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Swinging up of the spherical pendulum
SWINGING UP OF THE SPHERICAL PENDULUM...
14th World Congress of IFAC
E-2c-06-6
Copyright © 1999 IFAC 14th Triennial World ConJ;ress, Beijing, P.R. China
SW"INGING UP OF THE SPHERICAL PENDULUM A.S . Shiriaev* H . Ll1dvigsen ~ O. Egeland lie
* Department of Engineering Cybernetics Norulegian University of Science and Technology jV-7034 Trondhei·m, i\lORMlAY.
Anton.ShiriaevGitk.ntnu.no Halgeir.Ludvigsen~itk.ntnu.no
Olav.EgelandCitk.ntnu.no
Abstract: '""fhe stabilization problem for the upright position of the controlled spherical pendulum is solved in the sense that all solutions tend to the upright position. The suggested stabilizing regulator is a passive controller derived using the speed-gradient method in combination vlith a switching conditions. It is sho\vn that a straightforward energy-based passive controller of the speed-gradient type does not even solve the stabilization problem for the compact invariant set containing the uprjght position. Copyright© 1999 IFAC Keywords: passivity~ V-detectability, spherical pendulum 1 attract ability of hyperbolic equilibrium, control of first integrals
1. INTRODUCTION Passive nonlinear systems have attracted much attention from the nonlinear control community, and such interest has been motivated by both theoretical and practical reasons. Passive nonlinear systems are encountered in many applications, for instance, mechanical and electra-mechanical systems. At present, due to the efforts of many authors, in particular, due to Byrnes et al. (1991), the complete solution of the stabilization problem for passive nonlinear systems \lI,lith a unique
equilibrium point stabilization problem has been obtained.
Another important problem concerning the passive nonlinear systems is the stabilization of a prescribed invariant set, which is possibly lloncompact. Some results in this direction can be found in (Shiriaev and Fradkov, 1998; Shiriaev, 1995a; Shiriaev~ 1998c). In this paper ,ve restrict our attention to a detailed analysis of the controlled spherical pendulum motions with the objective to render the upper equilibrium point to be an 'unique attractive point of
the closed loop system (and further to stabilize this point). The controlled spherical pendulum is a passive nonlinear systenl with two equilibrium points. In these points, in contrast to the equilibriums of the pendulum in a plane, the vector field of the unforced spherical pendulum does not have any linear approximations+ T'he last property does
not depend on the parametrization but originates from the conservation of the angular momentum around the vertical axis.
The unforced spherical pendulum has two independent first integrals (the. total energy Ho and the generalized moment Po, corresponding to rotation in the horizontal plane) and is a completely integrable nonlinear system. T'hc upright position belongs to the invariant subset of the phase space defined by Ho 2mgl and Po = 0, "\vhere m; 1 are mass and length of the pendulum; g is the acceleration of gravity. By proposition 1 this subset is a stable manifold of the upright position.
=
·VVc therefore propose to use the energy-based
speed-gradient algorithm (Fradkov, 1996; Astrom and Furuia, 1996; Fradkov et aL, 1997; Shiri-
aev, 1995c) to stabilize a COJTlpact subset of the
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ISBN: 0 08 043248 4
SWINGING UP OF THE SPHERICAL PENDULUM...
ey lindrical state space defined by Ho Po = O.
14th World Congress of IFAC
== 2mgl and
As it is shown in theorem 6 the straightforward passive controller based on energy considerat,lons
may not provide the stabilizability of the prescribed goal set. The solution of the closed loop system may tend to a cycle~ which does not belong to the goal set. Even if the closed loop system solution tends to the stable manifold of the upright position this feedback control does not provide the attractability of this point and the closed loop system solution may perform infinite rotations. This reflects the fact that in general the upright position ,"viII be non-unique w-lirnit point of the closed-loop system. It turned out 1 and this is the Inain contribution of the paper, that a simple modification of the speed-
gradient-energy method by introducing
s~"itching
leads to global attractability of the upright position of the spherical pendulum. The paper is organized as follows. Section 2 con-
tains the problem statement and the important auxiliary results concerning the detailed analysis
Fig. 1. The spherical pendulum
an unique attractive point for the closed loop system.
It is convenient to choose as a state space for the spherical pendulum the cylindrical space ~"ith a sphere of radius l in the base, i. e. 0 ::; ql ::; 'if, o ::; q2 < 27f. The unforced system (1), (2) IS completely integrable. Indeed~ the functions
of the unforced spherical pendulum motions. The main results of the paper is presented in section 3.
1 12 [(')2 H oq,q ( .) ==2"m ql
°
+mgl(l - cos ql)
The computer simulations are collected in section 4 and conclusions in section 5 °
2~
+ (')2 q2 SIll 2 ql 1+(4)
PO(ql q)
= rn1 2(42) sin
2
(5)
ql
are the independent first integrals of the unforced spherical pendulum with Ho(q, gJ ~ 0, Po(q, q) E R 1 - Take the constants Ho>< 2: 0, p* E R 1 and
PRE·LI1\1INARIES AND PROBLEM STATEMENT
The motions of the controlled spherical pendulum
introduce the scalar function V a.nd t.he dummy
are described by the Euler-Lagrange equations
output Y = (Yl, Y2)T of the system
(l)~
(2) as
follows
.:!... o£(q, q) dt
_ 8£(q, q) _
8rjl
8ql
(
.)
