UAV Glider Control System Based on Dynamic ... - Semantic Scholar
Proc. of 17th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje (Poland), August 27-30, 2012, pp.114-118
UAV glider control system based on dynamic contraction method. M. J. Bachuta1 , R. Czyba1 , W. Janusz1 , and V. D. Yurkevich2 1
2
Silesian University of Technology Novosibirsk State Technical University
Abstract— In this article two control structures for UAV (Unmanned Aerial Vehicle) glider are presented. Both structures are using control law based on dynamic contraction method, both of these are of cascade structure. Presented control systems allow to control aircraft yaw angle and velocity with respect to air. One of them also allows to stabilize sideslip angle on values close to zero. Proper operation of suggested structures is checked by using numerical simulation prepared in Matlab environment.
I. I NTRODUCTION The interest in unmanned aerial vehicles is growing nowadays. Despite general military usage where such objects are used in observation or strike tasks, unmanned vehicles are also used in civil areas. Operation with these objects takes place on distances where their direct observation by human operator is excluded. Therefore there is a need for some level of their autonomy, however higher levels of autonomy can not exist without control system of lower level that will stabilize flight of UAV. Such system should allow to track desired yaw angles and aircraft velocity with respect to air. In this article, for synthesis of control system, the control law based on dynamic contraction method was used. It allows for model reference control under unknown disturbances and unknown plant dynamics. Reference model is represented by a differential equation, of which parameters are chosen to provide desired behavior of closed loop system. Usage of controllers based on dynamic contraction method in control of aerial vehicles is also presented in: [1], [2]. Considered plant is small glider unmanned vehicle. In spite of not achieving large velocities by these types of aircrafts, dynamic model which describes them is highly nonlinear. Nonlinearities of this model, comes from nonlinear differential equations of rigid body motion and from equations describing aerodynamic forces and moments. II. UAV N ONLINEAR MODEL Nonlinear model of aircraft can be divided into two parts. First of them is a set of twelve nonlinear state equations, the second is a set of equations describing aerodynamic forces and moments. These forces and moments are connected with states of aircraft and with control surfaces (aileron, rudder and elevator) deflections. The aircraft model was implemented in Matlab - Simulink environment.
978-1-4673-2124-2/12/$31.00 ©2012 IEEE
A. Nonlinear state equations Nonlinear state equations, describing 6 - DOF aircraft motion in body axes can be found in [4] . These state equations define the following state vector: Xb = [u v w p q r φ θ ψ xe ye he ]T
(1)
In such defined state vector, u, v, w are linear velocities along x, y, z axes of aircraft body frame. Velocities p, q, r are angular velocities along x, y, z axes of aircraft body frame. Angles φ θ ψ are Euler angles. To describe aircraft position in NED (North East Down) frame quantities xe ye he are used, meaning aircraft position in north - south direction, east - west direction and height respectively. Equations describing translational motion: m(u˙ + qw − rv) = Fx − mg sin θ
(2)
m(v˙ + rv − pw) = Fy + mg cos θ sin φ
(3)
m(w˙ + pv − qu) = Fz + mg cos θ cos φ
(4)
Standard acceleration is marked as g and Fx , Fy , Fz are aerodynamic forces acting along x, y, z axes of aircraft body frame. Equations describing rotational motion: p˙
=
q˙
= c5 pr − c6 (p2 − r2 ) + c7 m
(c1 r + c2 p)q + c3 l + c4 n
r˙
=
(c8 p − c2 r)q + c4 l + c9 n
(5) (6) (7)
