1' dia

1
Fixed Point Transformations as Simple Geometric
Alternatives in Adaptive Control
József K. TarInstitute of Intelligent
Engineering Systems John von Neumann Faculty of
Informatics Budapest Tech Polytechnical
Institution
Plenary Talk at the 5th IEEE International
Conference on Computational Cybernetics (ICCC
2007), October 19-21, 2007, Gammarth,
Tunisia Saturday, October 20th, 2007
2
Brief Contents of the Presentation

• 1. Comments on the history and significance of
  geometric way of thinking in natural
  sciences
• 2. Geometrical interpretation of the state
  propagation equation of dynamical systems
  geometrization of Classical Mechanics
• Approaches based on the concept of classical
  analytical modeling geometrical
  interpretation of Lyapunovs 2nd Method
• Soft Computing as alternative modeling
• The adaptive approach elaborated at Budapest
  Tech a) for positive definite MIMO systems, b)
  increasing and decreasing SISO systems, and c)
  generalization for MIMO systems using the
  geometric interpretation of SVD
• Simulation examples for various physical systems

3
Comments on the history and significance of
geometric way of thinking in natural sciences
The foundations of strict, formal mathematical
thinking historicallywas in coincidence of the
axiomatic formulation of Geometry that happened
a long time ago (300 BC) by Euclid of
Alexandria. The effects of geometrical way of
thinking obtained real significancein natural
sciences only from the end of the 18th century
when various levels of abstraction came into
fashion. Geometry obtained application at
various levels of abstraction.
Its main advantage is that it makes it possible
to apply lucid and simple problem formulation and
argumentation (we became familiar with in our
childhood) on these very abstract fields
generating the feeling of cosy familiarity.
4
Giuseppe Lodovico Lagrangia (1736
1813) Mécanique Analytique (1788) the basis
for all later works in this field. Lagrange
Multipliers, Variation Calculus
Sir William Rowan Hamilton (1805
1865) Introduction of directed quantities
(quaternions) Hamilton Principle Canonical
equations of motion in Classical Mechanics
Euclid of Alexandria Method of proving
mathematical theorems by logical
reasoning. Axioms in Geometry etc. (300
BC) (Justus van Ghent's 15th-century depiction.)
5
Hermann Günther Grassmann (1809 1877) German
polymath Theory of Extensive Magnitudes
(1832-1844) the first known appearance of
linear algebra and the notion of vector space.
Marius Sophus Lie (1842 1899) On a Class of
Geometric TransformationsStudied the
continuous transformation groups by considering
their tangent space and their generating vector
fields.
David Hilbert (1862 1943) Generalized the
Euclidean Geometry to complex vectors spaces of
infinite number of dimensions (separable and
non-separable Hilbert Spaces)
He realized the insignificance of the three
dimensions in the generalization.
6
Vladimir Igorevich Arnold (???????? ????????
??????? (1937-) made contributions in a number
of areas as e.g. symplectic topology, algebraic
geometry, classical mechanics and singularity
theory.
William Kingdon Clifford (1845 - 1879) English
mathematician and philosopher. With Hermann
Grassmann, he invented Geometric Algebra,
Clifford Algebras playing a role in
contemporary mathematical physics, technical
applications.
Clifford was the first to suggest that
gravitation might be a manifestation of an
underlying Riemannian Geometry.
Georg Friedrich Bernhard Riemann (1826 -
1866) Opened up research areas combining
analysis with geometry. By using Riemannian
Geometry physical quantities can be described in
curved spaces. (Tensors.)
7
Geometrical interpretation of the state
propagation equation of dynamical systems
geometrization of Classical Mechanics
If certain part of Q can be expressed by the
gradient of certain potential energy term as
-?V(q)/?q, instead of T L(q,dq/dt)T-V can
evidently be used.
This restriction establishes connection between
the formal analytical and the phenomenology-close
original approach by Galilei and Newton, and
gives physical meaning to Q!
Introduction of the concept of Kinetic Energy
with respect to Inertial Systems ofReference
This form does not offer immediately a lucid
geometric interpretation! For this a Legendre
Transformation has to be executed on T to
eliminate the second time-derivatives of q.
Variational Principle (Hamilton principle)
Solution Euler-Lagrange Equations of Motion
8
The Legendre Transformation
The replaced quantity appears as partial
derivative of the new function by the new
variable. The partial derivative by the variables
not replaced simply change sign.
Certain independent arguments of function f may
be substituted by partial derivatives of f
according to that variables in a new function.
Cancelling terms
Apply this for T or LT-V if some potential
energy V(q) can be also introduced in order to
eliminate the appearance of the 1st
time-derivative before the 2nd derivation in the
Euler-Lagrange equations of motion!
Cancelling terms
9
Cancelling terms
Cancelling terms
10
The present form has lucid geometric
interpretation
Canonical Equations of Motion
Normal vector of the hypersurface Hconst.
The state-propagation vector is within the
hypersurface.
Tangent space of the hypersurface Hconst.
11
Further geometric analogies for Q0
Consider any physical quantity characteristic to
the system that exclusively depends on its
physical state! Its time-derivative can be
expressed by a typical quadratic structure
Poisson Brackets
A fictitious flow generatedby the physical
quantity F is defined as
12
The quantity F is symmetry of H if this
fictitious flow leaves thevalue of H invariant,
i.e.
To each symmetry of H belongs a conserved
physical quantity and vice versa.
The set of the possible physical states can be
regarded as a smooth geometric object to the
same point of which differentx, x(x) coordinate
values may belong like in the case of
representing the same point of the surface of
the Earth in various maps
13
The time-derivatives of the state-dependent
physical quantities obey similar equations if
the coordinates of the new map are used
14
The special case in which A?? by definition
corresponds to the so called canonical x(x)
transformations!
This special form is analogous with the
fundamental quadratic form of other well known
geometries used in various fields of Physics.
15

