On the Stabilization of the Flexible Manipulator

Liapunov based Design. Robustness.

Daniela Danciu, Dan Popescu and Vladimir R˘asvan

Department of Automation, Mechatronics and Electronics, University of Craiova,

A.I. Cuza str. No. 13, Craiova, RO-200585, Romania

Keywords:

Hyperbolic Partial Differential equations, Hamilton Variational Principle, Energy Identity, Liapunov Energy

Functional, Feedback Stabilization.

Abstract:

This work deals with dynamics and control of the ﬂexible manipulator viewed as a system with distributed

parameters. It is in fact described by a mixed problem (with initial and boundary conditions) for a hyperbolic

partial differential equation, the ﬂexible manipulator being assimilated to a rod. As a consequence of the

deduction of the model via the variational principle of Hamilton from Rational Mechanics, the boundary con-

ditions result as “derivative” in the sense that they contain time derivatives of higher order (in comparison with

the standard Neumann or Robin type ones). To the controlled model there is associated a control Liapunov

functional by using the energy identity which is well known in the theory of partial differential equations.

Using this functional the boundary stabilizing controller is synthesized; this controller ensures high precision

positioning and additional boundary damping. All this synthesis may remain at the formal level, mathemati-

cally speaking. The rigorous results are obtained by using a one to one correspondence between the solutions

of the boundary value problem and of an associated system of functional differential equations of neutral type.

This association allows to prove in a rigorous way existence, uniqueness and well posedness. Moreover, in

several cases there is obtained global asymptotic stability which is robust with respect to the class of nonlinear

controllers - being in fact absolute stability. The paper ends with conclusions and by pointing out possible

extensions of the results.

1 INTRODUCTION

A. In order to start a motivation for the present paper

and the reported research, we shall reproduce from

a survey paper in the ﬁeld (Dwivedy and Eberhard,

2006): “Robotic manipulators are widely used to help

in dangerous, monotonous, and tedious jobs. Most

of the existing robotic manipulators are designed and

build in a manner to maximize stiffness in an at-

tempt to minimize the vibration of the end-effector to

achieve a good position accuracy. This high stiffness

is achieved by using heavy material and a bulky de-

sign. Hence, the existing heavy rigid manipulators are

shown to be inefﬁcient in terms of power consumption

or speed with respect to the operating payload. Also,

the operation of high precision robots is severely lim-

ited by their dynamic deﬂection, which persists for a

period of time after a move is completed. The settling

time required for this residual vibration delays sub-

sequent operations, thus conﬂicting with the demand

of increased productivity. These conﬂicting require-

ments between high speed and high accuracy have

rendered the robotic assembly task a challenging re-

search problem. Also, many industrial manipulators

face the problem of arm vibrations during high speed

motion.

In order to improveindustrial productivity, it is re-

quired to reduce the weight of the arms and/or to in-

crease their speed of operation. For these purposes

it is very desirable to build ﬂexible robotic manipula-

tors. Compared to the conventional heavy and bulky

robots, ﬂexible link manipulators have the potential

advantage of lower cost, larger work volume, higher

operational speed, greater payload-to-manipulator-

weight ratio, smaller actuators, lower energy con-

sumption, better maneuverability, better transporta-

bility and safer operation due to reduced inertia. But

the greatest disadvantage of these manipulators is the

vibration problem due to low stiffness.”

In terms of Dynamical Systems, vibration quench-

ing is tightly connected with the basic problem of

stability and stabilization; both stabilization and vi-

bration quenching are achieved by feedback control

if,especially increased stiffness is to be avoided. At its

turn a good stabilizing structure may be achieved pro-

vided a sound mathematical model of the dynamics

508

Danciu D., Popescu D. and Rasvan V..

On the Stabilization of the Flexible Manipulator - Liapunov based Design. Robustness..

DOI: 10.5220/0005051705080518

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 508-518

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

is available. Or, the ﬂexible manipulator belongs to

the class of the controlled objects with distributed pa-

rameters having one space dimension for parameter

distribution. More speciﬁc, reduced stiffness results

in blocking the possibility of neglecting this param-

eter distribution along the length of the manipulator

arm. Consequently, the modeling of the manipulator

arm is made by assimilating it to a rod/beam. With

respect to this we would like to point out another sur-

vey (Russell, 1986), where various “energy conser-

vative and dissipative beam models” are described:

Euler- Bernoulli model, Rayleigh model, Timoshenko

model, models with Kelvin-Voight dissipation. If the

models are deduced using the generalized variational

principle of Hamilton, then considering one model or

another depends on the expressions adopted for the

potential energy and various forces involved in mod-

eling.

B. The Hamilton approach in modeling beams for

ﬂexible robot manipulators as well as for other en-

gineering devices (R˘asvan, 2014) leads to somehow

unusual mathematical structures i.e. initial bound-

ary value problems for hyperbolic partial differential

equations with derivative boundary conditions. From

these models it is obvious that the natural damping of

the systems consists of the quite weak distributed and

boundary dampings; moreover, stability improvement

which is in close connection with vibration quenching

has to be achieved by feedback control. At its turn

this feedback may be space distributed or boundary

or combined.

