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Application of the theory of combination systems to adaptive cybernetic systems
Application of the Theory of Combined Systems to Adaptive Cybernetic Systems A. G. IVAKHNENKO This paper deals with the latest development of the theory of invariance and also shows that this theory is applicable not only to ordinary systems with constant characteristics but also to cybernetic systems. Notwithstanding the great significance of the theory of invariance for automatic control, many scientists and technical engineers are not yet sufficiently acquainted with it. Therefore before we proceed to the main object of this paper, which is to discuss the possibility of applying the theory of invariance (and the connected theory of combined systems) to cybernetic or adaptive systems, a very brief introduction to this theory is necessary.
A Brief Introduction to the Theory of Invariance and the Theory of Combined Systems Conditions of invariance indicate methods of eliminating errors in control systems. By conditions of invariance we shall mean conditions applied to the coefficients of the differential equation of a dynamic system and the principal disturbing influences, which are such that the error of the system is identical to zero, ~ =
Table 1 Equation a3(p) = b3(P)L\ + bH3(P)L2, where p = (d /dt)
]c Forms of the equations J5~ of invariance --
Data on disturbances
I
Component of the error which is eliminated
L\(t) = 0, Lz(t) = 0
I
Has no practical application
(1)
S
0
t
-li
0
2
In arbitrary form with limitation for acceleration
«l ""'
~
Method of accomplishment
Reduces the error caused by all disturbances, including variations of the parameters in time·. Nonabsolute invariance
Conditional stability multi-loop systems with 'key' type frequency of the characteristic (see reference 4, p. 194; and reference 9). Non-unitary operational feedback in servomechanisms
The same
Eliminates all the error caused by disturbances L\ and Lz. Absolute invariance
Control by disturbances and their derivatives by time 4
In the general form in symbols
Only the steady component caused by all disturbances
The integral equation according to Minorski\O and Kulebakin z
The same
Only the steady component caused by one disturbance L\
Control by disturbances and their derivatives by time 4
In the form in numerals
All the error caused by the disturbance Device displacing L2(t) (forcing) and also artificial variation of data for programmed L\ systems (see reference 4, p. 293)
11
5":' ""'I~
00:0 N
'" <::s'"
IIII~
-Cl
~C:;: "'-~
I
~'"
""
-Cl-Cl
,-...
0
M ~
11 ,-...
:::r
0
-I~
11 ,-...
:::r
3
«l ""' 8
o~
~I~ ~~
11;;:
~~
~~
:::r
~
-4 I' b3(p)L\(t)
,£
+b"3(P)L2(t) = 0
(4)
i
• The validity of form 2a for systems whose parameters are variables in time has been pointed out and investigated by A. I. Kukhtenko.
568
578
APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
paid to the different properties of conditions 2a and 3a and of conditions 2b, 3b and 4. The first are connected with certain requirements pertaining to the left side of the equation of a dynamic system, and are called the conditions of non-absolute (i .e. incomplete) invariance. The second impose certain requirements on the right side of the equation of a dynamic system and are called the conditions of absolute invariance, since with certain limitations these conditions determine the absolute, i.e. the complete elimination of all the components of the error of control systems.
ated; (c) there is no necessity to use delta-function impulses in the manipulated variable to compensate errors; the last condition is satisfied if q
I'0
I
I
n-m
where q is the order of the equation of the dynamics of the object, n is the order of the left side of the equation of the system, and m is the order of the right side of the equation of the dynamics of the system. It follows from (b) that absolute invariance is achieved mainly in combined systems, which combine the principles of control 'by disturbance' and 'by output'. Control by
Oscillograms of the ·,oltoge at t h e ou t pu t of t h e p h ase discriminator 6 with 11 / rO' 2C 12
y2
~
. d ~x 0 f th e 0 f t he In extremum Cl>
Osc illogram
of lrror
Overcompens.ation
Overcompensiltion of error
15 Compromi se tuning
C ,0·25
-.......
+005 10
f.llillW
0·5
o~
re·!
t<>S
0
0
o ·~
~
0
S ro' 1.0.
-o·!
-{).!
fill ill lill WJ 05
..u
5
~
1-0- 5
-o~
WJ
r
?i'1 10 10 >
~
0
1'2
-OS
-05
0·5 10
!'
10
0
0
0
-0'5
-OS
05
-{)·S
-0·5
0
70 ,0'5
05
05 0
0
-o~
I
~ f--1+0
-
r
1 ·5
Shortest process
II
'08, ,15
Fulfil of tment he condl lions of inva riance
f
11
se;vomotor
does no t come i n
lL 1. iO' 2C12 70 '
O!
o!
0
0
rro0
o·5
IJj j
0
lIII/I
0 05
-0
05
lill
~S
Undercompensation of error
0·2
I
I
04
06
Undercompensation of error
1 10
0'25
0' 5
(a i
Figure I.
{"""
1·0
10
(b)
The effect of a feedforward link (coordinate Y2 /YO) and a f eedback loop (coordinate in a combined system with the equation
Some authors call the conditions of absolute invariance the conditions of disturbance compensation. It is clear that an error can be under-compensated, fully compensated or overcompensated (Figure I). In distinction to this, the conditions of non-absolute invariance 2a can only give satisfactorily sufficient (and only within limits, complete) reduction of the error. Here over-compensation is possible only for the separate components of error. For example, we can obtain negative steady-state error. The condition of absolute invariance 2b is valid, if: (a) for equal disturbances the forms of the transient processes in the controlled object are almost identical, i.e. the processes can be repeated; this is important for open cycle systems ; (b) the process is self-controlled or the system is a combined one so that zero drift, which upsets the initial zero conditions, is elimin-
e l2)
011
the form of the trallsiellf processes
disturbance is very often control by open-loop circuit (if t:.L #- f(t:.tJ») , where L is the principal disturbance and tJ> is the controlled variable; control by deviation (by the output variable) is, however, always control by a feedback closed cycle. Principal rules for combined systems setting
The principal condition for the setting of a single-loop system is what is called 'compromise setting' in which such a coefficient of amplification is selected that, when the rigidity is sufficient (degree of accuracy s = I + a p = ao), a sufficient stability is achieved. Statistical methods of selecting the optimum transfer function can also be used for the same purpose. In multi-loop systems, besides the conditions of compromise setting, the condition of absolute invariance 2a
569
579
A. G. IVAKHNENKO
(Table I) can also be used. This condition is closely bound up with the work of Meyerov 9 and others. The principal law for the setting of feedforward links by disturbances is to satisfy the conditions of absolute invariance 2b (Table I). The orthogonality (absence of correlation effect) of the two setting methods. If small deviations of the controlled variable do not affect the disturbance (!J.L =I=- f(!J.IP», and also when the conditions of invariance are satisfied ( == 0), methods of increasing accuracy and quickness of action by improving the feedforward links affect the selection of the coefficients of the right side of the equation of the dynamics of the system, and the methods of improving the feedbacks affect the selection of the coefficients of its left side (Figure I). The effect of feedforward link on stability is in practice small. The setting of combined systems can always be done 'by parts'; in this method the feedbacks are first improved as much as possible, and then the same is done for the feedforward links (see Appendix I).
