Global stabilization of the inverted pendulum via continuous state feedback David Angeli a
a
Dip. Sistemi e Informatica, Universita di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy
Abstract Continuous state feedback laws are presented for \almost" global stabilization of the planar and spherical inverted pendulums on a cart. Some of the new ideas employed could be possibly generalized to the problem of stabilization of systems with multiple equilibrium points. Key words: Stability properties, Lyapunov methods
1 Introduction The project of control laws for underactuated mechanical systems often presents a serious challenge for designers. This happens not only because of intrinsic diculties related with the low number of actuators, but also because mechanical systems typically evolve on manifolds which are not di eomorphic to the euclidean space. Obstructions to global stabilization with continuous feedback usually arise and make a whole body of theory, namely the Lyapunov second method in the large, not satisfactory to deal with, at least in its current and most widely accepted formulation. Such diculties are well-known already with the simplest \toy-example" of underactuated mechanical system: the planar inverted pendulum on a cart. One possibility to achieve global stabilization of the pendulum is by means of hybrid controllers. Tipically two di erent control laws are employed, one takes care of the local stabilization of the pendulum in the upright position and the other swings the pendulum up (see Astrom & Furuta (1996), Shiriaev, Ludvigsen & Egeland (1998) for 1
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works along this lines). An alternative way is to adopt a continuous feedback law and some passivity-based strategy to achieve asymptotic stabilization of the system on a region as large as possible. Several authors have proposed control laws for the inverted pendulum along this line (see Teel (1996), Mazenc & Praly (1996) ); particularly close to the approach we will follow in this paper, is the method of controlled Lagrangian presented in Bloch, Leonard & Marsden (1998). There, energy-shaping techniques are used as a tool for the synthesis of a controller. The control design is carried out in two steps: rst a preliminary feedback loop is given which preserves the hamiltonian structure of the system and suitably modi es the mechanical energy of the system; then dissipation is added in order to achieve asymptotic stability. Nevertheless, it was still an open problem whether a continuous control law existed with a region of asymptotic stability whose projection on the axis was larger than (?=2; =2), (the region where the pendulum is pointing upward). The present paper is concerned with the problem of exploring the intrinsic limits of continuous feedback for the inverted pendulum and, more in general, with the issue of \almost" global asymptotic stability for systems evolving on manifolds. We propose a smooth feedback law whose basin of attraction is an open dense set in the state-space; in particular, the set of initial conditions which do not converge to the upright position has zero measure, which is the best one can hope for using a continuous feedback. In the last section we show how to extend the result to the spherical inverted pendulum.
2 Pendulum on a cart Consider the planar inverted pendulum on a cart shown in Figure 1. If we neglect the e ects of friction we can model the system according to the following system of di erential equations:
H (q)q + C (q; q_)q_
2 = 64
3
F 7 5 mglc sin()
(1)
where q = [r; ]T . The inertia tensor H (q) and the Coriolis term C (q; q_) are de ned as 2 H (q) = 64
M +m mlc cos()
3 mlc cos() 7 5
J
2 0 C (q; q_) = 64
3
?mlc sin()_ 7 (2) 5 0 0
where M and m are respectively the mass of the cart and of the pendulum, lc is the distance between the joint and the center of mass of the pendulum and 2
θ
g
m F
M r
Fig. 1. Planar inverted pendulum on a cart
J is the rotational momentum of inertia of the pendulum. For instance for a uniform pendulum of lenght l and mass m we have: lc = l=2
J = ml2 =3:
(3)
The system is driven by the input force F acting on the cart along the horizontal plane. We are looking for a continuous state-feedback law which achieves global stabilization of the system at the origin, namely with the pendulum in the upright position and the cart at some prescribed location on the x axis that we might label zero without loss of generality. One way to address the problem is to exploit the feedforward nature of the system. In particular we can rewrite the equations as a rst-order system just by taking the inverse of the inertia tensor in the following way:
r_ = v 2 JF ? m2lc2 g sin() cos() v_ = mlcJ sin()! +det( H (q)) _ = ! 2 2 lc sin() cos()!2 ? mlc cos()F !_ = (M + m)mglc sin() ? mdet( H (q))
(4)