• Euclidean Geometry
• Scalar product of vectors aTIb
• Orthogonal Transformations aOa, OTIOI

The fundamental quadratic term is related to the
square of measurable distances.

• Symplectic Geometry
• Gradients in the Poisson Brackets aT ?b
• The Jacobian of the Canonical Transformations
  are Symplectic Matrices S?ST?

The fundamental quadratic term is related to the
time-derivative of physical quantities depending
only on the state of the system.
The fundamental quadratic term is related to the
constant speed of propagation of light with
respect to optically inertial systems of
reference (Michelson-Morley Experiment).

• Minkowski Geometry
• Scalar product with a metric tensor aTgb,
• Lorentz-transformations aLa, LTgLg

16
Definition of the Concept of Groups
Definition An abstract set G is a group if there
is defined on it a GG?G mapping with the
following properties (the so called group
product)

• The group product is associative, that is
  (gh)ug(hu) holds for arbitrary g,h,u?G.
• The group contains the unit element e?G for
  which eggeg for each g?G.
• Each element of G has a unique left and right
  side inverse g-1ggg-1e.

Sophus Lies method of introducing the linear
combinations of the elements of these symmetry
groups with the matrix product (the Group
Algebra) and considering near identity
transformations (i.e. studying the Tangent Space
of the group) was very fruitful
17
Infinitesimal transformations near the unit
element can generate the whole group.They can be
obtained by considering the fundamental quadratic
expression in the first order of a continuous
transformation parameter as e.g. In the case of
the Symplectic Group
The dimensions of the tangent space at I can
easily be determined. The tangent space at any
other element of the group has just the same
structure as that of the unit element since by
considering e.g. time-dependent continuous
parametric transformations it is obtained that
By definition this term must be a generator G(t)!
18
On the basis of formal considerations it holds
that
Compare it with the previous expression!
This term must be a generator that appears in a
simple differential equation!
In this manner each generator generates infinite
number of group elements that are parametrized by
G and a continuous real variable t!
19
This gives the possibility for lucid geometric
interpretation
The tangent space at the unit element
Generator H as the element of the tangent space
at the unit element
By considering time-dependent transformations
starting from the unit element it can be shown
that the generators form a linear vectors space
and an algebra, too!
The unit element having distinguished
significance
The set of group elements as a hypersurfaceembedd
ed in a higher dimensional space
The S(t) trajectory on the hypersurface generated
by generator H!
20
The linear combination of the generators is
generator, too, therefore they form a linear
space!
21
With the multiplication rule of using the
commutator a Lie Algebra is obtained!
It is neither commutative nor associative, but
plays very important role in Natural Sciences!
22
Lie-Cartan Structure Coefficients Clebsh-Gordan
Series
Application to Classical Mechanics in the
Canonical form of the equations of motion for a
short time interval ?t
The canonical equations of motion of the isolated
Classical Mechanical Systems canonically map the
state space onto itself!
?A is a generator of the Symplectic Group!
23
Liouvilles Theorem the state propagation of the
conservative system is similar to the flow of
incompressible fluids
The volume of a fluid cell around x0, determined
by the neighboringvertices x0?x(i)
24
Symmetric in (s,z)
Skew symmetric in (s,z)
25
In the theory of dynamical systems, and control
theory, Lyapunov functions are a family of
functions that can be used to demonstrate the
stability or instability of some state points of
a system. The demonstration of stability or
instability requires finding a Lyapunov function
for that system. There is no direct method to
obtain a Lyapunov function but there may be
tricks to simplify the task. The inability to
find a Lyapunov function is inconclusive with
respect to stability or instability.
The great advantage of the method is It is not
necessary to solve the equations of motion for
determining stability or instability! That was
especially important when no implementable
numerical techniques existed! It is extensively
used even in our days in theoretical proofs, too!
In 1892 he was awarded his Ph.D. with the thesis
A general task about the stability of motion
(????? ?????? ?? ???????????? ????????).
Aleksandr Mikhailovich Lyapunov (?????????
?????????? ???????) (June 6, 1857 November 3,
1918) was a Russian mathematician, mechanician
and physicist. Sometimes his name is also
written as Ljapunov, Liapunov or Ljapunow.
26
Approaches based on the concept of classical
analytical modeling geometrical
interpretation of Lyapunovs 2nd Method
Let f be a nonlinear vector function describing
the motion of a non-autonomous dynamic system in

