100 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, TO APPEAR. of general nonlinear systems. A. Problem Formulation

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On Global Output Feedback Regulation of

Euler-Lagrange Systems with Bounded Inputs

Antonio Loria, Rafael Kelly, Romeo Ortega and Victor Santiba~nez

Abstract| In this short paper we identify a class of Euler-Lagrange systems with bounded inputs that can be globally asymptotically stabilized via dynamic output feedback. In par-ticular we prove that if the system is fully actuated and the forces due to the potential eld can be \dominated" by the con-strained control signals then global output feedback regulation is still possible incorporating some suitable saturation functions in the controller.

I. Introduction

It is well known that in Euler-Lagrange (EL) systems the number and stability properties of the equilibria are univocally dened by its potential energy and dissipation functions, see e.g., [1]. This fundamental feature is at the core of passivity{ based control of EL systems, which hinges upon the (physically appealing principles) of shaping the systems energy and injec-tion of the required damping. These ideas {though well known in mechanics, were rst introduced in control in the seminal pa-per [2] and rigorously articulated in [3]{ have been successfully used to control various practical EL systems, e.g., robots [2], [4], motors [5] and power converters [6].

In [1] we observed that in feedback interconnections of EL systems (which preserve the EL structure), the energy and dis-sipation functions of the closed loop are thesum of the corre-sponding functions of each subsystem. Motivated by this fact we proposed in that paper to consider EL controllers. In this way we have a direct, systematic way to solve the energy shap-ing and dampshap-ing injection tasks, i.e., by addshap-ing up the plant and controller total energies and dissipation functions. These procedure was used in [1] to identify a class of Euler-Lagrange (EL) systems {which includes robot manipulators{ that can be globally asymptotically stabilized via nonlinear dynamic out-putfeedback. To inject the damping without velocity measure-ment we added a dynamic extension and invoked a dissipation propagation condition. This condition, also known as perva-sive damping [7], is connected with the detectability property of the system. An interesting particular case of the controllers derived in [1], is the one reported in [8] where it is shown that, in set-point control tasks, velocity can be replaced by its dirty derivative preserving global stability.

The main motivation of this paper is to extend the methodol-ogy of [1] to the practically important situation when the system is subject to input constraints. Our contribution is the proof that if the system is fully actuated and the forces due to the potential eld can be \dominated" by the constrained control signals then global output feedback regulation is still possible in-corporating some suitable saturation functions in the controller. Instrumental for the solution of the problem is the result of [9] where globalstatefeedback regulation of robots with saturated inputs is established. A corollary of this paper is therefore, the extension {to theoutput feedbackcase{ of the result in [9]. See also [10].

This work was supported in part by CONACyT{Mexico.

R. Ortega and A. Loria are with the HEUDIASYC lab. at the Universite de Technologie de Compiegne, URA C.N.R.S. 817, BP 649, 60206 Compiegne, FRANCE. E-mail: aloria@hds.utc.fr, rortega@hds.utc.fr. R. Kelly and V. San-tiba~nez are with the Applied Physics Division at CICESE, AP 2615, Adm. 1, Ensenada, BC, 22800, Mexico. E-mail: rkelly@cicese.mx, vsantiba@cicese.mx.

Shortened version inProc. IFAC World Congress, San Francsico, CA, 1996.

Besides the obvious interest of ensuring stability with bounded controls, there is another motivation to introduce sa-turations in the loop. This stems from the fact that the action of the approximate dierentiation lter can also be regarded as a linearhigh gainobserver (see [11]), which may introduce very large values of the state estimate over a short period of time. In-troducing saturated controls, as rst proposed in [12], is a way to overcome this diculty. It should also be mentioned that dirty derivative lters with saturations are also used (among other various tools) in [13] for semiglobal output feedback sta-bilization of general nonlinear systems.