The state is therefore the vector of the angular and linear positions and their derivatives, x = [r; v; ; !]T . It can be shown that for all det H (q) = [J (M + m) ? m2 lc2 ] + m2 lc2 sin2() > 0 3
(5)
and hence the right hand side of (4) is well de ned and analytic over the whole R 4 . As pointed out by Teel (1996), inverting the relationship between F and v_ , we can rewrite system (4) as
r_ = v v_ = u _ = ! c !_ = mlJcg sin() ? ml J cos()u
(6)
where the input signal u is de ned as: 2 JF ? m2 lc2g sin() cos() u = mlcJ sin()! +det( H (q))
(7)
With this new choice of the input variable the feed-forward nature of the system is clearly shown; we will focus our attention on the two-dimensional subsystem (; !). It is proved in Teel (1996) that if a stabilizing law is known for the (; !) subsystem with basin of attraction A, then it is possible to modify that law in order to obtain a stabilizing feedback which is attractive for the original system (6) on the whole R 2 A. Let us rewrite by simplicity of future reference the equations of the (; !) subsystem as:
_ = ! !_ = a sin() ? b cos()u
(8)
These equations can be seen as a model of the inverted pendulum on a massless cart (pure acceleration inputs). We would like rst of all to investigate what kind of obstructions to global stabilization with continuous feedback, if any, this system is subject to. Since in (8) only appears as an argument of periodic functions (of period 2), we can see (8) either as a system de ned over R 2 (interpreting the angle (t) as taking into account the number of loops done from the time 0 to t), or over the cylinder S R , (basically taking modulo 2, which is probably more appealing given its clear physical meaning). The di erence in that respect would be that if we think of the system as de ned on the cylinder we can only allow feedback laws which depend on sin() and cos() whereas looking at the system as de ned over the plane R 2 we can look for feedback laws which depend explicitly on . We will show that in both cases the system is subject to some obstruction to global stabilizability via continuous feedback. We then design a feedback law on S R , which stabilizes for all initial conditions except those in a \small" subset. 4
2.1 System on the plane: R 2
Let us denote with k(; !) an arbitrary continuous function from R 2 to R . In particular for ! = 0 and using k as a feedback law the system equations read
_ = 0 !_ = a sin() ? b cos()k(; 0):
(9)
Notice that the function of in the right hand side of equation (9) is continuous with respect to . Consider the points where cos() = 0; since at those points the function sin() takes the value ?1 and 1, which have di erent signs, by the intermediate value theorem and no matter how we choose k, there must be an equilibrium in between each couple of points where cos() = 0. 2.2 System on the cylinder: S R
Since S R is homotopy equivalent to S it is not a contractible set. Henceforth global stabilization is not possible (see for instance Sontag (1998) ) and there must exist at least another equilibrium point (we will allow this equilibrium to be the downward pendulum position). Furthermore, if we require that this equilibrium and the points with ! = 1 be anti-stable, we show that we must allow for some more equilibrium points in the state space. This further assumption basically amounts to require that bounded kinetic energy at initial time should imply bounded kinetic energy of the controlled system for all future t. Formally we would like to have:
8M > 0 9RM : j!(0)j M ) j!(t)j RM 8t 0
(10)
Notice that this kind of stability, usually called Lagrange stability (except that here it is only with respect to a subset of the state variables, the subset of variables for which speaking of a neighborhood of 1 makes sense), is implied by global asymptotic stability for systems evolving on an euclidean space; this is not true for general nonlinear systems on manifolds. Projecting the cylinder on a sphere, (for instance with the mapping (r; ) ! (atn(r); ) ) , we obtain two anti-stable singular points on the north and south pole of the sphere. Then, by the Poincare-Bendixson theorem we have that the sum of the indexes of the singular points of the vector eld is equal to the Euler characteristic of the 2-sphere, which is 2. Hence, since we have already shown that there are at least two equilibrium points, of which one is stable and the other is unstable (downward position), a way to get the relationship satis ed is by assuming 5
+ +1 ω −1 θ
+1
+1 −1 +1 −
Fig. 2. Singular points and indexes on the sphere
the downward position to be anti-stable and allowing two saddle points, (see Figure 2). A simpler way of having the index theorem satis ed would be to make the downward position of the pendulum a saddle. In this way there is no need for the two extra saddle points. The author was not able to nd a control law which achieved that task.
