three-time scale singular perturbation control for a radio...

Three-Time Scale Singular Perturbation Control for a

Radio-Control Helicopter on a Platform

Sergio Esteban ∗, Javier Aracil †and Francisco Gordillo ‡

Universidad de Sevilla, Sevilla, 41092, Spain

A three-time scale singular perturbation control is applied to a Radio/Control helicopteron a platform to regulate its vertical position. The proposed control law allows to achievethe desired altitude by either selecting a desired collective pitch angle or a desired angularvelocity of the blades.

Nomenclature

z height of the helicopter above the ground, metersω rotational speed of the rotor blades, rad/sθc collective pitch angle of the rotor blades, raduth throttle inputuθc collective servomechanismCT Main rotor thrust coefficientR/C Radio-Control

I. Introduction

Control of rotatory wing aircrafts represents a very challenging task due to the nonlinearities and inherentinstabilities present in such systems. The versatility of rotorcrafts allows them to perform almost any taskthat no conventional aircraft can do, but this ability is ultimately associated to the stability and controlcharacteristics obtained via automatic control design.1 These stability and control characteristics come at aprice of complex control designs in order to deal with these highly nonlinear aerospace systems.

Historically, linear classic control techniques such design via root locus techniques, frequency responsetechniques, state space techniques, PID controllers, or gain scheduling to name few,2 have been sufficientlyto obtain reasonable control responses of aerospace systems. The evolution of the aerospace industry and theconsequent improvement of technologies, have increased the performance requirements of all systems in gen-eral, which has called for better control designs that can deal with more complex systems, making linear con-trol techniques insufficient to cope with the industry demands. Specifically, in the area of aerospace systems,a wide range of different nonlinear controller techniques have been studied to deal with the nonlinear dy-namics of such systems. From singular perturbation,3 feedback linearization,4,5, 6 dynamic inversion,7,8, 9, 10

sliding mode control,11 or backstepping control methods12,13 to name few. Neural Networks (NN) are alsoincluded within the realm of nonlinear control techniques, and seem to provide improved robustness prop-erties under system uncertainties. Some of works include Adaptive Critic Neural Network (ACNN) based

∗PhD. Student, Departamento de Ingenierıa de Sistemas y Automatica, Sevilla, Professional AIAA Member.†Professor, Departamento de Ingenierıa de Sistemas y Automatica, Sevilla.‡Professor, Departamento de Ingenierıa de Sistemas y Automatica, Sevilla.

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American Institute of Aeronautics and Astronautics

controls, originally presented by Balakrishnan and Biega,14 and later extended to many other aerospacesystems.15,16,17 Calise and Kim18 used feedback linearization along with neural-networks as an alternativeto gain scheduling extended to a wide range of aerospace systems from helicopters to reusable launch vehi-cles.19,20,21,22,23,24,25,25 Haley and Soloway26 proposed a Neural Generalized Predictive Control (NGPC)algorithm capable of real-time control law reconfiguration, model adaptation, and the ability to identifyfailures in control effectiveness by using an innovative user define cost function that can be associated toeither the aircraft outputs or to the control inputs. Bull, Kaneshige, and Totah27 used an innovative genericneural flight control and autopilot system.

Throughout the history, helicopter control has been solved using mechanical stabilizers, electronic sta-bilization, digital systems or high-gain controls.1 A basic problem in control design is the mathematicalmodelling complexity of a physical system. The modelling of many systems calls for high-order dynamicequations, which for the case of rotorcraft systems, represents a unique challenge due to the rotatory parts.Generally, the presence of parasitic parameters such as small time constants, is often the source of a increasedorder and stiffness of these systems.28 The stiffness, attributed to the simultaneous occurrence of slow andfast phenomena, gives rise to time scales, and the suppression of the small parasitic variables results indegenerated, reduced order systems, called singularly perturbed systems, that can be stabilized separately,and thus simplifying the burden of control design of high-order systems. Naidu28,29 gives a extended surveyof the use of singular perturbed and time scales control methods in aerospace systems.

The motivation to this article comes from the work of Sira-Ramirez et al.11 that used a dynamicalmultivariable discontinuous feedback control strategy of the sliding mode type for the stabilization of anonlinear helicopter model in vertical flight which include the dynamics of the collective pitch actuators. Thesame helicopter model is used in this article, but the control law here proposed uses singular perturbationtheory, and as will be shown, the results here obtained outperform those presented by Sira-Ramirez et al.