A rather widespread method for control synthesis

is now the control Liapunov function approach. For

the aims of this paper a good reference survey might

be (de Queiroz et al., 2000). As it is known, ﬁnd-

ing a suitable Liapunov function(al) implies “guess-

ing”. With respect to this the energy identity repre-

sents a very useful hint in ﬁnding a good Liapunov

function which allows synthesis of a stabilizing con-

troller. Since this synthesis may be performed at the

formal level (from the mathematical point of view),

the closed loop structure should be considered as a

mathematical problem an und f

ur sich (in itself) and

treated as such (existence, uniqueness, well posed-

ness, stability).

Following some references e.g. (Icart et al., 1992;

Cherkaoui and Conrad, 1992; de Queiroz et al., 2000)

we shall consider here the rod model for the ﬂexi-

ble manipulator which coincides basically with the vi-

brating string equation. For this model we shall per-

form the synthesis of the stabilizing controllers, ob-

tain the closed loop model and discuss the theory it.

As a consequence what is left of this paper is orga-

nized as follows. First the corresponding expressions

for the kinetic and potential energies will be consid-

ered allowing to obtain exactly the rod model via the

Hamilton principle. The involved external or virtual

(accounting for constraints) forces are listed in order

to obtain the controlled mathematical model. To it we

associate the energy identity which will turn useful

for controller synthesis. Once the closed loop model

will be obtained, a section of the paper will be allo-

cated to the basic theory. The stability discussion will

be followed by the robust control issues and robust-

ness properties. Finally a section of conclusions and

hints for future research will end the exposition.

2 THE MATHEMATICAL MODEL

AND THE ENERGY IDENTITY

A. We shall consider the ﬂexible manipulator carrying

a payload and being rotated through a hub, as sug-

gested by (de Queiroz et al., 2000) and illustrated in

Figure 1

)(tu

( , )v s t

)(t

Figure 1: Flexible manipulator dynamics: s - current coor-

dinate on the ﬂexible arm; ν(s,t) - current local deﬂection of

the ﬂexible arm element including with respect to the inertia

axis of the rotating hub; J

- mass of the torque controlled

rotating hub; m - payload mass; u(t) - the control force at

the payload; τ(t) - the control torque at the hub.

In the following we reproduce in brief the model

deduction of (R˘asvan, 2014). In order to make use of

the variational principle of Hamilton we write down

The kinetic energy of the controlled hub with the

moment of inertia J

, the ﬂexible arm and the

payload mass m, given by

(t) =



θ(t)



+ m



y(L,t)



ρ(s)y

(s,t)ds



(1)

OntheStabilizationoftheFlexibleManipulator-LiapunovbasedDesign.Robustness.

509

where y(s,t) is the current position of the moving

ﬂexible cable at the local coordinate s (including

the elastic deﬂection) and at the moment of time

t, ρ(s) being the local mass density at the local

coordinate s. We shall have also

y(s,t) = sθ(t) + ν(s,t) (2)

where θ is angular position of the rotating hub and

ν(s,t) is position with respect to the rotating sys-

tem of coordinates

The potential energy due to the strain energy of

the ﬂexible cable given by

(t) =

T(s)ν

(s,t)ds (3)

where T(s) is the tension of the ﬂexible link of the

arm.Remark that (2) implies ν

(s,t) ≡ y

(s,t) −

θ(t).

The external “forces” acting on this mechanical

system are the following: the control torque τ(t),

the thrust regulating force applied at the payload

boundary, possible local and distributed perturba-

tions. Even if they might be negligible, we shall

include also the friction forces, in order to keep

the model structure as close as possible to the

most general case (R˘asvan, 2014). The work of

these forces is given by

(t) = (τ(t) + φ

(t) + χ

(t))θ(t)+

+(u(t) + f

(t) + χ

(t))y(L,t)+

( f(s,t) + χ(s,t))y(s,t)ds

(4)

Here χ

(t), χ

(t), χ(s,t) are the virtual forces ac-

counting for viscous damping forces given by

(t) = −c

θ(t) , χ

(t) = −c

(L,t) ,

χ(s,t) = −c(s)y

(s,t)

(5)

Deﬁne the functional

I(t

) :=

(t) −E

(t) +W

(t))dt (6)

and, following the approach of the variational calcu-

lus (Akhiezer, 1981), introduce the following varia-

tions

y(s,t) = ¯y(s,t) + εη(s,t) (7)

where ¯y(s,t) corresponds to an extremal. Let I

)

be the functional (6) written along the variations (7).