Definition Equilibrium State x is an
equilibrium state if
Definition Stable Equilibrium State the x
state is a stable equilibrium in tt0 if for
there exists
such that
Uniformly Stable states can be defined if in the
above definitions in
the value of t0 does not play significant role.
27
Definition Asymptotically Stable Equilibrium
State the x equilibrium state is
asymptotically stable at tt0 if it is stable
and there exists
such that
Definition Exponentially Stable Equilibrium
State the x equilibrium state is exponentially
stable at tt0 if there exist
and ?,?gt0
such that if
then
Definition Globally Asymptotically Stable
Equilibrium State the x equilibrium state is
globally asymptotically stable if
(Its basin of attraction is the whole space.)
28
Definition Function Class ? A continuous
function
is of class ? if ?(0)0 and ?(t) is strictly
increasing. (klt? or may be k?)
Theorems Jean-Jacques E. Slotine and Weiping Li
Applied Nonlinear Control. Prentice Hall
International, Inc., Englewood Cliffs, New
Jersey, 1991.
Let x0 be the equilibrium state of a system, and
let V(x,t) have continuous first order
derivatives around x0. Let ?(?), ?(?), and ?(?)
belong to the functions of class ?.
Theorem 1 If
and
then the equilibrium point x0 is stable.
Proof of Theorem 1 Consider the next figure!
Evidently
Here the initial error norm in t0 has
significance! In this case the allowable range in
V and x is bounded by the graph of ?(x)
from the right side, and by the V(x0,t0) line
from the top.
29
?(x)
Forbidden region for V
V(x0,t0)
Forbidden region for x
x
k
0
x0
x(t)??-1(V(x0,t0))
Allowed region for / drift of x
30
Theorem 2 If
and
and
then the equilibrium point x0 is uniformly
stable.
Proof of Theorem 2 Consider the next figure!
Evidently
?-1? (x0)x(t)
and
This estimation is independent of t0!
31
?(x)
?(x)
Forbidden region for V
V(x0,t0)
Forbidden region for x
Allowed region for / drift of x
x
0
k
?-1(V(x0,t0))
?-1(V(x0,t0))
x0
?-1? (x0)x(t)
32
Theorem 3 If
and
and
and
then the equilibrium point x0 is uniformly
asymptotically stable.
Proof of Theorem 3 Consider the next figure!
Evidently V cannot be stopped at finite x .
It can be stopped only in x0. The allowed
range is shrunk to x0 as the level of V
slides down to 0.
33
Forbidden region for V
?(x)
?(x)
Forbidden region for the drift of x
V(x0,t0)
Allowed region for / drift of x
Forbidden region for x
x
k
0
?-1(V(x0,t0))
?-1(V(x0,t0))
x0
?-1? (x0)x(t)
34
Application of Barbalats Lemma for Lyapunov
Functions
Definition The set of uniformly continuous
functions on 0,?)
is uniformly continuous if for any given ?gt0
there exists ?(?) that for each a, b ?0,?) for
which if ?b-a?lt? then ?f(b)-f(a)?lt?.
Lemma
If
and x(t) is uniformly continuous on 0,?), then
Petros A. Ioannou and Jing Sun Robust Adaptive
Control. Prentice Hall, Upper Slade River, NJ,
1996.
Indirect Proof of Barbalats Lemma
and x(t) is uniformly continuous
let us suppose that
on 0,?), but
. Since
means that for each ?gt0 ?Tgt0 so that for ? tgtT
35
the opposite of this statement is that there
exist ?gt0 for which for arbitrary Tgt0 there
exists tgtT that
. Due to the uniformly continuous nature of
x(t) there exists
so that
In other words this means that if
then
Consequently
or
This excludes the convergence of the integral of
x(t) to a finite value. It may diverge to ??, or
may be divergent with alternating positive
negative values. ?
The Barbalat Lemma is extensively used in the
control of non-autonomous systems in which the
uniform continuity of the time-derivative of the
V(x,t) Lyapunov function is guaranteed. Then
is upper and lower bounded, therefore dV/dt ? 0,
and normally V is defined as a quadratic
function of the errors,
36
consequently dV/dt is proportional with the
errors that converge to 0.
It is worthy of note that for a multiple times
differentiable function the uniform continuity
of dV/dt means that d2V/dt2 is bounded
If 0?Klt?
Trivial Case Study Damped Harmonic Oscillator
This is a dynamic system of the standard form!
Let
Lyapunov function candidate
37
From the equations of motion
Compare!
Can we properly choose P22 and P11 to make dV/dt
negative? Let
has to be chosen for findingdecreasing V. Is
dV/dt uniformlycontinuous?
From the equation of motion
Is this bounded?
38
It is certainly bounded since
q1, q2 is bounded
Barbalats Lemma ?
This means that xconst. , dx/dt0 state has to
be achieved. From the condition xconst. It
follows that d2x/dt20, too. That is the x?0
state is stable equilibrium point. The basin of
attraction of this equilibrium is the whole state
space. Comment It is worth observing that P11
and P22 cannot be independently chosen! This
situation used to be typical in the solution of
the Lyapunov equation! For instance for linear
systems in general
has to be achieved. Try to find a proper P for a
positive definite Q that
39
For a given A and Q this is a linear system of
equations for the matrix elements of P. The
solution is not necessarily unique, it may have
arbitrary parameters!
For the stability of the original system it is
evidently needed that the real part of the
eigenvectors of A must be negative!
40
Main drawbacks of precise Analytical Modeling
It needs precise analytical knowledge of the
model of the system to be controlled. This may
mean a lot of analytical calculations
Three persons worked for 5 weeks on establishing
the exact dynamic model of the PUMA 560 arm!
B. Armstrong, O, Khatib, J. Burdick The
Explicit Dynamic Model and Internal Parameters of
the PUMA 560 Arm, in Proc. IEEE Conf. On Robotics
and Automation, pp. 510.518, 1986.
Example Adaptive Inverse Dynamics Control
learning the exact parameters of an analytically
well known form by parameter tuning It is also
based on the Euler-Lagrange Equations of Motion
The exact form of Y is known in advance.
The scalar product form with the parameter vector
p is utilized, too.
The tracking error is
41
The driving torque / force is exerted on the
basis of an approximate dynamic model denoted by
the symbol . It also contain error-feedback
with positive definite symmetricmatrices K0, and
K1
The modeling errors are denoted by the symbol
tilde
Via introducing x consisting of the tracking
error and its time-derivative it is obtained that