Aerodynamic moments acting along x, y, z axes of aircraft body frame are marked as l, m, n respectively. Quantities c1 , c2 , c3 , c4 , c5 , c6 , c7 , c8 , c9 are functions of aircraft moments of inertia as stated below: where: 2 Γ = Ix Iz − Ixz , 2 (Iy −Iz )Iz −Ixz c1 = , Γ (I −I +I )I c2 = x yΓ z xz , c3 = IΓz , c4 = IΓxz , x) c5 = (IzI−I , y Ixz c6 = Iy ,
114
c7 = c8 = c9 =
1 Iy , 2 Ix (Ix −Iy )+Ixz , Γ Ix . Γ
F
Equations relating the rate of the Euler angles changes to the angular velocities: φ˙ θ˙ ψ˙
F = p + tan θ(q sin φ + r cos φ)
(8)
= q cos φ − r sin φ q sin φ + r cos φ = cos θ
(9)
= u cos ψ cos θ + v(cos ψ sin θ sin φ − sin ψ cos φ) + + w(cos ψ sin θ cos φ + sin ψ sin φ)
y˙ e
(11)
= u sin ψ cos θ + v(sin ψ sin θ sin φ + cos ψ cos φ) +
+ w(sin ψ sin θ cos φ − cos ψ sin φ) ˙h = u sin θ − v cos θ sin φ − w cos θ cos φ
(12)
cos α cos β S = − cos α sin β − sin α
mαV ˙ t cos β
g2 g3
sin β cos β 0
sin α cos β − sin α sin β cos α
(24)
Aerodynamic forces, acting on aircraft, are defined as product of dynamic pressure, wing area and dimensionless aerodynamic coefficient. In case of aerodynamic moments, additional term describing one of the wing dimensions appears (wingspan or chord length). Aerodynamic forces in wind axes: L = qS(CLα α + CLδe δe)
= −D + mg1
(15)
= Y − mVt rw + mg2
(16)
= −L + mVt qw + mg3
(17)
=
g(− cos α cos β sin θ + sin β sin φ cos θ
+
sin α cos β cos φ cos θ)
=
g(cos α sin β sin θ + cos β sin φ cos θ
−
sin α sin β cos φ cos θ)
(19)
=
g(sin α sin θ + cos α cos φ cos θ)
(20)
(18)
(25)
D
= qS(CD0 + CDδe δe + + CDi (CLα + CLδe ) + CDβ β)
(26)
Y
=
(27)
qSCYβ β
Aerodynamic moments:
(14)
Components of gravitational force in wind axes: g1
(23)
(13)
In such defined state vector, instead of linear velocities u, v, w states V t, β, α are used. Vt is an aircraft velocity with respect to air, β is sideslip angle and α is angle of attack. Using these states is useful because aerodynamic forces and moments are usually modeled as functions of β, α. New set of state equations consists of (5)-(13) and equations: mV˙ t ˙ t mβV
S T FW
(22)
B. Model of aerodynamic forces and moments
Equations (2)-(13) allow to describe motion of aircraft in body frame. Describing motion of aircraft with respect to air allows to define alternative state vector: Xw = [Vt β α p q r φ θ ψ xe ye he ]T
=
(21)
where:
(10)
Equations relating linear velocities expressed in body frame with rate of changes of velocities expressed in NED frame (navigational equations): x˙ e
FW
Fx = Fy Fz −D = Y −L
qSb(Clβ β + Clδa δa + b (28) + Clδr δr + (Clp p + Clr r)) + ∆l 2Vt m = qSc(Cmα α + Cmδe δe + c + (29) (Cmq q + Cmα˙ α)) ˙ + ∆m 2Vt n = qSb(Cnβ β + Cnδr δr + b + Cnδa δa + Cndi + Cnr r) + ∆n (30) 2V t Quantities Ci, i = D, L, Y, l, m, n are called aerodynamic dimensionless coefficients. In considered model part of them are scalars, the other ones are in the form of lookup tables. Quantities ∆l, ∆m, ∆n are additional moments, which are results of displacement of aero reference point with respect to center of mass. This displacement results in force arm, on which aerodynamic forces are acting. This force arm results in additional moments, which are described by the equations below:
In equations (15)-(17) quantities D, Y, L appear. These are aerodynamic drag, side force and lift. Relation of these forces, with forces expressed in body frames is following:
115
l
∆l ∆m
=
= −(cgz − aerpz )Fy =
∆n =
(31)
(cgz − aerpz )Fx − (cgx − aerpx )Fz
(32)
(cgx − aerpx )Fy
(33)
Quantities cgx , cgz are coordinates of center of mass, aerpx , aerpz are coordinates of aero reference point, that means point, on which aerodynamic forces are acting. III. C ONTROL SYSTEM A. Dynamic Contraction Method Control law used in presented control system is based on dynamic contraction method( additional references can be found in [3]). Let us consider nonlinear, nonstationary, continues, single input single output system, described by the following equation: x(n) = f (X, w) + g(X, w)u,
y = x, X(0) = X 0
(34)
where: y - measurable output, X(0) = X 0 - initial condition, u - input, w - signal representing effect of unknown external disturbances and varying parameters.