The necessary condition for the extremum is given by

′

(0) =

dε



ε=0

= 0 (8)

After some standard manipulation (R˘asvan, 2014) we

ﬁnd the following model boundary value problem

−ρ(s)y

−c(s)y

+ (T(s)(y

−θ))

+ f(s,t) = 0

(0,t) = θ(t)

−J

θ−c

θ+

T(s)(y

(s,t) −θ)ds+

+τ(t) + φ

(t) = 0

−my

(L,t) −c

(L,t) −T(L)(y

(L,t) −θ)+

(t) + d

(t) = 0

(9)

The model is somehow nonstandard due to the fact

that it contains both linear and angular motion coor-

dinates. However, if the ﬂexible rod material has ho-

mogeneous properties i.e. ρ(s), c(s) but especially

T(s) are constant i.e. independent of the space coor-

dinate then equations (9) become closer to the tradi-

tional ones, as follows

−ρy

−cy

+ Ty

+ f(s,t) = 0

(0,t) = θ

−J

θ−c

θ−TLθ+ T(y(L,t) −y(0,t))+

+τ(t) + φ

(t) = 0

−my

(L,t) −c

(L,t) + Tθ−Ty

(L,t)+

(t) + d

(t) = 0

(10)

The model is clearly described by a boundary value

problem for the string equation, the boundary condi-

tions being derivative.It is worth mentioning that in

standard cases c - the distributed damping - is negligi-

ble and there is no thrust control for the payload mass

at s = L i.e. u

(t) ≡0.

In the following we shall write down the so called

energy identity for (9): we multiply the ﬁrst equation

by y

and perform some integration by parts with re-

spect to s from 0 to L and take into account the bound-

ary conditions to obtain



(

θ(t))

+ m(y

(L,t))

[ρ(s)y

(s,t)

+ T(s)(y

(s,t) −θ(t))

]ds



(

θ(t))

+ c

(L,t))

c(s)y

(s,t)

ds−

−(τ(t) + φ

(t))

θ(t)−

−(u

(t) + f

(t))y

(L,t) −

f(s,t)y

(s,t)ds ≡ 0

(11)

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510

We shall end this subsection by some additional

comments and explanations concerning the premises

of the obtained model. Our starting references had

been at the beginning (Icart et al., 1992; Cherkaoui

and Conrad, 1992); the model used there appeared

to be more of pure mathematical interest while for

the stabilizing control displayed some drawbacks (at

least with respect to our purpose - “guess” and use of

a “natural” control Liapunov function); on the other

hand, the analogy of several models arising from var-

ious ﬁelds (overhead crane and marine riser, oilwell

drillstring) turn to be stimulating for seeking for an-

other model - obtained from physical premises. The

reference (de Queiroz et al., 2000) contained such

uniﬁed models under the framework of the control

Liapunov functionals and the model of the ﬂexible

arm was adopted but based on the Hamilton princi-

ple. As mentioned in (de Queiroz et al., 2000), the

model there strongly relies on the model in (Junk-

ins and Kim, 1993). It must be mentioned however

that the models of the aforementioned reference were

obtained mainly for aeronautical structures and the

physics might have been different e.g. centrifugal

effects of stiffening and softening. Neglecting such

“side effects” led e.g. to formula (2). There are also

other speciﬁc simplifying approaches in our model

but we discuss only the choice of the arm modeling as

a rod what gavethe string equation while there existed

other options as described in (Russell, 1986) among

which the Euler Bernoulli beam is mostly preferred.

Last but not least, we have been guided in our option

by the aim to obtain a rigorous ground for the bound-

ary value problems thus obtained: complicated mod-

els, due to their nonlinearities and/or discontinuities

are not easy to give a basic theory allowing to go be-

yond the formal level. It will become clear that even

our models cannot be treated in all cases and open

problems persist.

B. In the following we shall discuss the equilibria -

constant trajectories - of system (9) and their “inher-

ent stability”i.e. with blocked control signals; with re-

spect to this we shall consider all perturbations iden-

tically 0

f(s,t) ≡0 , φ

(t) ≡ 0 , d

(t) ≡0 (12)

and take τ(t) ≡

τ, u

(t) ≡ ¯u

. Letting the time deriva-

tives be identically 0 the following equations for the

steady state are obtained

(T(s)(¯y

−

θ))

≡ 0 ; ¯y

(0) −

θ = 0

T(s)( ¯y

(s) −

θ)ds+

τ = 0

−T(L)(¯y

(L) −

θ) + ¯u

= 0

(13)

to obtain ¯u

= 0,

τ = 0, ¯y(s) =

θs+ ¯y(0); taking into

account (2) and the signiﬁcance of ν(s,t) it follows

that ¯y(0) = 0 ; however

θ remains undetermined what

is quite natural since it is a cyclic coordinate.

The occurrence of a cyclic coordinate requires

a deeper investigation of the possible steady states.