- 91 q, q ul,
d 8£(q, q) (0) . dt 842 =g2Q;q u 2,
V(q, 4)
(1)
Yl
(2)
=
~ [Ho(q, q) - H*]2 + r; [Po - p*]2 .(6)
= r 1 . q1 . 91 ( q, q) . [Ho (q, q) -
Y2 == 92(Q, q)
+r2 . [Po(q, q) - p*]),
where q = (ql' Q2), 4 == (till 42) are the coordinates and velocities of the spherical pendulum; £(q, q) where rI,
is the Lagrangian function £(q, q)
= ~m12
[((11)2
+ (q2)2 sin 2 qt] -
(3)
-mgl(l - cos ql); rn, I are mass and length of the pendulum, 9 is the acceleration of gravity; 91 (q, ri), g2(Q, q) are scalar smooth functions; Ut ~ u2 are control functions; see ,,~ill
understand stabilization of the set S as
rendering S attractive; i. e. providing that all
tra-
jectories of closed loop system tend to S as t tends to +c.X). This paper is devoted to the problem: define a state feedback regulato'r' which stabilizes the upright position of the spherical pendulum, more precisely, rendering the upright position as
r2
are some posItive constants. The
functions V, y possess the important property: the controlled spherical pendulum (1), (2) "\vith
output function y is a passive system with nonnegative storage function V, see (V\lillems,. 1972). In particular the derivative of the function V along the solution of the system (1), (2) takes the form l
~ V(q, q) = Yt Ut + Y2
figure 1.
"re
H >It],
. (rl . q2 . [Ho(q, q) - 11*]+ (7)
U 2.
The solution of the upright position stabilization
problem will take advantage of the passivity, and the feedback regulator will be forrrled by the modified speed-gradient algorithm, see (Fradkov ~ 1990). The analysis of the qualitative behaviour of the closed loop system solutions with the feedback
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ISBN: 0 08 043248 4
SWINGING UP OF THE SPHERICAL PENDULUM...
14th World Congress of IFAC
regulator formed by the speed-gradient algorithm is essentially based on the detailed investigation
Deciphering of the equation (10) for the different values- of the constants H* ,p. leads to the follov·l-
of the lllotions of the unforced spherical pendu-
ing corollaries of theorem 2.
lum. The rest of this section is devoted to such ilnportant auxiliary results.
Corollary 3~ Given the prescribed constants H* RI and the set
~
O~ p" E
Proposition 1. Denote _~p as the maximal set of the trajectories (q, q) of the unforced spherical pendulum such that the point p of the cylindrical phase space is an w-limit point of any motion fronl Apo Then
1). the set l\{O)O,O,O) == {(O~ 0, O~ O)}, i. e. there is no other trajectory except the point (O~ 0,0,0), for which this equilibrium is its w-limit point;
2). the set A(1f:o o)O) has the forIll
V a J3 == {(q, rj) : IHo(q, q) - H-I 2: et, IPo(q~ q) - popt ::; t3} , suppose that the assumptions of theorem 2 are valid. Then for any a > 0, ;3 > 0 there exist positive constants rl, r2 such that for any solution
of the unforced spherical pendulum with initial conditions from V Q ,;3 consistent with yet) == 0 for all t ;::: 0 only case 2) of theorem 2 is possible~ 0
l
A(,..,o,o,O)
= {(.it (t), Q2, ~l(t), 0) } ,
where q2 is any constant fcoIn [0, 27T); the pair (ql(t), ql(t)) is a solution of the equation
m1 2 . ~1 (t)
==
CO'"ollary 4. Let H~ = 0, p* = 0 and suppose that the assumptions of theorem 2 are valid~ Let rl i r2 be any positive constants. Then for an arbitrary rnotion of the unforced spherical pendulum consistent ,vith y(t) 0 for all t ~ 0 only case 2) of theorem 2 is possible. 0
=
-mgl . sin ql(t)
subjected to the constraint " "21 m1 2(qI (t))2+mgl(1-cos ql(t))
~ 2mgl,
Remark 5. Theorem 2 essentially describes the
"It ~ O.
motions of the unforced spherical pendulum ".-hich
are V ~detectable, see (Shiriaev, 1995c). It is v.rorth also to mention that Corollary 4 states the local zero~state detectability of the spherical pendulum with respect to the dov\I'nward equilibrium and provides the local asymptotic stabilizability of this equilibrium, see also (Shiriaev, 1995b). D
Moreover, the upright position is unique w-limit point of any trajectory from A(7T~O,O,O). 0
Consider the motion (q, q) of the unforccd spherical pendulum ' lith 42 f:. O. By proposition 1 this trajectory does not belong to the sets A(O,O,OlU), 11(1£1 0 ,0 0). Therefore the smooth function ql (t) is strictly bounded away from zero and Jr. By this reason the trajectory (q} q) satisfies the equations lI
1
(8) (9) which is well defined for alIt
2::
o.
3.
= H * ~ Po ( q(t) ~ q(t )) = (7r~O,O~O) or
2). (q(t),q(t))
Taking the values H* = 2'mgl, p* = 0, V\re obtain that the set 1/0 of zeros of the function V, introduced in (6), corresponds the invariant subset of the spherical pendulum state space which contains the upright position. So the first step is to consider the st.abilizing problem of the set Vo.
Let 4>1 (z ),
p~ ;
(q(t),4(t))
(O,O,0~O);
3). (.j(t), ~(t))
=
RESlTLTS
"le \\.-ill investigate the feedback regulator suggested by the speed-gradient method to solve the upright position stabilization problem. This allo,vs for taking advantage of the passivity properties of the system_
Theorem, 2. Suppose that the functions 91,92, are not identically equal to zero along any solution of the unforced spherical pendululli+ Let (q, ej) be any motion of the unforced spherical pendulum consistent with y(t) = 0 for all t ~ 0, where y is defined by (7). Then only the follo","ing cases are possible:
1). Ho ( q( t), q(t»)
~IAIN
(qI' q2 . t, 0, Q2), where the constants ql 1 q2 satisfy the equations
where
£2
?:
Cl
> 0, and take the regulator of the
form
The follov\ring Theorem reflects the qualitative behaviour of the solutions of the closed loop system
2195
Copyright 1999 IFAC
ISBN: 0 08 043248 4
SWINGING UP OF THE SPHERICAL PENDULUM...
