Table Of ContentMATHEMATICAL ANALYSIS AND TECHNIQUE FOR
NON-LINEAR SERVOMECHANISM PROBLEMS
KOUAN FONG
B.S., National Central University, 1938
M.S., University of Illinois, 1948
THESIS
SUUM1TTED IN PARTIAL. FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ELECTRIOAL ENGINEERING
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS, 1951
URBANA, ILLINOIS
1^1
r I * UNIVERSITY OF ILLINOIS
THE GRADUATE COLLEGE
4lay 19th« 1951
#
I HEREBY RECOMMEND THAT THE THESIS I'REl'ARED UNDER MY
SUPERVISION BY K0I3AIT F0I3G-
ENTITLED MATHBMATIflAL AHft.KB3I3 AND EECHNIQ.TIE FOR NORgLIHEAR
SERVOMECHAHISM PROBLEMS
BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCgOB-Qg-£HILOSO?HY
^£-1- H,
In Charge of Thesis
ig£L^- -^7_ _ (^
Hfciul of Department
Recommendation concurred inj*
Committee
^t£eC-
on
Final Examination-)*
t Required for doctor's degree but not for master's.
M440
i
TABLE OF CONTENTS
PART I. MATHEMATICAL THEOREMS OF NON-LINEAR SERVOMECHANISMS
1.1 Introduction 1
1.2 Mesh Equations for Non-linear Servomechanism
k
1.3 Representation of Non-linear Elements • 5
1.4 An Example for Setting-up Mesh Equations of 7
Servomechanism
1.5 Green Function Responses and First Approximation 9
1.6 Pseudo-linear Analysis of a System with a Single ll
Non-linear Element
1.7 Convergence of the Successive Corrections 13
1.8 Servomechanism Involves an Unilateral Saturated 16
Non-linear Element
1.9 Convergence of Successive Approximations 18
l.lO Generalization to System with m Non-linear 21
Elements
1.11 The Exsistance Theorem of Asymptotic Stable 26
Solutions
1.12 The Exsistance Theorem of Bounded Solutions 31
1.13 The Applications of the Two General Theorems 38
In the Analysis of Servomeonanisms
'ART II. ANALYSIS OF NON-LINEAR OPEN AND CLOSE CYCLE
CONTROL SYSTEM WITH TWO-PHASE MOTOR
2.1 Introduction -4-7
2.2 Approximate Expression for "Torque/Maximum 51
Torque" Ratio
Torque at Motor Shaft as a Function of k and v. 52
2.3
£2
2.4 Case I. Viscous Damping Only
2.5 Case II. Zero Control Voltage &
2.6 Case III. Combined Coulomb and Viscous Damping 56
58
2.7 Case IV. Simple Close Cycle Control System
under step-displacement
Table of Contents (Continued) lj
2.8 The Effect of the Parameters upon the Equation 62
2.9 Condition on Convergence of Successive Corrections 66
2.10 Second Order Correction Terms 67
Nomenclature 8l
Bibliography 8J4.
Vita 87
ill
Acknowledgment is made for the encouragements and suggestions
given to the candidate by his present and former advisors, Dr.
G. H. Fett and Dr. L. T. DeVore. It should also extend to Dr.
C. Y. Lin for all the fruitful discussions given to the writer
in regard to the transient operations of motors.
L
PART I
MATHEMATICAL THEOREMS OF NON-LINEAR SERVOMECHANISMS
1.1 Introduction
The linear servomechanisms with fixed parameters are studied
from their differential equations of motion. The solutions of such
differential equations with constant coefficients can be found
easily by the routine method of operational calculus. A servomech
anism is a heteronomous system, whose responses to a variety of dif
ferent forcing functions are of interest. They may be obtained
more systematically by using either of the following two methods,
both of which are based upon the suitable superposition of some
fundamental set of solutions.
(1) The fundamental set of solutions adopted are the re
sponses to suddenly applied step or impulse disturbances. The re
sponse to an arbitrary forcing function f(t) is obtained by
Duhamal's*'1''"'^) superposition theory. This method is generally
called transient approach.
(2) The fundamental set of solutions adopted are the steady
state amplitude response and phase delays of sinusoidal forcing
functions which cover the complete frequency spectrum. This can be
expressed as an analytic function of p. The Fourier transform of
the arbitrary forcing function f(t) can be expressed as another
analytic function of p. The inverse Fourier transform of the
product of these two p-functions gives the actual response. This
method is generally called steady state approach.
Number in parenthesis refer to bibliography at the end of
this thesis.
, = — =_
Both approaches depend upon the principle of superposition,
that is, the sum of the responses of two forcing functions f (t)
and f (t) is equal to the response to the sum of the two functions
fl(t) and f (t)- And this particular property holds only for
2
linear systems.
