Design of Robust Pitch Controller for an Aircraft Autopilot

8/18/2019 Design of Robust Pitch Controller for an Aircraft Autopilot

1/23

Design of Robust Pitch Controller foran Aircraft Autopilot

Zeashan H. Khan, Faisal Saud,Iftikhar Makhdoom & Naveed Ur Rehman

N E S C O M


2/23

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)


3/23

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


4/23

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)


5/23

Singular Values of OL Plant


6/23

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


7/23

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)


8/23

H∞∞∞∞ robust stabilization problem

M ∆

1− N

K

N ∆

u y

+

+

+ -

φ


9/23


)()( 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


10/23

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


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)


11/23

The lowest achievable value of γ and the corresponding maxstability margin ‘ε ‘ are given by Glover and McFarlane as [1]:


{ } 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)


12/23

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


13/23

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


14/23

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 )


15/23

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


16/23

G(s)

W 1 K s(s) W 2

Step 3

Final controller K(s) = W 1 .K ∞∞∞∞ .W 2

K(s)


17/23

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


18/23

Closed loop SV

Figure: 3 SV of plant with weighting functions Figure: 4 SV of closed loop plant


19/23

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


20/23

Response with Disturbance

Figure: 7 Simulink model for disturbance injection


21/23

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


22/23

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


23/23

Thank you!

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Design of Robust Pitch Controller for an Aircraft Autopilot

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