[ieee >2007 4th international symposium on applied computational intelligence and informatics -...

Geometric Approach to Nonlinear Adaptive ControlJozsef K. Tar, Imre J. Rudas

Budapest Tech Polytechnical Institution, John von Neumann Faculty of Informatics,

Institute of Intelligent Engineering Systems

Becsi ut 96/B, Budapest, H–1034, Hungary

E-mail: [email protected], [email protected]

http://www.bmf.hu

Abstract— In this paper a brief survey is provided on a novelapproach to adaptive nonlinear control developed at BudapestTech in the past few years. Since this problem tackling is mainlybased on simple geometric and algebraic considerations a briefhistorical summary is given on the antecedents to exemplify theadvantages of geometric way of thinking in Natural Sciences. Fol-lowing that the most popular branches of the classical and novel,Soft Computing (SC) based robust and adaptive approaches areanalyzed with especial emphasis on the supposed need and theconsequences of obtaining complete, accurate, and permanentmodels either for the system to be controlled or to the controlsituation as a whole.

Following that, in comparison to the above mentioned moretraditional methods, our novel approach is summarized thathas the less ambitious goal of obtaining only partial, incomplete,temporal, and situation-dependent models that require continuousrefreshment via observing the behavior of the controlled system inthe given actual situation. It will be shown how simple geometricconsiderations can be used for developing a simple iterativelearning control for Single Input – Single Output (SISO) systemsthe conditions of the convergence of which is easy to satisfy bychoosing very simple and primitive initial system models androughly chosen control parameters.

Finally it will be shown how the most attractive mathematicalproperties of the fundamental symmetry groups of Natural Sci-ences can be utilized for the generalization of our approach tothe control of Multiple Input – Multiple Output (MIMO) systems.The already achieved results are exemplified via simulation, andthe possible directions of the future research are briefly outlined,too.

I. HISTORICAL ANTECEDENTS OF GEOMETRIC WAY OF

THINKING IN NATURAL SCIENCES

Until the 1st half of the 20th century the developmentof Mathematics aimed at serving the needs of natural andtechnical sciences. In the history of the ”quantitative sciences”geometric way of thinking always played a pioneering rule.

The principles of geometry first were reduced to a small setof axioms by Euclid of Alexandria, a Greek mathematicianwho worked during the reign of Ptolemy I (323-283 BC)in Egypt. His method of proving mathematical theorems bylogical reasoning from accepted first principles remained thebackbone of mathematics even in our days, and is responsiblefor that field’s characteristic rigor.

Following the pioneering work clarifying the phenomenol-ogy of Classical Mechanics by Galilei and Newton, in his fun-damental work entitled ”Mecanique Analytique” [1] Joseph-Louis Lagrange (1736-1813) solved various optimization prob-lems under constraints, introduced the concept of Reduced

Gradient and that of what we refer to nowadays as LagrangeMultipliers. It has to be noted that at that time the concept of”linear vector spaces” was not clarified at all.

The first mathematical means of describing quantities withdirection, i.e. the quaternions introduced by Sir William RowanHamilton (1805-1865) appeared not very long time afterLagrange’s death [2]. In the 19th century quaternions weregenerally used for such purposes. For instance, in the firstedition of Maxwell’s famous Treatise on Electricity and Mag-netism quaternions were used for describing the ”directed”magnetic and electric fields.

The first known appearance of what are now called lin-ear algebra and the notion of a vector space is relatedto Hermann Gunther Grassmann (1809–1877), who startedto work on the concept from 1832. In 1844, Grassmannpublished his masterpiece [3] that commonly is referred toas the ”Ausdehnungslehre”, (”theory of extension” or ”theoryof extensive magnitudes”). This work was mainly inspired byLagrange’s ”Mecanique analytique”. Grassmann showed thatonce geometry is put into the algebraic form he advocated,then the number three has no privileged role as the number ofspatial dimensions: the number of possible dimensions is infact unbounded.

The close relationship between geometry and algebra wasrealized and strongly utilized by William Kingdon Clifford(1845–1879) who introduced various associative algebras,the so called ”Clifford Algebras”. As special cases CliffordAlgebras contain the algebra of the real, the complex, the dualnumbers, the quaternion algebra, and the algebra of octonions(biquaternions) [4]. His Geometric Algebra is widely used intechnical sciences as e.g. in computer graphics, robotics, etc.

Equipped with the concepts of linear vector spaces MariusSophus Lie (1842–1899) in his PhD dissertation studied theproperties of geometric symmetry transformations [5]. One ofhis greatest achievements was the discovery that continuoustransformation groups (now called after him Lie groups) couldbe better understood by studying the properties of the tangentspace of the group elements, that form linear vector spaces(the vector space of the so-called infinitesimal generators), andwith the commutator as multiplication also form algebras, theso called Lie Algebras.

In the very fertile period of Mathematics, in the 19th centuryGeorg Friedrich Bernhard Riemann (1826–1866) elaboratedthe geometry of curved spaces in a special form that made

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it possible to study physical quantities as tensors even if thegeometry of the space differs from the Euclidean Geometry.This concept was very fruitfully used in the General Theoryof Relativity.

David Hilbert (1862–1943) extended the concept of theEuclidean Geometry to linear, normed, complete metric spacesin which the norm originates from a scalar product. Hisinvention had extreme advantages in Physics and technicalsciences since it makes it possible to apply a way of geometricthinking with which we became familiar from our childhoodin the daily experienced Euclidean Geometry of the realityaround us.

Stefan Banach (1892-1945) introduced the more generalconcept, the concept of Banach Spaces, that are linear, normed,complete metric spaces in which the norm not necessarilyoriginates from a scalar product. The great practical advantageof Banach’s invention is that by adding various norms tothe same mathematical set various complete, linear, normedmetric spaces can be obtained that offer a wide basis forelaborating diverse practical variants and solutions pertainingto the essentially same basic idea.

Vladimir Igorevich Arnold (1937–) studied the SymplecticGeometry and Symplectic Topology that are extremely usefulmeans of studying the behavior of various Mechanical andother physical systems.

Our aim with providing this brief historical survey was toshow that geometric way of thinking is a very useful andfruitful mode of problem–tackling in various fields as e.g. intechnical sciences. In the sequel its advantages will be shownin the field of nonlinear control.

II. CLASSICAL MODEL-BASED CONTROL

A plausible approach to solving control tasks would be toelaborate and use the ”exact dynamic model” of the system tobe controlled. In the case of the control of mechanical systemsas robots this approach can be referred to as ”ComputedTorque Control” since in this case the mechanical model estab-lishes mathematical relationships between the joint coordinatesaccelerations and the torques or forces acting on the systempartly by its own drives and / or by its environment with whichthe system may be in dynamic coupling. In the case of othersystems as e.g. chemical reactions considered in [6] the notionof ”Globally Linearizing Controllers” can be mentioned inwhich certain order time-derivative of the state variable of thesystem to be controlled or that of a well-defined function ofthe state variables can instantaneously be set by the controlsignal. In the sequel these typical cases are considered.

A. Computed Torque Control in Robotics

Before going into details it has to be noted that involvingthe model of the internal operation of the drives of a ClassicalMechanical System may considerably increase the complexityof the problem. However, even modeling the mechanical be-havior itself is a very complex task. As a result of such effortsthe Euler-Lagrange Equations of Motion can be obtained for

an open kinematic chain as follows:

H(q)q + h(q, q) = Q (1)

in which H(q) describes the configuration-dependent inertiamatrix of the system, a part of h(q, q) is quadratic in q anddescribes e.g. the Coriolis terms, while its other part dependingonly on q is responsible for the gravitational effects. It is worthnoting that due to physical reasons H is always symmetric andpositive definite, though it may be badly conditioned, too. Theterm Q stands for the generalized forces that partly originatefrom the robot’s own drives or from the environment. (Thisequation is valid only if the kinetic energy of the system isgiven with respect to an inertial frame of reference in whichcase the components of Q can be interpreted as forces forthe prismatic generalized co-ordinates, and torques for therotational axes.)

