ASYMPTOTIC STABILIZABILITY OF UNDERACTUATED HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 309-326. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190309

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 93D05, 93D20, 93C10

Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom

S. D. Grillo, L. M. Salomone, M. Zuccalli

For an underactuated (simple) Hamiltonian system with two degrees of freedom and one degree of underactuation, a rather general condition that ensures its stabilizability, by means of the existence of a (simple) Lyapunov function, was found in a recent paper by D.E. Chang within the context of the energy shaping method. Also, in the same paper, some additional assumptions were presented in order to ensure also asymptotic stabilizability. In this paper we extend these results by showing that the above-mentioned condition is not only sufficient, but also necessary. And, more importantly, we show that no additional assumption is needed to ensure asymptotic stabilizability.

Keywords: underactuated systems, Hamiltonian systems, asymptotic stability, Lyapunov functions

Received April 30, 2019 Accepted September 12, 2019

S. D. Grillo and L. M. Salomone thank CONICET for its financial support.

Sergio D. Grillo sergiog@cab.cnea.gov.ar

Instituto Balseiro, UNCuyo-CNEA

av. Bustillo 9500, San Carlos de Bariloche, Río Negro, República Argentina

Leandro M. Salomone salomone@mate.unlp.edu .ar

Marcela Zuccalli marcezuccalli@gmail .com

CMaLP, Fac. de Ciencias Exactas, UNLP

50 y 115, La Plata, Buenos Aires, República Argentina

1. Introduction

Consider an underactuated Hamiltonian system with two degrees of freedom and exactly one actuator (i.e., with one degree of underactuation). Such a system can be described by a pair (H,Y), where H is a Hamiltonian function on a 4-dimensional phase space and Y is a (vertical) vector field defining the direction of the actuator. Fix a critical point a0 for the related Hamiltonian vector field Xh and assume from now on that:

1. H is simple, i.e., in any canonical coordinate chart (x,y,px,py)

H {X, y, Px, Py) = ~ (Px,Py)

for some function h and with

H(x,y) :=

a(x,y) b(x,y) b(x,y) c(x,y)

a(x,y) b(x,y) b(x, y) c(x, y)

+ h(x,y)

a positive definite matrix;

2. there exist canonical coordinates, which we shall call adapted coordinates, in which:

(a) Y is given by the constant vector (0,0,0,1);

(b) the critical point a0 is represented by the vector 0 := (0,0,0,0).

A system satisfying the above conditions will be called underactuated simple Hamiltonian system with two degrees of freedom. In Ref. [7], D.E.Chang found, among other things, a sufficient condition that ensures the stabilizability1 of such systems at the given critical point 0. The above-mentioned condition can be written (in adapted coordinates) as

d2h d2h dx2 dxdy

(0,0) = 0 or

d2h dx2

(0,0) > 0.

(1.1)

In coordinate-free terms, according to [7], the above inequalities mean that: either the linearization of the system at the given critical point is controllable or it is uncontrollable with uncontrollable modes given by a purely imaginary pair.

Chang's work was done within the framework of the energy shaping method (see, for instance, [1-5,10, 14-17]), or more precisely, within his version of the method, developed in [6, 8, 9]. Let us briefly review such a method in the present context. Its main idea is to construct, for a given pair (H, Y) and a given critical point of XH, a state feedback controller u and a simple Lyapunov function H for the resulting closed-loop system. Note that H (to be simple) must have the form

1

H {x, y,Px,Py) = 2 (Px,Py)

f (x,y) g(x,y) g(x,y) l(x,y)

px py

+ h (x,y)

1 By stabilizable at a point a0 we mean that there exists a state feedback controller such that the

related closed-loop system is stable at a0.

with

f(x,y) g{x,y)

H(x,y) :=

g(x,y) l(x,y)

(1.2)

positive definite. To find the controller u, a set of partial differential equations (PDEs), known as matching conditions, must be solved. Such PDEs have the pair (H, Y) as datum and the aforementioned Lyapunov function H as their unknown. According to Ref. [11], for pairs (H, Y) as described above, and using adapted coordinates, the above-mentioned PDEs are

2

Y, (dkHij Hkl - dkHij Hkl) pipjpl = 0, (1.3)

i,j,k,l=\

the kinetic matching condition, and

2

Y (dkh Hkl - dkh Hkl^ pi = 0, (1.4)

k,l=l

the potential matching condition, and must be satisfied for all (x,y,px ,py) such that

Px g(x, y) + py l(x, y) = 0. (1.5)