This article is structured as follows: Section II presents the helicopter model used throughout this article,including an analysis of the equilibrium points of the model; Section III introduces the singular perturbationformulation, including the multi-time scale singular perturbation formulation used in this article and theconsequent control law formulation; simulation results are depicted in Section IV; conclusions are describedin Section V; Sections VI and VI describes future work and acknowledgments respectively, and figures of thecomputer simulations are shown in Section VII.

II. Model Definition

The helicopter model that will used throughout the remainder of this article is obtained from severaltechnical reports that were written in the at the University of Purdue in 199130,31 that describe the verticalmotion of a radio/control helicopter model mounted on a stand as seen in Fig. 1. Needs to be noted that inaddition to the nonlinear vertical motion of the helicopter, the model here used, includes the nonlinear dy-namics of the collective pitch actuators, which increases the complexity of the studied model. The differentialequation that describes the vertical motion of the X-Cell 5032 model miniature helicopter is

z = K1(1 + Geff )CT ω2 − g −K2z −K3z2 −K4, (1)

where CT is the thrust coefficient of the helicopter model, ω (radians) is the rotational speed of the rotorblades, z (meters) is the height above the ground, g (m/s2) is the gravitational acceleration, and Geff

models the ground effect, but during the remainder of this article it will be considered negligible (Geff = 0).The thrust coefficient and the dynamics of the angular velocity of the blades are modelled as33

CT = [−KC1 +√

K2C1 + KC2θc]2 (2)

ω = −K5ω −K6ω2 −K7ω

2 sin θc + K8uth + K9, (3)

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where θc (rad) is the collective pitch angle of the rotor blades. The dynamics of the collective pitch angleare defined as

θc = K10(−0.00031746uθc+ 0.5436− θc)−K11θc −K12ω

2 sin θc. (4)

where the inputs to the system are the throttle, uth, and the input to the collective servomechanism,uθc

. The nominal values of the parameters are K1 = 0.25, K2 = 0.1, K3 = 0.1, K4 = 7.86, K5 =0.7, K6 = 0.0028, K7 = 0.005, K8 = −0.1088, K9 = −13.92, K10 = 800, K11 = 65, K12 = 0.1, KC1 =0.03259, KC1 = 0.061456, and g = 9.81. Equations (1), (4), and (3) can be written in the state space formas

x = f(x) + g(x)u, (5)(6)

where the state space vector is defined as

x =

z

z

ω

θc

θc

=

x1

x2

x3

x4

x5

, (7)

and the control space vector is

u =

[u1

u2

]=

[K8uth

−0.00031746K10uθc

], (8)

being the resulting nonlinear equations of motion,

x1 = x2

x2 = x23(a1 + a2x4 −

√a3 + a4x4) + a5x2 + a6x

22 + a7

x3 = a8x3 + a10x23 sin x4 + a9x

23 + a11 + u1 (9)

x4 = x5

x5 = a13x4 + a14x23 sin x4 + a15x5 + a12 + u2,

where the constants are a1 = 5.31× 10−4, a2 = 1.5364× 10−2, a3 = 2.82× 10−7, a4 = 1.632× 10−5, a5 =−K2, a6 = −K2, a7 = −g −K4, a8 = −K5, a9 = −K6, a10 = −K6, a11 = K9, a12 = 0.5436K10, a13 =−K10, a14 = −K12, and a15 = −K11. Figure 2 depicts a simplified block diagram that helps to understandthe degree of coupling, and the dependence, between the three subsystems in which are organized the fivedifferential equations of time. These subsystems correspond to the position and velocity of the helicopter,[x1, x2], the main rotor angular velocity, [x3], and the collective pitch dynamics [x4, x5]. These subsystemsevolve in different time-scales as stated bellow.

A. Equilibrium Points Analysis of the Helicopter Model

In order to better understand the behavior of the system, an analysis of the equilibrium points is conducted.The equilibrium points are obtained by setting all the derivatives of system (9) to zero, thus yielding theequilibrium equations

x1 = 0 = x2

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x2 = 0 = x23(a1 + a2x4 −

√a3 + a4x4) + a5x2 + a6x

22 + a7

x3 = 0 = a8x3 + a10x23 sin x4 + a9x

23 + a11 + u1 (10)

x4 = 0 = x5

x5 = 0 = a13x4 + a14x23 sin x4 + a15x5 + a12 + u2.

