Maneuver Regulation from Trajectory Tracking: Feedback Linearizable Systems*

Copyright © IFAC Nonlinear Control Systems Design. Tahoe City. California. USA. 1995

Maneuver Regulation from Trajectory Tracking: Feedback Linearizable Systems· John Hauser t and Rick Hindman t ·Research supported in part by AFOSR under grant F-49620.9-4·1·0183 , by NSF under grant PYI ECS· 9396296, and by a grant from Hughes Aircraft Company . tElectrical and Computer Engineering, Univ ersity of Colorado , Boulder, [email protected], [email protected]. edu, (303)1,92.2758 fax

CO

80309·0-425,

Abstract. In this paper. we consider the maneuver regulation problem and techniques for converting a stable. trackmg control law to a stable maneuvenng control law. Roughly speaking, one designs a projection mappmg (onto the maneuver) that selects the appropriate trajectory time to be used in regulation. Using a Lyapunov ~rgument. we d~rlve a class of projection operators providing stable maneuver regulation for feedback hnearlzable nonhnear systems. This technique is illustrated using a simplified model of an experimental flight control apparatus . Key Words. Maneuver regulation, trajectory tracking, transverse stability. nonlinear control.

Introduction

for example,

((aCt) , lL(t)) E Rn

For an important class of nonlinear systems, one can obtain stable trajectory tracking of a family of trajec· tories using a static feedback control law

u

Consider the nonlinear control system

= I(x, u)

x = I(x, (3(x, 0' (t),IL(t))) is such that x(t) -+ O'(t) as t -+ 00 for all (0'( .), IL(' )) E T for IIx(O) - 0'(0)11 sufficiently small. Note that neither the time t nor knowledge of the future behavior of the specific trajectory (0'( .), IL( ' )) is available to the feedback law (3(.,. , .).

A maneuver is a curve in the state-control space that is consistent with the system dynamics. For example,

= {(0'(8) ,1L(8)) E Rn X R m : 8 E R}

We are interested in studying the following question . When is it possible to find a mapping

is a maneuver of (1) provided that 0"(8)

= 1(0'(8) , 1L(8)),

8ER

= (3(x, O'(t), lL(t))

where (3 : Rn X Rn X R m -+ Rm . That is, given the family of trajectories T, the closed loop system

(1)

with state x ERn and control u E R m. (Results on more general state and control manifolds will be similar.)

1/

R m : t 2: O}.

Note that an infinite collection of trajectories give rise to the same maneuver.

A number of important nonlinear control system objectives can be accomplished by providing a stable motion along a path through the system state space. This is true for systems ranging from aerospace flight vehicles and robotic manipulators to systems for the manufacture of sophisticated materials.

i;

X

(2)

where 0"(8) := !~ (8) . We will use 1/1 and '1/2 to denote the state and control curves (projections of 1/). The specific parametrization used to specify the maneuver is unimportant (though useful in calculations)-the maneuver 1/ is the curve. Thus consistency with the system dynamics (1) only requires that I(O'(O),IL(O)) E To(9)'1/1 for all O. For simplicity, in the sequel we will assume that the parametrization of maneuvers is consistent with a time parametrization, i.e., (2) is satisfied. Furthermore, we will assume that all maneuvers are such that 0'(.) E C 2 .

that can be used to convert a trajectory tracking controllaw into a maneuver regulation control law? The mapping 11"( ') is to be thought of as a projection onto the maneuver that selects the appropriate trajectory time to be used in regulation. Using 0 = 1I"(x) , we require that the closed loop maneuvering control system

be such that x(t) close to '1/1 .

