Global Stabilization of a PVTOL Aircraft with ... - Semantic Scholar
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
FrP03-2
Global stabilization of a PVTOL with bounded thrust
aircraft
I. Fantoni, A. Zavala 1, R. Lozano Heudiasyc, UMR CNRS 6599 UTC, BP 20529 60205 CompiSgne, France
{ifantoni/zavala/rlozano}@hds. u t Abstract
Global stabilizing control design is proposed for the planar vertical takeoff and landing (PVTOL) aircraft. The methodology is based on the use of nonlinear combinations of saturation functions bounding the thrust input to arbitrary saturation limits. The algorithm is simple and provides global convergence of the state to the origin. Keywords: Aircraft control, Non-linear control systems, Global stabilization, Saturated functions, Bounded control.
1 Introduction
Numerous design methods for the flight control of the Planar Vertical Take Off and Landing (PVTOL) aircraft model exist in the literature. Indeed, this particular system is a simplified aircraft with a minimal number of states and inputs but retains the main features that must be considered when designing control laws for a real aircraft. Since, the system possesses special properties such as, for instance, unstable zero dynamics and signed (thrust) input, several methodologies for controlling such a system have been proposed. Hauser et al. [21 in 1992 applied an approximate I-O linearization procedure which results in bounded tracking and asymptotic stability for the V / S T O L aircraft. In 1996, Andrew R. Teel [121 illustrated his central result of nonlinear small gain theorem using the example of the PVTOL aircraft with input corruption. His theorem provided a formalism for analyzing the behavior of control systems with saturation. He established a stabilization algorithm for nonlinear systems in so-called feedforward form which includes the PVTOL aircraft. The same 1On leave from Instituto Potosino de Investigacidn Cienfffica y Tecnoldgica, Mexico, e-rural: [email protected], his participation in this research work was supported by the Centre National de la Recherche Scientifique, France.
0-7803-7516-5/02/$17.00 ©2002 IEEE
4462
c. fr
year, Martin et al. [5] presented an extension of the result proposed by Hauser [2]. Their idea was to find a flat output for the system and to split the output tracking problem in two steps. Firstly, they designed a state tracker based on exact linearization by using the flat output and secondly, they designed a trajectory generator to feed the state tracker. They thus controlled the tracking output through the flat output. In contrast to the approximate-linearization based control method proposed by Hauser, their control scheme provided output tracking of nonminimum phase flat systems. They have also taken into account in the design the coupling between the rolling moment and the lateral acceleration of the aircraft (i.e. c ¢ 0). Sepulchre et al. [81 applied a linear high gain approximation of backstepping to the approximated model neglecting the coupling. In 1999, Lin et al. [3] studied robust hovering control of the PVTOL using nonlinear state feedback based on optimal control. Reza Olfati-Saber [6] proposed a configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback. M. Saeki et al. [7] offered a new design method which makes use of the center of oscillation and a two-step linearization. In fact, they designed a controller by applying a linear high gain approximation of backstepping to the model. A recent paper on an internal-model based approach for the autonomous vertical landing on an oscillating platform has been proposed by Marconi et al. [41. They presented an error-feedback dynamic regulator that is robust with respect to uncertainties of the model parameters and they provided global convergence to the zero-error manifold. In the present paper, we present a global stabilizing strategy for the control of the PVTOL aircraft. The proposed algorithm copes with a bounded thrust and takes into account its positive nature. The stability proof is simple. As far as we are aware, the previous works on the topic do not cover all these features simultaneously. The paper is organized as follows. In section
2, we recall the equations of motion for the P V T O L aircraft. In section 3, the global stabilizing control law is developed. Simulations are presented in section 4 and conclusions are finally given in section 5.
T h e system dynamics considering these new coordinates become -- sin(0)~21 ~)
aircraft m o d e l
2 The PVTOL
-
Cos(O)~
--
%2
1 --
1
(7)
where Ul - Ul - st) 2. Note t h a t this s t r u c t u r e (7) has the same form as (1) with c - 0. Out" control objective is to stabilize the P V T O L aircraft to the origin.