As seen in the previous section, the system is formed by five state variables, and two control signals,therefore two degrees of freedom are expected, but when conducted the equilibrium points analysis, only onedegree of freedom is observed, three equations with four unknowns (x3, x4, u1, u2), where the bar denotesthe value at equilibrium,

0 = x23(a1 + a2x4 −

√a3 + a4x4) + a7 (11)

0 = a8x3 + a10x23 sin x4 + a9x

23 + a11 + u1 (12)

0 = a13x4 + a14x23 sin x4 + a12 + u2, (13)

being the vertical velocity of the helicopter and the collective pitch rate of the blades equal to zero (x2 = x5 =0). This is caused because the helicopter equilibrium altitude (x1) does not show in any of the equilibriumequations, therefore, every equilibrium point can be attained at any altitude. This implies that there existsan infinitely number of equilibrium points, and one of the variables needs to be fixed in order to determinea single equilibrium point. The first equilibrium equation, (11), defines the equilibrium space by selectinga desired value for either x3 or x4, such that an expression can be determined as a function of the selecteddesired variable, defined from now on as x3D or x4D respectively. The last Eqs. (12-13), define the controlsignals required for achieving the selected equilibrium points. If the collective pitch angle (x4D ) is selectedas the fixed variable, the expressions to determine the values of the other three unknowns as a function ofde fixed variable (x3(x4D

), u1(x4D), u2(x4D

)) can be expressed as,

x3(x4D ) = ±√− a7

a1 + a2x4D−√a3 + a4x4D

(14)

u1(x4D) = −a8x3 − a10x

23 sin x4D

− a9x23 − a11 (15)

u2(x4D) = −a13x4D

− a14x23 sin x4 − a12. (16)

If the angular velocity of the blades (x3D ) is selected as the fixed variable, the expressions to determine thevalues of the other three unknowns as a function of de fixed variable (x4(x3D ), u1(x3D ), u2(x3D )) can beexpressed as

x4(x3D) =

a4x3D ±√

Cax23D

+ Cb

2a22x3D

+ Cc +Cd

x23D

, (17)

u1(x3D) = −a8x3D

− x23D

(a10 sin x4 + a9)− a11 (18)

u2(x3D) = −a13x4 − a14x

23D

sin x4 − a12, (19)

being the coefficients defined by

Ca = a24 − 4a2a1a4 + 4a2

2a3

Cb = −4a2a7a4

Cc = −a1

a2

Cd = −a7

a2.

It can be observed that Eq. (14) has two solutions for the equilibrium rotational speed of the blades (x3),but constrained by the physical rotation of the blades, only the positive solution is considered. It is also

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observed that Eq. (17) has two solutions for the equilibrium collective pitch angle of the blades (x4), but itcan be checked by substituting both solutions in the original equations (10), that the solution correspondingto the minus sign in front of the square root is a spurious solution introduced in the previous computations,therefore only the positive solution is considered.

Before continue defining the control method here proposed, it is necessary to define the range of thereachable equilibrium points, from now on considered as the desired final conditions (x3D

and x4D). For the

limits of the angular velocity of the blades, we assumed that the engine can physically generate a maximumangular velocity of x3max

= 180 rads/sec. For the range of collective pitch angles, a maximum collectivepitch angle of x4max = 0.25 ≈ 14.32◦ rads. is considered, and the minimum collective pitch angle can bedetermined analyzing the modelization of the thrust coefficient as seen in Eq. (2), where it can be observed

that only collective pitch angles x4 > −K2C1

KC1= −a3

a4will be defined. Analysis of x3(x4D

) and x4(x3D), shows

that there is a region within the collective pitch angle range, that it is not defined as an attainable desirefinal condition. This defines two distinctive regions of interest for the collective pitch angle

x4lim1> x4D

> −a3

a4

x4max > x4D> x4lim2

,

being x4lim1and x4lim2

the roots of the denominator of x3(x4D) equal to zero,

x4lim1=

a4 − 2a1a2 −√

a24 − 4a4a1a2 + 4a2

2a3

2a22

x4lim2=

a4 − 2a1a2 +√

a24 − 4a4a1a2 + 4a2

2a3

2a22

,

substituting the constants, the ranges are defined as

−0.3992× 10−3 rads > x4D > −0.1727× 10−1 rads

0.25 rads > x4D > 0.4138× 10−3 rads.