A trajectory is a specific time parametrized maneuver, 595

-+

'1/1 provided

x(O) is sufficiently

Suppose that (1) is feedback linearizable over a certain (open) region. We can design a trajectory tracking law fJ(-,·,·) for trajectories (0'(' )' It(·» belonging to the valid region as follows. Let «(.),11( '» be the trajectory expressed in (z, v) coordinates. The control law

In this paper, we give a constructive answer to this question for nonlinear systems that are feedback linearizable. For feedback linearizable systems, one can design a feedback law fJ( ·,· ,·) with the desired properties in a fairly routine manner. Using a Lyapunov argument, we derive a class of projection mappings 1!'0 providing maneuver regulation.

v The idea of maneuver regulation and its importance is easily understood since many of the aggressive tasks that we do each day are accomplished using a maneuver regulation approach. Indeed, suppose that you are driving you car on a curvy mountain road (e.g., to go skiing-another maneuvering task!). Using the recommended speed at each position along the road and a given starting time, one can easily determine a trajectory to travel the road. Suppose we now attempt to negotiate the road using a trajectory tracking controller. Unfortunately, our engine is not so powerful (we're spending that money on skiing!) and we get behind during an uphill section of the road. Immediately after the uphill section is a very curvy downhill section that must be negotiated with extreme care. To our dismay, the tracking controller attempts to catch up on the downhill section resulting in excessive speed that is dangerous if not fatal. Indeed, it is quite likely that the controller will take the car off-road in its attempt to catch up!

= .8(z,«t),II(t»:= lI(t) + K(z -

«t»

(4)

where Ac := A + BK is Hurwitz provides local stable trajectory tracking (global in (z, v) if there are no restrictions in these coordinates). This is an inherently linear control law. Indeed, it is easily seen that the tracking error e(t) := z(t) - «t) is governed by the linear dynamics e = Ace. Thus, provided (x(t), u(t» stays within the valid region, exponential convergence to the desired trajectory is guaranteed. Existence of an open set of converging initial conditions is guaranteed by continuity. One choice for fJ(·, ·,· ) is therefore given by

fJ(x, a, It) .- t/I(x,.B(4>(x),4>(O'),t/li(O',It))) = t/I(x, t/l i (a, It) + K( 4>(x) - 4>(0'))) .

(5)

2. From Trajectory Tracking to Maneuver Regulation

It is clear that this is not the way that a person would negotiate such a road. Indeed, a driver provides safe maneuvering by regulating her lateral position (and velocity) to keep the car in the center of the lane and by regulating the longitudinal velocity to the speed recommended by the longitudinal position on the road (so that she will not have excessive speed around dangerous hairpin curves) . We emphasize that longitudinal position (and not time) plays the key role in determining what state the system should be regulated to. Thus, in this case, we see that the projection 1!'(-) must effectively determine the longitudinal position of the car along the road.

Suppose now that instead of trajectory tracking, we are interested in regulating the system to a maneuver (0'(9), 1t(9», 9 E R. As shown above, we can easily track a specific trajectory on the maneuver. To convert the trajectory tracking law to a maneuver regulation law, we will, roughly speaking, redefine the trajectory time to be the value of the parameter 9 corresponding to the point on the maneuver closest to the current state. We first review some results on distance functions . Given a positive definite symmetric matrix P E R nxn (P > 0), we can define an inner product (with z, Zl, Z2 ERn)

1. Trajectory Tracking for Feedback Linearizable

Systems and a norm

Recall that the nonlinear system (1) is feedback linearizable (Hunt et al., 1983; Jakubczyk and Respondek, 1980) if there is a change of coordinates z = 4>( x) and a feedback u = 1/;(x, v) transforming (1) into a controllable linear system

z = Az +Bv.

Given a maneuver «(.),11('» and P> 0, define

lI'(z) := arg min

(3)

Under reasonable assumptions, the feedback 1/;(.,.) is invertible in the sense that there is a function 1/;i (., .) satisfying u = 1/;(X,1/;i(X,U» and = 1/;i(X, t/I(x, Using 4>(.) one can also express the feedback in terms of z. The conditions for feedback linearization are quite restrictive although a large class of noulinear systems is approximately feedback linearizable (Hauser, 1991). Furthermore, even when exact feedback linearization is possible, it may not be possible to define a global transformation due, e.g., to topological obstructions.

v

IIz -

«9)1I~ .