(2) If the noises and disturbances cannot be measured a closed loop system of feedback set according to the rules of compromise setting or by statistical criteria and, in addition, by the condition of non-absolute invariance 2a (Table 1), should be used. (3) If only part of the noises and disturbances can be measured, or if the characteristics of the amplifier are insufficiently stable, a combined system, set by the' parts' method, must be used. Setting includes both the selection of the parameter values and the synthesis of the scheme by the introduction of new elements (Appendix 1). Comparison of Ordinary and Cybernetic Adaptive Systems By cybernetic systems we understand systems which carry out more complicated tasks than the classical tasks of stabilization, schedule time systems and servomechanisms. In particular,
The advantages of combined systems in steady-state regimes (a) Power advantages-The greater the accuracy required from the system, the greater these are 4 • The power at the output of a feedback (corrector) can be approximately onefifth that of a system without feedforward link by disturbance, the magnitude of the latter being the same. The greater the accuracy of the feedforward links the less work is left to the corrector. The figure given above (a fifth) was found in practice in the control of synchronous generators with compounding by full current. Setting advantages-The setting of combined systems is simpler and covers a wider range. One can have a system with negative statism or a system with high rigidity and statics together. Negative statism is necessary to compensate the potential drop in the line, and great rigidity with high statism are required in order to get stable parallel performance of the controlled plant (generators, boilers, etc.).
(a)
Amplifier tJs (t)
Advantages of combination systems in dynamic regimes The total error of the system is !J. = !J.! +!J. 2 + !J. 4 , where !J.! is the static, !J. 2 the transient and !J. 4 the velocity error. Combination systems greatly increase the possibility of using the conditions of invariance which enable these errors to be eliminated or reduced, i.e. they increase the speed of action and accuracy of the system. In systems without feedforward links only conditions of incomplete invariance 2a and 3b (Table I) can be used to do this. Conditions of absolute invariance 2b, 3b and 4 can be used only with feedforward links. In distinction to systems which operate only by disturbances (and here also there are great possibilities for satisfying the conditions of invariance if the plant is selfaligning), it is possible in combined systems to have great rigidity; this eliminates the errors due to disturbances by which there are no feedforward links.
-;:::==1
( b) Figure 2. Examples of adaptive servomechanisms with self-variation of the coefficients of amplification: (a) on the' disturbance' prillciple; (b) on the 'output' principle (with reverse cOllnection). A is the amplitude measurer, f the frequency measurer, NP the 1I01l-linear converter, K the correlator
Rules for selecting control system schemes (I) If all noises and disturbances can be measured and the object is self-controlled and the characteristics of the amplifier are sufficiently stable, there is no necessity to use feedback. It is sufficient to use an open cycle control cycle constructed according to the condition of absolute invariance 2b (Table 1).
systems which automatically vary (or adapt) such characteristics as settings, programmes, parameters, non-linearities (from form S to N), algorisms, probability ratios, algorith, field of action, impulses, etc. 6 are such systems. If adaptiveness can be evaluated by one value function lP, then all these systems are made according to the same principles with little variation. It is possible to use either the' disturbance' or the 'output' principle (Figure 2). The most perfect is the combined system, in which both principles are combined. For example, tests of a combined controller of the steam/fuel ratio which acts on the air/fuel ratio, designed for steam boilers by the Electro-technical Institute of the Academy of Sciences of the Ukrainian S.S.R., showed that a
570
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APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
system which is non-combined is unworkable in this case (see Automatioll, No. 2, 1959). The theory of combined systems of stabilization evolved for ordinary non-adaptive systems can be applied with small changes to combined cybernetic systems. We shall investigate this thesis, the principal subject of this paper, taking as an example extremal systems, which are among the most highly developed of cybernetic systems today. Figure 2 shows how an automatic variation of amplification can be effected by means of extremal regulators. Servomechanisms with automatic variation of amplification according to the frequency and amplitude of a signal of varying sign were tested experimentally. Such signals are received by guidance systems and what are called correcting servomechanisms working in combination with open loop control systems. It has been established that cybernetic systems working by disturbance [Figure (2a)] can give a stable performance with a very high mean value of the coefficient of amplification, many times greater than that found for stability conditions with ordinary non-adaptive systems. As soon as the frequency and amplitude of the signal decrease, the system automatically returns from conditions of accurate movement
characteristic can be varied by means of the feedforward links 10 and /'0' By this it is possible to eliminate only the steady error LlI in certain •errorless conditions ' 4. In order to eliminate LlI in all conditions we must use a non-linear feedforward link with an •optimal compounding characteristic' M = f(L), which ensures the required value of fP5.6 . The main rules for the setting of combined adaptive systems are the same as for ordinary ones. Either the rule for compromise setting or statistical criteria may be used for single loop systems, and for multi-loop systems the conditions of non-absolute invariance 2a (Table 1) as well. In the case where the object of cybernetic control is an ordinary system of control (as in Figure 2), compromise setting or statistical criteria (or the entire by the' parts' method) can be applied twice- to the main system and to the cybernetic controller. The peculiarity of the synthesis of the feedback of a cybernetic regulator is that here it is easy to vary the input modulating signal, the feedback and its parameters being only slightly varied, whilst in ordinary systems on the other hand the layout and parameters of the feedback are changed (Appendix 2) .
Table 2 Equations of static~ For a stabilization system
For an extremal system
1:1 = -mo
Control law
1:1
= -mo
I
Regulator with proportional velocity
pM = (;(21: 1 = 0
d1:1
= 0
dM
Controlled process where
i
1:2
where
= M+l'oL
1:20
= co+ f3L;
1:21
= M+l'oL
The principal law for the setting of feedforward links is the conditions of absolute invariance 2c (Table I). These enable the error to be completely eliminated in ordinary systems, and a narrow oscillation around the extremum to be attained in extremal systems, the test error Ll3 remaining [Figure I (b)]. Orthogonality, discussed above, holds also for adaptive systems [Figure 1(b)]. Figure I (b) shows that the setting of the feedback of an extremal regulator (coordinate e12) does not affect the setting of the feedforward links (coordinate Y2 /YO) but on the other hand a feedforward link has no effect on oscillation damping. Thus the setting of combined extremal systems can be done by the' parts' method, as for ordinary systems. Is it possible to apply the conditions of invariance to extremal cybernetic systems, which we are accustomed to regard as nonlinear and as systems with variables whose parameters vary in time? Some types of extremal systems can be written as linear equations with constant coefficients. For example, a controlled plant with self-variations of setting has not got parameters varying in time . The system as a whole can be linear. If we use an extremal regulator with test signal modulation and a proportional servomotor, this is essentially a system of phase stabilization at the discriminator output (Figure 5 and Appendix 3). These linear equations are the first approximation for systems with variable (for example, self-varying) parameters, and the smaller the deviation the more accurate the approximation is.
with a high amplification coefficient to smoothing-out conditions with low amplification; the dynamic error is approximately halved. Comparison of static characteristics. It is sometimes stated that there is a very great difference between ordinary and cybernetic systems, since only fixed functions are encountered in the former, but statistical in the latter. In fact, however, all systems have statistical functions, but the fixed functions used in the analysis are a first approximation, which, the basis being equal, is also permissible in the analysis of cybernetic systems. Extremal systems very often have a servomotor. The nearest analogy to them is stabilization systems, which also include a servomotor. The geometrical position of the extremums in the space fPML can be regarded as an astatic characteristic (M is the manipulated variable), and the position of real operation (or the position of the centres of oscillation) as a static characteristic. Let us approximate the extremal characteristic by means of a parabola (Table 2); finding the equations of the static characteristic, we establish that (I) If 10 = 0, Co = 0, b l = 0, 110 = 0, then
and the system is astatic and the static and astatic characteristics coincide at all points; (2) The position of the static 571
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A. G. IVAKHNENKO
But what is more important is that the conditions of invariance are applicable not only to systems with a modulating test signal and a linear plant, but to all types of extremal controllers. Here, in order to satisfy the conditions of invariance in extremal systems with modulation , it is recommended to use both a measuring /(p) and a direct I'(p) feedforward link by disturbance (Figures 3 and 4), and in the re-
of noises are simplified in that it suffices to measure, not the noises themselves, but certain generalized parameters (the frequency of the signal, the ratio of the disturbances to the signal, etc.) . The signal can be cut out for an instant (the testing impulse method) in order to measure the interferences periodicallys.6 . Where a cybernetic controller is applied to an ordinary control system (Figure 2) it can be regarded as a secondary addition to the main system to improve its efficiency. It can act both on the feedback circuit and on the open loop part of the main system . A third controller, which adapts the characteristics of the secondary controller, can be added to a secondary cybernetic regulator and so on. Each such controller performs a given approximation in the iterated solution of the problem of control (the method of successive approximations) .