With respect to this we introduce the so called coordi-

nates of the symmetric Friedrichs form for the partial

differential equations as follows

(s,t) := v(s,t) , T(s)(y

(s,t) −θ(t)) := w(s,t)

(14)

also

θ = Ω to obtain the following boundary value

problem

ρ(s)v

+ c(s)v−w

= f(s,t)

−T(s)(v

(s,t) −Ω(t)) = 0

w(0,t) = 0

Ω+ c

Ω−

w(s,t)ds = τ(t) + φ

(t)

(L,t) + c

v(L,t) + w(L,t) = u

(t) + d

(t)

(15)

Taking again identically zero perturbations f (s,t),

(t), d

(t) and blocked (constant) input signals

τ and

¯u

, a new type of equilibria is obtained, namely

¯v(s) =

Ωs+ ¯v(0) , ¯w(s) =

(Ωσ+ ¯v(0))c(σ)dσ

(16)

where ¯v(0) and Ω are the unique solution of a linear

system. If these steady state variables are non-zero

this would signify a uniformly rotating arm (!) with

some steady state elastic deformations unless we im-

pose again ¯u

= 0,

τ = 0. This is the only interesting

case form the engineering point of view.

In order to obtain some information about the sta-

bility of the equilibrium thus determined we shall ex-

amine the free system (15); this is a linear boundary

value problem and the stability of the identically zero

equilibrium might be examined using the Laplace

transform at least at the formal level since we have no

information about its admissibility for the solutions

of (15). Taking into account the space varying pa-

rameters and the quite complicated boundary condi-

tions, the associated transforms might be not easy to

tackle. But we have at our disposal the energy identity

(11) which suggests the following quadratic Liapunov

functional (written in the variables of (15))

V (X,Y,φ(·),ψ(·)) =



+ mY

(ρ(s)φ

(s) + ψ

(s))ds



(17)

deﬁned on R ×R ×L

(0,L) ×L

(0,L) with the re-

striction Y = φ(L). The energy identity also gives the

OntheStabilizationoftheFlexibleManipulator-LiapunovbasedDesign.Robustness.

511

expression of the derivativeof (17) along the solutions

of the free system (15) i.e. with u

(t) ≡ 0, τ(t) ≡ 0,

(t) ≡ 0, d

(t) ≡0, f(s,t) ≡ 0

V (Ω(t),v(L,t),v(·,t),w(·,t)) =

= −c

Ω

(t) −c

(L,t) −

c(s)v

(s,t)ds

(18)

Since V is positive deﬁnite and its derivative - nega-

tive semi-deﬁnite, the equilibrium at the origin is sta-

ble (Liapunov stable in the sense of the norm deﬁned

by the Liapunov functional (17) itself). The Liapunov

functional being an energy, it is only natural to obtain

a derivative functional which is negative semi-deﬁnite

only - this is well known for energy like Liapunov

functions arising in the early stages of the stability

theory. We may use the Barbashin Krasovskii LaSalle

invariance principle (at least in a formal way, but we

may hope to be in one of the cases when it holds for

abstract dynamical systems also (Saperstone, 1981))

and ﬁnd that the only invariant set where the kernel

of the derivative function is contained is the zero so-

lution. This zero solution corresponding to the arm

with the payload stopped in some resulting position is

thus stable.

This result (still at the formal level) suggests the

following remarks

i) the free system (15) has a quite non-robust stabil-

ity since it depends on the natural dampings only

and these dampings are very weak even negligible

in practice;

ii) the system having a cyclic variable, it is but nat-

ural to have the arm with the payload stopped in

an arbitrary position depending on the initial one;

however this is not acceptable in the practical ap-

plications.

The above considerations show that, in order to

ensure positioning of the payload and an improved

stability for the ﬂexible manipulator, a feedback con-

troller must be used. In the following we shall con-

sider controller synthesis and the properties of the

closed loop system.

3 CONTROLLER DESIGN. THE

RESULTING CLOSED LOOP

SYSTEM

We shall realize the controller design based on the

c.l.f. (control Liapunov function) method by mak-

ing use of a modiﬁed version of the functional (17)

taking into account the two aforementioned tasks of

the controller: precise positioning of the payload and

improved stability. For the positioning we shall add

to (17) a quadratic term penalizing the error from a

given angular reference θ

A. Refer ﬁrst to the basic controlled system (9); since

we discuss stabilization, the persistent perturbations

f(s,t), φ

(t), d

(t) are again set to zero. The Lia-

punov functional suggested by (11) is as follows

V (X,Y, Z, φ(·),ψ(·)) =



+ J

+ mZ

(ρ(s)φ

(s) + T(s)(ψ(s) −X)

)ds



(19)

where a

> 0 is a free design parameter; as men-

tioned, the term (ψ(·) − θ

)

is penalizing the de-

viation from the prescribed angular position of the

payload. The functional is deﬁned on R ×R ×R ×

(0,L) ×L

(0,L) with the restriction Z = φ(L). On

the domain of deﬁnition the functional is positive def-

inite. Along the solutions of (9) this functional reads

as follows

V (θ(t) −θ

θ(t), y

(L,t),y

(·,t),y

(·,t)) =



(θ(t) −θ

)

+ J

(t) + my

(L,t) +

(ρ(s)y

(s,t) + T(s)(y

(s,t) −θ(t))

)ds



(20)

Its derivative along the solutions has the following

form

V (θ(t) −θ

θ(t),y

(L,t),y

(·,t),y

(·,t)) =

= a

(θ(t) −θ

)