3 Lyapunov functions for the inverted pendulum The feedback that we present is a passivity-based control law where a suitable sign for the dissipation is chosen depending on the state of the system. We start looking for functions, which, though not necessarily control Lyapunov functions, still enjoy the property that we can always decide the sign of their derivative along trajectories of the system through a suitable choice of the control. We believe that for systems evolving on manifolds, where in general we must allow for unstable invariant sets, and hence the derivative of the functions cannot be strictly decreasing everywhere, these kinds of functions could be an useful tool for the study of \almost" global stabilization. What we need in the following is a function V : S R ! 0, di erentiable everywhere, with the property that inf u Lf V (x) + Lg V (x)u 0
sup Lf V (x) + Lg V (x)u 0 u
(11)
where the system equations are taken to be x_ = f (x) + g(x)u, in order to emphasize the similarity with the de nition of control Lyapunov function. In particular we denote by x the vector [; !], where the angular position has to be interpreted modulo 2. Therefore we will refer only to angles with 6
amplitude between [?; +]. Since the downward position can be equally designated with + or ? we will use the notation .
Remark. We remark that this notion of Lyapunov function is weaker than
the usual clf in that strict negativity is not a requirement; on the other hand we also assume that V_ may be made nonnegative for a suitable choice of u; in this respect we have a stronger property with respect to the usual de nition of clf. An obvious choice for such a kind of function is clearly the mechanical energy of the system. In particular, letting
E (; !) = a[1 + cos()] + !2=2
(12)
and taking derivatives along trajectories of (8) we have
E_ = ?b! cos()u;
(13)
which can be made to be nonpositive or nonnegative by an appropriate choice of u. The basic idea is that, whenever (11) are satis ed, and under some additional technical assumptions, it is relatively easy to stabilize local minima (or maxima) of the Lyapunov function V (x). Unfortunately, the point [ = 0; ! = 0], is a saddle-point of the mechanical energy of the system. Therefore we need to nd some suitable term which makes the upright position a local minimum of V (x), while still preserving the validity of (11). This is not an easy task in general, especially since we are looking for a global ful llment of such conditions. Let us denote by p(; !) the following function: cos() p(; !) = 1 ? ea !2 =2
(14)
where is some strictly positive real to be chosen later. The quantity p(; !) will turn out to be essential in building a Lyapunov function for the system. Taking derivatives along trajectories of (8) we obtain "
#
) p_ = ! cos() bp(; !)u + a sin( 2 =2 : ! e 2 2
(15)
Take as a Lyapunov function
V (; !) = E (; !) + exp( E (; !))p(; !): 7
(16)
This function has two local minima, one in (0; 0) and another in (; 0). Taking the derivative of V along trajectorie yields by virtue of (13) and (15) (
)
E (;!) ) ? h1 + e E(;!)p(; !)( ? )i bu : V_ = ! cos() a2 2 e e!2sin( =2
Since we can choose a 1 (in order to make p(; !) positive semide nite) and , it is possible to consider the following preliminary feedback loop
E (;!)) sin() u~ + a22 exp(exp( 2 =2) u = [1 + exp( E (; !))p(! ; !)( ? )] b
(17)
where u~ is a new input signal; notice that taking = leads to a simpli ed expression for V ; we leave it in this general form since it is not clear which expression would be suitable for an extension to more general mechanical systems. This choice of u leads to the following simple expression for V_ : V_ = ?! cos()~u (18) As already pointed out V has more than one local minimum; hence the application of a passivity based feedback u~ = ! cos() would only provide a locally stable controller. In order to achieve a dense basin of attraction in R S we need to switch from positive to negative feedback according to the state of the system. Given a certain local minimum x0 of a di erentiable Lyapunov function V (x) : M ! R , consider the following family of sets depending on the real parameter c: o
Lc = fconnected component of fx 2 M; V (x) < cg containing x0 : (19) In particular, for a given x0 we de ne the critical level set Lc(x0 ) as the one corresponding to
c = sup fc > V (x0 ) : Lc is bounded and jrV (x)j 6= 0; 8x 6= x0 2 Lcg (20) When c = +1 we let Lc(x0 ) be the whole M. Notice that it may also happen c to be the supremum over an empty set. In that case Lc(x0 ) = ;. If we could make V_ strictly negative then the interior of Lc(x0 ) would be an estimate of the region of asymptotic stability (RAS ) of the equilibrium point. Since we only required negative semi-de nite derivatives we will need some additional zero-detectability notion in order to ensure convergence of the state to the equilibrium position. Let y be a scalar output of the system (the generalization to the multi-input 8
multi-output case is straightforward); in particular y = h(x), where h : M ! R is a continuous function. The following lemmas 1 and 2, refer to the closedloop system obtained applying the following passivity based feedback law:
u = ?k(y)p(x); with k() continuous, such that yk(y) > 0; 8y 6= 0(21) and p(x) is continuous and strictly positive for all x.