(b)
Ht)
- -~ ~
Step distributor
-- l Figure 4. Layout of all extremal control/er with illlegratioll by the 'sandwich' method
I
I I
B ---" _
I
4 : 1. 2. 3. 4. 5.
I I L ______ J
C }i L ______ _ __ _____ _ ____ J
I
Figure 3. Layout of temperature correctioll extremal regulators: (a) autocorreclioll, (b) reciprocal correctioll I. 2. 3. 4. 5.
Ampli fi er. R elay multi pli er. L ag line . Acc umul atin g fi lt er. Servomotor.
6. Rel ay. 7. Reversin g win din g.
Furnace . Thermocouple. Integrator. Reversing clutch . Step distributor.
8. Servomotor.
9. Fan. 10 . Atomizer.
The Technique of Eliminating the Effect of Noises
6. Operati ve dev ice.
The principal methods of receiving weak signals when disturbances are present , used in communications techniques, are also used in extremal systems (Figures 2, 3, 4 and 5), with the peculiarity that evaluations of the correlation functions of the , relay ' type are used:
7. A ir supply modulator.
8. At o mi zer. 9. Therm oco uple . 10. Amplifie r.
maining types of system, the direct connection l'(p). In order to satisfy the conditions of invariance it is sufficient that the ' characteristic of optimal compounding ' M = f(L) be linear, which is ensured by the linear static characteristic AA' A " (Table 3). With a direct feedforward link invariances can be applied to all types of extremal regulators. Rules for selecting adaptive sy stem schemes
The three rules given above apply to both ordinary and extremal systems in which the conditions for the measurement
I A 'tCT) = T
f Of(t)[A signf(t -
T)]dt
-T
fO
1 k '/",(T) = T _ /(t)[A sign .p(t-T)]dt
instead of the usual proportional ones s. The relay functions use only the -sign of one of the multiplied functions instead of the function itself. This enables the correlator to be
572
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APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
Table 3 Statical characteristic
In the coordinates M-L
I n the coordinates
Extremal system
Stabilizatioll system M = ~o+~IL,
M = ~o+~IL,
where
where
~o
ko .p = almO+IIO
~l
=
mo(f3- alI'O) + 10 almo+llo
= ao-alM,
~l
= f3-1'0 = ao-alM,
where k (f3- all'o)
o .p = mo(f3all'o)-lo
ao
lIo(f3-a,l'o)-a,lo a, = mo(f3-a l l'o)-/o
In the coordinates
= co+-2amo
where ao
110
~o
al
= =
bO~12-bl~0~I-a[c0~1-~0(f3-1'0)]2
6 bl~' +2a(~I- f3+l'o)[cog,-~o(f3-l'o)l ~12
= Yo+ylL,
where
= YO+YIL,
where
YO
~.p = almo+llo
Yl
IIO(f3-a,l'O)-all = almo+llo
YO
=
bo-a(~0-co)2
Yl
=
bl-2a(~I+1'0-f31T)(~o-co)
M
M
o
(a)
Figure 5.
(b)
Example 0/ a method 0/ the test sigllal complication: (a) system with harmonic modulation; (b) system with a more complicated modulating sigllal
simplified. The increase in noise stability achieved by various methods can be compared in the following table: Cross correlation method [Figure 3(b)] 1·0 0·8--0·9 Auto-correlation method [Figure 3(a)] Integration method (accumulation) (Figure 4) 1·0 Cross correlation method combined with filtering of the effective signal 1·1-1 ·2 over 1·5 Complex signal method This order of accuracy is maintained both for an open cycle and where a feedback loop is concerned. The
figures were confirmed by tests with an MN-7 type model. The corrector eliminates noises by the same principle as a synchronous detector. The advantage of the complex modulating signal method is explained by an increase in the amount of information; a complex signal is more easily distinguished from noise than a simple one. An example of a system which distinguishes the letters K and I (Morse code) is given in Figure 5. Another example is a system which has two frequencies of the modulating signal and two synchronous detectors, giving very high noise stability5. Since the complex signal method is as yet insufficiently developed and the integration method is rather complicated, the best method for practical use is that of
573
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A. G. IVAKHNENKO
cross correlation. The essential equipment for this method is being produced in quantity (VTI regulator, type ER)4. Extremal control systems based on this universal controller will be distinguished from each other only by the measuring element which produces a tension proportional to the 'value function or figure of merit' cI>.
which denotes that the rigidity
Appendix I-By The 'Parts' Method of Optimal Setting of Combined Systems We shall investigate the system described by the equations: control law
can be used by selecting the operator of the internal feedback n(p) = no+n\p+ n2pz... The magnitude Wo can be increased until the fluctuation of the coefficients 110n\n2 no longer has too great an effect on x and y. In the second stage of setting the coefficients of the right side must be selected using the condition of absolute invariance 2b (Table 1):
E
=
-m(p)cp-n(p)M-/(p)L;
s
=
x
cp = cI>-
=
az
(ao)t
b2 Wo
=
Y2=-=0
A+ Yz{p)M-f3(p)L+l'(p)L;
1=
Z(p)
m(p) = mo
where
(
ao
)t
Wo = (ao)! ---+ max
(3(p) + !'(p)+ Y1(p) Yz(p)! /(p) z(p) p = 0
The conditions of absolute invariance in the second form are written in the form
where D = d/dT. In the first stage of setting the coefficients of the left side of the equation must be selected. For a single-loop system [n(p) = 0] the condition of the maximum of the degree of stability can be used for compromise setting. We get the maximum when 2x 3 - 9xy+ 27 = 0, x < 3 (reference 4). For example, having selected x = 1'2, )' = 3, from this we determine the coefficient of amplification Ul u 2mO and the nominal value of the controlled variable cI>0. If, however, we have a two-loop system [n(p) of- 0] then there are further possibilities, i.e. we can select x = 1'2, y = 3 and use the condition of non-absolute invariance 2a (Table I) as well: in the form
Wo =
bo Yo = - = 0 Wo
1+ Yj(p) Yz(p)m(p)+
(D3+ x Dz+ yD+ I)cp = (y z D2+ y \D+Yo)L
x;
al
y = (a02)t'
'
b 3(p) = bzpz+b1P+bo
We shall introduce the non-dimensional time T = wot, where Wo = (ao)!. The equation of the system is written in the form
I - - -+ a3(p)
~,
Appendix 3-Selecting Coefficients of Feedforward Links by Conditions of Invariance In equations of the system discussed in Appendix 1 we get for the variable cI> the characteristic equation
then for the system as a whole we get
a3(p) = p3+a2pz+aIP+aO,
1."
Appendix 2-An Example of the Application of the Statistical Method We shall assume that an extremal controller with forced oscillations varies the manipulated variable M according to the law of rectangular impulses. The auto-correlation function of such a signal is A,.{T) = AZ e- 2kT , where k is the frequency. This variable passes together with noise of the 'blank' type, for which A,.{T) = a 2 through the plant circuit and measuring element, whose weight function (e.g. wet) = we- I / T ) is known (a plant of the first order). For the best noise stability of the system, the ratio of noise to signal must be as small as possible. Using Viner's formula in the form given in lones's articleS, we find: w(t) = A!'(T) or w < A2 and k ~ (I/2T). The practical conclusion to be drawn from this example is that it establishes the feasibility of determining the number of makes-and-breaks and the amplitude of the test signal by Viner's method.