θ(t) −c

(

θ(t))

−c

(L,t))

−

c(s)y

(s,t)

ds+ τ(t)

θ(t) + u

(t)y

(L,t)

(21)

A rather simple choice for the controller structure

would be

τ(t) = −a

(θ(t) −θ

) −g

(

θ(t)) ,

(t) = −g

(L,t))

(22)

where g

(σ), i := 0,L are sector restricted nonlinear

functions that is, subject to the following inequalities

≤ g

(σ)σ ≤ ¯g

, g

(0) = 0 , i := 0, L (23)

Observe that these nonlinear functions are arbitrary,

the only conditions being (23). This signiﬁes that we

assume some robustness properties of the stabilization

with respect to some uncertainty of the controllers.

In particular these functions can be taken linear thus

“pointing” to a PD (proportionalderivative) controller

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512

at the boundary s = 0 and a D (derivative)controller at

s = L. In the general case of the sector restricted non-

linearities, these are nonlinear controllers. Introduc-

ing the controllers’ equations (22) in the controlled

equations (9), the equations of the closed loop system

are obtained as follows (including now the non-zero

persistent perturbations

ρ(s)y

+ c(s)y

−(T(s)(y

−θ))

= f(s,t)

(0,t) = θ(t)

θ+ c

θ+ g

(

θ) + a

(θ(t) −θ

)−

−

T(s)(y

(s,t) −θ)ds = φ

(t)

(L,t) + c

(L,t) + g

(L,t))−

−T(L)(y

(L,t) −θ) = d

(t)

(24)

From these equations it is quite clear that the PD

controller at s = 0 “imposes” to the cyclic variable

θ the reference value θ

and introduces an additional

nonlinear/lineardamping.Theother controller - a non-

linear D controller - introduces an additional nonlin-

ear/linear damping.

B. We shall now discuss in brief some straightfor-

ward properties of system (24). Let again the persis-

tent perturbations be identically zero and check ﬁrst

the steady state solutions, which are static equilibria

(speaking the language of Rational Mechanics) sub-

ject to the steady state equations

(T(s)(¯y

(s) −

θ))

= 0 , ¯y

(0) =

(

θ−θ

) −

T(s)( ¯y

(s) −

θ)ds = 0

T(L)(¯y

(L) −

θ) = 0

(25)

It is a matter of simple and straightforward manip-

ulation to ﬁnd

θ = θ

, ¯y(s) = θ

hence the static equilibrium is unique and its signiﬁ-

cance is that positioning is without steady state error.

We have to prove however that this steady state is

a limit regime and it has asymptotic stability. With re-

spect to this we remind that to the closed loop system

(24) there is associated the c.l.f. (control Liapunov

functional) (19) which has the form (20) along sys-

tem (24) and is positive deﬁnite. If the persistent per-

turbations are again set to zero since the aim is equi-

librium stability, then the derivative of the Liapunov

functional that follows from (21) with the controller

choice (22) will be

V (θ(t) −θ

θ(t),y

(L,t),y

(·,t),y

(·,t)) =

= −

c(s)y

(s,t)

ds−(c

θ(t) + g

(

θ(t)))

θ(t)−

−(c

(L,t) + g

(L,t)))y

(L,t) ≤ 0

(26)

Let us ﬁrst mention that in the real case of a ﬂex-

ible manipulator there is no thrust controller on the

payload (unlike in the case of e.g. the marine riser)

hence g

(σ) ≡0. If the sector conditions (23) are such

that c

σ+ g

(σ) > 0 then the derivative function (26)

vanishes on the set

(s,t) ≡ 0 , c

θ(t) + g

(

θ(t)) ≡0 , y

(L,t) ≡ 0

On this set the trajectories of system (24) are deﬁned

exactly by the steady state equations (25) hence the

largest invariant set contained in the set where the

derivative function (26) vanishes is exactly the unique

equilibrium. Global asymptotic stability of this equi-

librium would follow provided the invariance princi-

ple Barbaˇsin Krasovskii LaSalle is valid in this case.

In some special cases this might be true, see (Saper-

stone, 1981); one of this cases which is some how

alike the considered here corresponds to the overhead

crane and is the subject of (d’Andr´ea Novel et al.,

1994), being mentioned in (R˘asvan, 2014). In the fol-

lowingthe problem will be analyzed within the frame-

work of the neutral functional differential equations

attached to the mixed initial boundary valued prob-

lems for hyperbolic partial differential equations.

4 THE BASIC PROPERTIES OF

THE DYNAMICAL SYSTEM

Starting from the 60ies of the previous century, some

authors who studied hyperbolic partial differential

equations in two dimensions, applied to physics and

engineering, discovered that integration of the Rie-

mann invariants along the characteristics allow asso-

ciation of a system of differential equations with de-

viated argument to the mixed initial boundary value

problem with unusual (in the sense of non standard

that is different from the Neumann or Robin) bound-

ary conditions containing higher order derivatives.