14th World Congress of IFAC
(1), (2), (12) \vith the objective to make all solutions converge to the stable manifold of the upright position of the unforced spherical pendulum~ which by Proposition 1 coi~cides with the compact set Vo =:: {[q, q]; Ho(q~) =:: 2mgl,po(q: q)
=
a}. Theorem 6. Let }[* = 2mgl, p* = 0 and Tl, :r2 be any positive constants, and the functions gl, 92 not identically equal to zero along any solution of the unforced spherical pendulum. Let
1>1, 4>2
be any scalar smooth functions satisfying the inequalities (11). Define the regulator by relation (12). Then a). for any solution (q~ q) of the closed loop system i ,".rhich does not coincide 'with the equi-
librium points
(O,O~O,O), (7r~O,O,O),
(12) the suggested feedback control ~l'ill contain several points of switching and at the first step provides the stabilization of the zero value of Po in finite time. This makes it possible to reduce the problem of the spherical pendulum upright point stabilization to the problem of the upright position of the pendul-um. 'l"he last problem as shown in (Shiriaev et al., 1998) can be successfully
solved. Theorem 8. Let 6 > 0 and 1J2(Z) be any smooth scalar function satisfying the inequality (11) and the Slllooth function g2 not identically equal to zero along any solution of the unforced spherical pendulum~
Ul(t) == 0
only the
following alternatives are possible: 1).
lim
t-+oo
Ho(q(t)~ q(t))
lim po(q(t), q(i))
t--++oo
:::::
H~,
= 0;
2). as t ~ +oo~ the trajectory (q(t), q(t)) tends to the unique cycle r of the unforced pendulum defined by Theorem 2; b). for any solution (q(t), q(t) of the closed loop systern (1), (2), (12), which corresponds to the case al), \/ E > 0 there exists the sequence of moments of time {Tn }~~, T n ---+ +00, such that Iql(7~)
- 141 < c~ !41(Tn )1 < c.
0
Remark 7~ Theorem 6 does not guarantee that even in the best case) \-vhen the solution (q, rj) of the closed loop system (1), (2), (12) tends to the set vQ = {(q~ q) : Ho(q, rj) = 2mgl, po(q, q) = O}, the solution (q, q) lvill be close to the upper equilibrium point for all sufficiently large moments of time. But this trajectory will be an infinite number of times in any neighbourhood of the upright position, see also the results of the computer simulations. 0 Thus the detailed analysis of the closed loop system (1), (2) with the feedback control (12) deri ved from the standard speed-gradient algorithlTI, shoV\Ts that even with the weakening goal of making the set Vo globally asymptotically stable, is not achieved. Below \ve describe the algorithm vvhich solves the problem of stabilizing of the upright point. T'he solution is based on a modification of the speedgradient lnethod and consists of two steps: first, construct a feedback control such that along the closed loop system trajectory (q, q) the variable q2(t) achieves the zero value in a finite time; second~ define the feedback control \vith U2 == 0 providing the stabilization of the pendulum upright position. Both steps will use the passivity of the controlled spherical pendulum, but in contrast to
0,
if 42
=0
Then any solution (q, q) of the closed loop system (1), (2), (13) achieves for finite time T > 0 the point (q(T»), q(T) with q2(T)"= o. 0 Thus any solution (q, q) of the closed loop sys-
tem (1), (2), (13) comes in finite time T == T(q(O),q{O)) to the point (ql(T»)~q2,ql(T),O), which differs from the origin, where qi is some constant from [Oi 27r). In this point the feedback control is s,vitched such that along the closed loop system solution for t 2:: T the important relation po(q(t), q(t)) == 0 (or equally q2(t) == 0) is preserved. Then the motion of the controlled spherical pendulum is reduced to
(q(t),q(t)) = (Ql(t),Q2,Qt(t)10), where (ql' ~l) satisfies the equation of the controlled pendulum
Theorem 9. Let 2Trtgl > £ > 0 and <1>1 (z) be any smooth scalar function satisfying the inequality (11) and the smooth scalar function 91 not identically equal to zero along any solution of the unforced pendulum. Then the regulator if q 1 == 0, q1 == 0 D, if Ho(q, qJ == 2mgl
'U *,
'Ul
==
-r/>1 (Q1g1(q, q)[Ho(q, q) - 2mgl
+ E]),
(15)
if Ho(q~ q) > 2mgl -4;1 (rllg1 (q q)[Ho(q, q) - 2mgl - £]) , if Ho(q; q) < 2mgl t
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Copyright 1999 IFAC
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SWINGING UP OF THE SPHERICAL PENDULUM...
14th World Congress of IFAC
,vhere u* is any nonzero number, globally makes the unstable equilibrium point (q, q] = [1r,0] of the
Figure 2 shows the motion of the unforced spherical pendulum
controlled pendulum (14) to be a unique w-limit
point of the closed-loop system.
0 30.···,····'·· .. · ..,··.. ······,··,·····:'r
Remark 10. The closed loop systeIll (14), (15) is a nonlinear systelll lvith discontinuous right side
g.20
in the s\vitching points of the control function.
C'"
=c
Thus the question of the definiteness of the solu0.2
proof of theorem 9 that for any initial condition
o
a piece-wise smooth functioll J which can have no more than one point of discontinuity. 0
Taking advantage of the theorem 8 and theorem 9 "re can write the final form of the upright position stabilizing regulator.
Theorem 11. Take positive constants 6:
f
,vith
> c, any nonzero constant u* and the scalar smooth functions q)t(z), q)2(Z) satisfying
-:,
.
.
10
tion of the system can arise. It follows from the the solution of the system (14)~ (15) is an absolutely continuous function, which is well defined on [0, +(0), and the derivative of this function is
:. .. ,
N
. ".
