Robustness of Variance Constrained Controllers For

Robustness of Variance Constrained Controllers For Complex, Control Oriented Helicopter Models Tugrul Oktay and Cornel Sultan, Senior Member, IEEE

Abstract—Complex helicopter models are used for variance constrained control design and robustness studies. The modeling process is briefly outlined and followed by output and input variance constrained control design. Extensive numerical investigations of closed loop systems eigenvalues variations and time responses indicate that these controllers are robustly stable for modeling uncertainties in flight speeds, inertial properties, initial conditions and noise intensities.

I. INTRODUCTION Helicopter control is a difficult problem. Technologically, aerodynamic forces and moments are generated using a sophisticated device called the swashplate (Fig. 1) which controls blade pitch angles. Blades’ motion must be correlated to achieve different pitch angles to compensate the “dissymmetry of lift” [1] and ensure straight level flight by counter-acting the tendency to roll and yaw if the blades have the same angle of attack. Blades’ attachments to the rotor hub must be soft to reduce the stress level and allow two fundamental motions: flapping and lead-lagging (Figs. 2, 3). Flapping (up-down motion) modulates the lift force according to the blade position in the rotor hub plane and alleviates the dissymmetry of lift, while lead-lagging (forward-backward motion) reduces the ground effect. The sophisticated aero-mechanics is very challenging for control because all the dynamics are coupled. Thus, for control design one should work with large, strongly coupled systems. Due to these difficulties, many helicopter control studies ignore the sophisticated modeling, using a single rigid body model [2, 3] and focusing on flight dynamics (fuselage) control with no attention paid to blade dynamics. Control design is very conservative to compensate for crude models. Further, this may lead to dangerous blade behavior due to the flight and blade dynamics coupling, which is not even captured by the model. This is especially the case for large helicopters whose blades are inherently flexible, their dynamics being neither decoupled nor ignorable. Here we use sophisticated models that capture the coupled fuselage and blade dynamics, are control oriented, physicsbased, and were extensively validated on realistic, Puma SA 330 helicopters [4]. We use these models for variance constrained controllers design, whose robustness is not guaranteed, so we perform a thorough robustness analysis.

Tugrul Oktay is with the Civil Aviation School, Erciyes University, Kayseri, Turkey, 38000, Turkey (e-mail: [email protected]). Cornel Sultan is with the Aerospace and Ocean Engineering Department, Virginia Tech, Blacksburg, VA, 24060 USA (e-mail: [email protected]).

Figure 1. Swashplate Mechanism

Figure 2. Blade Flapping Motion

Figure 3. Blade Lead-Lagging Motion

II. HELICOPTER MODEL Modeling is based on two key ideas. First, physics principles and appropriate assumptions are used to directly result in nonlinear ordinary differential equations (ODEs), without resorting to partial differential equations models. Second, flight and blade dynamics modes that are critical for safe and performant helicopter operation are captured. Multibody dynamics was used to include all helicopter components: fuselage, articulated main rotor with 4 blades, empennage, landing gear, tail rotor [4]. The key modeling steps are described next.

A. Fuselage Dynamics Newton-Euler equations were used to obtain the nondimensionalized helicopter force and moment equations (1), (2), where typical notations are used: uˆ , vˆ, wˆ and pˆ , qˆ , rˆ are nondimensional linear and angular fuselage velocities, IA , T A ,\ A are roll, pitch and yaw angles, : is main rotor’s angular speed, R and \ are blade length and

azimuth angle, I is inertia matrix, M a is helicopter’s mass. d d\ d d\ d d\

uˆ

qˆ wˆ

rˆ vˆ

vˆ

rˆ uˆ

pˆ wˆ

wˆ

pˆ vˆ

§ I yy pˆ qˆ rˆ ¨ d\ © I xx d\

qˆ

X

2

: R Ma

§ I zz

pˆ rˆ ¨ ¨

© I yy

§I rˆ pˆ qˆ ¨ xx d\ © I zz d

2

: R

g cos(T A ) sin(I A )

Y

2

: R Ma Z

2

: R Ma

I zz ·

· pˆ qˆ rˆ ¸ ¸ ¨ I xx ¹ I xx © d\ ¹ I xx ·

d

I

2 2 xz pˆ qˆ ¸¸ I yy ¹ I yy

I yy ·

I xz §

2

M

(2)

I yy :

· qˆ rˆ pˆ ¸ ¸ ¨ I zz ¹ I zz © d\ ¹ d

d LLF Faero

d LLF M aero

where U P

ª0 ¬

U U P UT ·¸ P ¹UT 2

T UT

§

1 ¨G a0 ¨ 0

©

U P UT

§

G 2 ¨T

©

2

º » 2 » · UP · » ¸ ¸» dx UT ¹ ¸»» ¹» » ¼ T

xR (dLLF Faero )III

and U T

4i (\ )