eF = F (X, R) − x(n)
We seek such control u, that will make upper expression for eF equal to zero. Combining equations (40) and (34), we get following expression: F (X, R) − f (X, w) − g(X, w)u = 0;
uN ID = {g(X, w)}−1 {F (X, R) − f (X, w)}
lim e(t) = 0
+
q−1 X
µi di u(i) = k0 eF
(43)
i=1
where k0 is gain. Expressing right side of upper equation, using (40) and (38) we get following expression for control signal u:
(35)
are fulfilled for all (X, w). 3) Influence of external disturbances and varying parameters is expressed by the dependence of the functions f (X, w), g(X, w) on w. Objective of designed control system, is to satisfy condition: t→∞
(42)
Upper expression for u is called solution of nonlinear inverse dynamics. To obtain u it is necessary to have complete information about states of the system, disturbances and parameters. Control law, that does not require this information, is control law based on dynamic contraction method. Control u is expressed by the following equation: µ u
|f (X, w)| ≤ fmax < ∞, 0 < |g(X, w)| < ∞
(41)
Control u, presented below, is a solution on former equation:
q (q)
Let us assume following assumptions: 1) Functions f (X, w) and g(X, w) are smooth for all (X, w) and their analytical form is unknown. 2) Conditions
(40)
(36)
where e(t) is the error of the reference input realization e(t) = r(t) − y(t), r(t) is the reference input. Reference model of the signal x(t) can be chosen as stable differential equation of the form:
µq u(q) + dq−1 µq−1 u(q−1) + ... + d1 µu(1) + d0 u k0 = n {−T n x(n) − adn−1 T n−1 x(n−1) − ... − ad1 T x(1) − x T +bdp τ p r(p) + bdp−1 τ p−1 r(p−1) + ... + bd1 τ r(1) + r} (44) Use of control law based on dynamic contraction method, makes model reference control possible, whithout having complete information about state, disturbances and functions describing nonlinear plant model. Necessary condition, is to have sufficiently small value of µ parameter. B. Control system architecture
Structure of control system, which is presented in this article, is of cascade structure. Stabilization of air velocity x(n) = F (x(n−1) , ..., x(1) , x, r(p) , ..., r(1) , r) (37) V is fulfilled by two controllers. Outer loop controller, which t where p < n and for r = const, x = r is equilibrium point input signals are measured value of Vt and its desired value, output signal is desired value for pitch angle θ. Task of inner of reference model. loop controller is to provide tracking for desired value of θ. Reference model can be also written in the lower form: Output signal of inner loop controller is elevator deflection. Due to the fact that gain in this loop has negative sign (positive T n x(n) + adn−1 T n−1 x(n−1) + ... + ad1 T x(1) + x (38) deflection of elevator yield to negative change in θ), output of = bdp T p r(p) + bdp−1 T p−1 r(p−1) + ... + bd1 T r(1) + r inner loop controller must by multiplied by -1. Next task for control system, is tracking for desired values Writing (37) in the form: of yaw angle ψ. This task is accomplished like before, by cascade control structure. Input signals for outer loop conx(n) = F (X, R), R = {r, r(1) , ..., r(p) }T (39) troller are desired value of ψ angle and measured value of Error of the reference input realization described by model this angle, output signal is desired value of roll angle φ. Inner loop controller is responsible for tracking desired values of φ, (37), can be expressed as:
116
output signal for this controller is simultaneously deflection of ailerons and rudder. An alternative to solution above, is structure where output of inner controller is deflection of ailerons, rudder is used to keep value of sideslip angle β on zero. Additional controller is responsible for stabilization of β. Controllers for Vt , θ, β and φ are described by the following differential equation:
Controller for ψ Controller for φ Controller forβ
d1 2 2 2
d0 0 0 0
µ 0.2 0.05 0.02
T 2 0.5 0.2
k0 0.5 1 0.2
TABLE III PARAMETERS OF ψ, φ AND β CONTROLLERS .