Figure 3 represents the relation of x3(x4D ), x4(x3D ) for the ranges of considered desired collective pitchangle and angular velocity of the blades. Analyzing the results of Figure 3 in detail, it can be seen that,despite the entire range of desired final conditions analyzed produce defined equilibrium points, it is illogicalto consider desired collective pitch angle values x4D

< 4.87◦, since it requires angular velocities above 180rads/sec to define this equilibrium conditions, thus the range of desired collective pitch angle is reduced to14.32◦ > x4D > 4.87◦. Results of the proposed control law will be discussed in Section IV

III. Singular Perturbation Formulation

The general two-time scale singular perturbation model formulation is described in3 as,

x = f(x, z, ε, t), x(t0) = x0, x ∈ Rn (20)εz = g(x, z, ε, t), z(t0) = z0, z ∈ Rm, (21)

and its quasi-steady-state condition obtained when ε = 0, thus reducing the dimension of the state spacedefined in Eqs. (20) and (21) from n+m to n. This quasy-steady state condition of the differential equation,that represents the ε-fast dynamics, degenerates into the algebraic equation

0 = g(x, z, 0, t), (22)

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where the bar denotes that the variables belong to a system with ε = 0. The new model is considered instandard form if and only if3 in a domain of interest Eq. (22) has k ≥ 1 distinct and unique real roots such

z = φi(x, t), i = 1, 2, ..., k. (23)

This assumption assures that a well defined n-dimensional reduced model will correspond to each root ofEq. (23). To obtain the ith reduced model, Eq. (23) is substituted into Eq. (20)

˙x = f(x, z, φi(x, t), 0, t), x(t0) = x0, (24)

and keep the same initial conditions for the state variable x(t) as for x(t). This model is called a quasi-steady-state model ,3 because z, whose velocity z = g/ε can be large when ε is small, may rapidly convergeto a root of (22), which is the quasi-steady-state form of (21). The slow response is approximated by thereduced model defined in Eq. (24), while the discrepancy between the response of the reduced model, (24),and that of the full model in Eqs. (20) and (21), is the fast transient. Singular perturbation techniquessimplify considerably the complexity of coupled dynamics such those present in aerospace systems. Thisallows generating nonlinear control laws that can be derived directly from the original nonlinear systemswithout the need of making unreasonable simplifications.

A. Multi-time Scale Singular Perturbation Model Formulation

After analyzing the open loop dynamics of the helicopter, it is observed that a three-time scale model isintuitively more precise than a two-time scale due to the treatment of the collective pitch angle as a statevariable, which is generally being treated as a control input. The general formulation of the three-time scalesingular perturbed systems requieres the system to posses three different time scales that are defined as

x = f(x, z1, z2)ε1z1 = g1(x, z1, z2) (25)ε2z2 = g2(x, z1, z2),

being x the slow variable, z1 fast variable, and z2 the ultra-fast variable and 0 < ε2 << ε1 << 1. Forthe problem here discussed, the slow variable is the angular speed of the blades, x3, the fast variable isthe vertical motion of the helicopter, z1 = [ x1 x2 ]T , and the ultra-fast variable is the collective pitchdynamics, z2 = [ x4 x5 ]T . The reason for choosing these time scales is the following one: from Eqs. 9, it isclear than x4 and x5 are faster than the rest. Variable x1 is not necessarily so fast but as a control objective,it is desired to posses fast maneuverability, therefore we can make x1 and x2 faster than x3 through thecontrol inputs. In order to express the original set of time differential equations (9) in the proper three-timescale singular perturbation formulation (25) a series of algebraic modifications are introduced such that theequations are written in the form

Ixx = f(x, z1, z2)Iz1 z1 = g1(x, z1, z2) (26)Iz2 z2 = g2(x, z1, z2),

where

Ix = − 1a9

= 357.142

Iz1 = − 1a5

= 10

Iz2 = − 1a13

= 0.00125,

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and

f(x, z1, z2) = Ixf(x, z1, z2)g1(x, z1, z2) = Iz1g1(x, z1, z2)g2(x, z1, z2) = Iz2g2(x, z1, z2),

where Ix, Iz1 , Iz2 represent the perturbation parameters of each of three-time scales. It can be seen thatIx >> Iz1 >> Iz2 , therefore in order to express the equations of the three time-scales in the correct multi-time singular perturbation formulation, all perturbation parameters are normalized by Ix, yielding

ε1 =Iz1

Ix=

a9

a5= 0.028

ε2 =Iz2

Ix=

a9

a13= 3.500× 10−6.