II. Problem Formulation and Background A. Problem Formulation

We consider in this paper plants described by Euler-Lagrange equations (in short, EL systems) of the form

d dt @Lp(q p;q_p) @q_p ? @Lp(q p;q_p) @qp =up ? @Fp( _q p) @q_p (1) where qp 2 IR

n are the generalized coordinates

as-sumed measurable, up

2 IR

n are the control inputs, L

p(qp;q_p) 4

=Tp(qp;q_p) ?V

p(qp) is the Lagrangian function,

Tp(qp;q_p) = 12q_ > pD p(qp) _qp; Dp(qp) = D > p(q p) > 0 is the

ki-netic energy and Vp(qp) is the potential energy which we

as-sume is twice dierentiable and bounded from below. Further, we assume that there exist some constants 0 < kg <

1 and 0< kv< 1such that kv 4 = sup q p 2IR n @Vp(qp) @qp (2) kg 4 = sup q p 2IR n @2_V p(qp) @qp i@q p j (3)

withkkthe Euclidean norm. F

p( _qp) is the Rayleigh dissipation

function which satises _ q> p @F p( _qp) @q_p pkq_pk 2 ₍₄₎ for all _qp 2 IR n with some p 0, and @Fp @qp_ (0) = 0. It is

important to remark that EL systems are fully characterized by theirEL parameters1: p: fT p(qp;q_p); Vp(qp); Fp( _q p) g:

In this paper we are interested in the problem of global asymptotic output feedback stabilization of EL systems (1) sub-ject toinput constraints

ju pi ju max p i ; i= 1;:::;n: (5)

In particular, we want to extend the design methodology pro-posed in [1] to the important bounded input case.

B. Background

The following facts, further elaborated in [1], are in order:

Fact 1(Proposition 2.1 of [1]).

EL systems denepassive operators[14] from the inputsup to 1Notice that, in contrast to [1] we consider here onlyfully actuatedEL systems

(i.e., number of degrees of freedom equal to the number of control actions). Therefore, only three EL parameters are needed.

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the generalized velocities _qp. This follows from integration of

the key energy balance equation _ Hp(t) = _q > pu p ?q_ > p @F p( _qp) @q_p and (4), where Hp(qp;q_p) 4 =Tp(qp;q_p) +Vp(qp) is the systems total energy. 222 Fact 2.

Let thecontrollerbe also an EL system with EL parameters c: fT c(qc;q_c); Vc(qc;qp); Fc( _q c) g

with generalized coordinates qc 2 IR

n. For simplicity we

con-sider here Tc(qc;q_c) = 12q_ > cD c(qc) _qc; Dc(qc) =D > c(q c) >0

al-though as explained in [1], with an obvious abuse of notation, we can also takeTc(qc;q_c)

0. We assumeV

c(qc;qp) is bounded

from below and @F c

@qc_ (0) = 0. Notice the dependence ofV c(qc;qp)

on the plant coordinatesqp.

The controller dynamics is given by

d dt @Tc(qc;q_c) @q_c ? @Tc(qc;q_c) @qc +@Vc(qc;qp) @qc +@F c( _qc) @q_c = 0 (6) while the feedback interconnection between plant and controller is established by up= ? @Vc(qc;qp) @qp : (7)

Then, the closed loop is also an EL system with gen-eralized coordinates q = [q> p;q > c] > and EL parameters fT(q;q_);V(q);F( _q)gdened as T(q;q_)4 =Tp(qp;q_p) +Tc(qc;q_c); V(q) 4 =Vp(qp) +Vc(qc;qp); F(_q) 4 =Fp( _q p) + Fc( _q c): 222 Fact 3(Proposition 2.3 of [1]).

The equilibria of the closed loop system (q;q_) = (q;0) are uniquely determined by the potential energy as the solutions

q of @V(q)

@q = 0. The equilibrium is unique and stableif q is

anuniqueandglobalminimum ofV(q) (see denitions B.3 and B.4 of Appendix B). Further, it is globally asymptotically stable (GAS) if _ q> c @F c( _qc) @q_c c kq_ c k 2 ₍₈₎

for some c >0, and the function @V(q)

@q

c = 0 has only isolated zerosinqpfor each givenqc.

222

Our motivation to consider EL controllers for regulation of EL systems stems from the facts above, and may be summa-rized as follows: Since the feedback interconnection of two EL systems is an EL system, and the dynamic behaviour of an EL system is fully characterized by the EL parameters, we propose to choose somedesiredclosed loop EL parameters and obtain from them the EL controller. In particular, forregulationtasks we chooseV(q) to have an unique and global minimum at the desired equilibrium, and pick Fc( _q

c) to inject the damping

re-quired for asymptotic stability.