Here, d1 (resp. d2) denotes the partial derivative w.r.t. x (resp. y), pi = px and p2 = py. Note that H = H is a solution of (1.3) and (1.4), but we also need Hi to be a Lyapunov function related to the point 0. In order to ensure this we need, besides H being a positive definite matrix, that the function /ibea positive definite function w.r.t. (0,0), i.e.,

h (0,0)=0 and h (x,y) > 0 for all (x,y) = (0,0). (1.6)

Once a solution H of the matching conditions is given, the method provides a concrete procedure to construct a state feedback controller. In the case under consideration, such a controller has the form (see Ref. [11])

u(x,y,px ,py) = (0,0,0,\(x,y,px ,py)), (1.7)

u + {H,H} \ (x,y,Px,py)

\{x,y,px,py) := —i--------. (1.8)

Px g(x,y)+ Py l(x,y)

Here {•, ■} denotes the canonical Poisson bracket and u is any nonnegative function such that

u(x,y,Px,Py)

Px g(x,y)+ Py l(x,y)

(1.9)

is smooth. Thus, if a solution H of the matching conditions (1.3), (1.4) and (1.5) is found, and satisfies the above-mentioned positivity requirements [see (1.6)], the system in question can be stabilized at the point 0 by means of the controller (1.7). And, as we said above, such stability is ensured by the existence of a Lyapunov function for the related closed-loop system: the solution H.

It was shown in [11] that Chang's version of the energy shaping method [6, 8, 9] is maximal among the so-called "Lyapunov based methods". More precisely, if an underactuated simple Hamiltonian system (with any number of degrees of freedom and any degree of underactuation) is stabilized by a method that gives rise to a closed-loop system with a simple Lyapunov function, then such a function must be a solution of the matching conditions and the related controller is exactly the one given by the energy shaping method.

Coming back to (1.1), what Chang showed in [7] was actually that (1.1) is a necessary and sufficient condition for finding a simple solution of the matching conditions (with the above mentioned positive requeriments), and consequently a sufficient condition for stabilizability. Moreover, in the same paper, two additional assumptions to ensure not only stabilizability, but also asymptotic stabilizability were presented.

In the present paper, we further study the stabilizability condition (1.1) and show:

a. A (slightly) different stabilizability characterization: (1.1) is a sufficient and a necessary

condition to stabilize an underactuated simple Hamiltonian system with two degrees of freedom by any method (not only the energy shaping) that guarantees such stability by exhibiting (or at least by ensuring the existence of) a simple Lyapunov function.

b. The main result of the paper: (1.1) not only ensures stabilizability, but also asymptotic

stabilizability. That is to say, no additional condition is needed, other than (1.1), in order to prove the asymptotic stabilizability for an underactuated simple Hamiltonian system with two degrees of freedom.

The paper is organized as follows. In §2 we write down a more convenient expression of the matching conditions (1.3), (1.4) and (1.5). Then, studying the existence of their solutions, we give an alternative derivation of condition (1.1) and show the point a above. We could show that point simply by combining the results of Chang in [7] and the above-mentioned maximal character of the energy shaping method (shown in Ref. [11]). Nevertheless, we decided to make a detailed proof because of the involved reasoning and calculations, which are necessary to prove the second result of the paper. The latter is done in §3, where, by combining the LaSalle invariance principle, a Dirac-like algorithm and the Morse lemma, we show that condition (1.1) also implies asymptotic stabilizability (i.e., we prove the claim of point b). Finally, we illustrate our results with an example.

2. Stabilizability of systems with two degrees of freedom

In this section we prove that, given an underactuated simple Hamiltonian system with two degrees of freedom, condition (1.1) ensures its stabilizability. Reciprocally, if such a system is stabilizable and that stability can be ensured by a simple Lyapunov function, we prove that condition (1.1) must hold. All that will be done by using the Chang version of the energy shaping method [6, 8, 9]. To begin with, we shall write down (1.3), (1.4) and (1.5) in a way which is more appropriate for our purposes.

We want to emphasize that the results of the present section represent a slight modification of those contained in the work of Chang [7] and, as we said in the Introduction, they can be proved just by using the maximal character of the energy shaping method [11]. However, we decide to give an alternative proof here because some of the intermediate steps are crucial for showing the main result of this paper, developed in §3.

2.1. Rewriting the matching conditions

Consider a pair (H,Y) and a set of adapted coordinates (x,y,px,py) as those described in the Introduction.

Remark 1. Since H is simple, it is easy to show that 0 is critical for the Hamiltonian vector field

XH = (—, —, - —, -—) (2.1)

V. Hp, ''/>/ dx ' dy J

if and only if (0,0) is critical for the function h.