42
Let P, ? be positive definite matrices!
Let the Lyapunov function be
Let
From
Let

, and
Parameter adaptation rule
43

• Possibilities for operation
• x?0 and exponential
  trajectory tracking is achieved without exactly
  learning the system model the nominal an
  realized trajectory may not yield satisfactory
  information on the complete dynamic model
• x?0 and exponential
  trajectory tracking with exactly learned dynamic
  model.
• It is impossible to have xgtEgt0 for
  arbitrarily long time because

The finite initial value Vini should decrease
with finite rate limit while V has a positive
lower limit. This would be contradiction.
This implies that if x remains finite for a
long time the parameter-error has to achieve 0,
then x has to decrease. In the worst case,
following some initial parameter learning phase
the conditions of achieving exponentially
decreasing tracking errorwill be met sooner or
later. Comment The restriction
is not
trivially soluble fora given strictly negative
definite U the real part of each eigenvalue of A
must be negative.This is needed for finding an
appropriate strictly positive definite P for a
given U and A.
44

• Main drawbacks of the method
• It needs the precise analytical form of H, and
  h, only the exact parameter values can be
  unknown This may mean a lot of analytical
  calculations e.g. for a 6 DOF robot arm
• It requires the calculation of the inverse of
  the model inertia matrix in ? within the cycle
  since it varies as the configuration of the robot
  arm varies

• It is supposed that the generalized force Q
  cannot have other components than the
  forces/torques exerted by the drives according to
  the control strategy because the essential
  equations were obtained on the basis of this
  supposition
• Only the increased tracking errors caused by
  temporal external forces acting for very
  limited time can be compensated by these methods
• In general the external generalized forces have
  to known by measurement
• The desired exponential tracking error
  relaxation cannot be realized while x does not
  well approach 0 This delay has the reason
  that in the case of certain trajectories at
  first the full dynamic model has to be learned,
  and in the most cases precise trajectory
  tracking can be achieved only in the possession
  of the precise model that is learned by the
  system.