k0 k0 ad d1 (1) d0 u + 2 u = − 2 x(2) − 2 1 x(1) − µ µ µ µ T k0 k0 (45) − 2 2x + 2 2r µ T µ T
u(2) +
Controllers for ψ angle are described by the following differential equation: d1 (1) d0 k0 k0 ad u + 2 u = − 2 x(2) − 2 1 x(1) − µ µ µ µ T k0 bd1 τ (1) k0 k0 (46) − 2 2x + 2 2 r + 2 2r µ T µ T µ T
u(2) +
It is assumed, that τ = T and bd1 = ad1 . C. Simulation results Parameters of controllers, responsible for realization of desired velocity with respect to air, are identical for both structures. These parameters are presented in Table I. Parameters of controllers, responsible for realization of desired yaw angle, for first structure (with simultaneously deflecting rudder and ailerons) are presented in Table II. Parameters of controllers, responsible for realization of desired yaw angle, for alternative structure are presented in Table III. Desired values of ψ angle was sinusoidal changing , in 30 second was step change in desired value of aircraft velocity with respect to air. Changes of Euler angles, for first structure are presented on Figure 1. On Figure 2 values of non stabilized sideslip angle are shown. Figures 3 and 4 shows Euler angles and stabilized sideslip angle respectively, for alternative structure. Controller for θ Controller for Vt
d1 2 2
d0 1 0
µ 0.01 0.1
T 0.2 1
k0 0.1 0.005
Fig. 1.
Euler angles changes for first structure.
ad1 2 2
TABLE I PARAMETERS OF θ, Vt CONTROLLERS .
Controller for ψ Controller for φ
d1 2 2
d0 0 0
µ 0.25 0.05
T 2.5 0.5
k0 2 0.75
ad1 2 2
TABLE II PARAMETERS OF ψ, φ CONTROLLERS . Fig. 2.
117
Sideslip angle changes for first structure.
ad1 2 2 2
R EFERENCES [1] M. J. Bachuta,V. D. Yurkevich, K. Wojciechowski, Design of analog and digital aircraft flight controllers based on dynamic contraction method, Proc. of the 1997 AIAA Guidance, Navigation and Control Conf., New Orleans, Louisiana, part 3, pp. 1719-1729, 1997. [2] M. J. Bachuta,V. D. Yurkevich, K. Wojciechowski, Design of aircraft 3D motion control using dynamic contraction method, 1rd ed. Proc. of IFAC-Workshop ”Motion Control”, Munich, pp. 323-330, 1995. [3] V. D. Yurkevich, Design of Nonlinear Control Systems with the Highest Derivative in Feedback, 1rd ed. World Scientific, Singapore, 2004. [4] B. L. Stevens, F. L. Lewis, Aircraft Control and Simulation, 2rd ed. Wiley-Interscience, New York, 2003.
Fig. 3.
Fig. 4.
Euler angles changes for alternative structure.
Sideslip angle changes for alternative structure.
IV. C ONCLUSION An aircraft, 6-DOF model was developed. Then two alternative control structures where proposed, both of them was cascade structures. Control system provided tracking for changes of desired values of yaw angle and aircraft velocity with respect to air. Simulation consisting of aircraft model and control system, was implemented in Matlab - Simulink environment and using developed simulation proper operation of control system was checked. Used control law was based on dynamic contraction method, which allowed to model reference control. One of the structures allowed to stabilize sideslip angle on values close to zero.