In the classical example in which Armstrong et al. devel-oped the dynamic model of a six degree of freedom PUMArobot arm three persons worked for five weeks [7]. This workinvolved the measurement of the appropriate data besidescoding the model in software blocks. In the possession of this”exact” model on the basis purely kinematic considerationssome desired qd can be computed in each control cycle toexert the necessary Qd.

However, a practical problem in the application of thismethod is that normally there are no sensors available thatcould exactly measure the external parts of Q. Their effectscan be observed only as their consequences in the actualmotion of the system, and in general cannot efficiently becompensated by simply prescribing some feedback correctionin qd. Such kind of feedback correction can work only if theunknown external perturbations are a) generally insignificant,or, if they are significant, b) they can be only instantaneousbut permanent.

It is worth noting that the kinematic structure of the robotarm itself determines the main mathematical ”skeleton” of (1):normally a parameter vector can be introduced that containsthe unknown dynamical information, while the elements ofthis vector in (1) are multiplied by known kinematic functions.This fact serves as a basis for developing the analytical modelbased controls toward adaptive solutions as e.g. in the case ofthe ”Adaptive Inverse Dynamics” approach.

B. Globally Linearizing Controllers

The concept of ”Globally Linearizing Controllers” as intro-duced e.g. by Khalil [8], Goodwine & Stepan, [9] are designedfor the following more or less ”canonical” form of equationsof motion:

x = f(x) + g(x)u,y = h(x) (2)

in which x ∈ �n denotes the state variable of the system,y ∈ �m denotes its observable output, u ∈ �k means themanipulated input (control signal). By applying the chain ruleof derivation the time-derivative of y can be obtained from (2)

SACI 2007 – 4th International Symposium on Applied Computational Intelligence and Informatics

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as

dy

dt=

n∑s=1

∂h

∂xs[fs(x) +

k∑z=1

gsz(x)uz] ≡ Lfh + (Lgh)u. (3)

in which the very condensed notation of the Lie–derivatives isapplied as Lfh, etc. If the lucky situation occurs in whichLgh �= 0 then y can simply be expressed as an affinefunction of u. If Lgh ≡ 0 then y can be expressed asy = L2

fh + uLgLfh. If in an unlucky manner LgLfh ≡ 0again, and in general if we have j > 0 so that LgL

sf ≡ 0 if

s = 0, 1, 2, ..., j − 1, but LgLjf �= 0 the dependence of the jth

time-derivative of y on u has an affine form as

y(j) = Ljfh(x) + LgL

j−1f h(x)u. (4)

In this case j is referred to as the relative degree of thenonlinear system. In the possession of the exact systemmodel the appropriate Lie-derivatives in (4) can be computed.Whenever (4) is able to uniquely determine the appropriatevalue of u that is needed for achieving a desired jth derivativeof the observable output y(j)d

determined on the basis ofsome ”kinematic” consideration, this formalism can evidentlybe successfully used for the control. The control signal u

evidently can be fed back in the form of u = p(x)+q(x)y(j)d

from which the name of the controller, i.e. the notion of ”Glob-ally Linearizing Control” originates. It is worth noting that inspite of the very ”special form” of the suppositions concerningthe identically zero values of certain Lie–derivatives in thepractice various physical systems meet these conditions. In[6] e.g. the temperature control of a ”Jacketed ContinuousStirred Tank Rector (JCSTR)” is considered in which the heatreleased in an exothermic reaction has to be extracted by thecooling system in the jacket, while in e.g. [10] a 4th orderClassical Mechanical system is considered.

It has to be noted again that even if we are not in thepossession of the exact model, the analytical form of (4) stillcan be a good basis for developing adaptive controllers as e.g.in the case of the control of a polymerization process in [11].

III. ROBUST SLIDING MODE / VARIABLE STRUCTURE

(SM/VS) CONTROLLERS

One of the most successful and most popular approaches inRobust Control is the application of the SM/VS control idea.It directly can be extended to the problems of the form as in(1) or in (4) whenever only a very approximate system modelis available. In this case the aim is to drop the very complexprogram of learning or identifying the accurate model. Thegoal rather is to apply a very rough bang-bang type controlfor the needs of which some rough over-estimation of thenecessary control signals can reliably be done on the basisof the rough model.

For this purpose it is plausible to introduce the linearoperator Λ := (d/dt+λ) with λ > 0, and apply its appropriatepower to the trajectory tracking error. Really, if for an integerconstant k > 0 and a function f(t) the situation of Λkf(t) = 0is achieved, the quantity Λk−1f(t) → 0 exponentially as

∝ e−λt. As soon as the situation of Λk−1f(t) ≈ 0 hasbeen approximated practically the quantity Λk−2f(t) starts todecay exponentially as ∝ e−λt, etc. Finally the situation off(t) → 0 is achieved. It is important to note that for thisconvergence it is not necessary to guarantee the exact valueof λ. Any positive λ ≈ λ works well with a little bit differentspeed of convergence. In the case of an mth order systemwhen dmx/dtm can directly be manipulated by the drive(s)the controller may have different ”ambitions”. Whenever onlya very rough system model is available dmx/dtm cannotexactly be prescribed. Instead of that by the use of Λ and thetrajectory tracking error h an ”error metrics” can be definedas S(t) := Λm−1h(t), and some attempt can be done to driveS to zero during finite time. Since dS/dt contains dmx/dtm aplausible choice is to approximate the ”desired” situation by

SDes ≈ −Ksign(S) (5)

with satisfactorily big positive constant K. Though a preciselyprescribed K value cannot be achieved in the lack of the exactsystem model, the situation 0 < Klim ≤ K can be achievedon the basis of a rough system model. Normally only somedrastic over-estimation of the necessary driver action can beobtained, therefore, as a consequence, the sign of the achievedS fluctuates that also makes the driving force/torque fluctuate.This fluctuation is the phenomenon of ”chattering” so typicalin the case of the SM/VS Control. A possibility for reducingchattering is smoothing the variation of S in (5) as

SDes = −K tanh(2S

w) (6)

in which w is a properly chosen ”width parameter” withinwhich the jump in (5) is smoothed. This smoothing evidentlydecreases the speed of the decay of S, therefore degrades theaccuracy of trajectory tracking.

To exemplify the operation of the SM/VS Controller let usconsider the adaptive control of a Ball-Beam system sketchedin Fig. 1. In the control of this system a ball or cylinder can rollon the surface of a beam the tilting angle of which is driven bysome actuator. The motion of the ball essentially is determinedby the tilting angle and the force of gravitation. This meansthat even if we are in the possession of a very strong actuator,the acceleration of the ball along the beam is limited by theabove two factors. Since the directly controllable quantity isthe torque determining the 2nd time-derivative of the angletilting the beam, this system acts as a 4th order one in the sensethat the 4th time-derivative of the ball’s position along thebeam is determined by the tilting torque. It has the followingparameters: the momentum of the beam ΘBeam = 2 (kg ×m2), the mass of the ball mBall = 2 (kg), the radius ofthe ball r = 0.05 (m), and the gravitational acceleration isg = 9.81 (m/s2). Via introducing the quantities A = ΘBeam,and B = ΘBall/r2+mBall, the following equations of motionare obtained:

Aϕ + mBallx cos ϕ − mBallrsinϕ = QBx + mBallgsinϕ = 0 (7)

J. K. Tar, I. J. Rudas • Geometric Approach to Nonlinear Adaptive Control

11

Fig. 1. The rough sketch of the Ball-Beam System

in which variable ϕ (rad) describes the rotation of the beamcounter-clockwisely with respect to the horizontal position,and x (m) denotes the distance of the ball from the center ofthe beam where it is supported. Variable Q (N ×m) describesthe torque at the axis rotating the beam. This quantity consistsof two different components: the torque directly exerted bythe drive, and the contribution by the friction forces actingat the surface of the axle. In this paper this latter componentis unknown by the controller, only the consequences of itsexistence in the trajectory tracking can be observed. From (7)x can be expressed as a function of ϕ. Since this angle cannotbe made abruptly vary, following two derivations by time d4x

dt4

can be expressed with ϕ as follows:

x(4) =mBallg

B

(sinϕϕ2 − cosϕϕ

). (8)