The P V T O L aircraft dynamics are modelled by the following equations [2] =
--
+
--
COS(0)%
I
%2
1 @ £sin(0)%
2 -- 1
(1)
3 Global stabilizing input Let us begin by explaining the background philosophy of the proposed control scheme. Notice t h a t the PVT O L dynamics (2)-(4) can be divided in two subsystems according to the n a t u r e of the motion: the rotational motion equation (4), and the translational motion dynamics (2)-(3). The first one consists of a double integrator with u2 as unique external input. It evolves independently of the translational motion variables. On the contrary, the second one consists of two (independent) double integrators with Ul and 0 as c o m m o n external forcing agents. Hence, as done in [8, 61, 0 can be viewed as an intermediate (fictitious) input to control, together with Ul, s u b s y s t e m (2)-(4). Under such perspective, let us define
where x, y denote the center of mass horizontal and vertical position and 0 is the roll angle of the aircraft with the horizon. T h e control inputs Ul and u2 are the t h r u s t (directed out the b o t t o m of the aircraft) and the angular acceleration (rolling m o m e n t ) . T h e p a r a m e t e r c is a small coefficient which characterizes the coupling between the rolling m o m e n t and the lateral acceleration of the aircraft. T h e constant " - 1 " is the normalized gravitational acceleration. Figure 1 provides a representation of the system. In general, c is negligible and
y
-- v / r 2 + (1 + r2) 2
(8)
e = a t ' c t a n ( - r l , 1 -[- r2)
(9)
gl
y
Note t h a t Ul > 0. 0 in (9) represents the (unique) angle such t h a t sin 0 = -~~ and cos0 = l+r2
.......
y/r~ + (l+r2) 2
y/r~ + (l+r2) 2"
By taking (8) and (9), it follows t h a t i i 0
--
>
x
X
--
Figure 1: The P V T O L aircraft (fi'ont view)
-- sin(O)u
-- Cos(O)tt
I
(2)
I -- 1
(3)
(4) Furthermore, several authors have shown t h a t by an appropriate change of coordinates, we can obtain a representation of the system without the t e r m due to c [6, 7, 91. For instance, R. Olfati-Saber [61 applied the following change of coordinates -
-
-
y +
t---+ o c
quence,
l)
lira 2(t) = rl and
t---+ c c
lira 9(t) = r2.
t---+cc
Now, rl
and r2 could be simply selected as linear stabilizing state feedbacks, i.e. rl = - k l l X k12k and r2 - k 2 1 y - k22~/, with kij > 0, Vi, j = 1,2, as is actually proposed in [71 and [81. Moreover, a similar
(5) (cos(O) -
(10)
with r 1 a n d 7"2 as free functions (auxiliary inputs) t h a t can be suitably defined to achieve our control objective. T h e convenient selection of rl and r2 actually constitutes the second step of the design methodology. Finally, the last step consists in the consideration of the rotational motion equation to determine an a p p r o p r i a t e u2 t h a t makes 0 follow the desired motion expressed in (9). Nevertheless, the secondorder dynamics (4) does not permit u2 to give directly any desired form to 0. T h e idea is, then, to achieve lira 0(t) = a I ' c t a n ( - r l , 1 + r2). As a conse-
not always well-known [2]. Therefore, it is possible to suppose t h a t c - 0, i.e. --
7"1 7"2
(6)
4463
tracking version could be considered for us in (4), i.e. us = Od -- k31(O -- Od) -- k32([~ -- [~d), with Od a r c t a n ( - - r l , 1 + r2) and k3i > 0, Vi = 1, 2, as exposed in [7] 1. Nevertheless, such approaches do not seem to be a p p r o p r i a t e whenever Ul a n d / o r us are (physically) b o u n d e d inputs. This constitutes the interest of the present study: to provide a solution to our control problem whenever the t h r u s t is furnished by an actuator with (output) s a t u r a t i o n limits (which is a realistic case). In other words, we take Ul _< U1 (recall t h a t the t h r u s t is by n a t u r e nonnegative) for some finite positive (constant) U1. We consider the proposed approach to constitute a first step towards a complete solution considering b o t h Ul and us bounded, which we are currently working on. Notice, from (3), t h a t U1 > 1 is a necessary condition for the P V T O L to be stabilizable at any desired position. Indeed, any static condition implies t h a t the aircraft's weight be compensated. In such scenario, the selected functions rl, r2, and us are based on the stabilization approach proposed in [11]. Therefore, they are defined in t e r m s of linear saturation functions, whose definition is recalled in [11]:
is
given by
~1 -- ~2, ~2 -- --Cr32(~2 q-Cr31(~1 q - ~ 2 ) ) .