Lemma 1
Let V (x) : M ! R be a di erentiable function such that along trajectories of x_ = f (x) + g(x)u
Lf V (x) + Lg V (x)u yu:
(22)
Then, for any given local minimum x0 2 M of V (x), the corresponding Lc(x0 ) is such that (21), makes x0 a locally asymptotically stable equilibrium position and
Lc(x0 ) RAS
(23)
provided that all solutions of the system belonging to Lc(x0 ) satisfy the following detectability condition
h(x(t)) 0 ) x(t) ! x0 :
(24)
Remark. Notice that given x0 , local minimum of V (x), the set Lc(x0 ) is open. Notice also that in general Lc need not be bounded even if c < +1. Proof. Local stability of x0 follows immediately by a Lyapunov argument
taking into account equation (22): V_ = Lf V (x) ? Lg V (x)k(y)p(x) ?yk(y)p(x) 0
(25)
Therefore we only need to show attractivity. Let x be in Lc(x0 ). By de nition we have V (x) < c, hence there exists " > 0 such that V (x) < c ? ". Then x 2 Lc?". It follows from (20) that the set Lc?" is bounded. Let X be its closure. It follows from (19) that X is a compact connected component of a sublevel set of V (x); further X is invariant by virtue of (25). By continuity we have that V (x) is lower bounded in X ; in particular, since V (x(t)) is non-increasing there exists the limit V (x(t)) for t ! +1 and by the Lasalle criterium of stability, x(t) tends to the largest invariant set included in the kernel of h(x)p(x). Since p(x) is always positive then y ! 0 and by assumption (24) it follows that x(t) ! x0 . 9
Lemma 1 gives an estimate of the region of asymptotic stability of an equilibrium point. Next we show that this region can be enlarged around all x where h(x) is di erent from 0. First of all notice that @ Lc(x0) \ fy 6= 0g belongs to the region of asymptotic stability of x0 . Let belong to the boundary of Lc and be such that h( ) 6= 0. By continuity, there exists an interval of time [0; ") such that y(t) is di erent from 0 on that interval and hence V (x(t)) < c for t 2 (0; "). Since by hypotesis h( ) 6= 0, we have by (22) that V_ < 0 and in particular jrV j 6= 0 in ; hence is not a singular point of V (x(t)) and the only way that V (x(t)) < c = V ( ) be satis ed when t 2 (0; "), is by allowing x(t) 2 Lc for all t > 0. Even more it is true as shown in the following lemma (see Figure 3).
Lemma 2 Let Lc be bounded. There exists a positive de nite function such that the closed-loop system achieved letting u = ?k(y)p(x) satis es d(; Lc(x0))) < (jh( )j) ) x(t; ) ! x0
(26)
where d(; S ) denotes the usual point-to-set distance inf fjz ? j; 8z 2 Sg .
Proof. For points in cl(Lc(x0 )) the result is already known. Let x^ belong to @ Lc(x0 ) and h(^x) 6= 0. We claim that there exists " > 0 such that j ? x^j " implies x(t; ) ! x0 . By lemma 1 we only need to show that for all in a neighborhood of x^ there is some T > 0 such that x(T; ) is in Lc(x0 ). Pick some t0 > 0 so that x(t0 ; x^) is in Lc(x0 ) (in fact, any t0 > 0 has this property as remarked before the lemma). By continuity of the map ! x(t0 ; ) and openness of Lc(x0 ), there is some " > 0 such that j ? x^j < " ) x(t0 ; ) 2 Lc(x0 ). Thus belongs to the region of asymptotic stability. Take a positive and strictly decreasing sequence ak , such that ak ! 0; then we can de ne according to the following procedure. The set Ck de ned as @ Lc \ fx : jh(x)j ak g is a compact set. For all x 2 Ck there exists "x such that j ? xj "x implies x(t; ) ! x0 . Consider for any given k, the family of sets n
o
Fxk2C = : j ? xj "x=2 : k
This is an in nite cover of Ck . Since Ck is compact each cover admits a nite subcover; let fxn; "xn gn=1:::N identify one such nite subcover: [ n
x2Ck
o
: j ? xj "x=2 10
[ n
i=1:::N
o
: j ? xn j "x =2 : n
y
Fig. 3. Estimates of the RAS : Lc solid line, enlarged basin dotted line
We de ne k as minn=1:::N "xn =2. It is clear by construction that for any x 2 Ck the ball of radius k centered in x belongs to the region of asymptotic stability. Without loss of generality assume k non-increasing. We de ne as follows 8 > ak :