In the possession of the desired x(4)dvalue by the use of (8)

and the 1st equation of the group (7) the necessary torque Qcan be computed in principle. Normally it can be supposed thatthe parameters of the actual system are not precisely known.Instead of the actual parameters some model values are usedas A , and B constructed of the model values of the otherparameters. On the basis of this rough model at first the desiredrotational acceleration of the beam ϕd is estimated as

ϕd = −Bx(4)d

˜mBallgcosϕ + tan ϕϕ2−

−Γϕ

∂ cosh2n(

βP otϕ

1.5

)∂ϕ − Γϕ

∂ cosh2n(

βP otϕ

3

)∂ϕ

(9)

in which the last two terms limiting potentials are introducedfor the rotational angle and for the rotational velocity of thebeam that nonlinearly curb the increase in |ϕ| around 1.5 (rad)and |ϕ| around 3 (rad/s), respectively to cancel slow driftsof these terms. (In the simulations βPot = 5 (dimensionless),and n = 1 (dimensionless) were used.) For describing thephenomenon of friction the LuGre model was used which thedeformation of the bristles of some ”brushes” are applied todescribe the deformation of the surfaces in dynamic contact,so friction is described as a dynamic coupling between two

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5−4

−2

0

2

4Phase space of x [10^−1 m/s vs 10^−1 m]

0 2 4 6 8 10−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5x vs. time [10^−1 m vs s]

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5−8

−6

−4

−2

0

2

4

6

8Phase space of fi [10^−1 rad/s vs 10^−1 rad]

0 2 4 6 8 10−18−16−14−12−10−8−6−4−2

02

Error metrics S [10^0 m/s^3] vs time [s]

Fig. 2. The operation of the VS/SM controller if the exact model is availableand no friction is present at the tilting axle: phase space of the translation ofthe ball along the beam [x vs. x](upper left), the nominal and the computedtrajectories of the ball vs. time (upper right), tilting angle of the beam [ϕ](lower left), and the error metrics S (lower right) vs. time.

systems having their own equations of motion as

dzdt = v − σ0|v|

FC+FS exp (−|v|/vs)z

F = σ0z + σ1dzdt + Fv × v

(10)

for which the proper direction of F has to be set in theapplications, Fv describes the viscous friction coefficient, andσ1 is a new parameter pertaining to the effect of the bendingbristles. This model is physically complete in the sense that noany velocity limit of dubious interpretation must be introducedfor its use. The behavior of the whole system is describedby the dynamic coupling between the hidden internal andthe observed degrees of freedom. Though the appropriatequantities in (10) were developed for linear motion and forces,it easily can be generalized for rotary motion in whin torquesappear in the role of the forces, and rotational velocity ispresent instead of linear motion’s velocity. The model givenin (7) evidently can be completed via adding the additionaltorque of the friction to Q in it.

To demonstrate the operation of the SM/VS controller forλ = 6 (s−1) and K = 140 (ms−4) in Fig. 2 simulation resultsare given for the case in which the exact dynamic model isavailable for the controller and no friction forces are presentat the axle tilting the beam. It is evident from Fig. 2 that onlysmooth and ”decent” variation happens in the phase space ofϕ, the error metrics remains very small, the phase trajectoryof x and x itself are nicely traced.

For simulating the effect of friction very drastic parametersettings was chosen as σ0 = 6000 (Nm/rad), σ1 = 2000(Nms/rad), FC = 10 (Nm), FS = 20 (Nm), Fv = 200(Nms/rad). For modeling errors A = 5A, and mBall =0.4mBall were applied (Fig. 3). The figures reveal that forachieving approximately comparable tracking accuracy forx in the phase space of ϕ drastic chattering appears as aconsequence of the drastic variation of S. The effect of themodeling error and external disturbances are well illustrated byFig. 4 in which an ”anti-chattering smoothing” in the switching


12

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5−4

−2

0

2


0 2 4 6 8 10−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5x vs. time [10^−1 m vs s]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5−8

−6

−4

−2

0

2

4

6


0 2 4 6 8 10−20

−15

−10

−5

0

5Error metrics S [10^0 m/s^3] vs time [s]

Fig. 3. The operation of the VS/SM controller using rough model in thepresence of drastic unmodeled friction: phase space of the translation of theball along the beam [x vs. x](upper left), the nominal and the computedtrajectories of the ball vs. time (upper right), tilting angle of the beam [ϕ](lower left), and the error metrics S (lower right) vs. time.

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5−4

−2

0

2


0 2 4 6 8 10−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5x vs. time [10^−1 m vs s]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5−6

−4

−2

0

2

4

6


0 2 4 6 8 10−20

−10

0

10Error metrics S [10^0 m/s^3] vs time [s]

Fig. 4. The operation of the VS/SM controller using rough model in thepresence of drastic unmodeled friction and anti-chattering parameter w = 6(ms−3): phase space of the translation of the ball along the beam [x vs.x](upper left), the nominal and the computed trajectories of the ball vs. time(upper right), tilting angle of the beam [ϕ] (lower left), and the error metricsS (lower right) vs. time.

law defined by w = 6 (ms−3) is applied. The chatteringevidently disappeared from the phase space of ϕ, instead ofit the certain jumps appeared due to the varying direction ofthe friction forces nearby the zero-transitions of ϕ. However,as a ”cost”, the tracking, the phase-tracking accuracy of x hasbeen corrupted, and the error metrics S increased, too.

To evade such deficiencies instead of the relatively simplyimplementable SM/VS controller more sophisticated adaptiveapproaches are needed. The traditional adaptive methods arebriefly considered in the sequel.

IV. ADAPTIVE CONTROL OF NONLINEAR DYNAMIC

SYSTEMS

The traditional adaptive control approaches normally arebased on certain well known formal properties of the analyticalmodel of the system to be controlled, and on the use of

Lyapunov 2nd Method to guarantee stability, uniform stability,or asymptotic stability of the system to be controlled. In thissection at first the main points of Lyapunov’s method aresummarized, then an example for its use, the Adaptive InverseDynamic Control is analyzed.

A. Lyapunov’s 2nd Method: Barbalat’s Lemma and the Func-tions of Class κ

Lyapunov’s 2nd Method is a widely used technique in theanalysis of the stability of the motion of the non-autonomousdynamic systems of equation of motion as x = f(x, t).The typical stability proofs provided by Lyapunov’s originalmethod published in 1892 [12] (and later on e.g. [13]) havethe great advantage that they do not require to analyticallysolve the equations of motion. Instead of that the uniformlycontinuous nature and non-positive time-derivative of a posi-tive definite Lyapunov-function V constructed of the trackingerrors and the modeling errors of the system’s parameters areassumed in the t ∈ [0,∞) domain from which the convergenceV → 0 can be concluded according to Barbalat’s lemma [14].This lemma states that if the integral of a uniformly continuousfunction (in this case the integral of V i.e. V ) in [0,∞) isbounded then this function has to converge to zero [14]. Asin the case of any self-consistent approach based method thisassumption has to be verified later. The uniform continuityof V used to be guaranteed by showing that V is bounded.Due to the positive definite nature of V from that it normallyfollows that the tracking errors have to remain bounded, or incertain special cases, has to converge to 0.

This technique can well be exemplified on the very simpleexample of the harmonic oscillator having the equation ofmotion x = −kx − bx, k, b > 0. By introducing q1 ≡ x,q2 ≡ x the equation of motion takes the ”canonical form”[

q1

q2

]=

[0 1−k −b

] [q1

q2

]= f(q). (11)

Let P11 > 0 and P22 > 0, then V (q1, q2) ≡ P11q21+P22q

22 ≥ 0

is an appropriate candidate for a Lyapunov function. Fromthat V (q1, q2) = 2P11q1q1 + 2P22q2q2, that due to (11) canbe written as V (q1, q2) = 2 (P11 − kP22) q1q2 − 2bP22q

22 . To

achieve negative derivative for V it would be advantageous tohave V = − (Aq1 + Bq2)

2 = − (A2q2

1 + B2q22 + 2ABq1q2

).