It can be seen that the new perturbation parameters fulfill 0 < ε2 << ε1 << 1. After normalizing by Ix,the slow dynamics remain the same as the original

x3 = a8x3 + a10x23 sinx4 + a9x

23 + a11 + u1 (27)

the fast dynamics are re-written

ε1x1 = c1x2

ε1x2 = x23(c2 + c3x4 −

√c4 + c5x4)− x2 − x2

2 + c6,

(28)

being

c1 =Iz1

Ix=

a9

a5= 0.028

c2 =a1a9

a5= c1a1 = 1.4868× 10−5

c3 =a2a9

a5= c1a2 = 1.5364× 10−1

c4 =a3a9

a5= c1a3 = 7.8960× 10−9

c5 =a4a9

a5= c1a4 = 4.5696× 10−7

c6 =a7a9

a5= c1a7 = −4.9476× 10−1

and the ultra-fast-dynamics can be re-written as

ε2x4 = c7x5

ε2x5 = −x4 + c8x23 sin x4 + c9x5 + c10 + c11u2, (29)

being

c7 =Iz2

Ix=

a9

a13= 3.500× 10−6

c8 =a9a14

a13= c7a14 = −3.500× 10−7

c9 =a9a15

a13= c7a15 = −2.2750× 10−4

c10 =a9a12

a13= c7a12 = 1.52208× 10−3

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The control strategy for the three-time scale singular perturbation formulation consists on treating thethree different scales as two distinct singular perturbed problems. The first problem considers the ultra-fastand fast dynamics, and obtains the associated control law that stabilizes the [x1−x2, x4−x5]-subsystem usingthe singular perturbation methodology described in the previous section, and the second problem considersthe fast and the slow dynamics to stabilize the angular velocity of the blades of the [x3, x1 − x2]-subsystem.

1. Control formulation for the singular perturbation [x1 − x2, x4 − x5]-subsystem

Prior to determine the control law for the singular perturbation [x1 − x2, x4 − x5]-subsystem, a change ofvariables is required in the ultra-fast dynamics, Eq. (28), to allow solving for the equilibria defined by themanifold 0 = g2(x, z1, z2, 0, t). A feedback transform is introduced such that

u2 = c8x23 sinx4 + c11u2, (30)

thus rewriting Eq. (28) into

ε2x4 = c7x5 (31)ε2x5 = −x4 + c9x5 + c10 + u2. (32)

Setting the perturbation parameter ε2 = 0, yields

0 = x5

0 = −x4 + c10 + u2,

being the equilibria for the ultra-fast dynamics

x5 = 0 (33)x4 = u2 + c10, (34)

substituting for x4 and x5 into the fast dynamics generates the reduced degenerated system,

ε1x1 = c1x2

ε1x2 = x23(c2 + c3x4 −

√c4 + c5x4)− x2 − x2

2 + c6 (35)

= x23(c2 + c3(u2 + c10)−

√c4 + c5(u2 + c10)− x2 − x2

2 + c6.

In order to obtain the control law that stabilizes the [x1, x2]-subsystem, a series of algebraic substitutionsare conducted. Let

w2 = c4 + c5(u2 + c10). (36)

From Eq. (36), a expression u2 as a function of w can be obtained such

u2 =w2 − c4 − c5c10

a4, (37)

and substituting Eq. (36) and (37) into (38) yields

ε1x1 = c1x2

ε1x2 = x23

(c2 + c3

(w2 − c4 − c5c10

c5+ c10

)− w

)− x2 − x2

2 + c6, (38)

which can be simplified into

ε1x1 = c1x2

ε1x2 = x23

(c12w

2 − w + Ka

)− x2 − x22 + c6, (39)

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being

c12 =c3

c5

Ka = c2 + c3c10 − c3(c4 + c5c10)c5

.

Let v = c12w2 − w + Ka, thus Eq. (39) becomes

ε1x1 = c1x2

ε1x2 = x23v − x2 − x2

2 + c6. (40)

We choose a stable target system of the form

ε1x1 = c1x2

ε1x2 = −b1(x1 − x1D)− b2x2, (41)

where b1, and b2 are control design parameters that determine the desired time response, and x1Drepresents

the desired altitude of the helicopter determined a priori by the designer. The control problem can be solvedif a control signal v is chosen such that system (40) behaves like the target system defined in (41). Thecontrol signal v is therefore chosen to be:

v =x2

2 − (1− b2)x2 − b1(x1 − x1D )− c6

x23

, (42)

where x1D represents the desired vertical position of the helicopter. It should be noted that this controlsignal is not defined for x3 = 0. The control law u2 can the obtained tracing back the algebraic substitutionsconducted from the final target system to the initial degenerated system such that

u2 =u2 − c8x

23 sin x4

c11, (43)

where x3 is treated by the moment as a constant, and u2 is

u2 =w2 − c4 − c5c10

c5, (44)

where w can be obtained solving the quadratic polynomial

c12w2 − w + Ka = v

c12w2 − w + Ka − v = 0, (45)

where v is defined by Eq. (42). Solving for the roots of the polynomial in Eq. (45) yields

w =1±

√1− 4c12(Ka − v)