Consider also the matching conditions (1.3), (1.4) and (1.5) for the unknowns H and h. Note first that, since H is positive definite (see (1.1) in the Introduction),

a,c> 0 and A := ac - b2 > 0. (2.2)

Analogously, regarding H, we have that

f,l> 0 and fl - g2 > 0. (2.3)

To further simplify the notation, define

4 - 7 - % (")

and

B := ax - 2 Ybx + Y2 Cx. (2.5)

(From now on, the subindices x and y denote partial differentiation). In terms of these new variables, it can be shown that the kinetic matching condition (1.3), combined with (1.5), and the positivity conditions (2.3) are equivalent to

(a - bY) Sx + (b - cy) = B5, 5> 0 (2.6)

and l > 0, while the potential matching condition (1.4), combined with (1.5), takes the form

(a - bY) hx + (b - cy) hy = hx S. (2.7)

Summing up, the matching and positivity conditions can be described by (2.6), (2.7) and (1.6) for the unknowns ^6,Y,hj, plus the condition l > 0. To go back to the original variables, we just must use the formulae [see (2.4)]

f = S + Iy2 and g = Iy. (2.8)

2.2. A sufficient condition for stabilizability

As we said in the Introduction about the energy shaping method, if we find a solution to (1.3), (1.4), (1.5) and (1.6), then we can construct a vector field u [see (1.7) and (1.8)] and a simple Lyapunov function that ensures the stability of the related closed-loop system at 00 (at least locally around 0). So, according to the last subsection, the stabilizability of (H,Y)

around 0 can be analyzed by studying the existence of solutions ^S, y, hj of (2.6), (2.7) and (1.6).

To do that, let us consider the next two lemmas. In what follows, we shall call U C R4 the neighborhood of 0 where the adapted coordinates take their values. Also, for simplicity, we shall write 0 := (0, 0).

Lemma 1. Given a function y satisfying

and

[(a - bj) hxx + (b — c y) hXy] (0) > 0, (2.10)

there exist functions 5 and h such that y5,Y,hj is a solution of (2.6), (2.7) and (1.6).

Proof. Let us begin with (2.6). This is a first-order PDE, so we can use the method of characteristics to find a solution around 0. But, in order for this to make sense, we need a suitable boundary condition on a noncharacteristic submanifold r. Let V be the projection of U onto the first two coordinates. Observe that the characteristic vector field is A = (a — bq,b — cy). Then we may take the submanifold r c R2 to be the x-axis, i.e., to take

r = {(x, 0): x e R} PI V,

so long as we ensure that the second component of A is nonzero around 0. But this amounts to choosing y such that (2.9) holds. Since we need 5 > 0, we can impose the boundary condition 5|r = s, where s: R ^ R is a function such that s(0) > 0. In this case, the theorem of characteristics states that there is a unique solution 5 of (2.6) such that 5(x, 0) = s(x), which implies, by continuity, that 5(x,y) > 0 around 0. We can shrink V (together with U), if necessary, in order to ensure that 5 > 0 along all of V. From now on, we shall use this shrinking process implicitly (finitely many times).

Let us continue with (2.7) and (1.6). The former is also a first-order PDE, and with the same characteristic vector field A. Assuming (2.9) again, the x-axis is a noncharacteristic submanifold

and we can impose h = r, where r : R ^ R is a smooth function such that r(0) = 0. This

implies that h (0) = 0, which is the first part of (1.6). The second part says that 0 is an isolated minimum for h, or equivalently, 0 is critical for h and the Hessian of h is positive definite at 0. Let us analyze these conditions. Since 0 is critical for h (see Remark 1), i.e., (hx(0),hy(0)) = 0, it follows from (2.7) that h must satisfy

(a — bY) hx + (b — cy) hj (0) = 0.

Thus, since b(0) — c(0) y(0) = 0 [see (2.9)], in order that [hx(0),hy(0)J = 0, it suffices to ask that hx(0) = r'(0) = 0. So far, we have that r and s must satisfy

s (0) > 0, r (0) = r'(0)=0. (2.11) On the other hand, the Hessian of h is positive definite at 0 if and only if

hxx(0) > 0 and (hxx hyy — hZ^) (0) > 0. (2.12)

It is easy to compute the second partial derivatives of h at 0 using (2.7) and the boundary conditions 5|r = s and h

= r. This gives, omitting the evaluation point 0,

' hxx = r" (0),

j _ hxx s (0) - (a - bY) r" (0) xy (b-cy) ' (2.13)

• _ hxy s (0) (a - b y) hxx s (0) + (a - b y)'2 r" (0)

ibqiqi —

yy (b-cy) (b-cy)2

Then we must have

hxx = r" (0) > 0

(2-14)

and

î Î Î 2 s (0) r" (0) (. , ,, h2xx s (0)\ „

hxx hyy - h2xy = _ ( (a - 67) xx + (6 - C7) /?,T?/ - I > 0.