Here the following designations are introduced: cp is the deviation of the controlled variable, '" is reference, cI> is the absolute value of the controlled variable, M is the manipulated variable, L is the principal disturbance (load on the controlled plant), E is the sum of the voltages at the input of the amplifier, Z(p) is the operator (impedance) of load of the controlled plant, m(p), n(p), /(p) are the operators of the measuring links, Y1(p) is the operator of the amplifier, Yz{p) is the operator of the controlled plant, l'(p) is the operator of the direct feedforward links, and 0 is the voltage on the servomotor of an extremal system with modulating test signal p = d /dt. If Yz(p) = Uz ,Y1(p) = _U_l_ (-'IP+ I)(,zp+ I) T3P+ I (3(p) = (30,
ao ---+ max
These conditions can be satisfied by selecting the compounding feedforward links.
plant
=
=
The equations
amplifier and servomotor
cI>
1 + up
Hence we get for
l'(p) = 0
for
/(p) = 0
I (3(p)+ Y\(p) Yz{p) - /(p) = 0
(2')
(3(p) + /'(p) = 0
(2")
p
We can say: (I) Conditions 2' and 2" can be fulfilled only in a system in which it is possible to select /(p) or /'(p), i.e. in combined systems with control by disturbance. (2) The feedforward links /(p) and /'(p) can be considered as open loop cycle and affect stability except in cases where the conditions of invariance are fulfilled or where
574
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APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
z(p) = 00 . Reference4discussesthecasewherez(p) = AO+C
L
amplifier
U = Y 4 (p)L
Then, in order that the servomotor will not come in when varies, the following correlations must be maintained:
M
control law
L
=
-8+/(p)L
amplifier
U
=
Y4(p)L
MI = Yip)[M + l'(p)L]
M = YI(p)! U p
servomotor
non-linear part of the plant
non-linear part of the plant
(
LzO
=
(MI-M IO ) = -a(M- L zo)2
co+13L where
the phase discriminator (correlator)
6
=
Y;(P) = i'O+I'IP+l'ZPZ+ ... , l'npn
I
linear part of the plant
where
=
P
M = YI(p) - U
servomotor
0, l'(p)
(2) In the arrangement for displacing the object , a nonlinear part acts first, and then a part with inertia. Here we get the following equations for a system with modulation:
-6+/(p)
=
=
Y 3(p)[M I - Lzo]
LzO = co+13o(L)+l'o(L)
There is a system of linear equations for the variables L,6, U, M, M I · For example, let Yo(p) = Y I Yz Y 3 Y 4 , (1/ Yop) = TP+ 1, rep) = 0 and Co = O. We then get
linear part of the plant
6
We shall introduce the non-dimensional time T = wot, where Wo = (az)t. As a result we get the equation
CI2
= {I/(Ta)t}
for a linear system
Solutions
Lzo
(1) where L = [1]:
=
=
};20 =
13 ZClZ Yo-Yz, b -- YOC12-YI+Y 1312 and 12 = (l-cli)l-
(2) where L = vt:
6 = YoT-[).4+e-c12T(a cos 1312T+b sin 13lzT) where
b = 2clz2yo- CIZYI +yz -Yo 1312 The conditions of absolute invariance here have the form:
Y2
co+13oL-l'oL = constant
130 = 1'0
or
for a non-linear system
where
a
Y 3(p)M I
Here it is sufficient for absolute invariance to use a rigid feedforward link selected in accordance with the characteristic of optimal compounding (where /(p) = 0) :
(DZ+2c12D+ 1)6 = (yzDz+yID+Yo)L where D = (d/dT),
=
=
Yo
and
Y = 2clzYo
The forms of the transient processes for [).4 = 0 are shown in Figure lea). The above example shows the selection of the coefficients of the measuring link /(p) . We shall also discuss a method of selecting the coefficients of the direct link {'(p). It is necessary to distinguish two cases: (I) The objects consist of parts with inertia and nonlinearity. The optimal characteristic of compounding is known:
or
130L = l'(L)
In references 5 and 6 a method of selecting a non-linear direct feedforward link is developed, and the substitution of a simpler arrangement of approximation links for the links for a few disturbances l' o(L1> L z. L3 ... L,J is discussed . The results of applying the conditions of invariance in an extremal system are given in Figure I (a) and (b). Attention is drawn tQ the fact that, as may be seen by comparing Figure I (a) and (b), the conditions of invariance for the variable 6 differ in no way from the conditions of invariance for the controlled variable
I SHC HPANOV, G. V.
I (I939)
575
585
co+13o(L)-l'o(L) = constant
A.
G.
IVAKHNENKO
KULEBAKIN, V. S. The applicability of the principle of absolute invariance to systems which can be physically realized. Dokl. Akad. Nauk S.S.S.R. 68 No. 5 (1949) 3 IVAKHNENKo, A. G. The theory of compounding regulators. Sbom. Trud. Inst. Elektrotekh . Akad. Nauk Ukr. S.S.R. No. 6 (195 I) and No. 10 (1953) 4IvAKHNENKO, A. G. Electro-automation. 1954. (2nd edition, 1957). Gostekhizdat of the Ukrainian S.S.R. 5 IVAKHNENKo, A. G. A series of articles on technical cybernetics. Avtomatika Nos. I, 2, 3, 4 (1958), Nos. I, 2, 3, 4 (1959), Nos. I, 2 (1960) 6IvAKHNENKO, A. G. Technical Cybernetics. 1959. Gostekhizdat of the Ukrainian S.S.R. 7 KRYZHANOVSKI, O. M. and KUNTSEVICH, V. M. Investigation transient conditions in extremal control systems. Avromatika No. 3 (1958) 8 Automatic Control. Collected papers of the conference at Crenfield . 1951. Foreign Publications, 1954
9
2
10 11
12 13
14
MEYEROV, V. M. Syllfhesis of the Arrangement of High Accuracy Automatic COllfrol Systems. 1958. Fizmatgiz MINORSKY, N . Directional stability of automatically controlled bodies. J . Amer. Soc. Ilav. Archit. 34 (1922). See also Autopilots, ed. Prof. G. V. Shchpanov. 1939. Oborongiz PETROV, B. N . The application of the conditions of invariance. TrailS . 11th AII-Unioll COil! on the Theory of Automatic Control, vol. 11, pp. 241-246. 1955. Academy of Sciences of the U .S.S.R. See also Petrov, B. N. Selbsttatige Regelungssysteme mit zusatzlicher Lastaufschaltung. Fachtagung Regelungstechnik, Heidelberg, 1956, Beitrag 92 MOORE, J. R. Combination open-cycle closed cycle control system. Proc. Inst. Radio Ellgrs, N. Y. 39 No. 11 (1951) DRENICK, R . F. and SHACHBENDER, R . A. Adaptive servomechanisms. TrallS. A mer. Inst. elect. Engrs 76, Pt. Il (1957) ASELTINE, J. A., MANCINI, A. R. and SARTURE, C. W. A survey of adaptive control systems. Inst. Radio Engrs, Trans . on Automatic Control PGAC-6 Dec. (1958)
DISCUSSION SHORRIN (U.S.S.R.) In the majority of real control systems the initial conditions are not zero. In this case what conditions of invariance are used to obtain zero or nearly zero dynamic error ? A. G. IVAKHNENKO, in reply. The real initial conditions should be distinguished from those which may be called equivalent initial conditions. We cannot change the real initial conditions. Equivalent conditions are those described by the homogeneous equation of motion, a3(p) q, = 0, solution of which does not differ from the
solutions of the complete equation a3 (P). q,= b3 (P) . A with real initial conditions. The condition of invariance (in the second form) corresponds to zero values of the equivalent initial conditions, defined by Kryzhanovski's formula. It is found that the open-loop disturbance coefficients enter into expressions defining the equivalent initial conditions in such a way that we may freely choose their magnitude. Consequently, taking zero values of the equivalent initial conditions, we can in this way completely eliminate errors, i.e. obtain adjustment corresponding to absolutc invariance q, - O.