118
UAV glider control system based on dynamic contraction method. M. J. Bachuta1 , R. Czyba1 , W. Janusz1 , and V. D. Yurkevich2 1
2
Silesian University of Technology Novosibirsk State Technical University
Abstract— In this article two control structures for UAV (Unmanned Aerial Vehicle) glider are presented. Both structures are using control law based on dynamic contraction method, both of these are of cascade structure. Presented control systems allow to control aircraft yaw angle and velocity with respect to air. One of them also allows to stabilize sideslip angle on values close to zero. Proper operation of suggested structures is checked by using numerical simulation prepared in Matlab environment.
I. I NTRODUCTION The interest in unmanned aerial vehicles is growing nowadays. Despite general military usage where such objects are used in observation or strike tasks, unmanned vehicles are also used in civil areas. Operation with these objects takes place on distances where their direct observation by human operator is excluded. Therefore there is a need for some level of their autonomy, however higher levels of autonomy can not exist without control system of lower level that will stabilize flight of UAV. Such system should allow to track desired yaw angles and aircraft velocity with respect to air. In this article, for synthesis of control system, the control law based on dynamic contraction method was used. It allows for model reference control under unknown disturbances and unknown plant dynamics. Reference model is represented by a differential equation, of which parameters are chosen to provide desired behavior of closed loop system. Usage of controllers based on dynamic contraction method in control of aerial vehicles is also presented in: [1], [2]. Considered plant is small glider unmanned vehicle. In spite of not achieving large velocities by these types of aircrafts, dynamic model which describes them is highly nonlinear. Nonlinearities of this model, comes from nonlinear differential equations of rigid body motion and from equations describing aerodynamic forces and moments. II. UAV N ONLINEAR MODEL Nonlinear model of aircraft can be divided into two parts. First of them is a set of twelve nonlinear state equations, the second is a set of equations describing aerodynamic forces and moments. These forces and moments are connected with states of aircraft and with control surfaces (aileron, rudder and elevator) deflections. The aircraft model was implemented in Matlab - Simulink environment.
978-1-4673-2124-2/12/$31.00 ©2012 IEEE
A. Nonlinear state equations Nonlinear state equations, describing 6 - DOF aircraft motion in body axes can be found in [4] . These state equations define the following state vector: Xb = [u v w p q r φ θ ψ xe ye he ]T
(1)
In such defined state vector, u, v, w are linear velocities along x, y, z axes of aircraft body frame. Velocities p, q, r are angular velocities along x, y, z axes of aircraft body frame. Angles φ θ ψ are Euler angles. To describe aircraft position in NED (North East Down) frame quantities xe ye he are used, meaning aircraft position in north - south direction, east - west direction and height respectively. Equations describing translational motion: m(u˙ + qw − rv) = Fx − mg sin θ
(2)
m(v˙ + rv − pw) = Fy + mg cos θ sin φ
(3)
m(w˙ + pv − qu) = Fz + mg cos θ cos φ
(4)
Standard acceleration is marked as g and Fx , Fy , Fz are aerodynamic forces acting along x, y, z axes of aircraft body frame. Equations describing rotational motion: p˙
=
q˙
= c5 pr − c6 (p2 − r2 ) + c7 m
(c1 r + c2 p)q + c3 l + c4 n
r˙
=
(c8 p − c2 r)q + c4 l + c9 n
(5) (6) (7)
Aerodynamic moments acting along x, y, z axes of aircraft body frame are marked as l, m, n respectively. Quantities c1 , c2 , c3 , c4 , c5 , c6 , c7 , c8 , c9 are functions of aircraft moments of inertia as stated below: where: 2 Γ = Ix Iz − Ixz , 2 (Iy −Iz )Iz −Ixz c1 = , Γ (I −I +I )I c2 = x yΓ z xz , c3 = IΓz , c4 = IΓxz , x) c5 = (IzI−I , y Ixz c6 = Iy ,