.. .... ,.. .. .... ~
~
o
4
~
10
6
6
4-
10
time (s)
ti'ne(s)
Fig. 2. 'The motion of the unforced spherical pendulum
By Theorem 6 the upright position is a relative (i. e. \vith respect to only the part coordinate Ql, ql of the state vector) w-limit point of any solution of the closed loop system along which the functions Ho, Po tend to the prescribed values 2mgl and o. But it is possible that there exist other ,:J.)limit points. The last means that the closed loop
2mgl
system solution may be rotating. Figure 3 presents
(11). Then -the regulator
such a motion of the controlled pendulum "l'ith Ul == -Yl, U2 == -Y2 as defined in (7).
ifq2 i= 0 or Ho(q~q) == 2mgl 'U ~, if ql = 0, 41 = 0 - ,p 1 (4191 ( q, q) [If0 ( q, q) - 2m 9 1 + c]) : if (Ho(q, q) > 2'mgl) & (42 == 0) -c/Jl (Q1g1(q, q)[Ho(q, q) - 2mgl- EJ), if (Ho(q, q) < 2m,gl) & (42 = 0) if q2 == 0 0, - 0 O~
Ul
15
~1a E 5
~
~~
40r-------.-----r--------, 30 .
12o
0
10
15
'0
15
30 ~ 20 .
.
I'>E 10
g
N
0
:J
globally makes the upright position of the controlled spherical pendulum (l)j (2) to be a unique w-limit point of the closed loop SystClll. 0
15
10
'
.....
-10
0
20 ,..........r--r
~ 15 ,
Remark 12. Since the submission of this paper the authors became aware of the interesting v,rork of Bloch et aL (1998), 'illhere using the different argum.ents the local asymptotic stabilizability of the upright position of the spherical pendulum on a cart was established. To emphasize the difference between the presented results and (Bloch e-t aI., 1998) it is necessary to mention that our results are global. Due to this advantage we can establish only the attractability of the upright position~
0
-5
,
I;Ij
;.
--e4
NE
F
~10
,
NE
,
.
Cl
"
:
~d
..
z
.,.,
,.. ,
,
.
Cl.
5L...-----------!
o
10
15
O'------'-~--------J
o
time(s]
10
15
time(sj
Fig. 3. The oscillations of the spherical pendulunl Figure 4 shows the motion with the regulator defined in Theorem Ilj using rPl(Z) tP2(Z) Z
=
=
0
5. CONCL"["SIONS 4. COMPUTER SIMULATIONS
The spherical pendulum is a nonlinear system
In all the sirnulations the following parameters
voted to the solution of stabilizing the upright
~"ith
have been used: 1Tt = lkg, I:::: Im, 9 :::: 9.81rn/s 2 'J Ql(O) = lrad, li1 (0) == lradl s~ Q2(O) == 1r j2rad,
t,vo equilibrium points. 'This paper is de-
position. The main idea of the solution is based on the facts that, first, the spherical pendulum is a passive system 'lIy~ith respect to some natural
42(0) = lrad/s.
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14th World Congress of IFAC
global stabilization of minimum phase nonlinear systems. IEEE Transactions on A utomatic Control 36, 1228-1240. Fradkov, A.L,. (1990). Adaptive control of complex
!-10 E ~~20
i" -30·
10
15
20
25
10
15
20
25
3C
1-0.:(' :'.:... ,... ,...... :! :... ;s"'"
-1 ."."""" .~- ."" .. , .....• " - .". ".""-" ..
Fig. 4. The stabilization of the spherical pendulum upright position defined storage function V and, second, the zero level Vo of this storage function V coincides with
the stable manifold of the upright position. So by stabilizing the compact set Vo one can hope that the upright position will be an unique attractive point of the closed loop system. rro this end it seems naturally to apply the speedgradient method '~lith the objective to stabilize the set Vo . However: it was sho\vn that for any passive controller the union of w-limit sets- of all closed loop system trajectories will consist of three
systems. h1oscow: Nauka. (in Russian). Fradkov, A.L. (1996). Swinging control of nonlinear oscillations. Interna.tional Journal of ControI64(6), 1189-1202. Fradkov, A.L.~ 1....4... Makarov~ A.S. Shiriaev and O.P. Tompchina (1997). Control of oscillations in Hamiltonian systems. In: Proceedings of the 4th Europea.n Control Conference. Brussels. J-\strom, K.J. and K. Furuta (1996). S,vinging up a pendulum by energy controL In: Proceedings of the 13th IFA C ~llorld Congress. Val. E. San Francisco. pp. 37-42. Shiriaev, A.S. (1998a). Control of oscillations in affine nonlinear systems. In: Proceedings of the IFAC Conference on System Structure and Control. \'01. 3. Nantes, France. pp. 789794. Shiriaev, A.S. (1998b). New tests of zero-state detectability. In: Proceeding,,, of the 4th IFA C Symposiu'm on .l\j'" onli-near Control Systems Design. Vol. 2. Enschede~ The Netherlands. pp. 533-538. Shiriaev, .~.S. (1998c). The notion of 1/detectability and stabilization of invariant sets of nonlinear systems. In: Proceedings of the 37th enc. Tanlpa. pp. 2509-2514. Shiriaev, _4..5. and A.L. Fradkov (1998). Stabilization of invariant set of nonlinear systems. In: N onlinear dynarnical systems. St. Petersburg
connected invariant sets: the downVv"ard position of the spherical pendulum, some cycle and the set
Vo. 'I'his means that any passive controller does not provide the global asymptotic stability of ~jo. Therefore, it does not provide the attractability
of the upright position. The main contribution of the paper is the modification of the speed-gradient method. It was shown that~
consecutively stabilizing the prescribed values of the first integrals of the unforced spherical pendulum and using the switching bet~veen the regulators, the upright position becomes a unique
State University. St. PetersbuTg, Russia. (in
Russian). A.S.~ O. Egeland and H. Ludvigsen (1998). Global stabilization of unstable equilibrium point of pendulum. In: Proceedings of S7th eDC. IEEE. Tampa. pp. 4584-4.585. "'illcms, J .C. (1972). Dissipative dynamic systems. Part 1. General Theory. Arch. Rat. -,-lfech. Anal. 45(5),321-351.
Shiriaev)
attractive point of the closed loop system.. The
obtained rigorous results are successfully tested by the simulations.