Remark 4. From the discussion above it is clear that the suc-cess of our approach hinges upon our ability to \dominate" the plant potential energy Vp(qp) withVc(qc;qp). Now, since upis

restricted by (5), this implies {via (7){ a bound on @V c( q c ;q p) @qp ,

which in its turn imposes a bound on the growth rate ofVp(qp).

This restriction onVp(qp) is inherent to the energy shaping

ap-proach taken here.

Remark 5. Since we are dealing with fully actuated systems the simplest way to \dominate" the plant potential energy is to cancel Vp(qp) and impose a desired shape to the closed loop

function. This however might entail some potential robustness problems, hence we favour a solution that does not rely on this cancellation. Interestingly enough, if we use a controller that does not cancel the vector of potential forces, the growth rate restriction on Vp(qp) mentioned above is imposed only at the

desired position. The price paid, however, is that in this case we need to use high gains inVc(qc;qp) to dominateVp(qp) and

this translates into stier requirements on the input saturation bound.

Instrumental for the solution of the stabilization problem is the utilization ofsaturation functionswhich we dene as follows

Denition 6. A saturation function sat(x) : IR! IRis a C

2

strictly increasing function2 _{that satises}

1. sat(0) = 0, 2. jsat(x)j<1, 3. @ 2 sat( x) @x 2 6 = 0 8x6= 02IR

For instance, we can take sat(x) 4

= tanh(!x); ! >0, as pro-posed in [9], [15].

It can be proven that saturation functions dened as in 6

satisfy the following properties, required for our further devel-opments. P1 R~ qp i 0 sat(x)dx12 sat(~q p i)~q p i 8q~ p i 2IR P2 For all" >0 we have that

sat(~qp i)~q p i sat(") " q~p2 i 8jq~ p i j< " (9) sat(~qp i)~q p i sat(")jq~ p i j 8jq~ p i j": (10)

III. Main Result

Proposition 7. Consider the EL system (1) with saturated inputs (5), measurableoutput qp, and a constantdesired

reference valueqpd 2IR

n.

(i) (Controllers with cancellation of potential forces). Assume that the systems potential energy veries the strict inequality sup q p 2IR n @Vp(qp) @qp i < u max p i ; 8i2[1;:::;n] (11) with ()

itheithcomponent of the vector. Under these

con-ditions, any EL controller (6), (7) with dissipation function

Fc( _q

c) satisfying (8) and potential energy

Vc(qc;qp) =Vc 2(q

c;qp) ?V

p(qp) 2As will become clear later, the strict qualier for sat(

x) is required to insure @ V(q )

@ qc = 0 has onlyisolated zerosin

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where Vc2(qc;qp) 4 = n X i=1 k2i bi Z ( qc i+ b i qp i) 0 sat(xi)dxi +k3i Z ~ qp i 0 sat(xi)dxi (12) where ~qp 4 =qp ?q p d,b i>0, andk2 i; k3i >0 suciently small, makes ( _qp;qp;q_c;qc) = (0;qpd;0;qcd) (13)

with qcd some constant, a GAS equilibrium point of the

closed loop.

(ii) (Controllers without cancellation of potential forces). Choose now the potential energy of the EL controller as

Vc(qc;qp) =Vc2(qc;qp) ?q > p @Vp @qp (qpd) (14) where Vc 2(q c;qp) is given by (12), we takek2 i suciently

small and mini

fk3ig> k

min 3i withk

min

3i some suitably

de-ned positive constant. Then (13) is a GAS equilibrium point of the closed loop (1), (5) { (7) provided3

umax p i > @Vp @qp (qpd) i +k3 i; i 2[1;:::;n]: (15)

In particular, if we take sat(x) = tanh(x) then

kmin₃ i 4 = 4kv tanh 4kv k g (16)

wherekvandkg are given by (2) and (3) respectively. 222 Remark 8. The proposition above characterizes, {in terms of the EL parameters Tc(qc;q_c);

Fc( _q

c) and Vc(qc;qp) {, a class

of output feedback GAS controllers for EL systems with satu-rated inputs. Thus, providing an extension to the constrained input case, of the result reported in [1]. Also, as a corollary of our proposition, we obtain an extension to theoutputfeedback case of the result in [9] where a full statefeedback solution to the problem of global regulation of rigid robots with saturated inputs was presented.