41c cos(\ i ) 4 sin(\ i ) 40 d

40

\

1s

1

i

(6)

(S / 2)(i 1) is the i-th blade azimuth angle,

E. Main Rotor Downwash, Inflow Modeling For control oriented modeling, linear static inflow model is selected. Its cyclic components, c , s , are [5] [p. 160]

15S

N

0

D. Multi-Blade Equations Blade flapping, lead-lagging and flapwise bending motions are described, ignoring higher harmonic terms, by

c

xR (dLLF Faero )II º¼

23

§F· ¸, ©2¹

tan ¨

s

0,F

1

§

· ¸ © wˆ O0 ¹

tan ¨

uˆ

(7)

component of the linear inflow, O0 , is computed numerically using the momentum theory (see [4]).

(3)

Figure 5. Wake Skew Angle

(4)

are perpendicular and tangential

components of air velocity to the blade leading edge, T the blade pitch angle, xR the location of a point on the blade, a0 the blade lift curve slope, G 0 , G 2 parasite and induced drag coefficients. By integrating (3) and (4) along the blade span and including flapping and lead-lagging spring and damper moments, single blade equations are obtained [4].

C. Blade Flexibility For flexibility modeling, blades are divided into rigid pieces connected by flapwise bending springs and dampers (Fig. 4).

Figure 4. Lumped System Modeling for Blade Flexibility

0

where F is the wake skew angle (Fig. 5). The uniform

B. Single Blade Aerodynamic Effects The infinitesimal aerodynamic force and moment acting on a blade strip in lead-lagging and flapping frame (LLF) are ª « « 2 JIb «« § ¨T UT 2 R3 « © « « « ¬

(5)

bending spring and E (\ ) the root flapping angle. An equivalent energy approach [4] was used to find spring stiffnesses for n = number of blade segments = 3.

2

I zz :

1, ..., n 1

k 1

4d are collective, two cyclic, and differential components.

L I xx :

¦ G (\ ) k , i

4 is any of the three angles mentioned above, 40 , 4c , 4s ,

2

I xz §

i

E (\ )

)

where G (\ ) k is the deflection angle of the k-th flapwise

where \ i

g cos(T A ) cos(I A )

: R

i 1 E (\

(1)

2

: R

qˆ uˆ

d

d

g sin(T A )

The flapping angle of the (i+1) blade segment is

F. Tail Rotor The tail rotor generates a control force, and the drag force due to tail rotor hub and shaft is also modeled [4]. After assembling all helicopter components, the nonlinear equations of motion were obtained in implicit form, 44

41

, x•\ the nonlinear state vector comprising fuselage states (linear and angular velocities, Euler angles) and blade states (flapping, leadf ( x , x ,u )

0 , with f • \

4

lagging, flexibility states), and u • \ the nonlinear control vector comprising two cyclic and a collective control for the main rotor and tail rotor force (see [4] for details). The nonlinear equations were further simplified using an ordering scheme [4], trimmed using “fsolve” in Matlab, linearized around these trims in Maple, and extensively validated for Puma SA 330 [4, 6]. See [7] for linear model matrices for some specific flight conditions and [8, 9] for more applications of our helicopter models.

III. VARIANCE CONSTRAINED CONTROLLERS Variance constrained controllers are improved linear quadratic Gaussian (LQG) controllers that guarantee output (OVC) or input (IVC) variance bounds satisfaction [10-12]. A. Output Variance Constrained Controller (OVC) For a continuous LTI, stabilizable and detectable plant x p Ap x p B p u w p , y C p x p , z M p x p v

(8)

and input penalty R ! 0 , a full order dynamic controller

xc

Ac xc

Fz , u

G xc

(9)

must be found to solve the problem

min

T

J

Ef u Ru

Ac , F ,G 2

2

subject to Ef yi d V i ,

(10)

i

1,...., n y

(11)

Here y is the output vector, z measurement vector, w p and v zero-mean uncorrelated process and measurement Gaussian white noises with intensities Wp and V, J the control energy, 2

V i the upper bound imposed on the i-th output variance, Ef

lim

E the expectation operator. t of

OVC solution reduces to a linear quadratic Gaussian (LQG) problem by selecting the output penalty Q using an algorithm given in [10, 11]. OVC matrices are given by T 1 1 T (12) Ac Ap B pG FM p , F XM p V , G R Bp K 0

XAp

T

Ap X

0

KAp

Ap K

T

T

1

XM p V

M pX T

1

KB p R B p K

Wp

(13a)

T

(13b)