6. REFERENCES
Bloch, A.M., N .E. Leonard and J .E. Ivlarsden (1998). ~Iatching and stabilization by the method of controlled Lagrangians. In: Proceedings of the 37th enc. IEEE. Tampa, USA. pp. 1446-1451. Byrnes; C.I., .A.. Isidori and J.C. Willems (1991). Passivity,
feedback
equivalence,
and
the
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
E-2c-06-6
Copyright © 1999 IFAC 14th Triennial World ConJ;ress, Beijing, P.R. China
SW"INGING UP OF THE SPHERICAL PENDULUM A.S . Shiriaev* H . Ll1dvigsen ~ O. Egeland lie
* Department of Engineering Cybernetics Norulegian University of Science and Technology jV-7034 Trondhei·m, i\lORMlAY.
Anton.ShiriaevGitk.ntnu.no Halgeir.Ludvigsen~itk.ntnu.no
Olav.EgelandCitk.ntnu.no
Abstract: '""fhe stabilization problem for the upright position of the controlled spherical pendulum is solved in the sense that all solutions tend to the upright position. The suggested stabilizing regulator is a passive controller derived using the speed-gradient method in combination vlith a switching conditions. It is sho\vn that a straightforward energy-based passive controller of the speed-gradient type does not even solve the stabilization problem for the compact invariant set containing the uprjght position. Copyright© 1999 IFAC Keywords: passivity~ V-detectability, spherical pendulum 1 attract ability of hyperbolic equilibrium, control of first integrals
1. INTRODUCTION Passive nonlinear systems have attracted much attention from the nonlinear control community, and such interest has been motivated by both theoretical and practical reasons. Passive nonlinear systems are encountered in many applications, for instance, mechanical and electra-mechanical systems. At present, due to the efforts of many authors, in particular, due to Byrnes et al. (1991), the complete solution of the stabilization problem for passive nonlinear systems \lI,lith a unique
equilibrium point stabilization problem has been obtained.
Another important problem concerning the passive nonlinear systems is the stabilization of a prescribed invariant set, which is possibly lloncompact. Some results in this direction can be found in (Shiriaev and Fradkov, 1998; Shiriaev, 1995a; Shiriaev~ 1998c). In this paper ,ve restrict our attention to a detailed analysis of the controlled spherical pendulum motions with the objective to render the upper equilibrium point to be an 'unique attractive point of
the closed loop system (and further to stabilize this point). The controlled spherical pendulum is a passive nonlinear systenl with two equilibrium points. In these points, in contrast to the equilibriums of the pendulum in a plane, the vector field of the unforced spherical pendulum does not have any linear approximations+ T'he last property does
not depend on the parametrization but originates from the conservation of the angular momentum around the vertical axis.
The unforced spherical pendulum has two independent first integrals (the. total energy Ho and the generalized moment Po, corresponding to rotation in the horizontal plane) and is a completely integrable nonlinear system. T'hc upright position belongs to the invariant subset of the phase space defined by Ho 2mgl and Po = 0, "\vhere m; 1 are mass and length of the pendulum; g is the acceleration of gravity. By proposition 1 this subset is a stable manifold of the upright position.
=
·VVc therefore propose to use the energy-based
speed-gradient algorithm (Fradkov, 1996; Astrom and Furuia, 1996; Fradkov et aL, 1997; Shiri-
aev, 1995c) to stabilize a COJTlpact subset of the
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ISBN: 0 08 043248 4
SWINGING UP OF THE SPHERICAL PENDULUM...
ey lindrical state space defined by Ho Po = O.
14th World Congress of IFAC
== 2mgl and
As it is shown in theorem 6 the straightforward passive controller based on energy considerat,lons
may not provide the stabilizability of the prescribed goal set. The solution of the closed loop system may tend to a cycle~ which does not belong to the goal set. Even if the closed loop system solution tends to the stable manifold of the upright position this feedback control does not provide the attractability of this point and the closed loop system solution may perform infinite rotations. This reflects the fact that in general the upright position ,"viII be non-unique w-lirnit point of the closed-loop system. It turned out 1 and this is the Inain contribution of the paper, that a simple modification of the speed-
gradient-energy method by introducing
s~"itching
leads to global attractability of the upright position of the spherical pendulum. The paper is organized as follows. Section 2 con-
tains the problem statement and the important auxiliary results concerning the detailed analysis
Fig. 1. The spherical pendulum
an unique attractive point for the closed loop system.
It is convenient to choose as a state space for the spherical pendulum the cylindrical space ~"ith a sphere of radius l in the base, i. e. 0 ::; ql ::; 'if, o ::; q2 < 27f. The unforced system (1), (2) IS completely integrable. Indeed~ the functions
of the unforced spherical pendulum motions. The main results of the paper is presented in section 3.
1 12 [(')2 H oq,q ( .) ==2"m ql
°
+mgl(l - cos ql)
The computer simulations are collected in section 4 and conclusions in section 5 °
2~
+ (')2 q2 SIll 2 ql 1+(4)
PO(ql q)
= rn1 2(42) sin
2
(5)
ql
are the independent first integrals of the unforced spherical pendulum with Ho(q, gJ ~ 0, Po(q, q) E R 1 - Take the constants Ho>< 2: 0, p* E R 1 and
PRE·LI1\1INARIES AND PROBLEM STATEMENT
The motions of the controlled spherical pendulum
introduce the scalar function V a.nd t.he dummy
are described by the Euler-Lagrange equations
output Y = (Yl, Y2)T of the system
(l)~
(2) as
follows
.:!... o£(q, q) dt
_ 8£(q, q) _
8rjl
8ql
(
.)