2c12. (46)

It can be checked by substituting in the original equations (10), that the solution corresponding to theminus sign in front of the square root is a spurious solution introduced in the previous computations. In thefollowing, only the positive root will be considered. The control law for the u2(x1, x1D, x2, x3, x4) is thereforedefined by

u2 = Kb

(1 +

√1− 4c12

(Ka +

−x22 + (1− b2)x2 + b1(x1 − x1D

) + c6

x23

))2

+ Kc + Kdx23 sin x4, (47)

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where Ka has been previously defined, and the rest of constants are

Ka = c2 + c3c10 − c3(c4 + c5c10)c5

Kb =1

4c5c11c212

Kc = −c4 + c5c10

c5c11

Kd = − c8

c11.

Results will be discussed in Section IV

2. Control formulation for the singular perturbation [x3, x1 − x2]-subsystem

Once the first control law for the [x1−x2, x4−x5]-subsystem is obtained, the second control law to stabilizethe [x3, x1 − x2]-subsystem needs to be determined. The root of the [x1 − x2]-manifold are determined bysetting 0 = g1(x, z1, z2, 0, t), yielding

0 = x2

0 = x23(c2 + c3x4 −

√c4 + c5x4) + c6, (48)

which represents the first two of the equilibrium Eqs. (10). The first equation yields that the vertical velocityof the helicopter is zero for the [x1 − x2]-manifold, which is expected if the helicopter model is to reach anequilibrium point, and the second equation yields an expression that defines the space of configuration forthe vertical motion as a function of both x3 and x4.

x23(c2 + c3x4 −

√c4 + c5x4) + c6 = 0. (49)

Solving Eq. (49) for both x3 and x4, and substituting the associated independent variable by x3D andx4D

, respectively, yields

x3(x4D) = φ1(x4D

) = ±√− a7

c2 + c3x4D−√c4 + x4D

c5(50)

x4(x4D) = φ2(x3D

) =c5x3D ±

√Kex2

3D+ Kf

2c23x3D

+ Kg +Kh

x23D

, (51)

with

Ke = c25 − 4c3c2c5 + 4c2

3c4

Kf = −4c3c6c5

Kg = −c2

c3

Kh = −c6

c3.

where φ1(x4D) represents the solution of the rotor blade angular velocity in the [x1 − x2]-manifold when a

desired collective pitch angle (x4D ) is selected, and φ2(x3D ) represents the solution of the collective pitchangle in the [x1 − x2]-manifold when a desired rotor blade angular velocity (x3D ) is selected. These twoexpression allow the designer to choose which one of the variables is considered as the second desired state,which is required to define the equilibrium points of the helicopter. Note that both Eqs. (50)-(51) have two

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distinct solutions depicted by the ± sign. As it was shown in Section A, it can be observed that Eq. (50)has two solutions for the rotor blade angular velocity in the [x1 − x2]-fast manifold, (φ1), but constrainedby the physical rotation of the blades, only the positive solution is considered. It is also observed that Eq.(51) has two solutions for the collective pitch angle in the [x1 − x2]-manifold, (φ2), but it can be checked bysubstituting both solutions in the original equations, (10), that the solution corresponding to the minus signin front of the square root is a spurious solution introduced in the previous computations, therefore in thefuture only the positive solution will be considered. Once the root of the manifold 0 = g1(x, z1, z2, 0, t) isdefined, the control laws can be obtained by substituting Eqs. (50) or (51) into the [x3]-dynamics dependingif the control law has to be solved for a desired collective pitch angle, (x4D

), or a desired angular velocity,(x3D

), respectively yielding:

x3 = a8φ1 + a10φ21 sin x4 + a9φ

21 + a11 + u1, (52)

orx3 = a8x3 + a10x

23 sin φ2 + a9x

23 + a11 + u1. (53)

The control laws are obtained by defining a target system of the form

x3 = −b3(x3 − x3D ), (54)

where b3 is a control design parameter that defines the desired dynamics of the angular velocity of the blades.The control law associated to Eq. (52) for a desired collective pitch angle (x4D ) is

u1(x4D ) = − (a8φ1 + a10φ

21 sin x4 + a9φ

21 + a11 + b3(x3 − φ1)