576
586
A Brief Introduction to the Theory of Invariance and the Theory of Combined Systems Conditions of invariance indicate methods of eliminating errors in control systems. By conditions of invariance we shall mean conditions applied to the coefficients of the differential equation of a dynamic system and the principal disturbing influences, which are such that the error of the system is identical to zero, ~ =
Table 1 Equation a3(p)
]c Forms of the equations J5~ of invariance --
Data on disturbances
I
Component of the error which is eliminated
L\(t) = 0, Lz(t) = 0
I
Has no practical application
(1)
S
0
t
-li
0
2
In arbitrary form with limitation for acceleration
«l ""'
~
Method of accomplishment
Reduces the error caused by all disturbances, including variations of the parameters in time·. Nonabsolute invariance
Conditional stability multi-loop systems with 'key' type frequency of the characteristic (see reference 4, p. 194; and reference 9). Non-unitary operational feedback in servomechanisms
The same
Eliminates all the error caused by disturbances L\ and Lz. Absolute invariance
Control by disturbances and their derivatives by time 4
In the general form in symbols
Only the steady component caused by all disturbances
The integral equation according to Minorski\O and Kulebakin z
The same
Only the steady component caused by one disturbance L\
Control by disturbances and their derivatives by time 4
In the form in numerals
All the error caused by the disturbance Device displacing L2(t) (forcing) and also artificial variation of data for programmed L\ systems (see reference 4, p. 293)
11
5":' ""'I~
00:0 N
'" <::s'"
IIII~
-Cl
~C:;: "'-~
I
~'"
""
-Cl-Cl
,-...
0
M ~
11 ,-...
:::r
0
-I~
11 ,-...
:::r
3
«l ""' 8
o~
~I~ ~~
11;;:
~~
~~
:::r
~
-4 I' b3(p)L\(t)
,£
+b"3(P)L2(t) = 0
(4)
i
• The validity of form 2a for systems whose parameters are variables in time has been pointed out and investigated by A. I. Kukhtenko.
568
578
APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
paid to the different properties of conditions 2a and 3a and of conditions 2b, 3b and 4. The first are connected with certain requirements pertaining to the left side of the equation of a dynamic system, and are called the conditions of non-absolute (i .e. incomplete) invariance. The second impose certain requirements on the right side of the equation of a dynamic system and are called the conditions of absolute invariance, since with certain limitations these conditions determine the absolute, i.e. the complete elimination of all the components of the error of control systems.
ated; (c) there is no necessity to use delta-function impulses in the manipulated variable to compensate errors; the last condition is satisfied if q
I'0
I
I
n-m
where q is the order of the equation of the dynamics of the object, n is the order of the left side of the equation of the system, and m is the order of the right side of the equation of the dynamics of the system. It follows from (b) that absolute invariance is achieved mainly in combined systems, which combine the principles of control 'by disturbance' and 'by output'. Control by
Oscillograms of the ·,oltoge at t h e ou t pu t of t h e p h ase discriminator 6 with 11 / rO' 2C 12
y2
~
. d ~x 0 f th e 0 f t he In extremum Cl>
Osc illogram
of lrror
Overcompens.ation
Overcompensiltion of error
15 Compromi se tuning
C ,0·25
-.......
+005 10
f.llillW
0·5
o~
re·!
t<>S
0
0
o ·~
~
0
S ro' 1.0.
-o·!
-{).!
fill ill lill WJ 05
..u
5
~
1-0- 5
-o~
WJ
r
?i'1 10 10 >
~
0
1'2
-OS
-05
0·5 10
!'
10
0
0
0
-0'5
-OS
05
-{)·S
-0·5
0
70 ,0'5
05
05 0
0
-o~
I
~ f--1+0
-
r
1 ·5
Shortest process
II
'08, ,15
Fulfil of tment he condl lions of inva riance
f
11
se;vomotor
does no t come i n
lL 1. iO' 2C12 70 '
O!
o!
0
0
rro0
o·5
IJj j
0
lIII/I
0 05
-0
05
lill
~S
Undercompensation of error
0·2
I
I
04
06
Undercompensation of error
1 10
0'25
0' 5
(a i
Figure I.
{"""
1·0
10
(b)
The effect of a feedforward link (coordinate Y2 /YO) and a f eedback loop (coordinate in a combined system with the equation
Some authors call the conditions of absolute invariance the conditions of disturbance compensation. It is clear that an error can be under-compensated, fully compensated or overcompensated (Figure I). In distinction to this, the conditions of non-absolute invariance 2a can only give satisfactorily sufficient (and only within limits, complete) reduction of the error. Here over-compensation is possible only for the separate components of error. For example, we can obtain negative steady-state error. The condition of absolute invariance 2b is valid, if: (a) for equal disturbances the forms of the transient processes in the controlled object are almost identical, i.e. the processes can be repeated; this is important for open cycle systems ; (b) the process is self-controlled or the system is a combined one so that zero drift, which upsets the initial zero conditions, is elimin-
e l2)
011
the form of the trallsiellf processes
disturbance is very often control by open-loop circuit (if t:.L #- f(t:.tJ») , where L is the principal disturbance and tJ> is the controlled variable; control by deviation (by the output variable) is, however, always control by a feedback closed cycle. Principal rules for combined systems setting
The principal condition for the setting of a single-loop system is what is called 'compromise setting' in which such a coefficient of amplification is selected that, when the rigidity is sufficient (degree of accuracy s = I + a p = ao), a sufficient stability is achieved. Statistical methods of selecting the optimum transfer function can also be used for the same purpose. In multi-loop systems, besides the conditions of compromise setting, the condition of absolute invariance 2a
569
579
A. G. IVAKHNENKO
(Table I) can also be used. This condition is closely bound up with the work of Meyerov 9 and others. The principal law for the setting of feedforward links by disturbances is to satisfy the conditions of absolute invariance 2b (Table I). The orthogonality (absence of correlation effect) of the two setting methods. If small deviations of the controlled variable do not affect the disturbance (!J.L =I=- f(!J.IP», and also when the conditions of invariance are satisfied ( == 0), methods of increasing accuracy and quickness of action by improving the feedforward links affect the selection of the coefficients of the right side of the equation of the dynamics of the system, and the methods of improving the feedbacks affect the selection of the coefficients of its left side (Figure I). The effect of feedforward link on stability is in practice small. The setting of combined systems can always be done 'by parts'; in this method the feedbacks are first improved as much as possible, and then the same is done for the feedforward links (see Appendix I).
(2) If the noises and disturbances cannot be measured a closed loop system of feedback set according to the rules of compromise setting or by statistical criteria and, in addition, by the condition of non-absolute invariance 2a (Table 1), should be used. (3) If only part of the noises and disturbances can be measured, or if the characteristics of the amplifier are insufficiently stable, a combined system, set by the' parts' method, must be used. Setting includes both the selection of the parameter values and the synthesis of the scheme by the introduction of new elements (Appendix 1). Comparison of Ordinary and Cybernetic Adaptive Systems By cybernetic systems we understand systems which carry out more complicated tasks than the classical tasks of stabilization, schedule time systems and servomechanisms. In particular,
The advantages of combined systems in steady-state regimes (a) Power advantages-The greater the accuracy required from the system, the greater these are 4 • The power at the output of a feedback (corrector) can be approximately onefifth that of a system without feedforward link by disturbance, the magnitude of the latter being the same. The greater the accuracy of the feedforward links the less work is left to the corrector. The figure given above (a fifth) was found in practice in the control of synchronous generators with compounding by full current. Setting advantages-The setting of combined systems is simpler and covers a wider range. One can have a system with negative statism or a system with high rigidity and statics together. Negative statism is necessary to compensate the potential drop in the line, and great rigidity with high statism are required in order to get stable parallel performance of the controlled plant (generators, boilers, etc.).