- 91 q, q ul,
d 8£(q, q) (0) . dt 842 =g2Q;q u 2,
V(q, 4)
(1)
Yl
(2)
=
~ [Ho(q, q) - H*]2 + r; [Po - p*]2 .(6)
= r 1 . q1 . 91 ( q, q) . [Ho (q, q) -
Y2 == 92(Q, q)
+r2 . [Po(q, q) - p*]),
where q = (ql' Q2), 4 == (till 42) are the coordinates and velocities of the spherical pendulum; £(q, q) where rI,
is the Lagrangian function £(q, q)
= ~m12
[((11)2
+ (q2)2 sin 2 qt] -
(3)
-mgl(l - cos ql); rn, I are mass and length of the pendulum, 9 is the acceleration of gravity; 91 (q, ri), g2(Q, q) are scalar smooth functions; Ut ~ u2 are control functions; see ,,~ill
understand stabilization of the set S as
rendering S attractive; i. e. providing that all
tra-
jectories of closed loop system tend to S as t tends to +c.X). This paper is devoted to the problem: define a state feedback regulato'r' which stabilizes the upright position of the spherical pendulum, more precisely, rendering the upright position as
r2
are some posItive constants. The
functions V, y possess the important property: the controlled spherical pendulum (1), (2) "\vith
output function y is a passive system with nonnegative storage function V, see (V\lillems,. 1972). In particular the derivative of the function V along the solution of the system (1), (2) takes the form l
~ V(q, q) = Yt Ut + Y2
figure 1.
"re
H >It],
. (rl . q2 . [Ho(q, q) - 11*]+ (7)
U 2.
The solution of the upright position stabilization
problem will take advantage of the passivity, and the feedback regulator will be forrrled by the modified speed-gradient algorithm, see (Fradkov ~ 1990). The analysis of the qualitative behaviour of the closed loop system solutions with the feedback
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regulator formed by the speed-gradient algorithm is essentially based on the detailed investigation
Deciphering of the equation (10) for the different values- of the constants H* ,p. leads to the follov·l-
of the lllotions of the unforced spherical pendu-
ing corollaries of theorem 2.
lum. The rest of this section is devoted to such ilnportant auxiliary results.
Corollary 3~ Given the prescribed constants H* RI and the set
~
O~ p" E
Proposition 1. Denote _~p as the maximal set of the trajectories (q, q) of the unforced spherical pendulum such that the point p of the cylindrical phase space is an w-limit point of any motion fronl Apo Then
1). the set l\{O)O,O,O) == {(O~ 0, O~ O)}, i. e. there is no other trajectory except the point (O~ 0,0,0), for which this equilibrium is its w-limit point;
2). the set A(1f:o o)O) has the forIll
V a J3 == {(q, rj) : IHo(q, q) - H-I 2: et, IPo(q~ q) - popt ::; t3} , suppose that the assumptions of theorem 2 are valid. Then for any a > 0, ;3 > 0 there exist positive constants rl, r2 such that for any solution
of the unforced spherical pendulum with initial conditions from V Q ,;3 consistent with yet) == 0 for all t ;::: 0 only case 2) of theorem 2 is possible~ 0
l
A(,..,o,o,O)
= {(.it (t), Q2, ~l(t), 0) } ,
where q2 is any constant fcoIn [0, 27T); the pair (ql(t), ql(t)) is a solution of the equation
m1 2 . ~1 (t)
==
CO'"ollary 4. Let H~ = 0, p* = 0 and suppose that the assumptions of theorem 2 are valid~ Let rl i r2 be any positive constants. Then for an arbitrary rnotion of the unforced spherical pendulum consistent ,vith y(t) 0 for all t ~ 0 only case 2) of theorem 2 is possible. 0
=
-mgl . sin ql(t)
subjected to the constraint " "21 m1 2(qI (t))2+mgl(1-cos ql(t))
~ 2mgl,
Remark 5. Theorem 2 essentially describes the
"It ~ O.
motions of the unforced spherical pendulum ".-hich
are V ~detectable, see (Shiriaev, 1995c). It is v.rorth also to mention that Corollary 4 states the local zero~state detectability of the spherical pendulum with respect to the dov\I'nward equilibrium and provides the local asymptotic stabilizability of this equilibrium, see also (Shiriaev, 1995b). D
Moreover, the upright position is unique w-limit point of any trajectory from A(7T~O,O,O). 0
Consider the motion (q, q) of the unforccd spherical pendulum ' lith 42 f:. O. By proposition 1 this trajectory does not belong to the sets A(O,O,OlU), 11(1£1 0 ,0 0). Therefore the smooth function ql (t) is strictly bounded away from zero and Jr. By this reason the trajectory (q} q) satisfies the equations lI
1
(8) (9) which is well defined for alIt
2::
o.
3.
= H * ~ Po ( q(t) ~ q(t )) = (7r~O,O~O) or
2). (q(t),q(t))
Taking the values H* = 2'mgl, p* = 0, V\re obtain that the set 1/0 of zeros of the function V, introduced in (6), corresponds the invariant subset of the spherical pendulum state space which contains the upright position. So the first step is to consider the st.abilizing problem of the set Vo.
Let 4>1 (z ),
p~ ;
(q(t),4(t))
(O,O,0~O);
3). (.j(t), ~(t))
=
RESlTLTS
"le \\.-ill investigate the feedback regulator suggested by the speed-gradient method to solve the upright position stabilization problem. This allo,vs for taking advantage of the passivity properties of the system_
Theorem, 2. Suppose that the functions 91,92, are not identically equal to zero along any solution of the unforced spherical pendululli+ Let (q, ej) be any motion of the unforced spherical pendulum consistent with y(t) = 0 for all t ~ 0, where y is defined by (7). Then only the follo","ing cases are possible:
1). Ho ( q( t), q(t»)
~IAIN
(qI' q2 . t, 0, Q2), where the constants ql 1 q2 satisfy the equations
where
£2
?:
Cl
> 0, and take the regulator of the
form
The follov\ring Theorem reflects the qualitative behaviour of the solutions of the closed loop system
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(1), (2), (12) \vith the objective to make all solutions converge to the stable manifold of the upright position of the unforced spherical pendulum~ which by Proposition 1 coi~cides with the compact set Vo =:: {[q, q]; Ho(q~) =:: 2mgl,po(q: q)
=
a}. Theorem 6. Let }[* = 2mgl, p* = 0 and Tl, :r2 be any positive constants, and the functions gl, 92 not identically equal to zero along any solution of the unforced spherical pendulum. Let
1>1, 4>2
be any scalar smooth functions satisfying the inequalities (11). Define the regulator by relation (12). Then a). for any solution (q~ q) of the closed loop system i ,".rhich does not coincide 'with the equi-
librium points
(O,O~O,O), (7r~O,O,O),
(12) the suggested feedback control ~l'ill contain several points of switching and at the first step provides the stabilization of the zero value of Po in finite time. This makes it possible to reduce the problem of the spherical pendulum upright point stabilization to the problem of the upright position of the pendul-um. 'l"he last problem as shown in (Shiriaev et al., 1998) can be successfully
solved. Theorem 8. Let 6 > 0 and 1J2(Z) be any smooth scalar function satisfying the inequality (11) and the Slllooth function g2 not identically equal to zero along any solution of the unforced spherical pendulum~
Ul(t) == 0
only the
following alternatives are possible: 1).