By comparing the coefficients this can be achieved if A = 0,B2 = 2bP22, and P22 = P11/k. In this special case V =−4bP11q2q2

k = −4bP11q2(−kq1−bq2)k that evidently is bounded if

q1, q2 are bounded, too. If V < 0 these variables certainly arebounded, therefore, due to Barbalat’s lemma V = −B2x2 →0, therefore x → 0. This necessarily corresponds to thex = const equilibrium point that according to the equationsof motion only the x = 0 sate can be.

An alternative possibility for utilizing Lyapunov’s theoremis the use of the so-called special ”function class κ” certainelements of which can serve as upper and lower bounds ofV so evading the direct application of Barbalat’s lemma toshow uniform stability of the system according to Fig. 5. Bydefinition a continuous function κ : [0, k) → [0,∞) is of class


13

Fig. 5. Application of functions of class κ in Lyapunov’s stability theorem

κ if it is strictly increasing, and κ(0) = 0. (k < ∞ and k = ∞can also occur in the definition.)

From Fig. 5 it is evident that if a Lyapunov function isconstructed for that it is true that V (0, 0) = 0, V (x, t) ≥α(||x||), α is of class κ, and V (x, t) ≤ 0 then the systemis stable. Really, only the ||x|| ≤ α−1 (V (x0, t0)) values areallowed due to the stagnating or decreasing nature of V andthe criterion that V ≥ α. In this proof the initial t0 value hassignificant role.

If the above conditions can further be restricted by theexistence of a function of class κ β(||x||) so that V (x, t) ≤β(||x||) a not so strict estimation of ||x(t)|| ≤ α−1(β(||x0||))can be obtained in which t0 does not play any role. That meansuniform stability in time.

If it can also be shown that besides all of the above definedconditions V < 0 is significant enough for driving V tozero, uniform asymptotic stability can be shown, too. Really,according to the figure while V → 0 the allowable ||x|| valuescan only the projections of the cross section of the ”curvedfunnel” in the figure and the V = const line on the ||x|| axis.As the level of V sinks to 0 the funnel also narrows to zeroso ||x|| → 0.

In the great majority of the applications V used to be aquadratic term defined by a positive definite symmetric matrix.Exponential functions made of the maximal and minimaleigenvalues of positive definite matrices serve as β(||x||) andα(||x|| functions of class κ for the special case k = ∞.In spite of its lucid and simple geometric interpretation theapplication of Lyapunov’s 2nd method in the practice is ratheran ”art” than some mechanically applicable tool. It requires notonly great practice but also needs good intuition. Furthermore,the computational need of realizing the method is not alwaysnegligible. As a typical example, the method of AdaptiveInverse Dynamic control is considered in the sequel.

B. Example: Adaptive Inverse Dynamics

This approach is based on a more detailed form of (1) andsupposes that at lest the kinematic model of the system is

precisely known. On this basis a parameter vector p represent-ing the dynamical parameters and an array built up of wellknown kinematic functions Y (q, q, q) can be introduced in thedynamic model as follows:

H(q)q + h(q, q) ≡ Y T (q, q, q)p = Q. (12)

It is also supposed that some approximate model built up ofthe functions H(q), h(q, q) also is available with the modelparameters p on the basis of which the generalized forcesare calculated and exerted. The exerted forces ab ovo containfeedback–correction depending on the tracking error and itsderivatives e := q − qN , e = q − qN with symmetric positivedefinite gain matrices K(0) and K(1) as

H(q)(qN − K(0)e − K(1)e

)+ h(q, q) = Q. (13)

It is worth noting that in this method it is a supposition ofcrucial importance that the validity of (13) is supposed, i.e.it is supposed that Q is originates from the drives and doesnot contain unknown external components. On the basis ofthis assumption (13) can be subtracted from (12) to obtainHq − Hqd − H(−K(0)e − K(1)e) + h − h = 0. By addingand subtracting Hq at the right hand side it is obtained that

[H − H]q − H(−e − K(0)e − K(1)e

)+ (h− h) = 0. (14)

This equation evidently can be so rearranged that one s ide ofit contains the model data, while the other side contains themodeling errors defined as H := H − H , h := h − h:

−H(e + K(0)e + K(1)e

)=

= −H(q)q − h(q, q) = Y T (p − p) ≡ −Y T p.(15)

Via multiplying both sides of (15) with the inverse of theknown model H(q) and formally introducing the array x :=[e, e]T an equation of motion can be obtained for the systemwith error-feedback that corresponds to the ”standardizedform” of that of the non–autonomous dynamic systems:[

ee

]−

[0 I

−K(0) −K(1)

] [ee

]=

[0

H−1Y T p

](16)

or

x − Ax =[

0I

]H−1(q)Y (q, q, q)T p ≡ Z. (17)

Now let us try to construct a Lyapunov function of the trackingerror and its 1st time–derivative and of p as V := xT Px +pT Rp where P and R are constant, symmetric positive definitematrices od proper dimensions! Then evidently V = xT Px +xT P x + ˙pT Rp + pR ˙p. Due to the symmetry of P and R ithas the form of

V = xT (PA + AT P )x + 2ZT Px + 2pT R ˙p. (18)

To guarantee V < 0 the following restrictions can be pre-scribed: let S be a positive definite symmetric matrix, and let

PA + AT P = −S,

pT R ˙p + ZT Px = 0(19)


14

The first equation of the group (19) is referred to as theLyapunov equation. Normally an appropriate S is prescribedand the task is to find a proper P for this S by solvingthe Lyapunov equation. The Lyapunov equation evidently setslinear functional connection between the elements of P andS that may or may not have solution. (For the existence ofa solution the real part of each eigenvalue of A must benegative.) Since A = const. the Lyapunov equation has tobe solved only on times in order to find a proper P forthe prescribed S. (Each common software package as e.g.SCILAB immediately yields the solution of this equation in asingle command.) To satisfy the second element of the set (19)ZT has to be expressed from its definition (16). It is obtainedthat

pT(Y H−1[0, I] + R ˙p

)= 0. (20)

Since during the adaptation p �= 0, it is expedient to prescribethe parameter adaptation rule

˙p = −R−1Y H(q)−1[0, I]Px (21)

in which the computational burden mainly consists in the needfor inverting the model H(q) that must have the exact, intricateform determined by the concrete kinematic model of the givenrobot arm.

If the adaptation rule is applied by the controller for theconvergence of this method the following can be stated. V < 0can be estimated as

|V | = | − xT Sx| ≥ SEigmin||x||2 > 0 if x �= 0 , (22)

and in similar manner

|V | ≥ PEigmin||x||2 + REig

min||p||2 (23)

in which the minimal eigenvalues of P and R are selected. Inprinciple the very special case ||x|| → 0 may happen while premains finite. In this case the robot has special trajectory forthe proper tracking of which no full dynamic information isneeded. Other, and more probable possibility is that ||x|| → 0and ||p|| → 0: then the dynamic model is fully learned andthe tracking error asymptotically converges to zero. It an alsobe shown via an indirect proof that it is impossible to havea finite tracking error limit for arbitrarily long time. Really,suppose that ∃E > 0 so that ||x|| ≥ E holds. According to(22) then |V | ≥ SEig

minE2 > 0 that is a finite rate of decay fora finite positive initial V0 > 0 would be set for a non-negativevalue V . This would be a contradiction, therefore the trackingerror and its derivative must asymptotically converge to 0.