(a)
Amplifier tJs (t)
Advantages of combination systems in dynamic regimes The total error of the system is !J. = !J.! +!J. 2 + !J. 4 , where !J.! is the static, !J. 2 the transient and !J. 4 the velocity error. Combination systems greatly increase the possibility of using the conditions of invariance which enable these errors to be eliminated or reduced, i.e. they increase the speed of action and accuracy of the system. In systems without feedforward links only conditions of incomplete invariance 2a and 3b (Table I) can be used to do this. Conditions of absolute invariance 2b, 3b and 4 can be used only with feedforward links. In distinction to systems which operate only by disturbances (and here also there are great possibilities for satisfying the conditions of invariance if the plant is selfaligning), it is possible in combined systems to have great rigidity; this eliminates the errors due to disturbances by which there are no feedforward links.
-;:::==1
( b) Figure 2. Examples of adaptive servomechanisms with self-variation of the coefficients of amplification: (a) on the' disturbance' prillciple; (b) on the 'output' principle (with reverse cOllnection). A is the amplitude measurer, f the frequency measurer, NP the 1I01l-linear converter, K the correlator
Rules for selecting control system schemes (I) If all noises and disturbances can be measured and the object is self-controlled and the characteristics of the amplifier are sufficiently stable, there is no necessity to use feedback. It is sufficient to use an open cycle control cycle constructed according to the condition of absolute invariance 2b (Table 1).
systems which automatically vary (or adapt) such characteristics as settings, programmes, parameters, non-linearities (from form S to N), algorisms, probability ratios, algorith, field of action, impulses, etc. 6 are such systems. If adaptiveness can be evaluated by one value function lP, then all these systems are made according to the same principles with little variation. It is possible to use either the' disturbance' or the 'output' principle (Figure 2). The most perfect is the combined system, in which both principles are combined. For example, tests of a combined controller of the steam/fuel ratio which acts on the air/fuel ratio, designed for steam boilers by the Electro-technical Institute of the Academy of Sciences of the Ukrainian S.S.R., showed that a
570
580
APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
system which is non-combined is unworkable in this case (see Automatioll, No. 2, 1959). The theory of combined systems of stabilization evolved for ordinary non-adaptive systems can be applied with small changes to combined cybernetic systems. We shall investigate this thesis, the principal subject of this paper, taking as an example extremal systems, which are among the most highly developed of cybernetic systems today. Figure 2 shows how an automatic variation of amplification can be effected by means of extremal regulators. Servomechanisms with automatic variation of amplification according to the frequency and amplitude of a signal of varying sign were tested experimentally. Such signals are received by guidance systems and what are called correcting servomechanisms working in combination with open loop control systems. It has been established that cybernetic systems working by disturbance [Figure (2a)] can give a stable performance with a very high mean value of the coefficient of amplification, many times greater than that found for stability conditions with ordinary non-adaptive systems. As soon as the frequency and amplitude of the signal decrease, the system automatically returns from conditions of accurate movement
characteristic can be varied by means of the feedforward links 10 and /'0' By this it is possible to eliminate only the steady error LlI in certain •errorless conditions ' 4. In order to eliminate LlI in all conditions we must use a non-linear feedforward link with an •optimal compounding characteristic' M = f(L), which ensures the required value of fP5.6 . The main rules for the setting of combined adaptive systems are the same as for ordinary ones. Either the rule for compromise setting or statistical criteria may be used for single loop systems, and for multi-loop systems the conditions of non-absolute invariance 2a (Table 1) as well. In the case where the object of cybernetic control is an ordinary system of control (as in Figure 2), compromise setting or statistical criteria (or the entire by the' parts' method) can be applied twice- to the main system and to the cybernetic controller. The peculiarity of the synthesis of the feedback of a cybernetic regulator is that here it is easy to vary the input modulating signal, the feedback and its parameters being only slightly varied, whilst in ordinary systems on the other hand the layout and parameters of the feedback are changed (Appendix 2) .
Table 2 Equations of static~ For a stabilization system
For an extremal system
1:1 = -mo
Control law
1:1
= -mo
I
Regulator with proportional velocity
pM = (;(21: 1 = 0
d1:1
= 0
dM
Controlled process where
i
1:2
where
= M+l'oL
1:20
= co+ f3L;
1:21
= M+l'oL
The principal law for the setting of feedforward links is the conditions of absolute invariance 2c (Table I). These enable the error to be completely eliminated in ordinary systems, and a narrow oscillation around the extremum to be attained in extremal systems, the test error Ll3 remaining [Figure I (b)]. Orthogonality, discussed above, holds also for adaptive systems [Figure 1(b)]. Figure I (b) shows that the setting of the feedback of an extremal regulator (coordinate e12) does not affect the setting of the feedforward links (coordinate Y2 /YO) but on the other hand a feedforward link has no effect on oscillation damping. Thus the setting of combined extremal systems can be done by the' parts' method, as for ordinary systems. Is it possible to apply the conditions of invariance to extremal cybernetic systems, which we are accustomed to regard as nonlinear and as systems with variables whose parameters vary in time? Some types of extremal systems can be written as linear equations with constant coefficients. For example, a controlled plant with self-variations of setting has not got parameters varying in time . The system as a whole can be linear. If we use an extremal regulator with test signal modulation and a proportional servomotor, this is essentially a system of phase stabilization at the discriminator output (Figure 5 and Appendix 3). These linear equations are the first approximation for systems with variable (for example, self-varying) parameters, and the smaller the deviation the more accurate the approximation is.
with a high amplification coefficient to smoothing-out conditions with low amplification; the dynamic error is approximately halved. Comparison of static characteristics. It is sometimes stated that there is a very great difference between ordinary and cybernetic systems, since only fixed functions are encountered in the former, but statistical in the latter. In fact, however, all systems have statistical functions, but the fixed functions used in the analysis are a first approximation, which, the basis being equal, is also permissible in the analysis of cybernetic systems. Extremal systems very often have a servomotor. The nearest analogy to them is stabilization systems, which also include a servomotor. The geometrical position of the extremums in the space fPML can be regarded as an astatic characteristic (M is the manipulated variable), and the position of real operation (or the position of the centres of oscillation) as a static characteristic. Let us approximate the extremal characteristic by means of a parabola (Table 2); finding the equations of the static characteristic, we establish that (I) If 10 = 0, Co = 0, b l = 0, 110 = 0, then
and the system is astatic and the static and astatic characteristics coincide at all points; (2) The position of the static 571
581
A. G. IVAKHNENKO
But what is more important is that the conditions of invariance are applicable not only to systems with a modulating test signal and a linear plant, but to all types of extremal controllers. Here, in order to satisfy the conditions of invariance in extremal systems with modulation , it is recommended to use both a measuring /(p) and a direct I'(p) feedforward link by disturbance (Figures 3 and 4), and in the re-
of noises are simplified in that it suffices to measure, not the noises themselves, but certain generalized parameters (the frequency of the signal, the ratio of the disturbances to the signal, etc.) . The signal can be cut out for an instant (the testing impulse method) in order to measure the interferences periodicallys.6 . Where a cybernetic controller is applied to an ordinary control system (Figure 2) it can be regarded as a secondary addition to the main system to improve its efficiency. It can act both on the feedback circuit and on the open loop part of the main system . A third controller, which adapts the characteristics of the secondary controller, can be added to a secondary cybernetic regulator and so on. Each such controller performs a given approximation in the iterated solution of the problem of control (the method of successive approximations) .