The above methods served as paradigms describing
the essential way of thinkingused in the case of
the possession of the analytical white box, and
analytical grey box model and the technique of
using some Lyapunov function for the parameter
learning based adaptation. This mathematical
approach is used in many classical techniques
(e.g. MRAC), and in soft computing to prove
stability.
45
The Necessary Mapping
Analytical Construction
Soft Computing

• It is content with the measurability /
  physical interpretation only of the input
  (excitation) and output (response) signals of
  the system/situation The internal
  components of the model used need not have
  physical interpretation or describe some
  physically existing sub- system.
• The particular formulae of the model
  originate from some general approximation
  theory instead of particular physical laws.

• It is based on a detailed model in which each
  component has well defined physical
  interpretation (measurability)
• The particular formulae of the model
  originate from / express / apply physical
  laws.

• The approach needs the construction of a
  detailed model in each particular case
• This is difficult, time-consuming, and in
  many cases may be impossible either on
  theoretical (e.g. parameter identification
  problems) or practical reasons
• It may require complicated and laborious
  computations that not necessarily can be
  executed in parallel manner
• The numerical result of very complicated and
  laborious analytical calculations may be quite
  negligible.
• It does not offer plausible way for handling
  formal / qualitative uncertainties.

• It offers well defined uniform structures for
  well identified problem classes
• the parameters of the structures need not be
  physically interpreted and identified
• Normally it offers possibility for realizing
  parallel computing
• It offers plausible way for handling
  uncertainties

46
The Necessary Mapping (continued)
Analytical Construction
Soft Computing

• Its realization can be carried out by using
  universally programmable computers as well as
  special hardware solutions or even by
  embedded systems

• Its realization mainly can be carried out by
  using universally programmable computers due
  to the special mathematical forms applied

In both cases the mapping applied may be of
static or dynamic nature.(Dynamic nature means
that some internal iteration may happen in
real-time for finding the proper mapping.)
Mathematical Basis for Static Soft Computing
Antecedents (1900-1957)
D. Hilbert Mathematische Probleme. In 2nd
International Congress of Mathematicians, Paris,
France, 1900. listed 23 conjectures, hypotheses
concerning unsolved problems which he considered
would be the most important ones to solve by the
mathematicians of the 20th century. According to
the 13th conjecture there exist such continuous
multi-variable functions, which cannot be
decomposed as the finite superposition of
continuous functions of less variables.
47
Breakthrough via a constructive rebuttal by
Arnold Kolmogorov in 1957
V. I. Arnold. On functions of three variables.
Doklady Akademii Nauk USSR, 114679681, 1957.A.
N. Kolmogorov. On the representation of
continuous functions of many variables by
superpositions of continuous functions of one
variable and addition. Dokl. Akad. SSSR,
114953956, 1957. (In Russian)
Further refinements of Kolomogorovs constructive
proof G. G. Lorentz. Approximation of functions.
Holt, Reinhard and Winston, 1966. New York. D. A.
Sprecher. On the structure of continuous
functions of several variables. Trans. Amer.
Math. Soc., 115340355, 1965.
For any continuous real function f of n?2
variables on the domain 0 1 there exist
n(2n1) continuous, monotone increasing
single-variable functions on 0 1, by which f
can be reconstructed according to the following
equation
Here functions ?pq are universal for the given
dimension n, and are independent of f. Only
functions ?q depend on f.
However, these functions are often very
complicated and highly non-smooth, so their
construction is difficult. This result was later
improved by decreasing the number of functions in
the decomposition, but these functions remained
highly nonlinear and difficult to calculate with.
Practical Breakthrough by De Figueiredo in 1980
R. J. P. De Figueiredo Implications and
applications of Kolmogorov's superposition
theorem," IEEE Tr. Autom. Control, Vol. 25 Issue
6, pp. 1227-1230, 1980, ISSN 0018-9286.
He showed that Kolmogorov's theorem could be
generalized for multilayer feedforwardneural
networks, and so, these could be considered as
universal approximators. On this basis soft
computing presently can be considered as follows
48
Neural Networks
Learning Methods
From the late 80s several authors proved that
different types of neural networks possessed the
universal approximation property.