lim
t-+oo
Ho(q(t)~ q(t))
lim po(q(t), q(i))
t--++oo
:::::
H~,
= 0;
2). as t ~ +oo~ the trajectory (q(t), q(t)) tends to the unique cycle r of the unforced pendulum defined by Theorem 2; b). for any solution (q(t), q(t) of the closed loop systern (1), (2), (12), which corresponds to the case al), \/ E > 0 there exists the sequence of moments of time {Tn }~~, T n ---+ +00, such that Iql(7~)
- 141 < c~ !41(Tn )1 < c.
0
Remark 7~ Theorem 6 does not guarantee that even in the best case) \-vhen the solution (q, rj) of the closed loop system (1), (2), (12) tends to the set vQ = {(q~ q) : Ho(q, rj) = 2mgl, po(q, q) = O}, the solution (q, q) lvill be close to the upper equilibrium point for all sufficiently large moments of time. But this trajectory will be an infinite number of times in any neighbourhood of the upright position, see also the results of the computer simulations. 0 Thus the detailed analysis of the closed loop system (1), (2) with the feedback control (12) deri ved from the standard speed-gradient algorithlTI, shoV\Ts that even with the weakening goal of making the set Vo globally asymptotically stable, is not achieved. Below \ve describe the algorithm vvhich solves the problem of stabilizing of the upright point. T'he solution is based on a modification of the speedgradient lnethod and consists of two steps: first, construct a feedback control such that along the closed loop system trajectory (q, q) the variable q2(t) achieves the zero value in a finite time; second~ define the feedback control \vith U2 == 0 providing the stabilization of the pendulum upright position. Both steps will use the passivity of the controlled spherical pendulum, but in contrast to
0,
if 42
=0
Then any solution (q, q) of the closed loop system (1), (2), (13) achieves for finite time T > 0 the point (q(T»), q(T) with q2(T)"= o. 0 Thus any solution (q, q) of the closed loop sys-
tem (1), (2), (13) comes in finite time T == T(q(O),q{O)) to the point (ql(T»)~q2,ql(T),O), which differs from the origin, where qi is some constant from [Oi 27r). In this point the feedback control is s,vitched such that along the closed loop system solution for t 2:: T the important relation po(q(t), q(t)) == 0 (or equally q2(t) == 0) is preserved. Then the motion of the controlled spherical pendulum is reduced to
(q(t),q(t)) = (Ql(t),Q2,Qt(t)10), where (ql' ~l) satisfies the equation of the controlled pendulum
Theorem 9. Let 2Trtgl > £ > 0 and <1>1 (z) be any smooth scalar function satisfying the inequality (11) and the smooth scalar function 91 not identically equal to zero along any solution of the unforced pendulum. Then the regulator if q 1 == 0, q1 == 0 D, if Ho(q, qJ == 2mgl
'U *,
'Ul
==
-r/>1 (Q1g1(q, q)[Ho(q, q) - 2mgl
+ E]),
(15)
if Ho(q~ q) > 2mgl -4;1 (rllg1 (q q)[Ho(q, q) - 2mgl - £]) , if Ho(q; q) < 2mgl t
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,vhere u* is any nonzero number, globally makes the unstable equilibrium point (q, q] = [1r,0] of the
Figure 2 shows the motion of the unforced spherical pendulum
controlled pendulum (14) to be a unique w-limit
point of the closed-loop system.
0 30.···,····'·· .. · ..,··.. ······,··,·····:'r
Remark 10. The closed loop systeIll (14), (15) is a nonlinear systelll lvith discontinuous right side
g.20
in the s\vitching points of the control function.
C'"
=c
Thus the question of the definiteness of the solu0.2
proof of theorem 9 that for any initial condition
o
a piece-wise smooth functioll J which can have no more than one point of discontinuity. 0
Taking advantage of the theorem 8 and theorem 9 "re can write the final form of the upright position stabilizing regulator.
Theorem 11. Take positive constants 6:
f
,vith
> c, any nonzero constant u* and the scalar smooth functions q)t(z), q)2(Z) satisfying
-:,
.
.
10
tion of the system can arise. It follows from the the solution of the system (14)~ (15) is an absolutely continuous function, which is well defined on [0, +(0), and the derivative of this function is
:. .. ,
N
. ".
.. .... ,.. .. .... ~
~
o
4
~
10
6
6
4-
10
time (s)
ti'ne(s)
Fig. 2. 'The motion of the unforced spherical pendulum
By Theorem 6 the upright position is a relative (i. e. \vith respect to only the part coordinate Ql, ql of the state vector) w-limit point of any solution of the closed loop system along which the functions Ho, Po tend to the prescribed values 2mgl and o. But it is possible that there exist other ,:J.)limit points. The last means that the closed loop
2mgl
system solution may be rotating. Figure 3 presents
(11). Then -the regulator
such a motion of the controlled pendulum "l'ith Ul == -Yl, U2 == -Y2 as defined in (7).
ifq2 i= 0 or Ho(q~q) == 2mgl 'U ~, if ql = 0, 41 = 0 - ,p 1 (4191 ( q, q) [If0 ( q, q) - 2m 9 1 + c]) : if (Ho(q, q) > 2'mgl) & (42 == 0) -c/Jl (Q1g1(q, q)[Ho(q, q) - 2mgl- EJ), if (Ho(q, q) < 2m,gl) & (42 = 0) if q2 == 0 0, - 0 O~
Ul
15
~1a E 5
~
~~
40r-------.-----r--------, 30 .