This “association” must be understood in the sense

that the aforementioned system of differential equa-

tions with deviated argument is constructed starting

from a solution of the boundary value problem and

the structure of the boundary conditions. Moreover,

a one to one correspondence between the solutions

of the two aforementioned mathematical objects may

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513

be established in a rigorous way (see Theorem 1 and

all mathematical results obtained for one of these ob-

jects may be projected on the other one. Among the

papers contributing to this approach, which are enu-

merated in the Reference list of (R˘asvan, 2014), it is

worth mentioning the ﬁrst of that list (Abolinia and

Myshkis, 1960) and the less circulated but the most

useful for our context (Cooke, 1970) (where the term

“derivative boundary conditions” is used).We shall

not reproduce here the results of the aforementioned

reference but rather apply them to our case.

A. The ﬁrst step will be to associate the system of

functional differential equations, starting from the

considered basic system since it has been already

mentioned that this method does not apply but to par-

ticular cases. Consider the closed loop system (24)

under the following assumptions

c(s) ≡ 0 ; ρ(s) ≡ ρ = const , T(s) ≡ T = const

We assume also, to simplify the computation formu-

lae, that the distributed perturbations are identically

zero i.e. f(s,t) ≡ 0. This assumption does not af-

fect generality of the analysis since we are interested

mainly in the Liapunov stability of the equilibria. Un-

der these circumstances (24) becomes

ρy

−Ty

= 0 ; y

(0,t) = θ

θ+ c

θ+ g

(

θ) + TLθ+ a

(θ−θ

)−

−T

(s,t)ds = φ

(t)

(L,t) + c

(L,t) + T(y

(L,t) −θ) = d

(t)

(27)

Introduce now the variables for the symmetric

Friedrichs form, with their initial conditions deduced

from those of (24)

v(s,t) := y

(s,t) , w(s,t) := y

(s,t) ;

(y(s,0) = y

(s) , y

(s,0) = v

(s)) ⇒

⇒ v(s,0) = v

(s) , w(s,0) = y

′

(s)

(28)

to obtain the new system

= v

, ρv

−Tw

= 0 ; w(0,t) = θ

θ+ c

θ+ g

(

θ) + TLθ+ a

(θ−θ

)−

−T

w(s,t)ds = φ

(t)

(L,t) + c

v(L,t) + T(w(L,t) −θ) = d

(t)

(29)

Introduce now the Riemann invariants as follows

(s,t) =

(v(s,t) ∓

T/ρw(s,t)) (30)

with the corresponding initial conditions

(s,0) =

(s) ∓

T/ρy

′

(s)) (31)

Here u

(s,t) stands for the progressive wave and

−

(s,t) for the reﬂected one. Consequently the initial

boundary value problem for the Riemann invariants

reads as below

T/ρu

2ρ

f(s,t)

ρ/T(u

−

(0,t) −u

(0,t)) = θ

θ+ c

θ+ g

(

θ) + TLθ+ a

(θ−θ

)−

−

ρT

−

(s,t) −u

(s,t))ds = φ

(t)

−

(L,t) + u

(L,t)) + c

−

(L,t) −u

(L,t))+

√

ρT(u

−

(L,t) −u

(L,t)) −Tθ = d

(t)

(32)

We are now in position to apply the algorithm

suggested in (Cooke, 1970) and fully described and

proved in (R˘asvan, 2014). Integration along the char-

acteristics of the Riemann invariants will give

(t) := u

(L,t) ⇒u

(0,t) = y

(t + L

ρ/T) ;

−

(t) := u

−

(0,t) ⇒u

−

(L,t) = y

−

(t + L

ρ/T)

(33)

and suggests the following representation formulae

(s,t) = y

(t + s

ρ/T) ,

−

(s,t) = y

−

(t + (L−s)

ρ/T)

(34)

Introducing the new functions

(t) = y

(t + L

ρ/T) (35)

we obtain, after some manipulation the following sys-

tem of functional equations

θ+ c

θ+ g

(

θ) + TLθ+ a

(θ−θ

)−

−T

−L

√

ρ/T

(η

−

(t + λ) −η

(t + λ))dλ = φ

(t)

ρ/T(η

(t) −η

−

(t −L

ρ/T)) + θ = 0

(η

−

(t) + η

(t −L

ρ/T))+

(η

−

(t) + η

(t −L

ρ/T))+

√

ρT(η

−

(t) −η

(t −L

ρ/T)) −Tθ = d

(t)

(36)

Following the same procedure (Cooke, 1970;

R˘asvan, 2014), we associate to (36) the initial con-

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514

ditions

(t) = u

(−t

T/ρ,0) ,

−

(t) = u

−

(L+ t

T/ρ,0) ; −L

T/ρ ≤t ≤ 0

(37)

The converse procedure, based on the representa-

tion formulae deduced from (35) namely

(s,t) = η

(t + (s−L)

ρ/T) ,

−

(s,t) = η

−

(t −s

ρ/T)

(38)

allows the statement of the following theorem estab-

lishing a one-to-one correspondence between the so-

lutions of (32) and the solutions of (36)

Theorem 1. Let (u

(s,t),θ(t)) be a solution of (32)

with some initial conditions (u

(s,0),θ(0),

θ(0)), 0≤

s ≤ L. Then η

(t),θ(t) is a solution of (36) with the

initial conditions η

(t),θ(0),

θ(0)), where η

(t) are

deﬁned on −L

T/ρ ≤t ≤ 0 by (37).