• Steepest Descent (Backpropagation)
• Complex / Simplex Algorithm
• Simulated Annealing
• Genetic Algorithms
• Particle Swarm Optimization,
• etc.

E. K. Blum and L. K. Li Approximation theory
and feedforward networks," Neural Networks, vol.
4, no. 4, pp. 511-515, 1991. K. Hornik, M.
Stinchcombe, and H. White Multilayer
feedforward networks are universal
approximators," Neural Networks, vol. 2, pp.
359-366, 1989. V. Kurková Kolmogorov's theorem
and multilayer neural networks," Neural Networks,
vol. 5, pp. 501-506, 1992.
Classical SC
Fuzzy Systems
Suffers from the Curse of Dimensionality
or Wrong scalability
B. Moser Sugeno controllers with a bounded
number of rules are nowhere dense," Fuzzy Sets
and Systems, vol. 104, no. 2, pp. 269-277,
1999. D. Tikk On nowhere denseness of certain
fuzzy controllers containing prerestricted
number of rules," Tatra Mountains Math. Publ.,
vol. 16, pp. 369-377, 1999. E. P. Klement, L. T.
Kóczy, and B. Moser Are fuzzy systems universal
approximators?," Int. J. General Systems, vol.
28, no. 2-3, pp. 259-282, 1999. D. Tikk, P.
Baranyi and R. J. Patton Polytopic and TS model
are nowhere dense in the approximation model
space. In Proc. of IEEE Int. Conf. on Systems,
Man and Cybernetics SMC 2002, pp. 150153,
Hammamet, Tunisia, October 69, 2002.

• Problems
• The approximating models have exponential
  complexity in terms of the number of
  components, i.e. the number of components grows
  exponentially as the approximation error
• tends to zero.
• If the number of the components is bounded,
  the resulting set of models is nowhere dense in
  the space of approximated functions, i.e. this
  is an almost discrete set.