- C7)

r" (0)

(2.15)

Accordingly, since (2.10) holds by hypothesis, in order to ensure (2.12) it is enough to take

r" (0) >

hxx (0) s (0)

[(a - 67) hxx + (c - 67) hxy] (0)

(2.16)

□

The next lemma gives a necessary and sufficient condition, in terms of H, for the existence of a function 7 fulfilling (2.10). The proof can be found in the Appendix.

Lemma 2. There exists a function 7 such that (2.10) holds if and only if

[bhxx + chxy ](0) = 0 or hxx (0) > 0. Moreover, in such a case, 7 (0) can be chosen such that

a hxx + b hxy

(2.17)

^(0)l> bhxx+chxyi

[7 (bh xx + chxy)] (0) < 0

(0)

(2.18)

if [b hxx + c hxy] (0) = 0, and using the following table

hXy (0) = 0 hXy (0) < 0 hXy (0) > 0

6(0) = 0 any 7(0) > 0 7(0) < 0

6(0) > 0 , , a, (0) 7(0) < KO) 6(0) a (0) < 7(0) < c(0) ,lUJ 6(0) ■ Mo) 6(0)\

6(0) < 0 a (0) ^(0)>6(0) (a{0) 6(0)\ a(0)<7(0)<6(0) 6(0) <71 c(0)

(2.19)

if hxx (0) > 0. All of these conditions are compatible with (2.9).

Summarizing, if (2.17) holds, in order to find a solution of (2.6), (2.7) and (1.6),

it is enough to take 7 satisfying (2.9) and also (2.18) or (2.19), as explained in the last lemma. Thus, we have proved the following.

Theorem 1. Consider an underactuated simple Hamiltonian system with two degrees of freedom and a set of adapted coordinates related to it. Then, if (2.17) holds, the system is stabilizable at 0, i.e., there exists a state feedback controller u, defined at least around 0, such that the related closed-loop system is stable at 0. Moreover, such a stability can be ensured by the existence of a simple Lyapunov function.

2.3. A necessary condition for the existence of a simple Lyapunov function

Using the same notation as above, suppose that an underactuated simple Hamiltonian system with two degrees of freedom (H, Y) can be stabilized at 0, and that such stabilization is ensured by the existence of a simple Lyapunov function. More precisely, suppose that there exists a controller u = XY and a simple Lyapunov function H, both of them defined at least around 0, ensuring the stability of the related closed-loop system. Then it was shown in [11] that u and H must be given by the energy shaping method (or more precisely, by the so-called simple CH method). In particular, H must be locally given, in adapted coordinates, by a solution (5,Y,h^ of (2.6), (2.7) and (1.6) (and by (1.2) and (2.8) and some function l > 0). We want to show from this fact that (2.17) must be satisfied. To do that, let us consider two cases.

1. y does not satisfy (2.9): If y(0) = b(0)/c(0), then [recall (2.2)]

|o - N (0) = a(0) - 6(0) M = > 0.

On the other hand, if we differentiate (2.7) and evaluate the result at 0, we obtain

(a(0) — b(0) y(0)) hxx(0) = hxx(0) 5(0). (Recall that 0 is critical for h and h). As a consequence, using that 5(0) > 0 and hxx(0) > 0,

_ [a — &7] (0) hxx(0)

ll,xx{\J) — ^^ ^ u.

In other words, condition (2.17) must hold.

2. y satisfies (2.9): Let us call V the neighborhood of 0 where the functions 5, y, h are defined. Define r (x) := h(x, 0) and s(x) := 5(x, 0) for all x such that (x, 0) e V. It is clear that the domain of the last functions is an open neighborhood of 0. Then, as we saw in the previous section, differentiating (2.7) and evaluating at 0 (and using that 0 is critical for h), we arrive at (2.13). Thus, the positivity conditions (2.12) for h can be studied in terms of (2.14) and (2.15). From the latter, and from the fact that s (0) > 0 (since 5 must be positive), it easily follows that (2.10) is a necessary condition. But according to Lemma 2, this says again that condition (2.17) must be satisfied.

Combining the above discussion with Theorem 1, we have the following characterization.

Theorem 2. Under the conditions of Theorem 1, (H,Y) is stabilizable at 0, and such stability can be ensured by the existence of a simple Lyapunov function if and only if (2.17) holds.

3. Asymptotic stabilizability

In Ref. [7], it was shown that the condition2 [bhxx + chxy] (0) = 0 also implies asymptotic stability (as previously affirmed in [12], without a proof). In any other case, in the same reference, an additional condition is proposed to ensure this kind of stability. We show in the next subsection that no condition other than (2.17) is needed to this end.

2Actually, a weaker condition is considered there (see Theorem III.3).