+ akr+1?a?kak (ak+1 ? ak ) for r 2 [ak?1; ak ) 0 for r = 0
(27)
The function is continuous and positive de nite; by construction we have that (26) holds. We can now go back to the pendulum problem; we already mentioned that the Lyapunov function (16) has more than one local minima; in particular the points [0; 0] and [; 0] are local minima of the function as it can be veri ed zeroing the Jacobian and studying the de niteness of the Hessian. Between them there is :a couple of saddle points, symmetric with respect to the ! axis, at some xs = [s ; 0], with 0 < s < =2. These are the only singular points of V (x). The idea is to apply a negative feedback everywhere except in the region corresponding to Lc([; 0]). This sort of patched feedback law need not be discontinuous; we can use a type of continuous switching between positive and negative feedback, for instance making use of a continuous or even smooth bump function. In order to write down a regular feedback let b(r) be a smooth function such that: 8 > > > > >
c^ for all t 0. Since V is lower-bounded V admits a nite limit and, being level curves compact, by invariance, there exists a limit set and !(t) cos((t))b? (x) ! 0. Since V (x) > c implies b?(x) > 0, either ! ! 0 or cos() ! 0. In this last case particularly we would have an !-limit set with cos() 0 which cannot happen. As a consequence necessarily ! ! 0, but this contradicts the assumption that V (x) > c^ for all t 0. Consider now 2 Lc(). We claim that, for almost all initial conditions, the trajectories corresponding to (31) are such that x(t) would leave the set cl(Lc()) de nitely. Notice that b?(x) = 0 for x 2 Lc(). As a consequence the derivative of the Lyapunov function reads V_ = !2 cos2 ()b+(x) (32) Rewriting (32) in backwards time the point turns out to be anti-stable. More in general starting with 6= we may assume by a contradiction argument that x(t) belongs to cl(Lc()) for all future time. This implies by monotonicity of V the existence of the limit of V (x(t)) as t ! +1, and, again by compactness of trajectories, that !2 cos2()b+ (x) ! 0. Since the largest invariant set included in the kernel of ! cos()b+ (x) is the couple of saddle points f[s; 0]g we have that the only way x(t) can be in cl(Lc()) for all time t 0 is when belongs to the stable manifold of one of the two 13
ω
θ
Fig. 5. Stable manifold of the saddle point
saddle points. Since in a neighborhood of those points we know that u~ = 0, trajectories of the system are along level curves of V (x); in particular the stable manifold is locally one of the branches of the set V (x) = c. By unicity of solutions, going backwards in time, we have that the stable manifolds of the saddle points belonging to Lc() can only have zero measure (see Figure 5). In order to prove \almost" global stability, we only need to show that all trajectories starting in the set fc < V (x) c^g will eventually hit the region of asymptotic stability. Notice, by the previous arguments, that the set fV (x) c^g ? Lc() is invariant; hence, either trajectories will eventually hit Lc(0) or they will stay in fc < V (x) c^g for all future time. In the second case in particular we have that ! cannot change sign. Hence (t) is a monotonous function; assume without loss of generality ! 0. There are two possibilities, either x(t) hits the basin of attraction of the upright equilibrium or (t) is upperbounded, and by monotonicity it admits a limit: (t) ! 0 ; if this is the case, ! ! 0 and as a consequence the only way this can happen is that x(t) tends to the saddle point in between upright and downwards position; but this is a contradiction because the stable manifolds of the saddle points lies within cl(Lc()). See Figure 6 for a graphical exempli cation of the discussion.