Remark 9. A key feature of the controller given in part (ii)

of Proposition 7 is that, to enhance its robustness, we avoid

explicit cancellations of the plant dynamics. In this respect our result supersedes that of [10] where the following GAS con-troller, which relies on exact cancellation of potential forces, was proposed _ qc i = ?k1 iq c i ?k2 i sat(q c i ?q~ p i) up i = k2i sat(q c i ?q~ p i) + @Vp(qp) @qp i

wherek2i >0 andk2i is taken suciently small. Remark that

this is an EL controller with EL parameters

Tc(qc;q_c) = 0; Fc( _q

c) = 12 kq_ck

2

3Notice that the gradient of the systems potential energy is evaluated hereat

the desiredreference.

Vc(qc;qp) = n X i=1 1 2k1iq 2 ci ?V p(qp)+ n X i=1 k2i (qc i ?qp~ i) 0 sat(xi)dxi: Remark 10. The price paid for the robustness enhancement in

(ii)is that higher gains have to be injected into the loop through

k3i. As seen from the proposition, this imposes an additional

requirement of suciently large input constraints for stability. The condition on k3i stems from the fact that, to impose a

desired minimum point to the closed loop potential energy, now we have to dominate (and not to cancel) the systems potential energy.

IV. Proof of main results

For ease of presentation we will consider for both parts of the proposition the EL controller parameters

Tc(qc;q_c) = 0; F c( _qc) = 12 _q > cK2B ?1A?1q_ c withA 4 = diagfa i g>0; B 4 = diagfb i g>0; K2 4 = diagfk2 i g>

0. This choice together with equations (6) and (7) yield the EL controller _ qc i= ?a isat(qc i+b iqp i) upi= ? k2i ai _ qci ?k3 i sat(~q pi) + @Vp(qp) @qp i (17) for the rst part of the proposition, while for the second con-troller we only changeup

i to upi = ?k 2i ai _ qci ?k3 i sat(~q pi) + @Vp @qp (qpd) i : (18) Notice the dierence in the third right hand terms of the control signals.

It is interesting to remark that if in the controllers above we write () instead of sat() weexactly recoverthe (approximate

dierentiation) output feedback GAS controllers of [8], see also [16]. The proposition then shows that by simply including the saturations we can preserve GAS even under input constraints. As pointed out inFact 2of section 1 the closed loop system is an EL system with potential energyV(q) =Vc(qc;qp) +Vp(qp),

hence the proof of both results is carried out by proving the conditions of Fact 3. Notice that the diculty lies in proving that V(q) has a global minimum at (13) with qcd

4

=?B ?1q

pd.

For this we use the following fact4_.

Fact 11. Letf(x) :IRn !IRbe aC1 function. Assume, 1. f(x)>0, for allx6= 02IR n andf(0) = 0 2. @f @x(x) >0, for allx 6 = 02IR n 3. f(x)!1ifkxk!1.

Then the function f(x) is globally positive denite and radially unbounded with an unique and global minimum atx= 0 according to denitions of appendix B.

222 Remark 12. Condition 1 implies that f(x) is positive denite with 0, a strict global minimum. Nevertheless, it is important to remark that this condition alone does not imply the unique-ness of the minimum. Condition 2 implies that 0 is the only critical point, hence that 0 is also an unique minimum off(x). Finally, the third condition corresponds to the denition of ra-dially unboundedness.

4It is worth mentioning that this fact is a generalization of the ideas exposed

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A. Proof of (i)

Notice rst that for this simpler case we have that V(q) =

Vc 2(q

c;qp). Using properties P1andP2it is easy to show that

this function satises all conditions of Fact 11above5_{. To}

es-tablish that the equilibrium is GAS we use again Fact 3and notice that for eachqc, the function

@V(q)

@qc

=K2B?1

sat(qc+Bqp)

has onlyisolated zerosinqp (sinceB is full rank).

Now, applying the triangle inequality to (17), and using the fact thatjsat(x)j<1, we have the bound

ju p i j< k2 i+k3i+ @Vp(qp) @qp i :

Thus, under assumption (11), we can always choose suciently small k2i; k3i >0 such that (5) holds.