Conversely, let η

(t),θ(t) be a solution of (36)

with some initial conditions η

(t),θ(0),

θ(0)), where

(t) are deﬁned on −L

T/ρ ≤ t ≤ 0. Then

(s,t),θ(t)) is a solution of (32) with the initial

conditions (u

(s,0),θ(0),

θ(0)), where u

(s,t) are

deﬁned by the representation formulae (38) and the

initial conditions (u

(s,0),θ(0),

θ(0)) with u

(s,0)

deﬁned accordingly also from (38).

B. Having Theorem 1 at our disposal, we may focus

now on system (36) since all results concerning its

solutions are automatically projected back on the so-

lutions of (32). Observe ﬁrst that system (36) is a sys-

tem of coupled delay differential and difference equa-

tions in continuous time. Such systems belong to the

broader class of functional differential equations of

neutral type see e.g. (Hale and Lunel, 1993). It can be

seen that system (36) is quasi-linear: it contains a sin-

gle nonlinear function which is sector restricted - see

(23). For such systems existence, uniqueness and well

posedness of the Cauchy initial value problem holds -

see (Hale and Lunel, 1993), Section 2.8. From these

results on neutral functional differential equations we

deduce existence, uniqueness and well posedness of

the initial boundary value problem (32) at the level

of continuous/discontinuous classical solutions. This

is a ﬁrst step in overcoming the formal level of the

results concerning system (28) and of (27) also, due

to various representation formulae leading from these

systems to (32) and further, to (36) and conversely.

We focus in the following on the stability problem

for the equilibria of (36). The unique equilibrium of

(36) corresponding to φ

(t) ≡ 0, d

(t) ≡ 0 is given by

θ = θ

;

= −

−

T/ρθ

The next step is to write down the Liapunov func-

tional (20) in the language of the variables of sys-

tem (36). Some straightforward manipulation based

on (30),(33)-(35) will give the following form of the

Liapunov functional

V (θ(t) −θ

θ(t),η

(·),η

−

(·)) =



(θ(t) −θ

)

(t) + m(η

−

(t −L

ρ/T) + η

(t))

[(η

−

(t −s

ρ/T) + η

(t + (s−L)

ρ/T))

(η

−

(t −s

ρ/T) + η

−

(t + (s−L)

ρ/T)−

θ(t))

]ds



(39)

which is positive deﬁnite. In the same way we re-

write the derivative (26) as follows, taking also into

account that c(s) ≡ 0 and g

(σ) ≡ 0

V (θ(t) −θ

θ(t),y

(L,t),y

(·,t),y

(·,t))

= −(c

θ(t) + g

(

θ(t)))

θ(t)−

−c

(η

(t −L

√

ρT) + η

−

(t))

≤ 0

(40)

The set where this derivative vanishes is deﬁned

θ(t) ≡0 , η

(t −L

ρT) + η

−

(t) ≡ 0 (41)

From these conditions and from equations (36) we de-

duce that the largest invariant set contained in (41) is

exactly the unique equilibrium of (36) namely

θ = θ

;

= −

−

= (1/2)

T/ρ

Application of the Barbaˇsin Krasovskii LaSalle in-

varianceprinciple can thus giveglobal asymptotic sta-

bility of this equilibrium of system (36) and, via The-

orem 1, global asymptotic stability of the time invari-

ant solution of (32); taking into account the converse

expressions of (30), the same property holds for the

equilibrium of (29).

Now, the invariance principle for neutral func-

tional differential equations is given by Theorem

12.7.2 of (Hale and Lunel, 1993). However this theo-

rem is valid only for those neutral functional differen-

tial equations with strongly stable difference operator.

In the case of (36) this difference operator is deﬁned

−

(·) =

(0)

−

(0)

−

0 1

−1 0

(−L

ρ/T)

−

(−L

ρ/T)

(42)

OntheStabilizationoftheFlexibleManipulator-LiapunovbasedDesign.Robustness.

515

and its stability is established by the location of the

eigenvalues of the matrix

D =

0 1

−1 0

with respect to the unit disk of the complex plane.

Since these eigenvalues are ±ı, the difference oper-

ator is only stable and Theorem 12.7.2 cannot be ap-

plied. As mentioned in (Saperstone, 1981) the strong

stability condition might be replaced by an additional

condition of smoothing the trajectories of the dynam-

ical system, but such a condition has not been pointed

out yet.