Hence, from T h e o r e m 2.1 in [11], it follows that, for any ~5(0) C lg 2, lira ~5(t) - 0. T h a t is 0 --+ Od as t --+ co. In /~---+oo
the limit when 0 - Od, the (x, k) and (y, ~)) s u b s y s t e m s are given by ,~1
--
Z2
,~2 -- --0"12(Z2 @ 0"11(Z1 @ Z2))
~3 - z4 ,~4 - -022(z4 + 021(z3 + z4)) T h e stability of the overall system during the transient when 0 --+ Od can be g u a r a n t e e d by the result of Sontag [10]. The stability of the system when 0 - Od is again obtained fi'om T h e o r e m 2.1 in [11]. It then follows that, for any z(0) C //~a, lira z ( t ) - O, with rx(t) _< MI2 t--+ oo
and r2(t) < M22, Vt > O. Now, from the definitions of rl, r2, crij (i - 1, 2, 3, j - 1, 2), and Od, it is not difficult to check t h a t such a s y m p t o t i c a l stabilization of z implies lira R ( t ) - O, where R A ( r l , r 2 , ? l , ? 2 ) T , which t---+oc
in t u r n entails lira O d ( t ) t---+oc
D e f i n i t i o n 3.1 Given two positive constants L, M with L
FrP03-2
Global stabilization of a PVTOL with bounded thrust
aircraft
I. Fantoni, A. Zavala 1, R. Lozano Heudiasyc, UMR CNRS 6599 UTC, BP 20529 60205 CompiSgne, France
{ifantoni/zavala/rlozano}@hds. u t Abstract
Global stabilizing control design is proposed for the planar vertical takeoff and landing (PVTOL) aircraft. The methodology is based on the use of nonlinear combinations of saturation functions bounding the thrust input to arbitrary saturation limits. The algorithm is simple and provides global convergence of the state to the origin. Keywords: Aircraft control, Non-linear control systems, Global stabilization, Saturated functions, Bounded control.
1 Introduction
Numerous design methods for the flight control of the Planar Vertical Take Off and Landing (PVTOL) aircraft model exist in the literature. Indeed, this particular system is a simplified aircraft with a minimal number of states and inputs but retains the main features that must be considered when designing control laws for a real aircraft. Since, the system possesses special properties such as, for instance, unstable zero dynamics and signed (thrust) input, several methodologies for controlling such a system have been proposed. Hauser et al. [21 in 1992 applied an approximate I-O linearization procedure which results in bounded tracking and asymptotic stability for the V / S T O L aircraft. In 1996, Andrew R. Teel [121 illustrated his central result of nonlinear small gain theorem using the example of the PVTOL aircraft with input corruption. His theorem provided a formalism for analyzing the behavior of control systems with saturation. He established a stabilization algorithm for nonlinear systems in so-called feedforward form which includes the PVTOL aircraft. The same 1On leave from Instituto Potosino de Investigacidn Cienfffica y Tecnoldgica, Mexico, e-rural: [email protected], his participation in this research work was supported by the Centre National de la Recherche Scientifique, France.