C p QC p

B. Input Variance Constrained Controller (IVC) The IVC problem is the dual of OVC: for (8) and given Q ! 0 find a full order controller (9) to solve

min Ac , F ,G

T

Ef y Qy 2

2

subject to E f ui d Pi ,

(14)

i

1, ...., nu

tolerance. Further, all closed loop states displayed very good, non-catastrophic behavior (see [4]) because of the exponentially stabilizing effect of OVC/IVC (note that the open loop system is unstable). Controls also displayed variations within allowable limits [4]. Robustness is particularly important for OVC/IVC controllers, because as LQG based controllers they do not have guaranteed stability margins. Phase and gain margins typically used for SISO systems robustness study are not useful for MIMO systems, characterized by transfer matrices. Generalized stability margins (GSM) also proved to be inadequate for our models, yielding unrealistic results. For example, the gap and v-gap between models linearized around 40 kts and 80 kts were 0.8484 and 0.8483, respectively. However, the GSM is very small for the model 4 linearized around 40 kts, i.e. 8.9297. 10 . This would indicate poor robustness properties, which is not the case. The discrepancy between the small GSM and good robustness properties of OVC/IVC controllers is explained by the fact that GSM assumes that all matrix elements are perturbed. However, in our helicopter models matrix Ap has many elements which are fixed to zero [4]. Therefore, for robustness studies in this paper we resorted to extensive closed loop analysis and simulations, considering four modeling uncertainties: variation of flight conditions (e.g. flight speed), inertial properties, initial conditions, noise intensities (see [4] for more details). Extensive analysis [4] indicated that OVC and IVC controllers are robustly stable over a large range of helicopter speeds. For example, Fig. 6 gives the variation of relevant closed loop poles obtained using the nominal OVC and linear models for straight level flights at different speeds. All closed loops are exponentially stable and, in general, closed loop poles display small relative variations. Similarly, the nominal IVC is robustly stable for helicopter speeds between 0 to 80 kts (see Fig. 7). Note that the same units of measure for the imaginary and real parts of the eigenvalues are used due to the nondimensionalization process which is typical in helicopter dynamics (see [6]).

(15)

2

where Pi is the upper bound variance on the i-th input. For the results reported next sensor measurements were linear and angular velocities and Euler angles. The inputs were all helicopter controls (3 main rotor controls and 1 tail rotor control) and the outputs were Euler angles. IV. EXAMPLES The helicopter model was linearized around 40 kts straight level flight (“nominal” model) and OVC and IVC controllers (“nominal” controllers) were designed for

V 10 >1 1 0.1@ and P 10 >1 1 1 1@ . OVC and IVC control design was possible and fast even for this large dimensional system and tight constraints. The OVC 6 algorithm converged in 4 iterations with 10 tolerance 4

5

while the IVC algorithm converged in 12 with 10

7

Figure 6. OVC Stability Robustness w.r.t. Flight Speed

Figure 7. IVC Stability Robustness w.r.t. Flight Speed

Figure 9. IVC Stability Robustness w.r.t. Inertial Properties

To investigate robustness with respect to (w.r.t.) inertial properties, we perturbed the mass and inertia matrix from their nominal values, linearized the resulting models around straight level flight at 40 kts and formed closed loop systems using nominal OVC/IVC controllers. As in the previous case, numerous numerical experiments indicated closed loop robust stability for a large range of perturbations. For example Figs. 8, 9, which give the variations of relevant closed loop poles when all inertial properties are perturbed by up to 25% show that closed loops are exponentially stable and in general poles experience relatively small variations. In Figs. 10-17 we also present some simulation results. For the 1st study, nominal OVC and IVC controllers were evaluated on the helicopter model linearized around 80 kts. In Figs. 10, 11 closed loop responses of some states (helicopter roll angle, IA , collective blade flapping, E 0 ) and

For the 2nd study, nominal OVC and IVC controllers were evaluated on the helicopter model linearized around 40 kts with 25% reduction in all helicopter inertial quantities. Figs. 12, 13 give the corresponding closed loop responses of helicopter roll angle and main rotor collective pitch control to white noise perturbations. The 3rd study illustrates robustness of nominal OVC and IVC w.r.t. initial conditions. Fig. 14, 15 show some closed loop simulations when all initial Euler angle states were different than zero (equal to 0.1 rad). Helicopter roll angle state and main rotor collective pitch control using the nominal OVC and IVC, and nominal model are given. For the 4th study, the nominal OVC and IVC were evaluated for the nominal helicopter model. However, process noise on the helicopter model was changed (it is increased 10 times w.r.t the value used in OVC/IVC design). Figs. 16, 17 show the closed loop responses of helicopter roll angle state and main rotor collective pitch control using the nominal OVC and IVC, respectively, and nominal model experiencing different process noise. All closed loop simulations (Figs. 10-17) confirm the fact that OVC and IVC are robust w.r.t. all modeling uncertainties considered in this paper, from flight condition variations, to noise intensity variations. From these simulations it can be observed that helicopter Euler angles do not experience fast and large variations. Our extensive analysis also showed that all other states (i.e. helicopter linear and angular velocity states and blade states) do not experience dangerous behavior in closed loop [5]. It can also be observed that helicopter controls (e.g. T and T c ) do not

controls ( T 0 : main rotor collective pitch control, TT : tail rotor control) to white noise perturbations are given.