For the non-linear system, both of the elementary~.toe$hp*,ds £ai3
because the fundamentally important principle of superposition no
longer holds. This makes the beautiful and well organized method
developed along the linear analysis unapplicable. In analysing the
problem of such complexity as in a servomechanism system, under
such circumstances it is necessary to introduce certain artifices
to bring the linearity back to the system. This approach is fre
quently called pseudo-linear analysis. (1)(3)(5)
In pseudo-linear analysis, all the non-linear elements are
replaced by external applied forcing functions of time which are
equal to the force consumed by these elements at any instant. Such
I
changes cannot produce any change in the system. The system will
give exactly the same response. The system now consists of linear
elements only, but with some extra forcing functions of time. The
linearity has been thus restored. Then the equation of motion may
be solved by first omitting all the extra forcing functions, which
are the non-linear elements in the original system. The solution
obtained is called the first approximation. The remaining portion
of response may be called corrections, which are obtained by using
superposition theorem of operation calculus. Because the extra
forcing functions are not known before the exact solutions are ob
tained, a non-linear Volterra integral equation of the second kind
will be involved instead of the usual Duhamal integral.
3
In servomechanism problems, the properties in the large
(for semi-infinite time interval) of the boundness of solutions,
especially the errors, and the uniqueness of solutions are very
important considerations. The conditions connected to the first
requirement may be called stability criteria and the conditions
connected to the second requirement may be called reliability cri
teria. In Part I of this thesis, by using pseudo-linear analysis,
the famous Liapounoff stability theorem(12) relating, to the non
linear autonomous system has been extended to the case of non
linear heteronomous systems in several forms. These theorems are
constructed with a view of their applications to the stability and
reliability criteria of non-linear servomeonanisms.
1,2 Mesh Equations for Non-linear Servomechanism *"
In analysis of linear servomechanisms the system equations
in general are set-up in two ways. The first way by using transfer
functions,(14) takes advantage of the unilateral impedances presented
by the system. The second way is by using the standard technique in
feedback amplifier analysis, from the ratios of ••return ratio" and
"return difference"(8), (14). Both methods have the advantage that
the relation between the input and output or error is given immediate
ly, and these are the only responses of interest in case of a linear
servomechanism. In analysis of non-linear servomechanisms the situ
ation differs a little, because the responses at intermediate points
where the non-linear elements are located will be required in evalu
ating the solutions for output and error. In such cases the general
second order differential equation of motion with matrix coeffic
ients may be most convenient to use, this type of formulation has
been used by Kron in his Tensorial Analysis of Servomechanism.(10)
Such equations can be formulated more systematically, especially for
electrical engineers if the equivalent electrical mesh network or
node network is adopted.
More specifically, the differential equation of motion for
the hon-linear servomechanism will be set up on mesh bases, and the
following set of equivalence between electrical and mechanical quan
tities will be used.
Angular moments of inertia J -~ Inductance L
Viscous damping R <*- Resistance B
Stiffness S ~ Inverse of capacitance(I/O
Angular position 9 ~- Flow of electrical charge q
Angular velocity 9 or y ^ Flow of current q or i
Angular acceleration 8 or << ~ Time rate of change of cur
rent q or i
Torque T -—' Electro motive force E
Because the angular position is of prime importance in a servosystem,
we should formulate the voltage equibrium equation for each mesh and,
use the flow of charge q as the dependent variable. The symbol
Z(p) used in the analysis is not always impedance in the electrical
sence, but p times impedance when used before q is equal to electri
cal impedance if it is used before i. However, for simplicity, Z(p)
will be called impedance in general.
If the matrix equation of motion appeared as
11 Ell * II Z II || qll or II T II •*= I! Z II II 911 (1.2,1)
then the impedance function of the network should appear in the fol
lowing form
where p represents differentiation in respect to t.
In labeling the meshes, a definite scheme is adopted in
order to secure more precise definitions of the quantities discussed.
Let the first mesh be the circuit immediately following the differ
ential. The second mesh is the mechanical elements connected to the
motor shaft. The reference signal 9.(t) modified by the transfer
impedance Z of the differential will be a forcing function applied
t
to the first mesh, and the external load torque*-T (t) will be the
Ii
forcing function applied to the second mesh. These two are the only
external forcing functions applied in ordinary servomechanism.
For simplicity, dependent variables used are q. and i. and
the index * refers to the number of the mesh, for both electrical and
quantities. This means the q« is the output shaft position 9 and
2
i is the output shaft velocity v and furthermore if dq-j/dt.— PQj*
2 2
1.3 Representation of Non-linear Elements
The analysis is started by assuming that in each system there
is only a single non-linear element. After completing these simpler
cases, then this restriction is removed and the more general system
|having any number of non-linear elements is analyzed. Two types of^ag