3.1. No additional assumptions are needed

Let yd,Y,hj be a solution of (2.6), (2.7) and (1.6) defined around 0, with y satisfying (2.9) and (2.10) and with boundary conditions given by functions s and r, as described in the proof of Lemma 1. That is to say, 5 and h must satisfy

5 (x, 0) = s (x) and h (x, 0) = r (x), (3.1)

with s and r fulfilling (2.11) and (2.16). To ensure the existence of such a solution, we only must ask that (2.17) hold. Let H be given by (1.2) and (2.8), i.e.,

H(x,y,Px ,Py) =

2 I 5(x,y) 2 ! \ \ 1 \ 2

Px 77-7 - 7 [X, y) + 2 7 (x, y) pxpy + p

l(x,y)

l (x,y)+ h(x,y), (3.2)

for some positive function l. To write down an explicit expression for the controller, we must choose a nonnegative function i fulfilling (1.9). To that effect, it suffices to take

I (x,y,Px ,Py) = K (7 (x,y) Px + Py )2 l2 (x,y),

for some positive constant k.

Remark 2. According to the results of Ref. [11], the subset ¡¡-1 (0), which in this case is given by

I-1 (°) = {(x,y,Px,Py) : Px Y (x,y) + Py = °} , (3.3)

is the LaSalle surface related to the Lyapunov function H (see [13]). Note also that 0 G ¡¡-1 (0). With all these elements, the state feedback controller u adopts the form

u = (0,0,0,A), (3.4)

where A is locally given as [see (1.8)]

H,H| (x,y,Px,Py)

A {x,y,Px,Py) = -X (7 {x,y) Px+ Py) l{x,y) - -r-;--J-,-r. (3.5)

y y (Y (x,y) Px + Py) l (x,y)

Our next step will be to prove that the functions 5, y, l and h can be chosen such that the closed-loop system defined by u is asymptotically stable around 0, i.e., the origin 0 is an asymptotically stable equilibrium point of the vector field X = XH + u. More precisely, we are going to show, without any additional assumption other than (2.17), that boundary conditions s and r [see (3.1)] and an open subset T containing 0 can be chosen in such a way that the largest X-invariant3 submanifold of S0 := I-1 (0) nT is the singleton {0}. Taking into account Remark 2, this would imply, via the LaSalle invariance principle, that 0 is (locally) asymptotically stabilizable for X (see [13]). The proof will be based on the next two lemmas (the proof of the first one is easy to derive, so we omit it for brevity).

3Recall that, given a manifold P and a vector field X on P, a subset S C P is X-invariant if every integral curve of X with initial condition in S is contained in S.

Lemma 3. Given a manifold P, a vector field X on P, a critical point a0 of X, and a submanifold S0 C P containing a0, let us define4

Sn := {a e Sn-i: X(a) e TSn-i}, n e N,

(3-6)

where we are assuming that each Sn is a .submanifold of Sn-1. Then the largest X-invariant subset I of So satisfies

{ao} CI C p| Sn .

neN

In particular, if Sk = {a0} for some k e N, then I = {a0}.

Lemma 4. There exist boundary conditions s and r, a function y and an open subset T 3 0 such that (see (3.6)):

• the subset S1 corresponding to S0 = ¡i-1 (0) n T is a submanifold of S0;

• S2 is a submanifold of S1;

• S3 = {0}.

It is enough to take s and r such that (besides (2.11) and (2.16))

8(0)

= "

2 (6-cy) A

bx - Y cx -

Bc

2 (b - cy)

(0),

(3.7)

and choose y(0) according to (2.9), (2.10) and the additional restriction

'b (0)\

(a (0) ,b (0)) M

Y (0) =

c (0),

(b(0),c(0)) M

fb (0)N ,c (0),

(3.8)

where M is a positive definite matrix given by

M=

r" (0)

hxx s (0) — (a — 6 7) r" (0) (b~c 7)

hxx s (0)-(a - bY) r" (0) hxy s (0) (b - cy)-(a - bY)hxxS (0) + (a - bY)2r" (0)

(b - cY)

(b - cy)2

(3.9)

Proof. According to (3.3), ¡1 1 (0) can be described as the zero set of the function

F(x,y,Px,py) := Y(x,y) Px + Py.

We shall proceed in three steps.

1. Let us consider the subset Z1 C ¡-1 (0) such that F*(X)(x,y,px,py) = 0, where F* is the tangent map of F and (see (2.1) and (3.4))

X-X +u-(—— — (— A

" H \ dpx' dpy' dx' \dy

(3.10)

4Given a manifold P, by TP we denote, as usual, its tangent bundle.