0-7803-7516-5/02/$17.00 ©2002 IEEE
4462
c. fr
year, Martin et al. [5] presented an extension of the result proposed by Hauser [2]. Their idea was to find a flat output for the system and to split the output tracking problem in two steps. Firstly, they designed a state tracker based on exact linearization by using the flat output and secondly, they designed a trajectory generator to feed the state tracker. They thus controlled the tracking output through the flat output. In contrast to the approximate-linearization based control method proposed by Hauser, their control scheme provided output tracking of nonminimum phase flat systems. They have also taken into account in the design the coupling between the rolling moment and the lateral acceleration of the aircraft (i.e. c ¢ 0). Sepulchre et al. [81 applied a linear high gain approximation of backstepping to the approximated model neglecting the coupling. In 1999, Lin et al. [3] studied robust hovering control of the PVTOL using nonlinear state feedback based on optimal control. Reza Olfati-Saber [6] proposed a configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback. M. Saeki et al. [7] offered a new design method which makes use of the center of oscillation and a two-step linearization. In fact, they designed a controller by applying a linear high gain approximation of backstepping to the model. A recent paper on an internal-model based approach for the autonomous vertical landing on an oscillating platform has been proposed by Marconi et al. [41. They presented an error-feedback dynamic regulator that is robust with respect to uncertainties of the model parameters and they provided global convergence to the zero-error manifold. In the present paper, we present a global stabilizing strategy for the control of the PVTOL aircraft. The proposed algorithm copes with a bounded thrust and takes into account its positive nature. The stability proof is simple. As far as we are aware, the previous works on the topic do not cover all these features simultaneously. The paper is organized as follows. In section
2, we recall the equations of motion for the P V T O L aircraft. In section 3, the global stabilizing control law is developed. Simulations are presented in section 4 and conclusions are finally given in section 5.
T h e system dynamics considering these new coordinates become -- sin(0)~21 ~)
aircraft m o d e l
2 The PVTOL
-
Cos(O)~
--
%2
1 --
1
(7)
where Ul - Ul - st) 2. Note t h a t this s t r u c t u r e (7) has the same form as (1) with c - 0. Out" control objective is to stabilize the P V T O L aircraft to the origin.
The P V T O L aircraft dynamics are modelled by the following equations [2] =
--
+
--
COS(0)%
I
%2
1 @ £sin(0)%
2 -- 1
(1)
3 Global stabilizing input Let us begin by explaining the background philosophy of the proposed control scheme. Notice t h a t the PVT O L dynamics (2)-(4) can be divided in two subsystems according to the n a t u r e of the motion: the rotational motion equation (4), and the translational motion dynamics (2)-(3). The first one consists of a double integrator with u2 as unique external input. It evolves independently of the translational motion variables. On the contrary, the second one consists of two (independent) double integrators with Ul and 0 as c o m m o n external forcing agents. Hence, as done in [8, 61, 0 can be viewed as an intermediate (fictitious) input to control, together with Ul, s u b s y s t e m (2)-(4). Under such perspective, let us define
where x, y denote the center of mass horizontal and vertical position and 0 is the roll angle of the aircraft with the horizon. T h e control inputs Ul and u2 are the t h r u s t (directed out the b o t t o m of the aircraft) and the angular acceleration (rolling m o m e n t ) . T h e p a r a m e t e r c is a small coefficient which characterizes the coupling between the rolling m o m e n t and the lateral acceleration of the aircraft. T h e constant " - 1 " is the normalized gravitational acceleration. Figure 1 provides a representation of the system. In general, c is negligible and
y
-- v / r 2 + (1 + r2) 2
(8)
e = a t ' c t a n ( - r l , 1 -[- r2)
(9)
gl
y
Note t h a t Ul > 0. 0 in (9) represents the (unique) angle such t h a t sin 0 = -~~ and cos0 = l+r2
.......