J. L. Castro Fuzzy logic controllers are
universal approximators, IEEE Trans. on SMC,
vol. 25, pp. 629-635, 1995. B. Kosko Fuzzy
systems as universal approximators, in Proc. of
the IEEE Int. Conf. on Fuzzy Systems, San Diego,
1992, pp. 1153-1162. L. X. Wang Fuzzy systems
are universal approximators," in Proc. of the
IEEE Int. Conf. on Fuzzy Systems, San Diego,
1992, pp. 1163-1169.
Similar results have been established from the
early 90s in fuzzy theory claiming that different
fuzzy reasoning methods are capable to
approximate arbitrary continuous function on a
compact domain with any specified accuracy.
49
Classical example by Weierstraß for an everywhere
continuous and nowhere differentiable real
function
This example implies that the continuous
functions my be very crazy from engineering
point of view. Using better behaving functions
in the models may provide more easily treatable
results.
0 lt a lt 1, b is a positive odd integer,
K. Weierstraß, Über continuirliche Functionen
eines reellen Arguments, die für keinen Werth
des letzeren einen bestimmten Differentialquotient
en besitzen. A paper presented to the
'Königliche Akademie der Wissenschaften' on 18 of
July 1872. English translation available in On
continuous functions of a real argument that do
not have a well-defined differential quotient,
in G.A. Edgar, Classics on Fractals,
Addison-Wesley Publishing Company, 1993, 3--9.
Another example an everywhere continuous
function that almost everywhere can be
differentiated
50
The construction of a continuous function that is
not differentiable in countable ? number of
points
1
The functions fn are continuousand
non-differentiable only indiscrete points
(break points). Apart from these points these
functions have well defined derivatives
f0
f1
f2
f3
0
1
Let 0lt?lt1, and az2zlt?z, z0,1,2,. Let
Evidently f?C0,1
51
According to Riemanns rearranging theorem for
this absolute convergent series holds that
Observation if x is a break point of fn, it
also is a break point of fm (mgtn). For small ?
if x is a breakpoint of fn, for m?n
fm(x-?)fm(x?) as ??0.
If we try to calculate the derivative of f(x) in
a break point from the right side we obtain
that ?gt0, and
From the left side the derivative can be
approximated as
52
For ??0
The calculation of the derivative from the left
and the right hand sides yields different values.
This continuous function cannot be differentiated
in the break points. We have countable Infinite
many break points in 0,1 evenly distributed.
This function cannot be smooth andcannot
geometrically interpreted in a usual way. There
are very extreme continuous functions!
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54
The Adaptive Control Approach Developed at
Budapest Tech
Desired kinematic tracking has to be
prescribed. It may be some PD-type tracking as
or can be based on some error metrics as
This requires either an exact permanent model
available from the onslaught or obtained via
learning, or the here described adaptive model.
55
How to realize the kinematic strategy? The idea
of the adaptive control.
Calculatedexcitation
Desired response
Actual Systems Response
Rough systemmodel
Realizedresponse
Unknown function with known input and measurable
output values.
56
Modified Renormalization Transformation for
Monotone Increasing Systems
57
y
With further parameter D
x
x0
x1
x2
Transformation with two parameters for
MonotoneDecreasing Systems
58
yf(x)
0ltf(x)
xd
f(x0)
?D-
x1
x0
D-
For monotone increasing systems
59
?D
x1
x0
D-
xd
f(x0)
yf(x)
For monotone increasing systems
60
0gtf(x)
f(x0)
xd
yf(x)
x0
x1
D-
?D-
For monotone decreasing systems
61
0gtf(x)
?D
x0
x1
D-
yf(x)
xd
For monotone decreasing systems
62
Convergence Issues
Seeking the Fixed Point of the function g(x)
via iteration in the case of a contractive
mapping
Cauchy Sequence in a Complete Metric Space! It is
convergent!
The fixed point is the limit of the iteration
63
The Fixed Point and Convergence
64
Application Example for SISO SystemThe
Ball-Beam System.
g
r
A?Beam, and B?Ball/r2mBall
x
Dynamic friction at the axle (unknown by the
controller).
?
In variable x it is 4th order system because
onlyd2?/dt2 can directly be set by the torque
tilting the beam (Q).
65
The LuGre Model of Dynamic Friction
Viscous Term
Spring Forces
Dynamic Friction Force according to the LuGre
model
The hidden internal deformation variable of the
contacting surfaces
Bending Bristles
66
The Control Algorithm
67
Adaptive D--1e-3, ?--8e4
Using g
Adaptive D--1e-3, ?--8e4
Non-adaptive
Non-adaptive
68
AdaptiveD--1e-3, ?--8e4
Adaptive D--1e-3, ?--8e4
Using g
Adaptive D--1e-3, ?--8e4
Adaptive D--1e-3, ?--8e4
69
Adaptive D--1e-3, ?--8e4
Adaptive D--1e-3, ?--8e4
Using g
Adaptive D--1e-3, ?--8e4
Non-adaptive
Non-adaptive
70
Adaptive D--1e-3, ?8e4
Adaptive D--1e-3, ?8e4
Using h
Adaptive D--1e-3, ?8e4
Adaptive D--1e-3, ?8e4
71
Adaptive D--1e-3, ?8e4
Adaptive D--1e-3, ?8e4
Using h
Adaptive D--1e-3, ?8e4
Adaptive D--1e-3, ?8e4
72
x1,x2,x3,x4input
uinput initiator
Example Polymerization in a ContinuousStirring
Tank Reactor with Jacket
Stirring
Equations of Motion

• The state variables x1 x4 denote
  dimensionless concentrations of various
  chemical components taking part in the
  reaction.
• x1 denotes the monomer concentration
• The number-average molecular weight is
  denoted by y it is the prescribed output of
  this system.

F.J. Doyle, B.K. Ogunnaike, and R.K. Pearson
Nonlinear Model-based Control using Second-order
Volterra Models, Automatica, 31697, 1995. J.
Madár Application of a priori Knowledge in
Chemical Process Engineering, Doctoral (PhD)
Thesis, Department of Process Engineering,
University of Veszprém, Hungary, 2005.
x1,x2,x3,x4output,uout
Heating or cooling viathe jacket
A 10B 6C 24568D 80E 101022 F
0024121G 0112191H 10 I 245978 and J
10 are constants in the model. In this case no
heating/cooling via the jacket is needed.
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Adaptive D0.5, ?0.08
Adaptive D0.5, ?0.08
Adaptive D0.5, ?0.08
Non-adaptive
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Adaptive with g D--7e4, ?1e6
Adaptive with g D--7e4, ?1e6
Adaptive with h D--7e4, ?--1e6
Adaptive with h D--7e4, ?--1e6
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Example Adaptive Control of Convoys
Connected rigid body model by artificial damped
springs as an artificial train the preceding
cart is automatically tracked by the next one
The 1st carts motion is kinematically prescribed
according to the nominal motion of the last one.
The load-cart connection is modeled by viscous
friction model and a spring constant. The loads
are unknown for the controller.
in which
76
Certain results are given for Decentralized
Control in the forthcoming slides.
77
Non-adaptive distance and velocity tracking
Adaptive acceleration tracking
78
Non-adaptive distance and velocity tracking
Adaptive acceleration tracking
79
Non-adaptive distance and velocity tracking
Adaptive acceleration tracking
80
Non-adaptive distance and velocity tracking
Adaptive acceleration tracking
81
Algebraic considerations for MIMO systems