12o
0
10
15
'0
15
30 ~ 20 .
.
I'>E 10
g
N
0
:J
globally makes the upright position of the controlled spherical pendulum (l)j (2) to be a unique w-limit point of the closed loop SystClll. 0
15
10
'
.....
-10
0
20 ,..........r--r
~ 15 ,
Remark 12. Since the submission of this paper the authors became aware of the interesting v,rork of Bloch et aL (1998), 'illhere using the different argum.ents the local asymptotic stabilizability of the upright position of the spherical pendulum on a cart was established. To emphasize the difference between the presented results and (Bloch e-t aI., 1998) it is necessary to mention that our results are global. Due to this advantage we can establish only the attractability of the upright position~
0
-5
,
I;Ij
;.
--e4
NE
F
~10
,
NE
,
.
Cl
"
:
~d
..
z
.,.,
,.. ,
,
.
Cl.
5L...-----------!
o
10
15
O'------'-~--------J
o
time(s]
10
15
time(sj
Fig. 3. The oscillations of the spherical pendulunl Figure 4 shows the motion with the regulator defined in Theorem Ilj using rPl(Z) tP2(Z) Z
=
=
0
5. CONCL"["SIONS 4. COMPUTER SIMULATIONS
The spherical pendulum is a nonlinear system
In all the sirnulations the following parameters
voted to the solution of stabilizing the upright
~"ith
have been used: 1Tt = lkg, I:::: Im, 9 :::: 9.81rn/s 2 'J Ql(O) = lrad, li1 (0) == lradl s~ Q2(O) == 1r j2rad,
t,vo equilibrium points. 'This paper is de-
position. The main idea of the solution is based on the facts that, first, the spherical pendulum is a passive system 'lIy~ith respect to some natural
42(0) = lrad/s.
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global stabilization of minimum phase nonlinear systems. IEEE Transactions on A utomatic Control 36, 1228-1240. Fradkov, A.L,. (1990). Adaptive control of complex
!-10 E ~~20
i" -30·
10
15
20
25
10
15
20
25
3C
1-0.:(' :'.:... ,... ,...... :! :... ;s"'"
-1 ."."""" .~- ."" .. , .....• " - .". ".""-" ..
Fig. 4. The stabilization of the spherical pendulum upright position defined storage function V and, second, the zero level Vo of this storage function V coincides with
the stable manifold of the upright position. So by stabilizing the compact set Vo one can hope that the upright position will be an unique attractive point of the closed loop system. rro this end it seems naturally to apply the speedgradient method '~lith the objective to stabilize the set Vo . However: it was sho\vn that for any passive controller the union of w-limit sets- of all closed loop system trajectories will consist of three
systems. h1oscow: Nauka. (in Russian). Fradkov, A.L. (1996). Swinging control of nonlinear oscillations. Interna.tional Journal of ControI64(6), 1189-1202. Fradkov, A.L.~ 1....4... Makarov~ A.S. Shiriaev and O.P. Tompchina (1997). Control of oscillations in Hamiltonian systems. In: Proceedings of the 4th Europea.n Control Conference. Brussels. J-\strom, K.J. and K. Furuta (1996). S,vinging up a pendulum by energy controL In: Proceedings of the 13th IFA C ~llorld Congress. Val. E. San Francisco. pp. 37-42. Shiriaev, A.S. (1998a). Control of oscillations in affine nonlinear systems. In: Proceedings of the IFAC Conference on System Structure and Control. \'01. 3. Nantes, France. pp. 789794. Shiriaev, A.S. (1998b). New tests of zero-state detectability. In: Proceeding,,, of the 4th IFA C Symposiu'm on .l\j'" onli-near Control Systems Design. Vol. 2. Enschede~ The Netherlands. pp. 533-538. Shiriaev, .~.S. (1998c). The notion of 1/detectability and stabilization of invariant sets of nonlinear systems. In: Proceedings of the 37th enc. Tanlpa. pp. 2509-2514. Shiriaev, _4..5. and A.L. Fradkov (1998). Stabilization of invariant set of nonlinear systems. In: N onlinear dynarnical systems. St. Petersburg
connected invariant sets: the downVv"ard position of the spherical pendulum, some cycle and the set
Vo. 'I'his means that any passive controller does not provide the global asymptotic stability of ~jo. Therefore, it does not provide the attractability
of the upright position. The main contribution of the paper is the modification of the speed-gradient method. It was shown that~
consecutively stabilizing the prescribed values of the first integrals of the unforced spherical pendulum and using the switching bet~veen the regulators, the upright position becomes a unique
State University. St. PetersbuTg, Russia. (in
Russian). A.S.~ O. Egeland and H. Ludvigsen (1998). Global stabilization of unstable equilibrium point of pendulum. In: Proceedings of S7th eDC. IEEE. Tampa. pp. 4584-4.585. "'illcms, J .C. (1972). Dissipative dynamic systems. Part 1. General Theory. Arch. Rat. -,-lfech. Anal. 45(5),321-351.
Shiriaev)
attractive point of the closed loop system.. The
obtained rigorous results are successfully tested by the simulations.
6. REFERENCES
Bloch, A.M., N .E. Leonard and J .E. Ivlarsden (1998). ~Iatching and stabilization by the method of controlled Lagrangians. In: Proceedings of the 37th enc. IEEE. Tampa, USA. pp. 1446-1451. Byrnes; C.I., .A.. Isidori and J.C. Willems (1991). Passivity,
feedback
equivalence,
and
the
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