(6)

9ER

Note that lI'(z) will be well defined (i.e., the minimizing 8 will be unique) provided (0 is smooth enough and nonintersecting and z is close enough to «(-). Fix z and « .) and consider the function

v» .

h(9)

= IIz -

«9)1I~ .

For 9 to be a local minimizer of h(.) it is necessary that

(z - «9), ('(9)}p

596

=0

(7)

and sufficient that

(:5

V ( .) satisfies

is, in fact, necessary)

(z - (B), ("(B))p < II('(B)II~ .

(8)

V(z):5 -AV(Z) where A = Amin(Q)/Ama%(P), we see that V(z(t))stable maneuver regulation. •

We see therefore that, provided (.) is a nonintersecting C Z curve with nonvanishing (,(.), 11"0 will be a well defined mapping on some open neighborhood of (0. One might expect to get a neighborhood of uniform radius for the definition of 11"(') by requiring that (' (.) be bounded away from the origin and that ("0 be bounded. This will certainly result in a radius of curvature that is bounded away from zero but such local conditions do not prevent (Bt) from passing arbitrarily close to ( Bo) for some B1 > Bo.

o exponentially fast which proves exponentially

We remark that the maneuvering control law requires knowledge of the entire maneuver. It is an inherently nonlinear control law. In practice, one may wish to define the projection 'Ir(') using an approach based on local rather than global minimization. That is, since we are using B(t) = 'Ir(z(t)) to continually update the desired trajectory time, the value of B should only depend on the local nature of the maneuver. Indeed, by enforcing the necessary condition for a local minimum (7), we see that B( t), t ~ 0, satisfies

We can capture the above local conditions in: Assumption A The maneuver «((.),v(.)) is such that (' (.) is bounded away from zero and (,/(.) is bounded.

For simplicity of presentation, we will make use of the following global assumption:

iJ

= II(/(B)II~

Assumption B The maneuver «((.), v(.)) is such that given P > 0, there is a c > 0 such that 11"(') is well defined on

ne := {z E Rn: IIz - (B)II~

=

Note that any maneuver that satifies Assumption B must necessarily satisfy Assumption A. Theorem 2.1 Suppose that «((·),v(·)) is a maneuver satisfying Assumption B and suppose that K has been chosen so that the feedback law (4) results in stable trajectory tracking of «((t), vet)), t ~ O. Let P > 0 be such that Q := -(A~ P + PAe) > 0 and let 'Ir(') be given by (6). Then the control law

iJ

This local approach highlights some interesting properties of the maneuver regulation problem. First, we see to what extent future (or preview) information about the maneuver is needed. Essentially, in addition to the current desired maneuver state and input, we require the instantaneous curvature «("(B)) of the maneuver. No additional preview information is required. Second, by (11), we can view the resulting maneuver regulating controller as a nonlinear dynamic corn pensator .

defined on a neighborhood (e.g., ne) of the maneuver and note that V (.) is positive definite around the maneuver. Computing, we see that, on such a neighborhood, -lIz - ('Ir(z))il~

=

-lIz - ('Ir(z))lI~

(11)

(9)

V(z) = IIz - ('Ir(z))il~

+ 2(1 -

(B), ('(B))p - (z - (B),(II(B))p

which is less sensitive to initial conditions and numerical noise. Note that the curvature of the maneuver, ("(B), is a function of (B), v(B), and v'(B) (and the system dynamics).

Proof: First note that, by Assumption B, there is a constant c > 0 such that the control law is well defined on the set ne. Consider the Lyapunov function

=

= (oi, ('(B))p + A(Z 1I(1(8)1I~

provides exponentially stable maneuver regulation.