• Add arbitrary parameters to the problem by
  completing the vectors to obtain matrices

• Easily invertible matrix is obtained if it is
  the element of some Lie groups

• E.g. for the Generalized Lorentz and the
  Symplectic Groups

82
Generalized Lorentz matrices
Symplectic Matrices of Minimal Transformations
83
Special Symplectic matrices
84
In each particular case the e(i) pairwisely
orthogonal unit vectors can be obtained by the
rigid rotation of an orthonormal basis modifying
only the elements of a 2 D linear subspace.
Ancillary transformations can guarantee that the
adaptive law can be in the vicinity of the unit
transformation.
85
Extension for MIMO Systems of the Modified
Renormalization Transformation based method
Proof of convergence
Let f(x) be differentiable, invertible, and
?0, xd ?0, and
let be positive definite and of norm ltlt 1. If x
is near
solution with perturbation calculus and a Lie
group
Present error.
Error of next step.
In 1st order
86
If there exists 0ltKlt1 the approximation error
relaxes as
The solution converges
Via substitution
That has simple geometric interpretation
87
Acute angles
Allowed set of vectors of reduced norm!
88
Ancillary linear interpolation and extrapolation
Adaptive PID error relaxation with 3 comparable
real damping.
89
The model of coupled cart double pendulum
asymmetric systems
m1,L1,q1
m2,L2,q2
q3
M
Coupling spring
System A
System B
The linear axis and one of the rotational axes is
driven, the other rotational axis moves freely
in both sub-systems. Coupling by the spring is
not modeled.
The uncontrolled andunmodeled degree of
freedom is q2. It can rotatewithout limitation.
Two such carts of different parameters are
coupled by a spring.
The rough approximate dynamic model of the
system used by the controller was
90
The exact Euler-Lagrange equations of motion of
the system are
This inertia matrix can be badly conditioned!
Zero generalized force component for the axle
not driven (partial modeling)
91
Simulation results
m/s, rad/s vs m, rad
Adaptive
Adaptive
m/s, rad/s vs m, rad
Adaptive
Adaptive
m, rad vs s
m, rad vs s
92
Adaptive
Adaptive
rad/s vs rad
rad/s vs rad
Adaptive
Adaptive
dimless vs s
dimless vs s
93
m/s, rad/s vs m, rad
Non-adaptive
Non-adaptive
m/s, rad/s vs m, rad
m, rad vs s
Non-adaptive
Non-adaptive
m, rad vs s
94
Non-adaptive
Non-adaptive
rad/s vs rad
rad/s vs rad
95
SVD-Based Multiple Dimensional Generalization
of an Adaptive Control of Geometric Interpretation
CREATION OF CAUCHY SEQUENCES FOR MIMO SYSTEMS
It is given
Proposed translation from an initial probe
value of x
It has to move the functions output value
towards the desired one.
96
The generated sequence of the points is as
follows.
For ??(0,2) it reduces the error for exact
direction of translation.
Convergence issues
For small ? that can be realized if an
approximate model of the Jacobian of f is
available in SVD processed form
97
Associativity
Associativity
98
Apply SVD for the approximation of the real
Jacobian as
Use the columns of U and V as orthonormal basis
vectors in theinput and the output spaces of the
Jacobian
Since
Let the maximum allowable step length in the x
space be K ?
99
Simulation Results for the Cart plus Double
Pendulum System
Non-Adaptive
Adaptive
K2 m/s2 or rad/s2
100
Adaptive
Non-Adaptive
101
Conclusions

• 1. In general geometric way of thinking makes
  life far more convenient than any other
  approaches lacking its lucidity
• 2. It consists of a kind of combination of using
  the lucid form of the equations, and
• building up a kind of associative memory in
  which certain figures are associated with some
  abstract operations
• The adaptive approach elaborated at Budapest
  Tech is also based on geometric interpretations
• Simulation examples substantiate the expected
  applicability of the method.

Thank you for your attention!!!