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CHAPTSR I .
I N T R O D U C T I O N .
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During the last decade , the theory of control system has advanced quite rapidly. Thia is largely beoause of the dominance of automatic control in every sphere of life*
Starting with the automatic toaster, the alarm clock, the thermostat and similar things, the area of automatic control extends upto robots and space vehicles for travelling into outer 3pace and planets. The successful operation of such
• sophisticated system depends on the proper functioning of a large number of control systems, which often have desired and undesired couplings between them. Thus, the present control
engineers have been called upon to deal with increasingly complex problems.
A3 the problems become more complex and demands on the system performances become more stringent, the theory of modern oontrol systems has received increasing attention both by engineers and mathematicians. In general, the design of a
3 4 5 control system may be divided into a number of steps * * • These are (1) the establishment of a set of performance specifications, (2 ) the existence of a control problem as a result of the performance specifications, (3) the formulation
of a set of differen tial equations or sane other suitable mathematical representation to describe the physical system,
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( 4 ) the design ( i f necessary with suitable compensating networks) of the system so as to satisfy the desired performance specifica
tions, (5) the evaluations of the system performance by experi
ments, simulations and/or mathematical analysis to find out whether the given specifications are satisfied, ( 6 ) the system
optimization or the achievement of optimal performance for the required system response specifications, and (7) the determina
tion of the stability characteristics of the system.
A control problem is created in order to maintain the actual performance of a system close to the desired performance*
The neoessary basic equipment is then assembled into a system to perform the desired control function* To a varied extent, most systems are nonlinear* In many oases the nonlinearity
is so small that it can be negleoted, or the limits of operation are small enough that a linear analysis can be made* I f a
system under investigation consists of a few simple sub-systems, it is more advantageous to break it up into these simple
g
sub-systems and to determine the mathematical description of each of them separately, instead of looking directly into the whole complex system. Sometimes, the system consists of a few simple sub-systems independent of each other. In all suoh cases, the problem of undesired inherent couplings may arise.
This problem arises due to certain coupling signals going from
«
one sub-system to the others. The presence of coupling signals may be due to mechanical defect in the structure of the system or it may be because of stray pick-ups by the electrical network or it may be inherent in the system structure ’ .8 9
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Tile main objective of the present thesis is the analysis and design of a typioal control system with suoh inherent oouplings and optimization of the same. Some simple tools of analysis have been designed, and new techniques of
optimization of its performance have been developed. In addition, the well known Lurie's method is modified for determining the stability characteristics under certain ciroumstances.
An autofollower system far light signals was 7 considered as the typical oontrol system. A3 the aim was
( 1 ) to design and successfully develop the system, ( 2 ) to study the desired and undesired effects of inherent coupling signals
on its response, and (3) to optimize the system response by selecting the parameters (unknown) of the system, the design and construction of the system is first described in Chapter I I . The small model of the autofollower system, which was successfully developed for the experimental study of the problem, is expected to track light signals in all directions and hence the system consists of two independent servo systems. One servo system is used for the azimuth axis and tte other one for the elevation axis. The servo system for the elevation axis is the load for the azimuth axis servo system. Photo tubes were used as sensing element of light signals and they were placed on a table rotated by the elevation axis for three-dimensional scanning.
<* »
For investigation of any physical system, we generally 5 ,1 0
describe it in terms of some mathematical formulation .
A mathematical description is often referred to as mathematical
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model, and is always determined with idealization of the system.
Any correct mathematical description gives an accurate evaluation of the behaviour of the system and specify all its properties.
The mathematical model of the whole system was found out by
separating out the linear and nonlinear blocks and by measurement of their input-output characteristics. The mathematical model so obtained has been described in the later part of Chapter I I . The two servo systems were represented mathematically by two sets
of differential equations of the farm
where x , and are the an<l state vectors and and are the ( ) and 0 * * 0 vector functions of the two servo systems respectively and * t ’ is the running time.
by unintentional or inherent couplings from one sub-system to the other one. Regarding the inherent couplings, a little qualitative analysis suggesting names like pick-up has been reported by previous workers. But a systematic study is lacking so f a r . In Chapter I I I , a detail study of the couplings has been done. A completely new nomenclature has been suggested in which these coupling signals were classified as load, velooity
and acceleration couplings, depending on whether the unwanted signal is proportional to the output, the first derivative of the output and the second derivative of the output.