That is to say, Z1 is given by the equation

dH_ d$_ dH_d$_ dH_d$_ ( dH_ \ d$_

dpx dx dpy dy dx dpx \ dy J dpy

or equivalently by

dH dH dH dH .

and py = -y(x, y) px. Using the explicit forms of H and H, it is easy to see that, on ¡i 1 (0),

dH 1 „ o . , dH 1., ^

-y thy j

(3.12)

where B is given by (2.5) and

C := ay - 2^by + y2Cy.

On the other hand, and according to (3.5), the values of A at points of the form (x,y,px, -Y (x,y) px) G I-1 (0) are given by

{H,H} (x,y,Px,Py) 1 r){H H}

^ ~~ — lim —--=---—-— i^^ViPxi —/y V) Px)

Py(y (x,y) Px + Py) l (x,y) l (x, y) dpy x ' x'

So, by lengthy, but straightforward calculations, from (3.2), (3.5) and (3.12) we have that A =

±(BY + C)-Yx (a-bY)-Yy (b-cry) (3-13)

2 (bx -7 Cx)S -b ôx - c ôy 21

2 . . . . bhx + chy

px + 7 hx + hy----.

We omit, for simplicity, the evaluation point for the involved functions. Finally, combining (3.11), (3.12) and (3.13), we have at ¡-1 (0) that

MX) =\

y Cx) 5 -

Thus, Z1 is given by the equations

b5x + c5y

2 b hx + chy Px J •

where

and

Y(x,y) Px + Py = 0, K(x,y) px - L(x,y) = 0,

b5x + c5y

(3.14)

K = (bx - 7 cx) 6--(3.15)

L = b hx + chy. (3.16)

2

As a consequence, Z1 can be defined by the zero set of the function G(x, y,Px ,Py ) := (©1 (x,y,Px ,Py), ©2 (x,y,Px ,Py))

with

©1 (x,y, Px, Py) := Y(x,y) Px + Py, ©2(x,y, Px, Py) := K(x,y) Px - L(x,y).

We want to see that its related tangent map (omitting the evaluation point of the involved functions)

Yx Px

Yy Px

Y

Kx Px - Lx Ky Px - Ly 2 K'Px 0

(3.17)

has maximal rank around 0. In that case, the implicit function theorem would ensure that this zero locus is the graph of a smooth function (and hence a submanifold) when restricted to some open neighborhood of Such a tangent map is given at 0 by

© ,0

0 0 y(0) 1

-Lx(0) -Ly(0) 0 0

Note that the gradient of L can be written as

Lx

L

y

hxx hxy hxy hyy

V

+

bx cx Kby cy,

hx hy

(3.18)

(3.19)

Since 0 is critical for h, it follows that, at 0,

(L

L

hh

hxx hxy

y

hh

hxy hyy

But we know that the Hessian matrix of h is positive definite and the function c is always positive. So, the gradient of L cannot vanish at 0. This implies that (3.18) has maximal rank at 0. Consequently, there exists an open subset T1 containing 0 such that Z1 n T1 is a submanifold of ¡-1 (0) n T1.

2. Consider now the subset Z2 C Z1 nT1 given by ©*(X)(x,y,Px,Py) = 0. Easy calculations show that Z2 is given by the points of Z1 n T1 such that

(a - 67) (Kx pI ~ Lx) + (b - c7) (Ky p'i - Ly) -2 K[ ^Bpg + h

2

Px = 0. (3.20)

We only need to evaluate the second row of ©* (see (3.17)) on the components of X (see (3.10), (3.12) and (3.13)). In the following, we assume that K (0) = 0. Observe that, since 5(x, 0) = s(x), we have 5x(0) = s'(0), and using (2.7) at the origin

So (see (3.15))

K (0)

5y (0)

bx - YCx -

Bs{0) - (a — b s' (0) b - cy

(0).

Bc \ . 1( A

s (0)

(0),

1

c

and consequently, the condition K (0) = 0 is equivalent to (3.7). Under such an assumption,

we can replace p2x by in (3.20) (see (3.14)), and we get

(.a - 67) ( Kx ^-Lx)+(b- c7) ( Kt

K

L

y K y

-L,,\ - 2K

-2B± + hx

Px =0, (3.21) 0

on some open subset containing 0 (where K is nonvanishing). Moreover, since L(0) (see (3.16)) and hx (0) = 0, we have at 0 that

Lx K — L Kx

Lx

Ly K — LKy

L

y

K '

K2 K' K2

and then the bracketed expression in (3.21) takes the following form at 0:

(a - 67) Lx + (b - cy) Ly.