114
c7 = c8 = c9 =
1 Iy , 2 Ix (Ix −Iy )+Ixz , Γ Ix . Γ
F
Equations relating the rate of the Euler angles changes to the angular velocities: φ˙ θ˙ ψ˙
F = p + tan θ(q sin φ + r cos φ)
(8)
= q cos φ − r sin φ q sin φ + r cos φ = cos θ
(9)
= u cos ψ cos θ + v(cos ψ sin θ sin φ − sin ψ cos φ) + + w(cos ψ sin θ cos φ + sin ψ sin φ)
y˙ e
(11)
= u sin ψ cos θ + v(sin ψ sin θ sin φ + cos ψ cos φ) +
+ w(sin ψ sin θ cos φ − cos ψ sin φ) ˙h = u sin θ − v cos θ sin φ − w cos θ cos φ
(12)
cos α cos β S = − cos α sin β − sin α
mαV ˙ t cos β
g2 g3
sin β cos β 0
sin α cos β − sin α sin β cos α
(24)
Aerodynamic forces, acting on aircraft, are defined as product of dynamic pressure, wing area and dimensionless aerodynamic coefficient. In case of aerodynamic moments, additional term describing one of the wing dimensions appears (wingspan or chord length). Aerodynamic forces in wind axes: L = qS(CLα α + CLδe δe)
= −D + mg1
(15)
= Y − mVt rw + mg2
(16)
= −L + mVt qw + mg3
(17)
=
g(− cos α cos β sin θ + sin β sin φ cos θ
+
sin α cos β cos φ cos θ)
=
g(cos α sin β sin θ + cos β sin φ cos θ
−
sin α sin β cos φ cos θ)
(19)
=
g(sin α sin θ + cos α cos φ cos θ)
(20)
(18)
(25)
D
= qS(CD0 + CDδe δe + + CDi (CLα + CLδe ) + CDβ β)
(26)
Y
=
(27)
qSCYβ β
Aerodynamic moments:
(14)
Components of gravitational force in wind axes: g1
(23)
(13)
In such defined state vector, instead of linear velocities u, v, w states V t, β, α are used. Vt is an aircraft velocity with respect to air, β is sideslip angle and α is angle of attack. Using these states is useful because aerodynamic forces and moments are usually modeled as functions of β, α. New set of state equations consists of (5)-(13) and equations: mV˙ t ˙ t mβV
S T FW
(22)
B. Model of aerodynamic forces and moments
Equations (2)-(13) allow to describe motion of aircraft in body frame. Describing motion of aircraft with respect to air allows to define alternative state vector: Xw = [Vt β α p q r φ θ ψ xe ye he ]T
=
(21)
where:
(10)
Equations relating linear velocities expressed in body frame with rate of changes of velocities expressed in NED frame (navigational equations): x˙ e
FW
Fx = Fy Fz −D = Y −L
qSb(Clβ β + Clδa δa + b (28) + Clδr δr + (Clp p + Clr r)) + ∆l 2Vt m = qSc(Cmα α + Cmδe δe + c + (29) (Cmq q + Cmα˙ α)) ˙ + ∆m 2Vt n = qSb(Cnβ β + Cnδr δr + b + Cnδa δa + Cndi + Cnr r) + ∆n (30) 2V t Quantities Ci, i = D, L, Y, l, m, n are called aerodynamic dimensionless coefficients. In considered model part of them are scalars, the other ones are in the form of lookup tables. Quantities ∆l, ∆m, ∆n are additional moments, which are results of displacement of aero reference point with respect to center of mass. This displacement results in force arm, on which aerodynamic forces are acting. This force arm results in additional moments, which are described by the equations below:
In equations (15)-(17) quantities D, Y, L appear. These are aerodynamic drag, side force and lift. Relation of these forces, with forces expressed in body frames is following:
115
l
∆l ∆m
=
= −(cgz − aerpz )Fy =
∆n =
(31)
(cgz − aerpz )Fx − (cgx − aerpx )Fz
(32)
(cgx − aerpx )Fy
(33)
Quantities cgx , cgz are coordinates of center of mass, aerpx , aerpz are coordinates of aero reference point, that means point, on which aerodynamic forces are acting. III. C ONTROL SYSTEM A. Dynamic Contraction Method Control law used in presented control system is based on dynamic contraction method( additional references can be found in [3]). Let us consider nonlinear, nonstationary, continues, single input single output system, described by the following equation: x(n) = f (X, w) + g(X, w)u,
y = x, X(0) = X 0
(34)
where: y - measurable output, X(0) = X 0 - initial condition, u - input, w - signal representing effect of unknown external disturbances and varying parameters.