V(z)

(10)

with initial condition B(O) = 'Ir(z(O)). Note that (10) is not sufficient for a practical implementation since numerical errors (or an inaccurate initial condition) will result in long term drift of B. A stabilized version can be accomplished as follows. Defining s (z (B),('(B))p, we can stabilize the desired condition s = 0 using a differential equation such as S = -AS, A > O. Solving this equation for iJ we obtain

< c,B E R} .

v = .B(z,('Ir(z)),v('Ir(z)))

(.i, (/(8))p - (z - (B), (II(B))p

D'Ir(z).i)(z - ('Ir(z)),('('Ir(z)))p 3. Flight Control Example

where the second equality follows from the fact that, by the definition of 'Ir('), the error z - ('Ir(z)) is perpendicular (w.r.t. P) to the curve (0 at ('Ir(z)). It is clear that ne is a positively invariant set since V(z) :5 0 on this set. Thus, trajectories starting in this tube are such that 'Ir(z(t)) (hence the control) will be well defined for all t ~ O. Furthermore, since

As an example, we apply the proposed technique to the design of a maneuvering control law for a simplified model of the Champagne Flyer (Lemon and Hauser, 1994) depicted in Figure 1. The Champagne Flyer is an experimental three degree-of-freedom tethered aircraft system capable of very aggressive maneuvering. A (highly) simplified model of the aircraft

597

system

~

periment a failure. That~, such an initial condition would not be considered within the realizable domain of attraction.

given by

=

=

= = =

-sin6T -mg + cos 6 T -w~sin6+k.TD'2(U2)

In (Hauser et al., 1992), using the PVTOL (Planar Vertical Take-Off and Landing aircraft) it was shown that systems having a form similar to (12) are actually feedback linearizable. Indeed, it is easy to see that, provided the saturations are not active, x and y

(12)

-kt(T - D'l(Td)) Ul

where x, y are the lateral and vertical position (in meters) of the aircraft, 6 ~ the pitch angle, T ~ the propellor thrust (in Newtons and controlled through a second order dynamics by the input ud, and U2 ~ the stabilator input used to generate a pitching moment. The constants in (12) consist of the effective Control Systems Device Configuration Azimuth. Elevation ond Pitch Defined Positive As Shown Elevation Axis

sat~fy

[

=

+ [ Choosing

Aircraft

Ul, U2

-sin 0 cos 0

-cosOT -sinOT

]

] [! ].

so that

[! ] [ +[ =

--

2 sin 6 8 .T - 2 cos 6 8~ . - cos 0 02T - 2 sin 0 OT

8 2T -28T/T

- sin 6 -cos6/T

] cos 6 - sin 6/T

][ ] Vl

V2

we obtain the linear behavior

Fig. 1. Schematic of the Champagne Flyer.

In other words, the trajectories of the system are parametrized by a family of two scalar functions x(t), y(t), t ER that are four times differentiable. T~ ~ a very useful property of feedback linearizable systems.

mass of the aircraft, m ::::: 2.1Skg, the effective gravity seen by the aircraft, g ::::: 4.8m/ S2, the natural frequency in pitch, Wn ::::: .j33/ s, the stabilator constant, k. ::::: SA/kg m, and the thrust constant, k t ::::: 4/s. The thrust is liDlited to the range ON :s; T :s; 16N by the hard liDliter

The coordinates in which the system behavior can be made linear are thus

Td < 0 o :s; Td :s; 16 Td > 16

and the linearizing coordinate change

(the aircraft can actually produce approximately 2SN) and the stabilator input is limited to IU21 :s; 1 by U2

-1 U2

(x,x,y,y,0,8,T,Td )

f-+ Z,

valid outside of saturation, is given by

<-1

x x

:s; U2 :s; 1 >

1

The thrust limiter not only keeps the thrust in its range, it also provides a limit on the thrust rate.