(b)
Ht)
- -~ ~
Step distributor
-- l Figure 4. Layout of all extremal control/er with illlegratioll by the 'sandwich' method
I
I I
B ---" _
I
4 : 1. 2. 3. 4. 5.
I I L ______ J
C }i L ______ _ __ _____ _ ____ J
I
Figure 3. Layout of temperature correctioll extremal regulators: (a) autocorreclioll, (b) reciprocal correctioll I. 2. 3. 4. 5.
Ampli fi er. R elay multi pli er. L ag line . Acc umul atin g fi lt er. Servomotor.
6. Rel ay. 7. Reversin g win din g.
Furnace . Thermocouple. Integrator. Reversing clutch . Step distributor.
8. Servomotor.
9. Fan. 10 . Atomizer.
The Technique of Eliminating the Effect of Noises
6. Operati ve dev ice.
The principal methods of receiving weak signals when disturbances are present , used in communications techniques, are also used in extremal systems (Figures 2, 3, 4 and 5), with the peculiarity that evaluations of the correlation functions of the , relay ' type are used:
7. A ir supply modulator.
8. At o mi zer. 9. Therm oco uple . 10. Amplifie r.
maining types of system, the direct connection l'(p). In order to satisfy the conditions of invariance it is sufficient that the ' characteristic of optimal compounding ' M = f(L) be linear, which is ensured by the linear static characteristic AA' A " (Table 3). With a direct feedforward link invariances can be applied to all types of extremal regulators. Rules for selecting adaptive sy stem schemes
The three rules given above apply to both ordinary and extremal systems in which the conditions for the measurement
I A 'tCT) = T
f Of(t)[A signf(t -
T)]dt
-T
fO
1 k '/",(T) = T _ /(t)[A sign .p(t-T)]dt
instead of the usual proportional ones s. The relay functions use only the -sign of one of the multiplied functions instead of the function itself. This enables the correlator to be
572
582
APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
Table 3 Statical characteristic
In the coordinates M-L
I n the coordinates
Extremal system
Stabilizatioll system M = ~o+~IL,
M = ~o+~IL,
where
where
~o
ko .p = almO+IIO
~l
=
mo(f3- alI'O) + 10 almo+llo
= ao-alM,
~l
= f3-1'0 = ao-alM,
where k (f3- all'o)
o .p = mo(f3all'o)-lo
ao
lIo(f3-a,l'o)-a,lo a, = mo(f3-a l l'o)-/o
In the coordinates
= co+-2amo
where ao
110
~o
al
= =
bO~12-bl~0~I-a[c0~1-~0(f3-1'0)]2
6 bl~' +2a(~I- f3+l'o)[cog,-~o(f3-l'o)l ~12
= Yo+ylL,
where
= YO+YIL,
where
YO
~.p = almo+llo
Yl
IIO(f3-a,l'O)-all = almo+llo
YO
=
bo-a(~0-co)2
Yl
=
bl-2a(~I+1'0-f31T)(~o-co)
M
M
o
(a)
Figure 5.
(b)
Example 0/ a method 0/ the test sigllal complication: (a) system with harmonic modulation; (b) system with a more complicated modulating sigllal
simplified. The increase in noise stability achieved by various methods can be compared in the following table: Cross correlation method [Figure 3(b)] 1·0 0·8--0·9 Auto-correlation method [Figure 3(a)] Integration method (accumulation) (Figure 4) 1·0 Cross correlation method combined with filtering of the effective signal 1·1-1 ·2 over 1·5 Complex signal method This order of accuracy is maintained both for an open cycle and where a feedback loop is concerned. The
figures were confirmed by tests with an MN-7 type model. The corrector eliminates noises by the same principle as a synchronous detector. The advantage of the complex modulating signal method is explained by an increase in the amount of information; a complex signal is more easily distinguished from noise than a simple one. An example of a system which distinguishes the letters K and I (Morse code) is given in Figure 5. Another example is a system which has two frequencies of the modulating signal and two synchronous detectors, giving very high noise stability5. Since the complex signal method is as yet insufficiently developed and the integration method is rather complicated, the best method for practical use is that of
573
583
A. G. IVAKHNENKO
cross correlation. The essential equipment for this method is being produced in quantity (VTI regulator, type ER)4. Extremal control systems based on this universal controller will be distinguished from each other only by the measuring element which produces a tension proportional to the 'value function or figure of merit' cI>.
which denotes that the rigidity
Appendix I-By The 'Parts' Method of Optimal Setting of Combined Systems We shall investigate the system described by the equations: control law
can be used by selecting the operator of the internal feedback n(p) = no+n\p+ n2pz... The magnitude Wo can be increased until the fluctuation of the coefficients 110n\n2 no longer has too great an effect on x and y. In the second stage of setting the coefficients of the right side must be selected using the condition of absolute invariance 2b (Table 1):
E
=
-m(p)cp-n(p)M-/(p)L;
s
=
x
cp = cI>-
=
az
(ao)t
b2 Wo
=
Y2=-=0
A+ Yz{p)M-f3(p)L+l'(p)L;
1=
Z(p)
m(p) = mo
where
(
ao
)t
Wo = (ao)! ---+ max
(3(p) + !'(p)+ Y1(p) Yz(p)! /(p) z(p) p = 0
The conditions of absolute invariance in the second form are written in the form
where D = d/dT. In the first stage of setting the coefficients of the left side of the equation must be selected. For a single-loop system [n(p) = 0] the condition of the maximum of the degree of stability can be used for compromise setting. We get the maximum when 2x 3 - 9xy+ 27 = 0, x < 3 (reference 4). For example, having selected x = 1'2, )' = 3, from this we determine the coefficient of amplification Ul u 2mO and the nominal value of the controlled variable cI>0. If, however, we have a two-loop system [n(p) of- 0] then there are further possibilities, i.e. we can select x = 1'2, y = 3 and use the condition of non-absolute invariance 2a (Table I) as well: in the form
Wo =
bo Yo = - = 0 Wo
1+ Yj(p) Yz(p)m(p)+
(D3+ x Dz+ yD+ I)cp = (y z D2+ y \D+Yo)L
x;
al
y = (a02)t'
'
b 3(p) = bzpz+b1P+bo
We shall introduce the non-dimensional time T = wot, where Wo = (ao)!. The equation of the system is written in the form
I - - -+ a3(p)
~,
Appendix 3-Selecting Coefficients of Feedforward Links by Conditions of Invariance In equations of the system discussed in Appendix 1 we get for the variable cI> the characteristic equation
then for the system as a whole we get
a3(p) = p3+a2pz+aIP+aO,
1."
Appendix 2-An Example of the Application of the Statistical Method We shall assume that an extremal controller with forced oscillations varies the manipulated variable M according to the law of rectangular impulses. The auto-correlation function of such a signal is A,.{T) = AZ e- 2kT , where k is the frequency. This variable passes together with noise of the 'blank' type, for which A,.{T) = a 2 through the plant circuit and measuring element, whose weight function (e.g. wet) = we- I / T ) is known (a plant of the first order). For the best noise stability of the system, the ratio of noise to signal must be as small as possible. Using Viner's formula in the form given in lones's articleS, we find: w(t) = A!'(T) or w < A2 and k ~ (I/2T). The practical conclusion to be drawn from this example is that it establishes the feasibility of determining the number of makes-and-breaks and the amplitude of the test signal by Viner's method.