V. OBSERVATIONS ON THE TRACKING PROPERTIES OF THE

CLASSICAL METHODS

In contrast to the practice prevailing in the conventionalapproaches in which the prescribed error-relaxation is an in-tegral part of the control problem, in this approach the desiredresponse can freely be defined on the basis of purely kinematicconsiderations quite independently of the dynamic behaviorof the system under consideration. For instance, in the caseof an nth order system in which the nth time-derivative of

the coordinates x(n) can instantaneously be manipulated bythe drives, as it was earlier detailed, the prescription of thedesired x(n)d

(t) can be obtained by driving to zero an errormetrics defined by S(t) = ( d

dt + λ)n[xN (t) − x(t)], λ > 0where xN denotes the nominal, and x the actual coordinates.This requirement used to be typical in the case of nth ordernonlinear systems the exact model of which is available. Forinstance in the case of the ”Globally Linearizing Controllers”as depicted in [6] such ambitions can be realized. In the caseof a 2nd order system e.g. the requirement of xd(t) = xN (t)+P [xN (t) − x(t)] + I

∫ t

t0[xN (τ) − x(τ)]dτ + D[xN (t) − x(t)]

can also be prescribed with appropriate P, I,D parametersguaranteeing the asymptotic convergence x(t) → xN (t) ast → ∞.

We also have to note that similar ”deficiencies” also occur inthe case of the Lyapunov function based techniques. Though inthese techniques the tracking errors form an integral part of theLyapunov function the details of their relaxation cannot exactlybe ”prescribed”. The Barbalat lemma or the manipulationswith the function of ”Class κ” can guarantee only the stabilityor the asymptotic convergence without revealing too muchdetails of this convergence.

VI. SOFT COMPUTING AND UNIVERSAL APPROXIMATORS

The mathematical foundation of the modern Soft Computing(SC) techniques goes back to the middle of the 20th century,namely to the first rebuttal of David Hilbert’s 13th conjecture[15] that was delivered by Arnold [16] and Kolmogorov [17]in 1957. Hilbert supposed that there exist such continuousmulti-variable functions that cannot be decomposed as thefinite superposition of continuous functions of less variables.Kolmogorov provided a constructive proof stating that ar-bitrary continuous function on a compact domain can beapproximated with arbitrary accuracy by the composition ofsingle-variable continuous functions. Though the constructionof Kolmogorov’s functions that are used in this theorem isdifficult, his theorem later was found to be the mathematicalbasis of the present SC techniques.

From the late eighties several authors proved that differenttypes of neural networks possessed the universal approxima-tion property [18],[19],[20],[21]. Similar results have beenpublished from the early nineties in fuzzy theory claimingthat different fuzzy reasoning methods are related to universalapproximators, too [22],[23],[24].

In spite of these theoretically inspiring and promising con-clusions, from the point of view of the practical applicabilityof these methods various theoretical doubts emerged. The mostsignificant problem was, and remained important problem evenin our days, the ”curse of dimensionality” that means that theapproximating models have exponential complexity in termsof the number of components i.e. the number of componentsgrows exponentially as the approximation error tends to zero.If the number of the components is bounded, the resulting setof models is nowhere dense in the space of the approximatedfunctions. These observations frequently were formulated in anegatory style, as e.g. in [25] stating that ”Sugeno controllers


15

with a bounded number of rules are nowhere dense”, andinitiated various investigations on the nowhere denseness ofcertain fuzzy controllers containing prerestricted number ofrules e.g. in [26], [27].

In general similar problems arise with the application of theTensor Product (TP) representation of multiple variable con-tinuous functions that was also extended to Linear Parameter-Varying (LPV) models [28]. The TP representation can beused for achieving polytopic decomposition of LPV models i.e.obtaining a linear combination of Linear Time-Invariant (LTI)models in which the coefficients of the linear combinationdepend on time. The application of the Higher Order SingularValue Decomposition (HOSVD) provides this result in anespecially convenient form [29], [30]. Such a preparationor preprocessing of the initial model is very attractive frompractical point of view since due to it the Lyapunov-functionsbased stability criteria generally used in the control of non-linear systems can be reformulated in the form of LinearMatrix Inequalities (LMI). Due to the pioneering work byGahinet, Apkarian, Chilai [31], Boyd [32], and Bokor e.g. [33],[34], the feasibility problem of Lyapunov-based criteria wasreinterpreted as a Convex Optimization Problem. J. Bokor andhis research group gave a very lucid geometrical interpretationof this new representation and methodology that was found tobe very fruitful in solving optimization problems, too, beyondstability issues. When a polytopic model decomposition isrealized and the appropriate control is designed by the use ofcommercially available softwares as e.g. MATLAB as in [35]the available finite computational capacity always seems to bea ”bottleneck”. Possible complexity reduction techniques ase.g. HOSVD have to be applied in order to remain withintreatable problem sizes. This technique reduces modelingaccuracy in a ”controlled” or at least well interpreted manner[36].

A. Observations on Scalability Problems

In contrast to these observations SC techniques obtainedvery wide range of real practical applications. As examplesimplementation of backward identification methods [37], thecontrol of a furnace testing various features of plastic threads[38], [39], sensor data fusion [40] can be mentioned. Themethodology of the SC techniques, partly concerning controlapplications, had fast theoretical development in recent years,too. Various operators concerning the operation of the fuzzyinference processes were investigated in [41],[42], minimumand maximum fuzziness generalized operators were invented[43], and new parametric operator families were introduced[44], etc.

To resolve the seemingly ”antagonistic” contradiction be-tween the successful practical applications and the theoreti-cally proved ”nowhere denseness properties” of SC methodsone became apt to arrive at the conclusion that the problemroots in the fact that Kolmogorov’s approximation theoremis valid for the very wide class of continuous functions thatcontains even very ”extreme” elements at least from the pointof view of the technical applications. (The first example

0.00 0.25 0.50 0.75 1.0010.00

11.67

13.33

15.00

16.67f(x) [10^−1]

0.0315 0.0608 0.0901 0.1194 0.148611.500

11.829

12.157

12.486

12.815f(x) [10^−1]

0.05147 0.05596 0.06044 0.06493 0.0694112.040

12.144

12.247

12.350

12.453f(x) [10^−1]

0.2964 0.3369 0.3775 0.4180 0.458615.685

15.870

16.055

16.240

16.426f(x) [10^−1]

Fig. 6. The fractal-like graphs of the fξ(x) functions for increasing ξ ∈{0, 0.02, ..., 0.98} values (increasing zooming ratios from left to right andfrom up to down)

of a function that everywhere is continuous but nowhere isdifferentiable was given by Weierstraß in 1872 [45].)

To highlight the ”generality” of the concept of continuousfunctions another examples were given e.g. in [46] where aset of functions that are everywhere continuous in the openinterval (0, 1), and almost everywhere are differentiable withinthis interval were presented. These functions are defined as aseries built up from a set of ”triangular functions” in whichϕ0(x) := 1, ϕ1 := 2x if x ∈ (0, 0.5), 1 − 2(x − 0.5) ifx ∈ [0.5, 1), etc. Each {ϕi|i ≥ 1} function takes the value 0at x = 0, oscillates between 0 and 1 with constant derivativesof ±2i, and cannot be differentiated at the halving (”break”)points of the (0, 1) interval that can be expressed in the formof x = m

2n , where m,n are positive integers. The functionsunder consideration are defined as fξ(x) =

∑∞s=0(ξ/2)sϕs(x)

for the parameter ξ ∈ [0, 1). The ”graphs” of these functionsare ”visualized” in Fig. 6 for increasing zooming ratios. Thegraphs evidently show fractal structure since in each zoomingratio they have some ”zigzags”. All this indicates that theproblem of ”dimensionality” may mean serious difficultieseven in the case of one dimension within the class of thecontinuous functions.

Intuitively it can be expected that if we restrict our models tothe far better behaving ”everywhere differentiable” functionsthe problems with the dimensionality ab ovo could be evadedor at least reduced.

VII. THE NOVEL GEOMETRIC APPROACH

The first efforts in the direction of applying uniform struc-tures and procedures in quite different way as it is donein the classic SC applications were summarized in [47] inwhich the sizes of the necessary uniform structures used fordeveloping partial, temporal, and situation-dependent modelsthat needed continuous maintaining were definitely determinedby the degree of freedom of the system to be controlled.These considerations were based on the modification of theRenormalization Transformation, and were valid only for


16

”increasing systems” in which the ”increase” in the necessaryresponse could be achieved by also increasing the necessaryexcitation, and vice versa.