■*« = h ’ * 0 ( 1 . 1 )
• • t (1.2)
In actual case, the system performance is effected
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The most convenient cipproach to the study of the effect of various unwanted signals is definitely the experimental
investigation of the physical system. However, in moat cases the direct,investigation method or any adjustment or compensation on the actual system will be tedius, expensive and cumbersome.
Technical and engineering practice often uses models in such cases. The most common and widely used models of physical systems are those simulated in an Electronic Analog or Digital computers, which are based upon analogies between processes of different physical origin. Thus, in order to reveal the perfor
mance of the autofollower system with different amount of inter
£xis couplings, and to predict the effects of different parameters on its response, the system was simulated in an analog computer* 1 (Berkeley Ease) as discussed in Chapter IV . The output of the simulated, system in analog computer was then observed by ohanging different parameters, ( 1 ) in the absence of any coupling signal, (2) in the presence of load coupling signals, (3) in the presence of velocity coupling signals, (4) in the presence of acceleration coupling signals and (5) in the presence of two or more variety of the coupling signals. In every case, the parameters of the system were determined for optimum response. All the results were qualitatively verified by theoretical calculations on a
linearized model of the system 13.
The design of a cnntrol system will be considered as incomplete unless its performance is optimum. The performance is in general judged by the transient and steady state responses©
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The principal difficulty in the design is the establishment of 14 15
a criterion for optimum performance * • A oriterion function should have three basic properties : reliability, ease of appli
cation and selectivity. In this particular system, the perfor- •
«
mance criterion selected was based on the time response to a step function. IT AS and ISTS1 criterion were selected satisfying the requirement of the criterion funotion. The parameters of the system are then chosen so as to optimize the criterion function. This was achieved by two methods i . e . by (l) random search method and (2) inverse sensitivity method. Digital Computer (IBM 1620 Model I) was used for doing so . A very simple method of simulation was developed for this purpose and this is described in Chapter V . This method of simulation is similar to Digital-Analog Simulator, which has been used in
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standard programming languages liice MID AS , PACTOUJS ,
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MIMIC , DSL/90 etc. The proposed method consists of ( 1 ) construction of block diagrams usually accepted in analog computer simulation, ( 2 ) replacement of the block diagram by various equations relating the output of the blocks to the inputs, ( 3 ) programming of the subroutines for the blocks used in the analog simulation diagram and using the subroutines in the same way as the plug-in modules in electronic cirouits, (4) translation of the various equations into FORTRAN language, ( 5 ) calculation of the various quantities at different
intervals of time, ( 6 ) sequencing of the analog blooks for determining the order in which the blocJss are to be processed, and ( 7 ) error checking and correction for reducing computational errors*
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Random search method and inverse sensitivity method for the optimization of the system response are disoussed in Chapter VI* Both the methods are quite well known and have been used successfully by various workers for optimization of criterion functions under different constraints* But for the first time, in the present work, these approaches were proposed and success
fu lly implemented for the optimization of the response of the autofollower system when inherent coupling signals are present from one sub-system to the other* The random optimization method refers to an optimization method based on random experi
ments. In a pure random search method, computation of the
• criterion function is performed at a number of randomly chosen points in the parameter space and selection of the particular parameter values yielding smallest value of the criterion function. However, such a sequence of randomly selected para
meter vectors doSs not take advantage of the local continuity properties of most criterion function surfaces* Consequently, the strategy is modified so as to make successive steps in the successful direction 22. The adaptive step size random search (jlSSRS) 23 as proposed by Shummer and Steiglitz has been used and parameter optimization in presence of different types of coppling signals were achieved • The result was verified and checked by inverse sensitivity method of performance optimiza-
oc net
t£on . * * In this method the criterion function is expressed as a function of the sum of two variables c^-x. and
the difference of the output of the system from the predetermined ideal output and is the difference in the
output due to variations in the parameters
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be expressed in terms of and sensitivity coefficients.
t*
Once the performance criterion function is expressed in this fashion, is found out by solving the equations given by
n.
- ° • • • ( I * 3 )
i - I , -- TV .
ensitivity problems in optimal control has been referred by Dorato 28 in 1963. He has outlined a procedure for the sensitivity analysis and the calculation of the change in the value of the performance index of optimal oontrol systems in the case of plant parameter variations due to such thing as component inaccuracies, environmental effects, ageing eto.