222 B. Proof of (ii)

The closed loop potential energy is now

V(q) =Vc 2(q c;qp) +Vp(qp) ?q > p @Vp @qp (qpd): (19)

In Appendix I we show that, if there exists a kv such that

condition (2) there existskmin₃

i >0 such thatV(q) has a global

minimum at the desired equilibrium for all k3i kmin3

i . The

proof is completed observing that GAS follows along the lines of

(i)

and using (18) to get the bound

ju p i j< k2 i+k3i+ @Vp @qp (qpd) i :

From (15) it is easy to see that for a suciently small k2i,

(5) is satised. Finally, the value of kmin₃

i for the case when

sat(x) = tanh(x) is obtained inAppendixI.

222 V. Simulation results

Using SimulinkTM of

MatlabTM, we tested our algorithm

in the two link robot arm of [19] with a desired reference

qd = col[=2; =2]. We have imposed the input constraint

umax

pi = 320[Nm] in (5). To meet the conditions of

proposi-tion 7 we chose A = diagf100; 100g, B = diagf130; 130g

while the controller gains were set to K3 = diagf180; 180g,

K2= diagf125; 125gaccording to (15).

Then, in order to evaluate the performance of our con-troller, we tested as well the one proposed in [8] with ex-actly the same gain values and starting from initial conditions

qp

0 = col[=4; =4], and in accordance with the previous

dis-cussions we set qc0 = col[

?32:5; ?32:5] in order to make

#0 = 0.

In gure 1 we show the transient of the rst link position using the algorithm of [8], i.e. thenonsaturated controller and the control input signal yielded by this controller. In gure 2 we show the response of the same link driven by the saturated controller of proposition 7 and its control input.

On one hand, notice that the transient produced by the non saturated controller is much faster than the response using satu-rated controls. On the other hand, it must be remarked that the

5The proof of this claim is strictly contained in the proof of (ii). See appendix

A. 0 1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 3 3.5 Time [s] [rads]

First link response and reference

0 1 2 3 4 5 6 7 8 0 100 200 300 400 500 600 Time [s] [Nm]

NON saturated control input

<−− 503

Fig.1. ELControllerofKelly[8].

0 1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 3 3.5

First link response and reference

Time [s] [rads] 0 1 2 3 4 5 6 7 8 −100 0 100 200 300 Time [s] [Nm]

Saturated control input

<−− 216.65

Fig.2. SaturatedELcontroller.

control input yielded by the linear controller fails to satisfy the input constraint; in particular the maximum absolute value of

upis 503[Nm] for the rst link. In contrast to this the saturated

controller yields a control input withju pi

jmax= 211[Nm].

Thus we verify what is not surprising: that there is a com-promise between a fast transient and small control inputs.

VI. Concluding remarks

We have extended in this paper our results on output feed-back stabilization of EL systems to the practically important case of bounded inputs. As a corollary of our work we have improved in several directions the result of [9] on state feed-back global stabilization of robot manipulators with saturated inputs. First, we have removed the requirement of measurement of generalized velocities. In particular we have shown, that a suitably saturated approximate dierentiator can be used to es-timate these signals. Second, we have developed this theory for a wider class of EL systems, { that contains as a particular case

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robot manipulators {, and to general saturation functions. Fi-nally, we have identied a class of EL controllers, characterized in terms of their EL parameters, that achieve the global output feedback stabilization objective. A subject of current research is to devise a technique to select from this class one that optimizes a performance criterion.

It might be argued that limiting ourselves to EL controllers is unnecessarily restrictive. Our motivation for this choice is twofold: rst by preserving the EL structure of the closed loop the behaviour of the system is fully characterized by its energy and dissipation functions. From a Lyapunov perspective, this procedure naturally leads to the choice of an energy-based Lya-punov function candidate. Second, we have shown in [1] that the class of EL controllers contains several well known schemes which were derived from apparently unrelated perspectives, e.g. [11], [20], [4]. Henceforth, we provide a unied framework to compare them.

Finally we have illustrated in simulations the practical advan-tages of using bounded controls. In particular we compared the performance of the controller of [8] and its \saturated" equiv-alent. It was shown that for a particular case with non-zero initial conditions, the linear controller of [8] failed to satisfy the input constraints.

Acknowledgments

The authors would like to thank Ilya Burkov for providing them with early copies of his interesting paper [10], and grate-fully acknowledge Prof. Henk Nijmeijer for many fruitful dis-cussions in the topic of the present work.