In [48] this idea was systematically extended for SingleInput - Single Output (SISO) ”increasing” and ”decreas-ing” systems by developing various Parametric Fixed PointTransformations more or less akin to the RenormalizationTransformation.

In the sequel the basic idea, i.e. ”exciting” and ”observing”the response of the system to be controlled will be detailed.

A. The Idea of the Novel Adaptive Control: the Excitation –Response Scheme

Each control task can be formulated by using the conceptsof the appropriate ”excitation” Q of the controlled system towhich it is expected to respond by some prescribed or ”desiredresponse” rd. The appropriate excitation can be computed bythe use of some inverse dynamic model Q = ϕ(rd). Sincenormally this inverse model is neither complete nor exact,the actual response determined by the system’s dynamics, ψ,results in a realized response rr that differs from the desiredone: rr ≡ ψ(ϕ(rd)) ≡ f(rd) �= rd. It is worth notingthat the functions ϕ() and ψ() may contain various hiddenparameters that partly correspond to the dynamic model of thesystem, and partly pertain to unknown external dynamic forcesacting on it. Due to phenomenological reasons the controllercan manipulate or ”deform” the input value from rd so thatrr ≡ ψ(rd

∗). Other possibility is the manipulation of the outputof the rough model as rr ≡ ψ(ϕ∗(rd)). In the sequel it will beshown that for SISO systems the appropriate deformation canbe defined as some Parametric Fixed Point Transformation.

B. Possible Fixed–Point Transformations

The original idea for increasing systems is graphically in-terpreted in Fig. 7 suggesting the iteration xn+1 = xd

f(xn)xn ≡θ(xn|xd). Really, if f(x�) = xd then θ(x�|xd) = x�,and it can be expected that from an initial value x0 theiteration converges to the fixed point x�. Fixed point problemsin general have the advantageous feature that they can besolved via simple iteration provided that this iteration isconvergent. Really, consider the sequence of points {x0, x1 =Ψ(x0), ..., xn+1 = Ψ(xn), ...} obtained via iteration! Let ussuppose that this series converges to some xn → x∗. In orderto apply iterations let us consider the set of the real numbers� as a linear normed space with the common addition andmultiplication with real numbers, and with the absolute value|•| as a norm! It is well known that this space is complete, i.e. itis a Banach Space in which the Cauchy Series are convergent.Due to that, using the norm inequality it is obtained that

|Ψ(x∗) − x∗| ≤ |Ψ(x∗) − xn| + |xn − x∗| == |Ψ(x∗) − Ψ(xn−1)| + |xn − x∗|. (24)

It is evident from (24) that if Ψ is continuous then Ψ(x∗) =x∗, i.e. x∗ = x�, that is the desired fixed point is found bythe iteration because in the right hand side of (24) both termsconverge to 0 as xn → x∗. (It is worth noting that besides

Fig. 7. Geometric Interpretation of the Modified Renormalization Transfor-mation

Fig. 8. Possible false convergence of the Modified Renormalization Trans-formation: the iteration converges to 0 while the actual solution evidently isnegative

divergence, false convergence also may occur as it is indicatedon Fig. 8.)

The next question is giving the necessary or at least asatisfactory condition of this convergence. It also is evidentthat for this purpose contractivity of Ψ(•), i.e. the propertythat |Ψ(a)−Ψ(b)| ≤ K|a− b| with 0 ≤ K < 1 is satisfactorysince it leads to a Cauchy series (|xn+L − xn| → 0 ∀L ∈ N):

|xn+L − xn| = |Ψ(xn+L−1) − Ψ(xn−1)| ≤ ...≤ Kn|xL − x0| → 0 as n → ∞ (25)

For guaranteeing the contractivity of a differentiable � → �function Ψ(•) proper limitation on the absolute value of thederivative |Ψ′| ≤ K < 1 is satisfactory since

|Ψ(a) − Ψ(b)| =∣∣∣∫ b

aΨ′(x)dx

∣∣∣ ≤≤ ∫ b

a|Ψ′(x)|dx ≤ K|a − b|

(26)

that means that if Ψ is flat enough around the fixed point the


17

Fig. 9. Possible convergence for 0 < ζ < 1 for decreasing systems [thefigure belongs to Eq. (28)]

iteration will converge to it. Since

dθ

dx= −xdxf ′(x)

f(x)2+

xd

f(x)(27)

the situation of −1 < θ′(x�) = 1 − xxd f ′(x�) < 1 can

be achieved if x ≈ xd, and 0 < f ′(x�) is small enough.This latter condition can be satisfied by choosing very flatinitial model ϕ if ψ(•) is not singular. For f ′ < 0 thisiteration scarcely can converge. For negative system in [48] thefollowing iteration was proposed with small ζ > 0 describedin Fig. 9 and in (28):

xn+1 =f(xn) + ζ(f(xn) − xd)

f(xn)xn ≡ ϑ(xn|xd). (28)

Evidently ϑ(x�|xd) = x�, ϑ′ = 1 + ζ − ζ xd

f(x) + ζ xdxf(x)2 f ′, so

ϑ′(x�) = 1 + ζ xxd f ′(x�). If 0 < ζ < 1, x ≈ xd, f ′(x�) < 0,

and |f(x�)′| is small enough this iteration will converge tothe fixed point x�, too. To evade numerical problems withxd = 0 and f(x�) = 0 an additional ”shift parameter” D canbe introduced into (28) with the properties as follows:

ϑ(x|ζ,D) := f(x)+D+ζ(f(x)−xd)f(x)+D x,

ϑ(x�|ζ,D) = x�,dϑ(x|ζ,D)

dx = 1 + ζ f(x)−xd

f(x)+D +

+xζf ′(x) f(x)+D−(f(x)−xd)(f(x)+D)2,[

dϑ(x|ζ,D)dx

]x=x�

= 1 + ζf ′(x�)x�

xd+D

(29)

The convergence of (29) may happen under conditions similarto that of (28). However, the illustrative examples of thepossibility of convergence to a ”false value” as illustrated inFig. 8 and the relatively complicated derivatives in Eqs. (29)that makes not very easy to find proper D and ζ parametersmade it actual to find other transformations that have morelucid nature.

In the case of SISO systems this can be done by introducingthree shift parameters ad D−, Δ+, and Δ− as indicated byFigs. 10–13 in which the proposed iterative mappings are

Fig. 10. Fixed point transformation for f ′(x) > 0 with parameters D−and Δ− belonging to Eqs. (30); if x� > D−, xd > Δ−, f ′(x�) >0, and |f ′(x�)| is small enough, the iteration generated by the functionh(x|xd, D−, Δ−) converges to x�

Fig. 11. Fixed point transformation for f ′(x) > 0 with parameters D− andΔ+ belonging to Eq.(31); if x� > D−, xd < Δ+, f ′(x�) > 0, and |f ′(x�)|is small enough, the iteration generated by the function g(x|xd, D−, Δ+)converges to x�

defined by simply considering geometrically similar triangles.To be brief, the figure captions that refer to the appropriategroup of equations explain the conditions that are needed forthe convergence of the appropriate choice.

h(x|xd, D−,Δ−) :== (xd−Δ−)(x−D−)

f(x)−Δ−+ D−,

h(x�|xd, D−,Δ−) = x�,

h′ = (xd−Δ−)(f(x)−Δ−−f ′(x)(x−D−))(f(x)−Δ−)2 ,

h′(x�|xd, D−,Δ−) = 1 − f ′(x�)x�−D−xd−Δ−

(30)

g(x|xd, D−,Δ+) :== (f(x)−Δ+)(x−D−)

xd−Δ++ D−,

g′ = f ′(x) x−D−xd−Δ+

+ f(x)−Δ+xd−Δ+

,

g′(x�|xd, D−,Δ+) = 1 + f ′(x�)x�−D−xd−Δ+

(31)


