover and McFarlane as [1]:
H∞∞∞∞ robust stabilization problem
{ } 2/12/121maxmin ))(1(||][||1 XZ M N H ρ ε γ +=−== −−
where || . ||H denotes Hankel norm, denotes the spectralradius (maximum eigenvalue), and for a minimal state space
realization (A, B, C, D) of G, Z is the unique positive definite
solution to the algebraic Ricatti Equation (ARE).
(A-BS-1DTC)Z+Z(A-BS-1DTC)T-ZCTR-1CZ+BS-1BT = 0 --- (6)
where R = I+DDT, S = I+DTD
--- (5)
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H∞∞∞∞ robust stabilization problemwhere X is the unique positive definite solution of the
following ARE:
(A-BS-1DTC)TX+X(A-BS-1DTC)-XBS-1BTX+CTR-1C = 0 --- (7)
γ ≤−
∞
−− 11)( M GK I I
K
For a specified γ >γ min is given by
XZ I L
X BC DS F
D X B
ZC L DF C ZC L BF A K
T T
T T
T T T T S
+−=
+−=
−
+++=
−
−−
)1(
)(
)()()(
2
1
1212
γ
γ γ
Notice that the formulas simplify considerably for a strictly
proper plant, i.e. when D = 0. A controller (the "central"
controller in McFarlane and Glover), which guarantees that
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H∞∞∞∞ Loop ShapingThe loop shaping design procedure is based on robust stabilization combined
with the classical loop shaping, as proposed by McFarlane and Glover [1].
Step:1 Augmented the open loop plant with pre and postcompensators to give a desired shape to the singular valuesof the open loop frequency response
W1 G W2
Augmented System
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W1 and W2 chosen so weighted plant has “good” shape
high gain
at lowfreq
Low gain at
high freqSingular
values close
at cross over
Roll-off < 20 dB/dec
max sing. value
min sing. value
freq S i n g u l a r v a l u e s o f G
s ( d B )
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H∞∞∞∞ controller design
Step:2 The resulting shaped plant is robustly stabilized with
respect to coprime factor uncertainty using optimization. Animportant advantage is that no problem dependentuncertainty modeling or weight selection is required in thesecond step
G(s) W 2 (s)W 1(s)
K s(s)
optimal
controller
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G(s)
W 1 K s(s) W 2
Step 3
Final controller K(s) = W 1 .K ∞∞∞∞ .W 2
K(s)
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Controller
Design index value γ = 1.48, which indicates a good design
W1= (26 s + 5)/(52 s + 1)W2=1
H∞ controller is designed using MATLAB.
[ ] [ ]0& 0.5965 0.0000 0.9847- 0.0524- 0.0158 0.0004
0.3610- 1.4053
20.27 125.69-
1.3361 4.7656-
18.9098 3.3822-
13.7346 31.404-
383.8044-1.3361
0.1684- 0.0000- 1.6515 0.0131 0.0039- 0.3611-
0 0 94.8223- 0 0.9988- 20.3196
0 0 4.7656- 1.0000 0 1.3361
1.3720- 0.0000 17.7700- 10.5594- 0.9468- 23.6818
0.2401- 0.0000 34.3497- 28.4367 2.9540- 13.2486
0.2576 0.0000- 0.8414 0.2014- 0.0140 480.3555-
==
=
=
DC
B A
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Closed loop SV
Figure: 3 SV of plant with weighting functions Figure: 4 SV of closed loop plant
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Pitch angle controlPI controller
(G.M = 16.2 dB & P.M = 153 deg)
Figure: 5 Step response of closed loop Gthe2de Figure: 6 Bode plot of closed loop Gthe2de
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Response with Disturbance
Figure: 7 Simulink model for disturbance injection
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Comparison
Figure: 8 Comparison of PID and controller
∞ H 0 10 20 30 40 50
0
2
4
6
Time(sec)
d i s t u r b a n c e ( d e g )
0 10 20 30 40 50-20
-10
0
10
Time (sec)
y ( d e g )
PI controller
H∞ controller
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References
[1] D.C McFarlane and K. Glover, “A loop shaping designprocedure using synthesis”, 1992.
[2] Mangiacasale, Flight Mechanics of a µ Airplane, Milano,Italy.
[3] Magni, Bennani & Terlouw, Robust Flight Control: Adesign Challenge, Garteur.
[4] Robust Control Toolbox, Mathworks Inc.
[5] Ferreres, A practical approach to Robustness Analysiswith Aeronautical applications
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Thank you!