), (55)

and the control law associated to Eq. (54) for a desired angular velocity of the blades (x3D ) is

u1(x3D) = − (

a8x3 + a10x23 sin φ2 + a9x

23 + a11 + b3(x3 − x3D

)). (56)

Is to be noted that the fact that there are two possibilities for the control law used to stabilize the[x3, x1 − x2]-subsystem, as seen by Eqs. (55) and (56), allows to choose which method is used to stabilizethe helicopter at the desired altitude, by either selecting a desired final angular velocity (x4D ) or a desiredfinal collective pitch angle (x4D ). Generally, for radio/control helicopters, the angular velocity of the bladesis fixed to a desired nominal main rotor speed, which is generally controlled by means of a speed regulator,allowing the pilot to concentrate on the rest of the controllers, and specifically, for regulation of the verticalposition, it can do so by focusing only on the collective pitch angle of the blades. This is a reasonableapproach to regulate the vertical position of the helicopter, due to the fact that the collective pitch angle ofthe blades are in a much faster time-scale than the angular velocity of the blades, but from the aerodynamicpoint of view, and in order to optimize the drag produced by the blades, might be desirable to fix the pitchangle at a desired final condition and use the main rotor speed to reach the desired altitude. Future workwill be conducted to define which of the two control approaches is desirable to optimize the control effort.

IV. Simulation Results

Simulations are conducted for both proposed control laws, depending on which state variable is chosenas the second desired variable, x3 or x4. Several initial conditions and several desired final values areconsidered to analyze a greater range of possible conditions. The simulations are conducted using a fourthorder Runge-Kutta fixed step integration method with an integration step of 0.01 seconds.

To show the versatility of the two control laws to different desired states, the first simulation comparesthe results obtained when using the control laws for a x4D = 0.2 radss for the control law in Eq. (56) andx3D = 101.336 rads/sec for the control law in Eq. (55). These desired values correspond to solving Eq. (14)for a x4D = 0.2, thus allowing to compare both control laws. Figures 4 and 5 show the states and controlhistories respectively for both control laws, and can be seen that both generate the same control responses.

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A sensitivity study is conducted by selecting different initial conditions of initial altitude, x1(0), andcollective pitch angle, x4(0), for fixed desired final helicopter altitude x1D = 1.25, and desired collectivepitch angle of x4D

= 0.2. The range of the initial altitudes studied are −0.5m ≤ x1(0) ≤ 0.9m, and therange of initial collective pitch angles tested are −0.017rads ≤ x4(0) ≤ 0.2rads. The simulation results forvariable initial altitude are described in Fig. 6. The simulation results for variable initial collective pitchangle are described in Fig. 7. Fig. 6 is divided into four subfigures, where from left to right and top tobottom represents the helicopter altitude, x1, angular velocity of the blades, x3, collective pitch angle, x4,and both control signals, u1 and u2. The control law drives the system to the desired altitude and collectivepitch angle for the range of initial altitudes tested. The same can be observed in Fig. 7, where the controllerdrives the helicopter to the desired values for the set of different initial collective pitch angles simulated.

The sensitivity analysis is extended to variation in desired values. For this study, the initial conditionsof the helicopter are kept constant, x1(0) = 0.45 m, x2(0) = 0.1 m/sec, x3(0) = 70 rads/sec, x4(0) = 0.1rads and x5(0) = 0.5 rads/sec, while varying the desired final conditions, x1D

and x4D. Fig. 8 shows the

simulation results for desired final altitudes of 0m ≤ x1D≤ 1.25m, and Fig. 9 shows the simulation results

for desired final collective pitch angles of 0.075rads ≤ x4D≤ 0.2rads. The control laws perform well, and

the states are driven to the desired final states. A extended range of initial conditions will be studied andpresented on the final version of this article.

V. Conclusion

The results have shown that the proposed control laws using singular perturbation formulation have beenable to regulate the vertical motion of the helicopter. In addition to regulate the vertical position of thehelicopter, the proposed control laws allow the possibility of deciding which of the two, the collective pitchor the angular velocity of the blades, is chosen as the second desired state.

VI. Future Work

Future work that will be conducted will include the stability analysis of the three-time scale singularperturbation control methodology conducted in this article by studying the boundary layers stability, andthe interconnection conditions properties. A study of the actuators saturation and the robustness of thecontrol law to perturbations, both unmodeled dynamics and external disturbances, will also be conducted.Future work will also include the extension of this controller to a real system Radio/Control helicopter modelon a platform similar to the one presented in this study.