Using (3.19), this in turn may be written as

b

(a,b)

Then, if we assume that, at 0

h h

xx

xy

hh

h xy h yy

— Y (b, c)

hh

h xx h xy

hh

h xy h yy

(a, b) h h xx h h xy h h xy h 0

(b,c) h h xx h h xy h h xy h 0

Y =

it follows that (3.21) will hold only if px = 0 around (x,y) = 0. It is worth mentioning that this condition is compatible with (2.9) and (2.10). Note that, using (2.13), the condition above is given precisely by (3.8) and (3.9). In conclusion, there exists an open neighborhood TI (which contains the point 0) such that the subset Z2 n TI is given by

YPx + Py = 0, KpX - L = 0, Px = 0,

or equivalently

Px = Py = L = 0.

This means that Z2 n T'2 can be described as the zero set of the function

H(x,y,Px,Py) := (Py,Px,L(x,y)). The tangent map of H at 0 is given by

/0 0 0 A

(3.22)

H ,0 =

0 0 10 ^Lx(0) Ly(0) 0 0^

Again, since Lx(0) and Ly(0) cannot be both zero, we conclude that H*;o has maximal rank. Thus, there exists inside T'2 an open neighborhood T2 of 0 such that Z2 n T2 is a submanifold of Zi n T n T2.

c

3. Now consider the subset Z3 C Z2 n T2 defined by H*(X)(x,y,px,py) = 0. Using (3.10), (3.12) and (3.13), it follows that, along Z2 n T2 (see (3.22))

X = (0,0, -hx,Yhx),

so, in order for H*(X) to vanish, it is necessary that hx = 0. But, if this is the case, using the potential matching condition

(a - by) hx + (b - cy) hy = 5hx,

or equivalently, a hx + bhy - y L = 5 hx, we have on Z3 that

L = b hx + c hy = 0 and ahx + bhy = 0,

^ h ( hx) =0.

\hyj

Calling n the projection of R4 onto the first two components, we can say that the above identity holds if and only if all the points of n(Z3) are critical for h. By the Morse lemma, since 0 is a nondegenerate critical point of h (recall (1.6)), there exists a neighborhood V of 0 such that n ( Z3) n V = {0}. But px = py = 0 on Z3, which implies that Z3 n T3 = {0} for T3 := n-1 (V).

Summing up, if we define T := T1 n T2 n T3 and S0 := ¡¡-1 (0) n T, from (3.6) we obtain S1 = Z1 n T, which is a submanifold of S0, S2 = Z2 n T, which is a submanifold of S1, and S3 = {0}. Hence, the three points of the lemma follow. □

To conclude, if (2.17) holds, asymptotic stabilizability is ensured. Reciprocally, if we can ensure asymptotic stabilizability by the existence of a simple Lyapunov function, then we can also ensure stabilizability, and Theorem 2 implies that (2.17) holds. In other terms,

Theorem 3. Under the conditions of Theorem 1, (H, Y) is asymptotically stabilizable at 0, and such a stability can be ensured by the existence of a simple Lyapunov function if and only if H satisfies (2.17).

3.2. Example: the inertia wheel pendulum

Now, we apply our results to a concrete underactuated system (H, Y) with two degrees of freedom: the inertia wheel pendulum. This system has been well studied in different references. We include it here only with illustrative purposes.

The configuration space of the system is Q = S1 x S1, whose natural almost-global coordinates (see Fig. 1) will be denoted (d,^). The Hamiltonian is

a b b c

'pe \

+ M (1 + cos (

where a, b, c, M are constants and a, b, M, ac - b2 are strictly greater than zero. The space of actuators is given by the subbundle spanned by the constant vector field Y = (0,0,0,1).

We shall find, by using the energy shaping method, a state feedback controller u for this system and a related simple Lyapunov function H which make the closed-loop system XH + u asymptotically stable at (d,^,pe,p^) = (0,0,0,0) = 0.

Fig. 1. Inertia wheel pendulum with coordinates (6,-0).

Replacing x by 6 and y by (2.17) in this case says that (because he^ (0) = 0)

bhgg (0)=0 or hee (0) > 0,

which is equivalent to hee (0) = 0, since b = 0. And it does hold, because hee (0) = -M = 0. Then, as is well known, the inertia wheel pendulum can be asymptotically stabilized around 0. On the other hand, according to (2.5), we have that B = 0. So, the kinetic and potential matching conditions read [see (2.6) and (2.7)]

(a - bY) 5e + (b - cy) ^ = 0, (3.23)

and

(a - bY) he + (b - cy) h^ = -MS sin6, (3.24)

respectively. Let us construct a solution ^S, y, hj of the above equations, with S > 0 and h positive definite w.r.t. 0. We shall take y constant. Following the steps of §2.2, it is enough to take y such that y = b/c (see (2.9)) and, using (2.18) of Lemma 2 [since hee (0) = -M < 0], also ask that

a

|y| > — and — 7 6M < 0.