18

Fig. 12. Fixed point transformation for f ′(x) < 0 with parameters D−and Δ− belonging to Eqs. (32); if x� > D−, xd > Δ−, f ′(x�) <0, and |f ′(x�)| is small enough, the iteration generated by the functiong(x|xd, D−, Δ−) converges to x�

Fig. 13. Fixed point transformation for f ′(x) < 0 with parameters D−and Δ+ belonging to Eqs. (33); if x� > D−, xd < Δ+, f ′(x�) <0, and |f ′(x�)| is small enough, the iteration generated by the functionh(x|xd, D−, Δ+) converges to x�

g(x|xd, D−,Δ−) :== (f(x)−Δ−)(x−D−)

xd−Δ−+ D−,

g′ = f ′(x) x−D−xd−Δ−

+ f(x)−Δ−xd−Δ−

,

g′(x�|xd, D−,Δ−) = 1 + f ′(x�)x�−D−xd−Δ−

(32)

h(x|xd, D−,Δ+) :== (xd−Δ+)(x−D−)

f(x)−Δ++ D−,

h(x�|xd, D−,Δ+) = x�,

h′ = (xd−Δ+)(f(x)−Δ+−f ′(x)(x−D−))(f(x)−Δ+)2 ,

h′(x�|xd, D−,Δ+) = 1 − f ′(x�)x�−D−xd−Δ+

(33)

It is worth noting that is the rough dynamic model consistsof an affine function the desired response as ϕ(rd) = ard + b,in SISO systems, besides the parameters a and b the control isdefined by further simple parameters as D−, Δ−, or Δ+, it isvery easy to find the proper settings by computer simulations.

To show that it is enough to observe the derivatives un the fixedpoint x�. For instance, consider e.g. h′(x�|xd, D−,Δ+) = 1−f ′(x�)

x�−D−xd−Δ+

! If |x�| � |D−|, |xd| � |Δ+|, and D− and Δ+

are of the same order of magnitude then x�−D−xd−Δ+

≈ D−Delta+

=const., almost independently of xd and x�. Furthermore, if|a| is small enough the |h′| < 1 condition of convergencealmost surely can be met. So the control parameter ”experi-mentally” can be set via simulations by choosing small a, andcomparably big D−, Δ+. Later the absolute values of theselatter two parameters can be decreased to achieve more andmore sensitive control until reaching the limits of stability.If necessary, |a| can also be decreased. (Evidently similarconsiderations can be done for the parameters of the casesdescribed in Figs. 10–13.) To exemplify the operation of thesecontrol strategies in the sequel simulations will be presentedas the adaptive counterparts of the results in Fig. 3 that belongto the control of the Ball–Beam System.

C. Simulation Results for the Ball–Beam System

In the adaptive case the goal of the controller was to achievethe S := ( d

dt + λ)4(xN − x) = 0 condition for relaxingthe tracking error with λ = 6 (s−1) as in the case of Fig.3 of the robust VS/SM controller. The results depicted inFig. 14 belong to the non-adaptive control of a rough modelnot containing friction terms. The structure of the curves ofthe friction torque and the actuator torque well illustrate thatdue to the feedback policy expressed by S = 0 considerablerecurrent information was used by the controller: the actuatortorque seems to compensate the friction torques. However, thiscompensation is not precise enough to obtain precise tracking:the desired (d4x/dt4)d derivatives are very imprecisely set bythe controller.

The adaptive counterpart of the results depicted in Fig. 14obtained by the control using only function g are given inFig. 15. It has be noted that within each control cycle of theduration of 1 (ms) only one step was executed of the iteration.If the adaptation is faster than the dynamics of the system to becontrolled appropriate result can be expected even in his case,too. This approach is similar to the application of CellularNeural Networks (CNN) for image processing. In relation tothe operation of CNNs the concept of ”Complete Stability”can be introduced that means that a static input picture ismapped to a static output picture following a short dynamictransition of the physical state of the CNN. If the input pictureis not static but varies ”slowly” in comparison with the ”speed”of the CNN’s internal dynamics varying picture is mapped tovarying output [49]. In spite of using a single step duringone control cycle from each point of view the improvement isconsiderable. For the purpose of more precise comparison inthis latter figure the 4th line has been introduced to show thetracking error and the ”cumulative adaptive factor” as definedin the caption of the figure. It is evident that the adaptive factorvaries in correlation with the friction torque.

To show that the use of function g results in similar controlFig. 16 has been completed. It is exact counterpart of Fig. 15:the difference only consists in using g instead of h.


19

−3 −2 −1 0 1−6

−4

−2

0

2

4


0 2 4 6 8 10−3

−2

−1

0

1x vs. time [10^−1 m vs s]

−2.5−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0 2.5−8

−6

−4

−2

0

2

4

6


0 2 4 6 8 10−15

−10

−5

0

5

10

15The friction torque [10^1 N×m] vs time [s]

0 2 4 6 8 10−15

−10

−5

0

5

10

15The actuator torque [10^1 N×m] vs time [s]

0 2 4 6 8 10−50

−40

−30

−20

−10

0

10

20

30xD(4) and x(4) vs. time [10^1 m/s^4 vs s]

Fig. 14. Simulation results for the non-adaptive control of the Ball-BeamSystem: nominal and computed phase trajectory of x (upper left); nominaland computed x(t) (upper right); the phase trajectory of the tilting angle ϕ(middle left); the friction torque at the rotary axle (middle right); the actuator’storque (lower left); the desired and the computed d4x/dt4 (lower right)

It has to be noted that in contrast to the operation ofthe robust controller, in the adaptive control the variation ofthe control torque is as smooth as possible. Only the drasticvariation of the friction torques cause drastic but fluctuatingvariation in it. In the sequel the possibilities for extending thismethod for MIMO systems will briefly be outlined.

D. Possible Extension of the Method for MIMO Systems

With the introduction of the number sn+1 := xn+1/xn theoriginal iteration xn+1 = xd

f(xn)xn can be expressed in theform of

sn+1f(xn) = xd, xn+1 = sn+1xn etc. (34)

that immediately can be extended for MIMO systems if wesuppose that x, f ∈ �N , and the linear transformations aredescribed by the matrices sn of dimensions N × N . Eq.(34) does not yield unambiguous on the matrices sn, so thisambiguity can be utilized from practical point of view to obtaineasily treatable matrices. A plausible possibility is to put theknown vectors into some blocks of bigger matrices as

sn+1

[f(xn) b

d c

]=

[xd b′

d′ c′

](35)

in which the blocks b, d, c and b′, d′, c′ can be determined invarious manners. If the matrix containing f(xn) is nonsingulara plausible proposal for sn+1 is

sn+1 =[

xd b′

d′ c′

]×

[f(xn) b

d c

]−1

. (36)

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5−4

−2

0

2


0 2 4 6 8 10−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5x vs. time [10^−1 m vs s]

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5−6

−4

−2

0

2

4

6


0 2 4 6 8 10−15

−10

−5

0

5

10


0 2 4 6 8 10−15

−10

−5

0

5

10


0 2 4 6 8 10−9.8

−6.5

−3.2

0.1

3.4

6.7

10.0xD(4) and x(4) vs. time [10^1 m/s^4 vs s]

0 2 4 6 8 10−0.5

0.0

0.5

1.0

1.5

2.0

2.5Tracking error for x [10^−2 m] vs time [s]

0 2 4 6 8 109.6

9.7

9.8

9.9

10.0

10.1

10.2

10.3Adaptive factor s [10^−1 dimless] vs time [s]

Fig. 15. Simulation results for the adaptive control of the Ball-Beam Systemusing only function g: nominal and computed phase trajectory of x (1st lineleft); nominal and computed x(t) (1st line right); the phase trajectory ofthe tilting angle ϕ (2nd line left); the friction torque at the rotary axle (2ndline right); the actuator’s torque (3rd line left); the desired and the computedd4x/dt4 (3rd line right); tracking error of x (4th line left); the ”cumulativeadaptive factor” s(tn) :=