Figure 8. OVC Stability Robustness w.r.t. Inertial Properties

experience fast and large variations and are within physical limits typical of helicopter controls. The fact that these controllers are robust over a large range of uncertainties is crucial for practical implementation, because scheduling such controllers may be avoided: one controller will perform sufficiently well over a large uncertain flight envelope. However, systematic gain scheduling design may represent an area of future research.

IA(deg)

0.1 0.05 0 -0.05 1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 Time(s)

6

7

8

9

10

0

T0(deg)

-0.2

0.4 0.2 0 -0.2 -0.4

TT (deg)

E (deg) 0

0 0.2

0.2 0 -0.2 -0.4

IA(deg)

Figure 10. 1st OVC Robustness Example (Some Helicopter and Flapping States, and Controls)

0.1 0.05 0 -0.05 0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

E0 (deg)

0.05 0 -0.05 0

TT(deg)

T0(deg)

0.05 0 -0.05 0 0.04 0.02 0 -0.02 -0.04 0

Time(s)

T0(deg)

IA(deg)

Figure 11. 1st IVC Robustness Example (Some Helicopter and Flapping States, and Controls)

0.1 0.05 0 -0.05 0 0.4 0.2 0 -0.2 0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5 Time(s)

6

7

8

9

10

T0(deg)

IA(deg)

Figure 12. 2nd OVC Robustness Example (Roll Angle State and Collective Blade Pitch Control)

0.2 0.1 0 -0.1

0 0.04 0.02 0 -0.02 -0.04 0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5 Time(s)

6

7

8

9

10

Figure 13. 2nd IVC Robustness Example (Roll Angle State and Collective Blade Pitch Control)

IA(deg)

6 4 2 0 -2 -4

IA(deg)

T0(deg)

0 0 -20 -40 0

0

T (deg) A

I (deg)

3

4

5

6

7

8

9

10

1

2

3

4

5 6 7 8 9 Time(s) Figure 14. 3rd OVC Robustness Example (Roll Angle State and Collective Blade Pitch Control)

10

0

1

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3

4

0

1

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4

0

1

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4

5

6

7

8

9

10

0

1

2

3

4

5 Time(s)

6

7

8

9

10

9

10

0

1 0 -1 -2 -3

0.4 0.2 0 -0.2

0

2

5

-5

T (deg)

1

1 0 -1

5

6

7

8

9

10

5 6 7 8 9 Time(s) Figure 15. 3rd IVC Robustness Example (Roll Angle State and Collective Blade Pitch Control)

10

I (deg) A

0.6 0.4 0.2 0 -0.2

T (deg) 0

Figure 16. 4th OVC Robustness Example (Roll Angle State and Collective Blade Pitch Control)

0.2 0.1 0 -0.1

0

1

2

3

4

0

1

2

3

4

5

6

7

8

5 6 7 8 9 10 Time(s) Figure 17. 4th IVC Robustness Example (Roll Angle State and Collective Blade Pitch Control) [4] Oktay, T., Constrained control of complex helicopter models, Ph.D. Dissertation, Virginia Tech, Blacksburg, VA, 2012. V. CONCLUSIONS [5] Leishman, J.G., Principles of Helicopter Aerodynamics, Complex, large scale helicopter models that include blade Cambridge University Press, 2006. [6] Padfield, G.D, Helicopter Flight Dynamics, AIAA Education flexibility are successfully used in OVC and IVC control Series, 2007. design. Extensive numerical experiments indicate that [7] Oktay, T., and Sultan, C., “Constrained predictive control of these controllers are robust for large flight speed and helicopters,” Aircraft Engineering and Aerospace Technology, inertial properties variations, as well as initial conditions 2013, vol. 85, pp. 32-47. [8] Oktay, T., and Sultan, C., “Variance constrained control of and noise intensity variations. When flight and helicopter maneuvering helicopters with sensor failure,” J. of Aerospace parameters are varied significantly, fuselage and blade Engr. Part G, doi:10.1177/0954410012464002. states and controls do not display dangerous behavior, [9] Oktay T., and Sultan, C., “Simultaneous helicopter and control being within physical bounds of typical helicopters. system design, AIAA Journal of Aircraft, to appear. [10] Hsieh, C., Skelton, R.E., and Damra, F.M., “Minimum energy controllers with inequality constraints on output variances,” REFERENCES Optimal Control Application and Methods, 1989, vol. 10(4), pp. [1] Prouty, R. W., Helicopter Aerodynamics: Volume I, 2009. 347-366. [2]

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