The second inequality says that Y is positive, so, the above equations only impose the condition Y > a/b. Note also that, since a,b,c,ac - b2 > 0, we have that a/b > b/c. Hence, all the conditions on Y reduce to

a

Remark 3. Note that, for this system, we cannot take y = b/c. In fact, in such a case, according to the calculations we made in §2.3, the positivity of S and h would impose that hgg (0) > 0, which is not

Regarding the boundary conditions defining ô and h, i.e., the functions s and r, respectively, we must ask (see (2.11) and (2.16))

s (0) > 0,

r (0) = r'(0)=0 and

r" (0) >

h2d (0) s (0) Ms (0)

(3.25)

(a — bY) hee (0) a — bY'

And to ensure asymptotic stabilizability, according to Eqs. (3.7), (3.8) and (3.9) of Lemma 4, we ask that

s (0)

s (0)

and that Y satisfy

= 0, i.e., s' (0) = 0,

rj2 ab r" (0) - r? [M s (0) + ( r" (0)] (ac + b'2) + ( M s (0) + (2 r" (0) be 7 ^ r?2 b2 r" (0) -2r} [M s (0) + ( r" (0)] be + ( M s (0) + (2 r" (0) c2 ' (3'26)

where Z := a - bY and n := b - cy- Thus, take any number y > a/b, any function s such that s (0) > 0 and S (0) = 0, and any function5 r such that r (0) = r'(0) = 0 and r" (0) satisfying (3.25) and (3.26), and let us apply the method of characteristics to (3.23) and (3.24), with boundary conditions on ^ = 0 given by s and r. The characteristic equations for (3.23) are

'6 = a - bY, 6(0) = 60, ^ = b - cy, ^(0)=0, S = 0, S(0)= s(6o).

Then

6(t) = (a - bY) t + 6o, ^(t) = (b - cy) t, and defining Y := (a - bY) / (b - cy), we find

S(6,^)= s(6 - Y (3.27)

The characteristic equation for (3.24) (and for S given above) is

h = -M s(6o) sin ((a - bY) t + 6o),

and integrating we obtain

h{6,i)) =M (cos d - cos(d - Y tp)) + r{6- Y '0). (3.28)

Finally, with S and h given by (3.27) and (3.28), and considering any positive function l, we have from (3.2), (3.4) and (3.5) the controller u and the Lyapunov function H we are looking for.

5 Additionally, the functions s and r may be taken with period 2n in order to look for a quasi-global solution.

A. Proof of Lemma 2

Suppose first that hxx (0) ^ 0 and (bhxx + chxy) (0) = 0. Then, omitting the evaluation point 0,

(a - bY) hxx + (b - cy) hxy = ahxx + bhxy - Y (bhxx + chxy)

c xx c xx

for any function y. This proves the first implication of the lemma (by denying the second one). For the converse, suppose first that b hxx + c hxy = 0. Since

(a - bY) hxx + (b - cy) hxy = ahxx + bhxy - Y (bhxx + chxy), for any y with sign opposite to b hxx + c hxy and such that

a hxx + b hxy

>

b hxx + chx

xx xy

we have that (2.10) holds. This implies (2.18). Now suppose that hxx > 0. If hxy = 0, since a > 0, it is clear that it is enough to choose y such that a > bY. If instead hxy = 0, we distinguish three cases: b = 0, b > 0 and b < 0.

• If b = 0, then a - bY = a > 0 and b - cy = -cy, and consequently it is sufficient to choose y with opposite sign to hxy.

• If b > 0, we show that it is possible to take y so as to fulfill one of the following expressions

a - bY > 0 and b - cy > 0,

or

a - bY > 0 and b - cy < 0.

ab In order to make a — 67 > 0, we need 7 < -. If in addition b — ci > 0, then 7 < -.

b c

Hence, it suffices to take 7 < min . On the contrary, if b — C7 < 0, then we can

take - < 7 < p which is always possible because

ba

ac-b2> 0 and 6>0 - < -.

c b

aiid, if hxy < 0, we take - _ . _ , b c c b

cases we get the desired result.

ab

• If b < 0, then a — 67 > 0 implies 7 > —. If in addition b — ci > 0, then 7 < - and so we

bc

can take 7 such that — < 7 < -, which is always possible because

ab

ac - b2 > 0 and 6<0 T < 7 < -.

b c

On the contrary, if b — C7 < 0, then 7 > - and it is sufficient to choose 7 > max Again, both cases lead to the desired result. All these alternatives give Table (2.19). The last assertion of the lemma is immediate, because all conditions on y are inequalities.

Thus, if hxy > 0, we choose 7 < min ( -, - I and, if hxy < 0, we take - < 7 < -. In both

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