∏nj=0 srel(tj) constructed of the instantaneous

factors srel(tn) :=x(4)d

(tn)−Δ+x(4)(tn)−Δ+

for function h

It is evidently advantageous to have sn+1 = I (i.e. to havethe unit matrix) if xd = f(xn) that can simply be achievedif the appropriate matrices are constructed of xd and f(xn)according to the same procedure. It is worth noting thatthe lower blocks d, d′ serve to avoid zero columns for thexd = 0 and f(xn) = 0 cases. For the existence of theinverse the remaining columns must be linearly independentof each other and of the columns being in the precedingblocks. Furthermore, since the calculation of the inverse ofmatrices may mean computationally considerable burden, itis advantageous to chose simply invertible matrices. Suchmatrices occur in the fundamental quadratic expressions thatdefine the essential symmetries in Physics as

LT PL = P, det(P ) �= 0, [P−1LT P ]L = I (37)

that indicates that P−1LT P = L−1. For instance, the funda-mental symmetry of the Euclidean Geometry is the RotationGroup defined by P ≡ I . The fundamental symmetry ofElectrodynamics, and in more general, relativistic physics is


20

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5−4

−2

0

2


0 2 4 6 8 10−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5x vs. time [10^−1 m vs s]

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5−6

−4

−2

0

2

4

6


0 2 4 6 8 10−15

−10

−5

0

5

10


0 2 4 6 8 10−15

−10

−5

0

5

10


−0 2 4 6 8 10−10.00

−6.67

−3.33

0.00

3.33

6.67

10.00xD(4) and x(4) vs. time [10^1 m/s^4 vs s]

0 2 4 6 8 10−0.5

0.0

0.5

1.0

1.5

2.0

2.5Tracking error for x [10^−2 m] vs time [s]

0 2 4 6 8 109.6

9.7

9.8

9.9

10.0

10.1

10.2

10.3Adaptive factor s [10^−1 dimless] vs time [s]

Fig. 16. Simulation results for the adaptive control of the Ball-Beam Systemusing only function h: nominal and computed phase trajectory of x (1st lineleft); nominal and computed x(t) (1st line right); the phase trajectory ofthe tilting angle ϕ (2nd line left); the friction torque at the rotary axle (2ndline right); the actuator’s torque (3rd line left); the desired and the computedd4x/dt4 (3rd line right); tracking error of x (4th line left); the ”cumulativeadaptive factor” s(tn) :=

∏nj=0 srel(tj) constructed of the instantaneous

factors srel(tn) :=x(4)(tn)−Δ+

x(4)d(tn)−Δ+

for function g

the Lorentz group defined by P = g :=[

I 00 −c2

]in which

c denotes the velocity of light in vacuum. The appropriategeometry is called Minkowski Geometry having the metrictensor g. The fundamental internal symmetry of ClassicalMechanics is described by the Symplectic Group defined by

P = � :=[

0 I−I 0

]. All of these examples correspond

to Lie groups that contain near unity elements in the formof L ≈ I + ξG, in which ξ is some scalar parameter, andG is a generator of the appropriate group. This property isvery promising if we take it into account that the series of thematrices has to converge to I if the iteration is convergent.

On the basis of the above considerations various matriceswere constructed for control technical purposes. The Gen-eralized Lorentz Group contains e.g. the following specialelements:[

e(f) 2√|f |2/c2 + 1 e(2) ... eDOF f

|f |/c2 0 ... 0 2√|f |2/c2 + 1

](38)

in which DOF abbreviates the Degree of Freedom of thecontrolled system, c may have arbitrary nonzero real value,e(f) is a unit vector in the direction of f , e(2), ..., e(DOF ) arepairwisely orthogonal unit vectors in the orthogonal subspaceof f . They are made by rigidly so rotating the orthonormalcolumns of the unit matrix that the first column is rotated tothe direction of f , and the whole rotation happens in the 2Dsubspace spanned by e

(1)original and f .

Among others another practical choice is the use of specialsymplectic matrices built up of the columns of matrix M asfollows:

M =

⎡⎢⎣ f −f e(3)1 ... e

(DOF )1

d −d e(3)DOF+1 ... e

(DOF )DOF+1

D|f|2+d2

De(3)DOF+2 ... e

(DOF )DOF+2

⎤⎥⎦ , (39)

and

s =

[0 ... 0 −m(1)

σ... −m(DOF+2)

σm(1) ... m(DOF+2) 0 ... 0

](40)

It is not difficult to show that (40) is symplectic if D2 =|f |2+d2, and σ = 2D2. It is evident that the first two columnsof M in (39) are orthogonal to each other. The remainingunit vectors

{e(3),...,e(DOF )

}lie in the orthogonal subspace

of the first two columns and they are pairwisely orthogonalto each other, too. They are constructed from the orthonormalcolumns of the unit matrix by two successive rigid rotations.The first rotation moves e(1)original into m(1) while leaving theorthogonal subspace of these two vectors invariant and yieldse(1)′ of e(1)original . The next rotation turns e(2)′ to m(2) whileleaving the orthogonal subspace of these two vectors invariant.We note that the previously set vector e(1)′ is the elementof this latter set left invariant. These rotations can easily beconstructed on the basis of simple, elementary considerations.

At the time being we have successful results regardingthe application of the above matrices for Monotone Increas-ing MIMO Systems. In such systems the ”increase” in thenecessary response could be achieved by also ”increasing”the necessary excitation, and vice versa. The concepts as”increase” and ”decrease” were generalized from a singledimensional problem to a multiple dimensional one by usingthe concepts of ”acute angles” (replacing the term ”increase”with ”varies almost in the same direction”) and ”obtuseangles” (replacing the term ”decrease” with ”varies almost inthe opposite direction”), respectively, in real Hilbert spaces inwhich the angle between various vectors can be defined fromthe scalar product. The appropriate proof for such systems wasgiven in [47] by the use of perturbation calculus.

Regarding application issues in [50] adaptive control ofa 3 DOF SCARA robot arm with deformable axles alsosuffering from backlash was given. The role of adaptivity iswell exemplified by Fig. 17.

VIII. CONCLUSIONS

In this paper the most popular traditional nonlinear controlsolutions including soft computing issues were comparedto a novel approach based on simple and lucid geometric


21

qp1qp2qp3−6.03−4.87−3.72−2.56−1.41−0.250.90 2.06 3.21 4.37 5.52

−392

−266

−139

−13

114

240

367Phase Space

4.4

−0.6

−5.6−3.50

−0.003.50 3.49

−0.00

−3.49

4.4

−0.6

−5.6

z

−3.50−0.00

3.50y

3.49

−0.00

−3.49

x

Endpoint Trajectory in a Cartesian System

qp1qp2qp3−0.462−0.306−0.1490.0070.1640.3200.4760.6330.7890.9461.102

−0.947

−0.630

−0.312

0.006

0.323

0.641

0.959Phase Space

0.942

0.469

−0.0040.00

1.21

2.43 3.00

2.38

1.76

0.942

0.469

−0.004

z

0.00

1.21

2.43

y

3.00

2.38

1.76

x

Endpoint Trajectory in a Cartesian System

Fig. 17. Control of a 3 DOF SCARA robot arm with deformable axles alsosuffering from backlash: non-adaptive control: 1st line, adaptive control: 2ndline

principles. A brief history of geometric way of thinking innatural sciences was delivered, too. Certain statement werealso illustrated via simulation results.

It can be concluded the due to its simplicity the novelapproach seems to be very attractive and promising.

For not fully driven systems in which the functional relation-ship between the excitation and the response is not necessarily”increasing” or ”decreasing” development of certain observerseem to be necessary for guaranteeing the convergence of themethod.

It can also be expected that the here outlined simplerapproaches can be extended from the control of SISO systemsto that of MIMO systems. For instance, in the expressionh = xd−Δ+

f(x)−Δ+(x − D−) + D− the fractional part can be

substituted with matrix products the matrices of which cansimilarly constructed as in the detailed examples. This ideaneeds further elaboration and tests.

ACKNOWLEDGMENT

The authors gratefully acknowledge the support by the Hun-garian National Research Fund (OTKA) within the ProjectsNo. K063405 and T048756.

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