If nevertheless the payload mass effects are con-

sidered negligible, then the last equation of (36) is re-

placed by

(η

−

(t) + η

(t −L

ρ/T))+

√

ρT(η

−

(t) −η

(t −L

ρ/T)) −Tθ = d

(t)

Consequently the matrix D of the newly deﬁned dif-

ference operator will be replaced by

D =







0 1

−

c−

√

ρT

√

ρT







whose purely imaginary eigenvalues are now inside

the unit disk. The global asymptotic stability follows

now in a rigorous way being in fact valid for the equi-

librium of the following system

= v

, ρv

−Tw

= 0 ; w(0,t) = θ

θ+ c

θ+ g

(

θ) + TLθ+ a

(θ−θ

)−

−T

w(s,t)ds = φ

(t)

v(L,t) + T(w(L,t) −θ) = d

(t)

(43)

C. It is interesting and useful to make some com-

ments concerning the global asymptotic stability of

system (43). This control-theoretical result has been

obtained by using a naturally associated control Li-

apunov functional suggested by the energy identity.

The synthesized stabilizing controller has a twofold

role - exact positioning of the payload and increasing

the damping factor at the controlled boundary. This

simple controller has nevertheless a necessary robust

stability in the sense that the global asymptotic stabil-

ity property is valid for an entire class of linear and

nonlinear damping functions g

(σ) satisfying the sec-

tor condition

σ+ g

(σ) > 0

that is system (43) is absolutely stable. Moreover,

if the approach of K.P. Persidskii is carefully used

for the systems of neutral functional differential equa-

tions as in the case of delayed functional differential

equations (R˘asvan, 2012) then this asymptotic stabil-

ity may be shown as exponential - what is basic in

engineering systems stability.

5 CONCLUSIONS. OPEN

PROBLEMS AND

PERSPECTIVES

We have presented throughout this paper an attempt

to have a sound mathematical basis for the analysis

of the dynamics and the control of a ﬂexible manip-

ulator with distributed parameters - assimilated to a

rod. We overview brieﬂy the topics already presented

in the Abstract: writing down of the model as an ini-

tial boundary value problem with derivative bound-

ary conditions by applying the variational principle

of Hamilton; association of a “natural” control Li-

apunov functional issued from the energy identity;

synthesis of a positioning and stabilizing boundary

controller; association of a system of functional dif-

ferential equations of neutral type allowing the rig-

orous construction of the basic (existence, unique-

ness, well posedness) theory as well as of the stabil-

ity theory based on weak Liapunov functionals and

Barbaˇsin Krasovskii LaSalle principle. Worth men-

tioning that this association allows more, e.g., nu-

merically robust computational approaches based on

the method of lines (which turns to be within this

framework delay approximation by ordinary differ-

ential equations) which are implemented using struc-

tures belonging to the techniques of Artiﬁcial Intel-

ligence (Danciu, 2013a; Danciu, 2013b; Danciu and

R˘asvan, 2014).

A number of possible extensions and open prob-

lems have been pointed out even throughout the pa-

per exposition. We give here a brief account of some

of them. First, the approach via neutral functional

differential equations appears as feasible for systems

without distributed damping and with uniform param-

eters. In other cases they are much more compli-

cated (Abolinia and Myshkis, 1960) and an additional

analysis is necessary. With respect to this it is worth

recalling some aspects that have been discussed in

Section 2.

Mathematical modeling starting from Physics and

Theoretical Engineering is almost always associated

with the attempt to be as complete and exhaustive as

possible regardless how complicated the model re-

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sults. However, a good well ﬁtted simplifying as-

sumption may transform that primary model into a

tractable one (we have in mind the Prandtl assump-

tion that made the Navier Stokes equations solvable).

In the present case we have been guided by the simi-

larities between the models occurring in such various

engineering ﬁelds as ﬂexible manipulator arms, ris-

ers and overhead cranes, drillstrings (Bobas¸u et al.,

2012; Saldivar et al., 2013; R˘asvan, 2013; R˘asvan,

2014). These similarities at their turn arise from con-

sidering identical expressions for the energies and the

same type of forces or torques acting on these sys-

tems. Worth mentioning that such simplifying as-

sumptions as adoption of formula (2) have also their

role as well as the string model for the beams occur-

ring in the physical models; other models of beams

presented in (Russell, 1986) such as Euler Bernoulli,

Rayleigh, Timoshenko may provide other sources for

research development.

Even in the analyzed case a lot of problems oc-

curred when considering basic (existence and unique-

ness) as well as stability theory and we were able to

solve only some particular cases. Here the discussion

is quite interesting and deserves some attention. The

experience of the authors shows that all encountered

models arising from Mechanics and tackled by asso-

ciating neutral functional differential equations gener-

ate differenceoperators - see Section 4 - that are stable

but not strongly stable. On the other hand the standard

theory of these equations (Hale and Lunel, 1993) is

based on the strong stability assumption. Therefore

relaxation of this assumption is another urgent task.

Finally, development of the qualitative theory to in-

clude the case of persistent perturbations (dissipative-

ness) and stability of the forced oscillations is also

of obvious interest. The authors consider these asser-

tions as underlying a genuine research program since

the considered model of this paper applies to other

ﬁelds also.

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