-sin6T/m -cosO 8T/m + sin 0 kt(T - Td)/m

z=

y

(13)

y -g +cosO T/m

Note that the simplified model (12) does not account for many important physical effects including gyroscopic and aerodynamic phenomena. Nonetheless, the model (12) can be used to compute useful approximate trajectories.

- sin 6 eT/m - cos 0 kt(T - Td)/m In (z, v) coordinates, the linearized dynamics is Brunovsky canonical form (Brunovsky, 1970)

Furthermore, as we plan to use the lessons learned here to develop maneuvering control systems for the Champagne Flyer, we must also keep the aircraft within a small operating region so that, in particular, it does not crash into the ground. "Ground level" will be defined by yo -O.Sm. Thus even if a closed loop solution trajectory eventually converges to the desired maneuver or trajectory, if the air plane had to go underground to achieve it, we will consider that ex-

•

z

=

[ , = 0 0 0

~

0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0

In

0 0 0 0 0 1 0 0

The trajectory tracking feedback (3( .,., .) was designed

598

according to (5) with the feedback gain

results as can be seen in Figure 2. With trajectory tracking, the desired position at time zero is Xd(O) = 0 and moving away from its current position. In order to aggressively eliminate the error, the control law pitches the plane up to a very high angle requiring a large thrust to maintain altitude. Unfortunately, since the thrust is limited to 16N , the aircraft actually loses altitude for a few seconds before climbing back up to the desired altitude Yd == O. See Figure 3. It is clear that this initial condition is not within the realizable domain of attraction for this controller since the aircraft descended to nearly -1m, well below the ground level of -0.5m .

K = block diagonal{Ko,Ko} where

Ko

= [-3604

-2328

-509.25

-39]

chosen so that the system was decoupled in lateral and vertical modes. The pole locations (-16 ± 4j, -3.5 ± j) were chosen based on experience with the actual Champagne Flyer. To convert the tracking control law into a maneuvering control law, we define lrO according to (6) where

X and V vs. Tmo

Thrust vs. Time 30

= block diagonal {Po, Po} satisfies A~ P + P Ac + I = 0 so that P

~~[

436.39 281.17 53.41 0.00

281.17 189.77 39.22 0.12

53.41 39.22 10.46 0.07

S20

! 0.00 0.12 0.07 0.01

10

0 0

1

PI!ch vc. Tlmo

Tal vs. T1mo

50 .. _ 0.5

i

t

The chosen maneuver is a periodic lateral motion with constant vertical position (y == 0) . Specifically, the maneuver is generated by the desired lateral position trajectory

!

0

0

~~.5 -1

o

Xd(t) = Asinwt

15

15

Fig. 3. Trajectory tracking for Champagne Flyer model.

where A = 1.85h /2 and w = 211"/5. It is a simple matter to construct the functions (or a periodic table of them) comprising the complete maneuver «(( .),v(.)). For the Champagne Flyer, this maneuver corresponds to a periodic change in azimuth of lr radians with a period of 5 seconds. The maximum speed along this maneuver is 3.67m/ s or 8.25 miles per hour.

Now consider the maneuver regulation results shown in Figure 4. In contrast to trajectory tracking, the maneuver regulating closed loop system easily keeps the aircraft at the desired altitude Yd == O. This corresponds to the fact that the pitch response is much more reasonable so that both the thrust and tail remain well clear of saturation. All of the factors follow from the fact that the perceived error is initially much lower so that the extremely aggressive behavior seen in trajectory tracking is not necessary in maneuver regulation .

To contrast the performance of a maneuvering control system with that of the corresponding trajectory tracking control system, we present simulation results for each starting at the same initial condition. The initial condition is one that is meaningful from an experimental point of view. We start the aircraft from a hover with zero vertical position and a lateral position that falls within the range of the maneuver. In particular, the initial lateral position is x(O) = -1.5.

X and v ... Tmo

Thrust vs. Time

3O.---__

----~----,

2

i O'I=;t:::~=I=~::+i -2

Trlljocto
11.,....... Ae9Jlalion - X... V

j~~~f1$~~bkbd -3

-2

-1

0

1

2

j~~t~+'~t~J~~T~~t~l

i !