eF = F (X, R) − x(n)
We seek such control u, that will make upper expression for eF equal to zero. Combining equations (40) and (34), we get following expression: F (X, R) − f (X, w) − g(X, w)u = 0;
uN ID = {g(X, w)}−1 {F (X, R) − f (X, w)}
lim e(t) = 0
+
q−1 X
µi di u(i) = k0 eF
(43)
i=1
where k0 is gain. Expressing right side of upper equation, using (40) and (38) we get following expression for control signal u:
(35)
are fulfilled for all (X, w). 3) Influence of external disturbances and varying parameters is expressed by the dependence of the functions f (X, w), g(X, w) on w. Objective of designed control system, is to satisfy condition: t→∞
(42)
Upper expression for u is called solution of nonlinear inverse dynamics. To obtain u it is necessary to have complete information about states of the system, disturbances and parameters. Control law, that does not require this information, is control law based on dynamic contraction method. Control u is expressed by the following equation: µ u
|f (X, w)| ≤ fmax < ∞, 0 < |g(X, w)| < ∞
(41)
Control u, presented below, is a solution on former equation:
q (q)
Let us assume following assumptions: 1) Functions f (X, w) and g(X, w) are smooth for all (X, w) and their analytical form is unknown. 2) Conditions
(40)
(36)
where e(t) is the error of the reference input realization e(t) = r(t) − y(t), r(t) is the reference input. Reference model of the signal x(t) can be chosen as stable differential equation of the form:
µq u(q) + dq−1 µq−1 u(q−1) + ... + d1 µu(1) + d0 u k0 = n {−T n x(n) − adn−1 T n−1 x(n−1) − ... − ad1 T x(1) − x T +bdp τ p r(p) + bdp−1 τ p−1 r(p−1) + ... + bd1 τ r(1) + r} (44) Use of control law based on dynamic contraction method, makes model reference control possible, whithout having complete information about state, disturbances and functions describing nonlinear plant model. Necessary condition, is to have sufficiently small value of µ parameter. B. Control system architecture
Structure of control system, which is presented in this article, is of cascade structure. Stabilization of air velocity x(n) = F (x(n−1) , ..., x(1) , x, r(p) , ..., r(1) , r) (37) V is fulfilled by two controllers. Outer loop controller, which t where p < n and for r = const, x = r is equilibrium point input signals are measured value of Vt and its desired value, output signal is desired value for pitch angle θ. Task of inner of reference model. loop controller is to provide tracking for desired value of θ. Reference model can be also written in the lower form: Output signal of inner loop controller is elevator deflection. Due to the fact that gain in this loop has negative sign (positive T n x(n) + adn−1 T n−1 x(n−1) + ... + ad1 T x(1) + x (38) deflection of elevator yield to negative change in θ), output of = bdp T p r(p) + bdp−1 T p−1 r(p−1) + ... + bd1 T r(1) + r inner loop controller must by multiplied by -1. Next task for control system, is tracking for desired values Writing (37) in the form: of yaw angle ψ. This task is accomplished like before, by cascade control structure. Input signals for outer loop conx(n) = F (X, R), R = {r, r(1) , ..., r(p) }T (39) troller are desired value of ψ angle and measured value of Error of the reference input realization described by model this angle, output signal is desired value of roll angle φ. Inner loop controller is responsible for tracking desired values of φ, (37), can be expressed as:
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output signal for this controller is simultaneously deflection of ailerons and rudder. An alternative to solution above, is structure where output of inner controller is deflection of ailerons, rudder is used to keep value of sideslip angle β on zero. Additional controller is responsible for stabilization of β. Controllers for Vt , θ, β and φ are described by the following differential equation:
Controller for ψ Controller for φ Controller forβ
d1 2 2 2
d0 0 0 0
µ 0.2 0.05 0.02
T 2 0.5 0.2
k0 0.5 1 0.2
TABLE III PARAMETERS OF ψ, φ AND β CONTROLLERS .