References

[1] R. Ortega, A. Loria, R. Kelly, and L. Praly, \On passivity-based output feedback global stabilization of Euler-Lagrange systems,"Int. J. Robust and Nonlinear Control,special issue onControl of nonlinear mechanical systems, vol. 5, no. 4, pp. 313{325, 1995. H. Nijmeijer and J. van der Schaft eds. [2] M. Takegaki and S. Arimoto, \A new feedback method for dynamic control

of manipulators,"ASME J. Dyn. Syst. Meas. Contr., vol. 103, pp. 119{125, 1981.

[3] R. Ortega and M. Spong, \Adaptive motion control of rigid robots: A tutorial,"Automatica, vol. 25-6, pp. 877{888, 1989.

[4] A. Ailon and R. Ortega, \An observer-based set-point controller for robot manipulators with exible joints,"Syst. Contr. Letters, vol. 21, pp. 329{ 335, 1993.

[5] G. Espinosa and R. Ortega, \State observers are unnecessary for induction motor control,"Syst. Contr. Letters, vol. 23, no. 5, pp. 315{323, 1994. [6] H. Sira-Ramirez and R. Ortega, \Passivity-based control of DC to DC

converters,"Automatica, 1996. To appear.

[7] C. I. Byrnes and C. F. Martin, \An integral-invariance principle for non-linear systems,"IEEE Trans. Automat. Contr., vol. 40, pp. 983{994, June 1995.

[8] R. Kelly, \A simple set{point robot controller by using only position mea-surements," in Proc. 12th. IFAC World Congress, vol. 6, (Sydney, Aus-tralia), pp. 173{176, 1993.

[9] R. Kelly, V. Santiba~nez, and H. Berghuis, \Global regulation for robot manipulators under actuator constraints," tech. rep., CICESE, Ensenada, BC, Mexico., 1994.

[10] I. V. Burkov, \Stabilization of mechanical systems via bounded control and without velocity measurement," inProc. 2nd Russian-Swedish Control Conf., (St. Petersburg, Russia), pp. 37 { 41, 1995.

[11] H. Berghuis and H. Nijmeijer, \Global regulation of robots using only po-sition measurements,"Syst. Contr. Letters, vol. 21, pp. 289{293, 1993. [12] F. Esfandiari and H. K. Khalil, \Output feedback stabilization of fully

linearizable systems,"International Journal of Control, vol. 56, pp. 107{ 1037, 1992.

[13] A. Teel and L. Praly, \Tools for semiglobal stabilization by partial state and output feedback,"SIAM J. Contr. Opt., vol. 33, no. 5, pp. 1443{1488, 1995.

[14] C. Desoer and M. Vidyasagar,Feedback Systems: Input-Output Properties. New York: Academic Press, 1975.

[15] L. Cai and G. Song, \A smooth robust nonlinear controller for robot ma-nipulators with joint stick-slip friction," inProc. IEEE Conf. Robotics Au-tomat., (Atlanta, GA), pp. 449 { 454, 1993.

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Appendix

I. Proof of positive definiteness ofV(q)

First, notice that the rst right hand term of (19) is a non-negative function ofqc;qpwhich is zero atqc=B

?1q

p. Hence,

to prove that (19) has an unique global minimum at (13) it suces to show that the three remaining terms (i.e., the sec-ond term ofVc2 and the last two terms of (19)) have an unique

global minimum atqp=qpd, or equivalently, that the function

f(~qp) :IR n !IR; f(~qp) 4 = n X i=1 k3i Z ~ qp i 0 sat(xi)dxi +Vp(~qp+qpd) ? Vp(qpd) ?q~ > p @Vp @qp (qpd) (20)

has a global and unique minimum at zero. We will establish the proof verifying the conditions ofFact 11in order to dene akmin₃

i that insures this to be the case. Condition 1

. To prove thatf(x)>0 for allx6= 02IR

nwe shall prove

rst that for all" >0,

n X i=1 k3i Z ~ q p i 0 sat(x)dxmin i fk3 i g sat(") 2" kq~ p k 2 8kq~ p k< " (21) n X i=1 k3i Z ~ q p i 0 sat(x)dxmin i fk3 i g sat(") 2 kq~ p k 8kq~ p k" (22) where for the sake of clarity we consider two cases separately:

Case 1: kq~pk< ".Notice that in this case we have thatjq~ p

i j<

"8in, then usingP1and (9) we get n X i=1 k3i Z ~ qp i 0 sat(x)dx n X i=1 k3i sat(") 2" q~2p i min i fk3 i g sat(") 2" kq~ p k 2 Case 2: kq~ p

k ". Within this case we shall consider three

dierent cases:

case a: jq~ pi

j< " 8in

Again, usingP1and (9) we get

n X i=1 k3i Z ~ q p i 0 sat(x)dxmin i fk3 i g sat(") 2" kq~ p k2

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case b: jq~ p

i

j" 8in

FromP1and (10) notice that

n X i=1 k3i Z ~ q p i 0 sat(x)dx n X i=1 k3i sat(") 2 jq~ p i j min i fk3ig sat(") 2 n X i=1 jq~ p i j

Then (22) easily follows observing thatkq~ p k P n i=1 jq~ pi j. case c: jq~ p i j"; jq~ p j j< " 8i;jn; i6=j

Without loss of generality we can take i n=2 and 1

j < n=2, then a simple analysis along the lines of cases a and b, shows that (22) holds as well in this case.

Now we prove that for all" >0 there exist1(");2(") 2IR

such that Vp(qp) ?V p(qp d) ?q~ > p @V p( q p) @qp (q p d) 1kq~pk 2 8kq~pk< " 2kq~ p k 8kq~ p k": (23) On one hand, using (3) it can be proven that

Vp(qp) ?V p(qp d) ?q~ > p @Vp(qp) @qp (qp d) ?k g 2kq~ p k 2_;

and on the other, invoking the Mean Value Theorem we have that92IR nsuch that Vp(qp d) ?V p(qp) = @Vp(qp) @qp () (qp d ?q p) kvkq p d ?qpk

then using (2) we can write

Vp(qp) ?V p(qp d) ? @Vp(qp) @qp (qp d) > ~ qp ?2kvkq~pk:

We nally conclude from (21), (22), and (23) that

f(~qp) 8 < : mini fk3 i g sat (") 2" ? kg 2 kq~ p k2 8kq~ p k< " mini fk3 i g sat (") 2 ?2k v kq~ p k 8kq~ p k": (24) The proof of condition 1 is completed observing that (24) happens to hold provided (27) is satised.

Condition 2

Taking the partial derivative off(~q) we get

@f @q~p (~qp) = K3 2 6 6 4 sat(~qp 1) sat(~qp 2) ... sat(~qpn) 3 7 7 5+ @Vp @q~p (~qp) ?@V p @qp (qpd): (25)

Now, taking the norm and using the triangle inequality we ob-tain @f @q~p (~qp) K3 2 6 6 4 sat(~qp 1) sat(~qp 2) ... sat(~qp n) 3 7 7 5 ? @Vp @q~p (~qp) ? @Vp @qp (qpd) : (26) On one hand, from (3), (2) and using the Mean Value Theorem we have that for all " >0

? @Vp @q~p (~qp) ? @Vp @qp (qpd) ?k g kq~ p k ifkq~ p k< " ?2k v if kq~pk"

and on the other hand since K3 is diagonal and using P2, we

get K3 2 6 6 4 sat(~qp 1) sat(~qp 2) ... sat(~qpn) 3 7 7 5 mini fk3 i g sat (") " kq~pk ifkq~pk< " mini fk3 i g sat(") ifkq~pk":

Thus, we are able to write

@f @q~p (~qp) ( h mini fk3ig sat (") " ?k g i kq~pk ifkq~pk< " [mini fk3ig sat(")?2k v] if kq~pk":

From here it's easy to see that condition 2 is satised provided min i fk3 i gk min 3i >max "kg sat("); 4kv sat(") (27) holds with"as in P2,kg andkvdened by (2) and (3)

respec-tively.

The proof is completed observing from (23) thatf(~qp) is

ra-dially unbounded.

Notice that in the case of sat(x) = tanh(x) we have thatP2

is true with6 _"4

= 2k v

kg , then (16) immediately follows.

222 Remark.It is worth remarking that the proof for the simpler case when V(qp;qc) = Vc2(qc;qp) is strictly contained in the

proof above. See (21) and (22).

II. Definitions

For the sake of clarity we recall below some denitions bor-rowed from [21] and [22].

Let f :IRn