4 Spherical inverted pendulum Consider now the spherical inverted pendulum shown in Figure 7. The movable base is idealized to be a point of mass M and the pendulum a point of mass m. The free Langrangian for such a system is given by:
L(q; q_) = 21 M (y_ 2 + z_ 2 ) + 12 m y_ 2 + z_ 2 + r2_2 + n
+ r2 sin2() _ 2 + 2r cos()_[y_ cos( o) + z_ sin( )] + + 2r sin() _ [?y_ sin( ) + z_ cos( )] + mgr[1 ? cos()] 14
(33)
ω V(x) =^c
−π
−π/2
π/2
0
Bump function
π θ
Fig. 6. Hitting condition m
θ
z Fz y
M
Fy
φ
Fig. 7. Spherical inverted pendulum
where q = [y; z; ; ]0. Then, neglecting the e ect of friction, the equations of the system are given by :
d @L ? @L = F dt @ y_ @y y d @L ? @L = F dt @ z_ @z z d @L ? @L = 0 dt @ _ @ d @ L ? @L = 0 dt @ _ @
(34)
This is a system evolving on R 8 . It turns out, taking explicit derivative in (34), that the relationship between Fy ,Fz and y,z can be inverted. Hence it is possible, de ning a suitable internal feedback loop, to write the system as a parallel of a couple of double integrators and the (; )-subsystem which can be interpreted as an inverted spherical pendulum with massless cart. In order to get the explicit expression of the new acceleration inputs ay ,az one needs 15
to take the inverse of the inertia matrix H (q): 2 6 6 6 6 H (q) = mr 66 6 6 4
3
cos() cos( ) ? sin() sin( ) 7 7 7 M +m 0 cos( ) sin( ) sin( ) cos( ) 7 mr 7 7 7 cos() cos( ) cos() sin( ) r 0 7 5 2 ? sin() sin( ) sin() cos( ) 0 r sin () M +m mr
0
After some calculations it turns out that the systems equations in feedforward form can be written as follows:
y = ay z = az = cos() sin() _ 2 ? cos() cos( ) ary ? cos() sin( ) arz + gr sin() (35) sin2() = ?2 sin() cos()_ _ + sin() sin( ) ay ? sin() cos( ) az r r By applying the results in Teel (1996), we can focus our attention on the subsystem [; ; ;_ _ ], evolving in S2 R 2 . If a stabilizing feedback law is known for such a subsystem, attractive over A, then it is possible to modify that law in order to have R 4 A as a basin of attraction for (35). Notice that, by letting 2 3 6 u 7 4 5= u
2 16 4
cos( ) r ? sin( )
32 3 sin( ) 7 6 ay 7 54 5 cos( ) az
(36)
the equations of the (; ) subsystem read: i h = gr sin() ? cos() u ? sin() _ 2 sin2() = ?2 sin() cos()_ _ ? sin()u
(37)
Notice that the upright position of the inverted pendulum is not identi ed by a unique quadruple of cohordinates; in particular we want to make = 0 but we can allow for to be arbitrary. Though there might be ways of swinging up the pendulum exploiting its spherical nature, hereby we basically recast the problem as the one of stabilizing a planar inverted pendulum. In fact, taking u as: h
i
u = ? 2 cos()_ _ + K sin() _ 16
(38)
yields linear closed-loops dynamics for _ , = ?K _ , which are exponentially stable for all K > 0. As a matter of fact we are left with the problem of stabilizing the rst equation of (37); by letting u~ = u ? sin() _ 2 (39) it is easy to recognize that the equations of the subsystem are analogous to the equation of the planar massless inverted pendulum (8), then the same control law will provide \almost global" asymptotic stability of the spherical inverted pendulum as well.
5 Conclusions We have presented a smooth feedback for \almost global" stabilization of the planar and spherical inverted pendulums on a cart. The method followed in order to design the control was to nd some preliminary feedback law which would keep the system conservative with respect to a new energy function V . The fundamental idea was to ensure that the desired equilibrium position be a local minimum of V ; notice that this is not the case considering as a Lyapunov function the mechanical energy of the system since the upright pendulum position is a saddle point, (minimum of the kinetic energy but maximum of the potential). Then, the appropriate dissipation was added, with positive or negative sign, depending on the state space region. This approach, though some major technical diculties have still to be overcome, could be suitable for possible extensions to more general hamiltonian systems.
Acknowledgements. The author would like to thank Prof. Eduardo Sontag for having brought this problem to his attention and for the many useful discussions and comments during the writing up of the paper. 6 References Astrom K. J. & Furuta K., (1996), Swinging up a pendulum by energy control, Proceedings IFAC Conference, San Francisco, (1996) Bloch, A.M. , Leonard, N. E. & Marsden, J. E. (1998), Mathcing and Stabilization by the Method of Controlled Lagrangians, Proceedings of the 37th IEEE CDC, Tampa (FL) 17
Mazenc F. & Praly L. , (1996), Adding integrations, saturated controls and stabilizations for feedforward systems, IEEE Transactions on Automatic Control, 41, no.11, pp. 1559-1578 Shiriaev, A., Ludvigsen, H. & Egeland O., (1998) Global stabilization of unstable equilibrium point of pendulum, Proceedings of the 37th IEEE CDC, Tampa (FL) Sontag, E. D., (1998), Mathematical Control Theory, Springer, pp. 250-252 Teel, A. R., (1996), A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Transactions on Automatic Control, 41, no. 9, pp. 1256-1270
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