Acknowledgments

This work has been supported under MCyT-FEDER grants DPI2003-00429 and DPI2001-2424-C02-01.

References

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Academon Press, 1986.4Brockett, R. W., “Feedback Invariants for Nonlinear Systems,” Proceedings of the 7th IFAC Congress, IFAC, 1978.5Meyer, G., Su, R., and Hunt, L. R., “Application of Nonlinear Transformations to Automatic Flight Control,” 1984 IFAC

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High Angle of Attack Research Vehicle,” Proceeding of The AIAA Guidance, Navigation, and Control Conference, AIAA, 1990,pp. 20–22.

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10Snell, S. A., Enns, D. F., and Garrard, W. L., “Nonlinear Inversion Flight Control for a Supermaneuverable Aircraft,”AIAA Journal of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992, pp. 976–984.

11Sira-Ramirez, H., Zribi, M., and Ahmad, S., “Dynamical Sliding Mode Control Approach for Vertical Flight Regulationin Helicopters,” IEE Proceedings on Control Theory and Applications, Vol. 141.

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Guidance, Navigation and Control Conference, Boston MA, AIAA, 1998.17Balakrishnan, S. N. and Huang, Z., “Robust Adaptive Critic Based Neurocontrollers for Helicopter with Unmodeled

Uncertainties,” 2001 AIAA Guidance, Navigation and Control Conference, Montreal , AIAA, 2001, paper 2001-4258.18Kim, B. S. and Calise, A. J., “Nonlinear Flight Control Using Neural Networks,” AIAA Journal of Guidance, Control,

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Evaluation on an Unmanned Helicopter,” Proceedings to the IEEE International Conference on Control Applications, IEEE,1999, pp. 871–879.

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24Calise, A. J., Lee, S., and M. Sharma, M., “Development of a Reconfigurable Flight Control Law for the X-36 TaillesFighter Aircraft,” 2000 Proceeding of the AIAA Guidance Navigation and Control Conference, Denver , AIAA, August 2000.

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VII. Figures

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Figure 1. Helicopter mounted on a Stand

Figure 2. Block Diagram of the Helicopter Dynamics

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2 4 6 8 10 12 14

0

50

100

150

x4

D

− rads/sec

x 3 D

− d

egs

+φ1(x

4D

)

0 20 40 60 80 100 120 140 160 180

0

5

10

15

x3

D

− rads/sec

x 4 D

− d

egs

+φ2(x

3D

)

+φ1(x

4D

)

x3

max

−x3

max

+φ2(x

3D

)

x4

max

−x4

max

Figure 3. Relation of φ1(x4D ) and φ2(x3D ).

0 2 4 6 8 10−0.5

0

0.5

1

1.5

time

z −

m

0 2 4 6 8 10−1

0

1

2

time

zdot

− m

/sec

0 2 4 6 8 1070

80

90

100

110

time

ω −

rad

/s

0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

time

θ c − r

ad

0 2 4 6 8 10−1

0

1

2

3

time

θ c dot

− r

ad/s

0 2 4 6 8 100

0.005

0.01

0.015

time

CT

control law for x3

D

control law for x4

D

Figure 4. States Trajectories Comparison for u1(x3D ) and u1(x4D ) with x3D = 101.3359 rads/sec and x4D = 0.2rads

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0 2 4 6 8 10115

120

125

130

135

140

145

150

time0 2 4 6 8 10

−100

−50

0

50

100

time

u 2

0 2 4 6 8 10−1400

−1350

−1300

−1250

−1200

−1150

−1100

−1050

time0 2 4 6 8 10

−300

−200

−100

0

100

200

300

400

time

u θ c

control law for x3

D

control law for x4

D

Figure 5. Control Trajectories Comparison for u1(x3D ) and u1(x4D ) with x3D = 101.3359 rads/sec and x4D = 0.2rads

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(a) x1 (b) x3

(c) x4 (d) u1andu2

Figure 6. States and Control Histories For Variable Altitude Initial Condition.

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(a) x1 (b) x3

(c) x4 (d) u1andu2

Figure 7. States and Control Histories For Variable Collective Pitch Initial Condition.

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(a) x1 (b) x3

(c) x4 (d) u1andu2

Figure 8. States and Control Histories For Variable Desired Final Altitude.

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(a) x1 (b) x3

(c) x4 (d) u1andu2

Figure 9. States and Control Histories For Variable Desired Final Collective Pitch Angle.

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three-time scale singular perturbation control for a radio...

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