y/r~ + (l+r2) 2
y/r~ + (l+r2) 2"
By taking (8) and (9), it follows t h a t i i 0
--
>
x
X
--
Figure 1: The P V T O L aircraft (fi'ont view)
-- sin(O)u
-- Cos(O)tt
I
(2)
I -- 1
(3)
(4) Furthermore, several authors have shown t h a t by an appropriate change of coordinates, we can obtain a representation of the system without the t e r m due to c [6, 7, 91. For instance, R. Olfati-Saber [61 applied the following change of coordinates -
-
-
y +
t---+ o c
quence,
l)
lira 2(t) = rl and
t---+ c c
lira 9(t) = r2.
t---+cc
Now, rl
and r2 could be simply selected as linear stabilizing state feedbacks, i.e. rl = - k l l X k12k and r2 - k 2 1 y - k22~/, with kij > 0, Vi, j = 1,2, as is actually proposed in [71 and [81. Moreover, a similar
(5) (cos(O) -
(10)
with r 1 a n d 7"2 as free functions (auxiliary inputs) t h a t can be suitably defined to achieve our control objective. T h e convenient selection of rl and r2 actually constitutes the second step of the design methodology. Finally, the last step consists in the consideration of the rotational motion equation to determine an a p p r o p r i a t e u2 t h a t makes 0 follow the desired motion expressed in (9). Nevertheless, the secondorder dynamics (4) does not permit u2 to give directly any desired form to 0. T h e idea is, then, to achieve lira 0(t) = a I ' c t a n ( - r l , 1 + r2). As a conse-
not always well-known [2]. Therefore, it is possible to suppose t h a t c - 0, i.e. --
7"1 7"2
(6)
4463
tracking version could be considered for us in (4), i.e. us = Od -- k31(O -- Od) -- k32([~ -- [~d), with Od a r c t a n ( - - r l , 1 + r2) and k3i > 0, Vi = 1, 2, as exposed in [7] 1. Nevertheless, such approaches do not seem to be a p p r o p r i a t e whenever Ul a n d / o r us are (physically) b o u n d e d inputs. This constitutes the interest of the present study: to provide a solution to our control problem whenever the t h r u s t is furnished by an actuator with (output) s a t u r a t i o n limits (which is a realistic case). In other words, we take Ul _< U1 (recall t h a t the t h r u s t is by n a t u r e nonnegative) for some finite positive (constant) U1. We consider the proposed approach to constitute a first step towards a complete solution considering b o t h Ul and us bounded, which we are currently working on. Notice, from (3), t h a t U1 > 1 is a necessary condition for the P V T O L to be stabilizable at any desired position. Indeed, any static condition implies t h a t the aircraft's weight be compensated. In such scenario, the selected functions rl, r2, and us are based on the stabilization approach proposed in [11]. Therefore, they are defined in t e r m s of linear saturation functions, whose definition is recalled in [11]:
is
given by
~1 -- ~2, ~2 -- --Cr32(~2 q-Cr31(~1 q - ~ 2 ) ) .
Hence, from T h e o r e m 2.1 in [11], it follows that, for any ~5(0) C lg 2, lira ~5(t) - 0. T h a t is 0 --+ Od as t --+ co. In /~---+oo
the limit when 0 - Od, the (x, k) and (y, ~)) s u b s y s t e m s are given by ,~1
--
Z2
,~2 -- --0"12(Z2 @ 0"11(Z1 @ Z2))
~3 - z4 ,~4 - -022(z4 + 021(z3 + z4)) T h e stability of the overall system during the transient when 0 --+ Od can be g u a r a n t e e d by the result of Sontag [10]. The stability of the system when 0 - Od is again obtained fi'om T h e o r e m 2.1 in [11]. It then follows that, for any z(0) C //~a, lira z ( t ) - O, with rx(t) _< MI2 t--+ oo
and r2(t) < M22, Vt > O. Now, from the definitions of rl, r2, crij (i - 1, 2, 3, j - 1, 2), and Od, it is not difficult to check t h a t such a s y m p t o t i c a l stabilization of z implies lira R ( t ) - O, where R A ( r l , r 2 , ? l , ? 2 ) T , which t---+oc
in t u r n entails lira O d ( t ) t---+oc
D e f i n i t i o n 3.1 Given two positive constants L, M with L