k0 k0 ad d1 (1) d0 u + 2 u = − 2 x(2) − 2 1 x(1) − µ µ µ µ T k0 k0 (45) − 2 2x + 2 2r µ T µ T
u(2) +
Controllers for ψ angle are described by the following differential equation: d1 (1) d0 k0 k0 ad u + 2 u = − 2 x(2) − 2 1 x(1) − µ µ µ µ T k0 bd1 τ (1) k0 k0 (46) − 2 2x + 2 2 r + 2 2r µ T µ T µ T
u(2) +
It is assumed, that τ = T and bd1 = ad1 . C. Simulation results Parameters of controllers, responsible for realization of desired velocity with respect to air, are identical for both structures. These parameters are presented in Table I. Parameters of controllers, responsible for realization of desired yaw angle, for first structure (with simultaneously deflecting rudder and ailerons) are presented in Table II. Parameters of controllers, responsible for realization of desired yaw angle, for alternative structure are presented in Table III. Desired values of ψ angle was sinusoidal changing , in 30 second was step change in desired value of aircraft velocity with respect to air. Changes of Euler angles, for first structure are presented on Figure 1. On Figure 2 values of non stabilized sideslip angle are shown. Figures 3 and 4 shows Euler angles and stabilized sideslip angle respectively, for alternative structure. Controller for θ Controller for Vt
d1 2 2
d0 1 0
µ 0.01 0.1
T 0.2 1
k0 0.1 0.005
Fig. 1.
Euler angles changes for first structure.
ad1 2 2
TABLE I PARAMETERS OF θ, Vt CONTROLLERS .
Controller for ψ Controller for φ
d1 2 2
d0 0 0
µ 0.25 0.05
T 2.5 0.5
k0 2 0.75
ad1 2 2
TABLE II PARAMETERS OF ψ, φ CONTROLLERS . Fig. 2.
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Sideslip angle changes for first structure.
ad1 2 2 2
R EFERENCES [1] M. J. Bachuta,V. D. Yurkevich, K. Wojciechowski, Design of analog and digital aircraft flight controllers based on dynamic contraction method, Proc. of the 1997 AIAA Guidance, Navigation and Control Conf., New Orleans, Louisiana, part 3, pp. 1719-1729, 1997. [2] M. J. Bachuta,V. D. Yurkevich, K. Wojciechowski, Design of aircraft 3D motion control using dynamic contraction method, 1rd ed. Proc. of IFAC-Workshop ”Motion Control”, Munich, pp. 323-330, 1995. [3] V. D. Yurkevich, Design of Nonlinear Control Systems with the Highest Derivative in Feedback, 1rd ed. World Scientific, Singapore, 2004. [4] B. L. Stevens, F. L. Lewis, Aircraft Control and Simulation, 2rd ed. Wiley-Interscience, New York, 2003.
Fig. 3.
Fig. 4.
Euler angles changes for alternative structure.
Sideslip angle changes for alternative structure.
IV. C ONCLUSION An aircraft, 6-DOF model was developed. Then two alternative control structures where proposed, both of them was cascade structures. Control system provided tracking for changes of desired values of yaw angle and aircraft velocity with respect to air. Simulation consisting of aircraft model and control system, was implemented in Matlab - Simulink environment and using developed simulation proper operation of control system was checked. Used control law was based on dynamic contraction method, which allowed to model reference control. One of the structures allowed to stabilize sideslip angle on values close to zero.
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