Here the following designations are introduced: cp is the deviation of the controlled variable, '" is reference, cI> is the absolute value of the controlled variable, M is the manipulated variable, L is the principal disturbance (load on the controlled plant), E is the sum of the voltages at the input of the amplifier, Z(p) is the operator (impedance) of load of the controlled plant, m(p), n(p), /(p) are the operators of the measuring links, Y1(p) is the operator of the amplifier, Yz{p) is the operator of the controlled plant, l'(p) is the operator of the direct feedforward links, and 0 is the voltage on the servomotor of an extremal system with modulating test signal p = d /dt. If Yz(p) = Uz ,Y1(p) = _U_l_ (-'IP+ I)(,zp+ I) T3P+ I (3(p) = (30,
ao ---+ max
These conditions can be satisfied by selecting the compounding feedforward links.
plant
=
=
The equations
amplifier and servomotor
cI>
1 + up
Hence we get for
l'(p) = 0
for
/(p) = 0
I (3(p)+ Y\(p) Yz{p) - /(p) = 0
(2')
(3(p) + /'(p) = 0
(2")
p
We can say: (I) Conditions 2' and 2" can be fulfilled only in a system in which it is possible to select /(p) or /'(p), i.e. in combined systems with control by disturbance. (2) The feedforward links /(p) and /'(p) can be considered as open loop cycle and affect stability except in cases where the conditions of invariance are fulfilled or where
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APPLICATION OF THE THEORY OF COMBINED SYSTEMS TO ADAPTIVE CYBERNETIC SYSTEMS
z(p) = 00 . Reference4discussesthecasewherez(p) = AO+C
L
amplifier
U = Y 4 (p)L
Then, in order that the servomotor will not come in when varies, the following correlations must be maintained:
M
control law
L
=
-8+/(p)L
amplifier
U
=
Y4(p)L
MI = Yip)[M + l'(p)L]
M = YI(p)! U p
servomotor
non-linear part of the plant
non-linear part of the plant
(
LzO
=
(MI-M IO ) = -a(M- L zo)2
co+13L where
the phase discriminator (correlator)
6
=
Y;(P) = i'O+I'IP+l'ZPZ+ ... , l'npn
I
linear part of the plant
where
=
P
M = YI(p) - U
servomotor
0, l'(p)
(2) In the arrangement for displacing the object , a nonlinear part acts first, and then a part with inertia. Here we get the following equations for a system with modulation:
-6+/(p)
=
=
Y 3(p)[M I - Lzo]
LzO = co+13o(L)+l'o(L)
There is a system of linear equations for the variables L,6, U, M, M I · For example, let Yo(p) = Y I Yz Y 3 Y 4 , (1/ Yop) = TP+ 1, rep) = 0 and Co = O. We then get
linear part of the plant
6
We shall introduce the non-dimensional time T = wot, where Wo = (az)t. As a result we get the equation
CI2
= {I/(Ta)t}
for a linear system
Solutions
Lzo
(1) where L = [1]:
=
=
};20 =
13 ZClZ Yo-Yz, b -- YOC12-YI+Y 1312 and 12 = (l-cli)l-
(2) where L = vt:
6 = YoT-[).4+e-c12T(a cos 1312T+b sin 13lzT) where
b = 2clz2yo- CIZYI +yz -Yo 1312 The conditions of absolute invariance here have the form:
Y2
co+13oL-l'oL = constant
130 = 1'0
or
for a non-linear system
where
a
Y 3(p)M I
Here it is sufficient for absolute invariance to use a rigid feedforward link selected in accordance with the characteristic of optimal compounding (where /(p) = 0) :
(DZ+2c12D+ 1)6 = (yzDz+yID+Yo)L where D = (d/dT),
=
=
Yo
and
Y = 2clzYo
The forms of the transient processes for [).4 = 0 are shown in Figure lea). The above example shows the selection of the coefficients of the measuring link /(p) . We shall also discuss a method of selecting the coefficients of the direct link {'(p). It is necessary to distinguish two cases: (I) The objects consist of parts with inertia and nonlinearity. The optimal characteristic of compounding is known:
or
130L = l'(L)
In references 5 and 6 a method of selecting a non-linear direct feedforward link is developed, and the substitution of a simpler arrangement of approximation links for the links for a few disturbances l' o(L1> L z. L3 ... L,J is discussed . The results of applying the conditions of invariance in an extremal system are given in Figure I (a) and (b). Attention is drawn tQ the fact that, as may be seen by comparing Figure I (a) and (b), the conditions of invariance for the variable 6 differ in no way from the conditions of invariance for the controlled variable
I SHC HPANOV, G. V.
I (I939)
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co+13o(L)-l'o(L) = constant
A.
G.
IVAKHNENKO
KULEBAKIN, V. S. The applicability of the principle of absolute invariance to systems which can be physically realized. Dokl. Akad. Nauk S.S.S.R. 68 No. 5 (1949) 3 IVAKHNENKo, A. G. The theory of compounding regulators. Sbom. Trud. Inst. Elektrotekh . Akad. Nauk Ukr. S.S.R. No. 6 (195 I) and No. 10 (1953) 4IvAKHNENKO, A. G. Electro-automation. 1954. (2nd edition, 1957). Gostekhizdat of the Ukrainian S.S.R. 5 IVAKHNENKo, A. G. A series of articles on technical cybernetics. Avtomatika Nos. I, 2, 3, 4 (1958), Nos. I, 2, 3, 4 (1959), Nos. I, 2 (1960) 6IvAKHNENKO, A. G. Technical Cybernetics. 1959. Gostekhizdat of the Ukrainian S.S.R. 7 KRYZHANOVSKI, O. M. and KUNTSEVICH, V. M. Investigation transient conditions in extremal control systems. Avromatika No. 3 (1958) 8 Automatic Control. Collected papers of the conference at Crenfield . 1951. Foreign Publications, 1954
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MEYEROV, V. M. Syllfhesis of the Arrangement of High Accuracy Automatic COllfrol Systems. 1958. Fizmatgiz MINORSKY, N . Directional stability of automatically controlled bodies. J . Amer. Soc. Ilav. Archit. 34 (1922). See also Autopilots, ed. Prof. G. V. Shchpanov. 1939. Oborongiz PETROV, B. N . The application of the conditions of invariance. TrailS . 11th AII-Unioll COil! on the Theory of Automatic Control, vol. 11, pp. 241-246. 1955. Academy of Sciences of the U .S.S.R. See also Petrov, B. N. Selbsttatige Regelungssysteme mit zusatzlicher Lastaufschaltung. Fachtagung Regelungstechnik, Heidelberg, 1956, Beitrag 92 MOORE, J. R. Combination open-cycle closed cycle control system. Proc. Inst. Radio Ellgrs, N. Y. 39 No. 11 (1951) DRENICK, R . F. and SHACHBENDER, R . A. Adaptive servomechanisms. TrallS. A mer. Inst. elect. Engrs 76, Pt. Il (1957) ASELTINE, J. A., MANCINI, A. R. and SARTURE, C. W. A survey of adaptive control systems. Inst. Radio Engrs, Trans . on Automatic Control PGAC-6 Dec. (1958)
DISCUSSION SHORRIN (U.S.S.R.) In the majority of real control systems the initial conditions are not zero. In this case what conditions of invariance are used to obtain zero or nearly zero dynamic error ? A. G. IVAKHNENKO, in reply. The real initial conditions should be distinguished from those which may be called equivalent initial conditions. We cannot change the real initial conditions. Equivalent conditions are those described by the homogeneous equation of motion, a3(p) q, = 0, solution of which does not differ from the
solutions of the complete equation a3 (P). q,= b3 (P) . A with real initial conditions. The condition of invariance (in the second form) corresponds to zero values of the equivalent initial conditions, defined by Kryzhanovski's formula. It is found that the open-loop disturbance coefficients enter into expressions defining the equivalent initial conditions in such a way that we may freely choose their magnitude. Consequently, taking zero values of the equivalent initial conditions, we can in this way completely eliminate errors, i.e. obtain adjustment corresponding to absolutc invariance q, - O.
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