2f· ....... · .. ,': .. / .. ·i ..

i:i 0;..

-1

Tal vs. Time

L

.... L) L ·.· · · .-········ e .... ...

of .... '· .. ~f .. :'t ....

E

.§. ~-21.... · .... .1~ ; .. ·..

PI1ch vs. Time

0 1 2 3 X (m ....) Maneuver R.g.Aalicn - x ..... Xdot

3

X(mew.)

~

-2

-3

~~--~----------~

·:·

)(

~Li=~Z::=:::::J

:

i-2 ....

i...

x

~

~

.... ~ .. ~

:

:

:

:

:

:

:

~

.. ..... ·...... ; ........ ..

-1 .50L---~-----1:'::-O-----J15

... ..... ~ .... ..•...... ~ ...

-2

0 2 X (m."',)

Fig. 4 . Maneuver regulation for Champagne Flyer model. Figure 5 shows the error response for the two control schemes in both the standard 2-norm as well as the Pnorm. Note the large difference in the initial error seen

Fig. 2. Trajectory tracking and maneuver regulation. The two control laws produce dramatically different

599

by the trajectory tracking controller relative to the maneuver regulation controller. This is expected since the maneuver regulation error is the distance from the current state to the nearest point (in P-norm) on the maneuver. Clearly, the minimum distance will always be less than (or equal to) the distance to a specific point as required by trajectory tracking. IIElIOr11

the above results to a larger class of systems is under study.

4. REFERENCES Brunovsky, Pavol (1970) . A classification of linear controllable systems. Kybernetika 3, 173-188. Hauser, John (1991). Nonlinear control via uniform system approximation. Systems and Control Letters 17, 145-154. Hauser, John, Shankar Sastry and George Meyer (1992) . Nonlinear control design for slightly nonminimum phase systems: application to V /STOL aircraft. Automatica 28 , 665-679. Hunt, L. R., Renjeng Su and George Meyer (1983). Global transformations of nonlinear systems. IEEE Transactions on Automatic Control AC-28, 2431. Jakubczyk, Bronislaw and Witold Respondek (1980). On linearization of control systems. Bulletin de L 'Academie Polonaise des Sciences, Serie des sciences mathematiques XXVIII, 517-522. Lemon, Chris and John Hauser (1994) . Design and initial flight test of the Champagne Flyer. In: 99rd Conference on Decision and Control.

IIErrorJLP

g !!!. -100 .

15

Close up view

Close up view

6r--.~----~-----'

oL---~~~~~--~

o

5

10

15

Fig. 5. Norm of the error vectors.

Since the squared P-norm of the error is a valid Lyapunov function proving stable tracking and maneuvering, we expect the P-norm of the error to decrease monotonically to zero. However, as the right column of Figure 5 shows, the trajectory tracking error does not decrease monotonically. During periods of saturation, it is possible (as happened) for the P-norm of the error to increase since the system is outside of the region where the linearizing transformation is valid. In contrast, the P-norm of the maneuver regulation error decreases mono tonically since the system stayed well clear of saturation.

Conclusion In this paper, we have studied the maneuver regulation problem and techniques for converting trajectory tracking control laws into maneuver regulating control laws. For the class of feedback linearizable systems, we have shown that a standard family of trajectory tracking control laws are easily converted into maneuver regulating control laws that provide exponential regulation. The proposed technique has been illustrated using a simplified model of the experimental Champagne Flyer, a highly maneuverable tethered aircraft system. For this system, the maneuver regulation control system provided stable convergence to the desired maneuver without altitude loss while the corresponding trajectory tracking control law allowed the aircraft to descended below ground level effectively crashing the system. We believe that the maneuvering approach provides a powerful framework for analysis, design, and implementation of nonlinear control laws. Extensions of

600