> 29 30
Pagurek * has further broaden this analysis and has shown that the sensitivity analysis problem in optimal control systems closely parallels the original optimization problem. The same principle, with necessary modifications, has been applied here for the first time in control systems with inherent coupling problems* Proceeding on the same line as previous workers, the performance index sensitivity function of the system to inherent unintentional coupling signals were determined 31• This is
described in Chapter V I I • Two methods for the determination of the sensitivity function are described. These are done by
(i) using two system models with slightly different coupling signals, and (i i ) parameter influence coefficients method, using an analog computer. Once the sensitivity function due to each type of coupling signal is determined, the total change in performance index sensitivity function oan be determined, i f the awount of each type of coupling signal is known*
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Another important aspect in the study of the control system is the stability analysis. Stability 32 oocupies a special place
in the analysis of linear and nonlinear dynamic systems. It is of great practioal importance to know whether a systemf3 response ' is stable, and to what degree. Stability is thus necessarily
included in the set of characteristics which desoribe a dynamic system. Present day stability of a control system means stability in the Liapunov’ s 33 sense. Liapunov's definition of stability is based on whether or not a dynamic system, for small changes in the in it ia l conditions, returns to the same equilibrium state.
In Chapter V I I I , some attempt has been made to study the stability characteristics of the system in Liapunov’ s sense, under various
►
conditions. The real difficulty in applying Liapunov’ s second method lies in the synthesis of a suitable V-function. Although various methods are available for the determination of stability characteristics by generating V-functions, but one is faced with some d iffic u ltie s i f the methods are applied to some nonlinear control problems like hysterisis or backlash where the nonlinear characteristic extends over a ll the four quadrants. In most methods of stability analysis, it is assumed that the output of the nonlinear element, i . e . -f6r) for an input «r* must satisfy the condition o <*■$(«-) . In other words, one has to make an assumption that the output of the nonlinear element must lie in the first and third quadrants, in a rectangular coordinate
*
representation of f with • There are Beveral nonlinear elements which f a il to satisfy this strict condition. To avoid
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this lim itations, the usual V-function in Lurie’ s ’ method of stability analysis is modified in order to incorporate general class of nonlinearities extending in all four quadrants. Th©
i
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and the stability characteristics, were found out*
In Chapter IX , a control system analyzer is described.
A new instrument was designed and developed so as to determine 36 37
very quickly the transfer funotion ’ of control systems for its analysis, design and improvement. The instrument operates by evaluating the fourier expression for the inphase and
quadrature components of the output a*c. signal (system output) with respect to a related reference voltage. The instrument
consists of (i ) a multiplier using the principle of time division method for multiplication of two signals X and Y in
all four quadrants, ( i i ) a simple 4-diode gate having transmi
ssion of exactly one complete cycle of the reference input, ( i i i ) some switching networks for the operation of gate circuit,
(iv) an integrator using high gain d .o . amplifiers and computing resistances and condensers for integration purposes in the
evaluation of fourier components, (v) an osoillator used as a reference signal, and (vi) a zero-oentre d .c . meter as an indicating degice. The oscillator contains three operational amplifiers (two integrators and one sign changer) forming the frequency determining loop by which the differential equation
^ s st is solved to get sinusoidal oscillation. In addition, in order to stabilize amplitude and frequency of the
39
oscillator, an absolutely new technique has been proposed and successfully implemented in the oscillator circ u it. This novel
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technique is the introduction of a sub-loop in the oscillator circuit and this has been described in Appendix I I *
.In Appendix I , two now methods 38 of drift compensation of transistorized d .c . amplifiers are desorlbed. These methods were triad with success in the high gain d .c . amplifiers used in multiplier, integrator and oscillator. In the first method cros3-coupled feedback diodes are used to reduce d r ift , whereas in the second method, drift reduction is achieved with the help of temperature dependent current source.
Thus, on the whole, the thesis aims at drawing the attention to the importance of the inherent coupling problems in control systems through a thorough investigation on the analysis, simulation, performance optimization and stability analysis of a simple control system. Although the methods described and results obtained were always with reference to the autofollower system only, but these cannot be considered to be restricted to this particular system alone* They may be employed in any continuous data system wherever such coupling
problems exist*
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R 3 F 3 R B N C 3 S .
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1 9 . Syn, W.M. and Lineberger, R .N ., "DSL-90 - A Digital Simulation Program for Continuous System Modelling"*
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Chatterji, B .N . and Chatter jee, B ., "PerforinancQ
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of Electronics and Control.
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of Control, in course of publication.
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La S a lle , J .P . and Lefschetz, S . , "Stability by Liapunov's Direct Method with Applications", Academic Press I n c ., 1961.
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36. Henning, T ., "Testing for plant transfer functions in presence of noise and nonlinearity", Control Engg.,
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..s* f . V '.JL.
