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Synthesis of control systems operating linearly for small signals and approximately ‘bang-bang’ for large signals
Synthesis of Control Systems Operating Linearly for Small Signals and Approximately 'Bang-Bang' for Large Signals E. V. PERSSON Summary
Introduction
A non-linear controller operating linearly in the small-signal range and approximately 'bang-bang' in the large-signal range is proposed. Its application to a process comprising n integrators with limited nth derivative of the controlled output is treated. The non-linear feedback functions of the controller are given such a form that the response in principle is independent of the signal amplitude. The possibilities of giving the system a bang-bang character for large control deviations are investigated. Analogue-computer studies have been made on systems with limited second, third and fourth derivatives. These studies show that very satisfactory results are achieved for the large-signal response, which deviates only insignificantly from the bang-bang response. The proposed type of controller is very simple in application, since it is only necessary to make non-linear the normal feedbacks in a conventional controller.
Conventional control systems designed as linear systems operate satisfactorily within the limits of linearity, and the desired response and accuracy can be achieved by means of correct design for which effective methods of synthesis exist. If, however, a system of this kind is subjected to such large changes in the command input or such large disturbances that limiting is reached, the response may be unsatisfactory and many systems will even become unstable. Thus a suitable compromise must be made between large-signal and small-signal response, a compromise which may mean that neither will be fully satisfactory. If, on the other hand, one designs the system for the fastest possible response with the control effort available, this will lead to a bang-bang system. Such a system will be very complicated, however, and difficult to apply if the process is described by a differential equation of higher order than the second. Furthermore the bang-bang response is not desirable for small control errors, since very high-frequency oscillations will appear at steady state. It has been proposed 1 that the non-linear controller should be disconnected when the error has been corrected and that the steady state should be maintained by connecting up a new controlling device. It is hardly likely, however, that such a system would operate satisfactorily for small disturbances occurring frequently. This paper presents a system which operates mainly bangbang for large control. errors but linearly for smaJl errors. The transition between the two modes of operation takes place smoothly without any switching devices. The author has aimed at a system which is simple to apply, and for this reason the requirements as to an exact bang-bang response have been waived. This paper is limited to the case where the control loop comprises a number (11) of integrators and where the input to the first integrator is limited to its absolute value, i.e. the nth derivative of the controlled output is limited.
Sommaire
L'auteur propose un n!gulateur non-lineaire qui fonctionne lineairement dans la gamme des petits signaux et approximativemcnt «bangbang» dans la gamme des grands signaux. II traite de son application a un processus qui comprend n integrateurs avec limitation de la nieme derivee de la grandeur reglee. On donne aux asservissements non lineaires du regulateur une forme telle qu'en principe, la n!ponse soit independante de I'amplitude du signal. On examine les possibilites de donner au systemc un caractere «bang-bang» pour de grands ecarts de reglage. Des etudes au calculateur analogique ont ete faites sur des systemes avec limitation de la seconde, troisieme et quatrieme derivees. Ces etudes montrent qu'on obtient une reponse tres satisfaisante pour de grands signaux, reponse qui ne differe que d'une fa~on insignifiante de la reponse bang-bang. Le type de regulateur propose est tres simple a realiser en pratique de fait qu'il est seulement necessaire de rendre non-lineaires les reactions normales d'un regulateur classiqlle. Zusammenfassung
Es wird ein nichtlinearer Regler vorgeschlagen, der bei kleinen Signalamplituden ein lineares Verhalten, bei groBen Signalamplituden angenahert Zweipllnktverhalten aufweist. Seine Anwendung fur eine Regelstrecke, die alls n Integratoren mit beschrankter lit er Ableitung der RegelgroBe besteht, wird behandelt. Die nichtlinearen Ruckflihrungen sind so ausgeflihrt, daB die Ubergangsfunktion im Prinzip von der Signal amplitude unabhangig is!. Die Moglichkeiten, dem Regelsystem bei groBen Regelabweichungen Zweipunktverhalten zu geben, werden betrachtet. Untersuchungen von Systemen mit beschrankter zweiter, dritter und vierter Ableitung am Analogrechner zeigen, daB sich sehr befriedigende Ergebnisse erzielen lassen. Die Ubergangsfunktion bei groBen Signalamplituden weicht nur geringfligig vom Zweipunktverhalten ab. Der vorgeschlagene Regler laBt sich in der Praxis leicht verwirklichen, da es nur notwendig ist, die normalen Ruckfuhrungen eines konventionellen Reglers nichtlinear zu machen. 2\0
The Problem Let us start with a non-stabilized control system according to Figure 1, where it is assumed that the process is represented by 11 integrators.
Figure 1.
Non-stabilized system under studI'
ro = command input; Co = controlled output; M = limitil1g leeel for C n
SYNTHESIS OF
CO~rROL
SY5TE\B OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARG E SIGNALS
The following notation is used: = Command input ry = d y fold t y = vth derivative of the command input Co = Controlled output Cv = Variable equal to constant· dYco/d (' (see Figure 1) Cn = Input to first integrator M = Limiting value for Cn according to: - M < Cn < M e = Input to the non-linear function representing the limitation K,(l' = 1 ... n) = Gain of the integrators fa
l!~n
K
=
n K,
"~ I
Ko
=
s
= Time = Laplace operator
Gain of the non-linear function within the linear range
Figure 2. Normalized system Xo = control error in relative scale = Co - ro/M; x"
=
d" xo/dT";
n
where T =
yk
t
Since d n ro/dt = 0 according to eqn (2), all the variables x, at steady state will assume the value 0, and the behaviour of the system, according to Figure 1, can be studied instead in the system shown in Figure 2 by allowing the variable x, to assume the initial value
(1'.)'=0
It is now desired to design a practicable non-linear controller, operating linearly for small signals and functioning satisfactorily even for large signals. The bang-bang response is taken as the ideal for large-signal response. The command input is assumed to have the form
(1) where bo ... bn - 1 are constants. As is apparent, it has been assumed that:
r=dnro=o n dt n
M [ZlKJ Introduction of a Non-linear Controller A non-linear controller is now introduced into the system shown in Figure 2. The variable y is determined by the control error Xo and its derivatives Xl' X 2 ••. X n - l with regard to the normalized time T. In order to make this controller practicable, y is assumed to be the sum of the single-variable non-linear functions fa (xo), fl (Xl) .. .fn-1 (X"-l) as shown in Figure 3.
(2)
I
Normalizing the Equations to Reduce the Number of Parameters The following variables are now introduced instead of the original ones:
I ~f~o~(X~o~)------~~~
r=ZI K· t
(3)
dId 1 p=-=- - = - s dr ZlK dt ZlK
(4)
Figure 3. Introduction of /lon-linear control/er with feedback functions fv
(5)
The Large-signal Response of the System. Form of the Non-linear Feedback Functions
CO-fO
.\ 0
= -----;.:;rdx o
.\1=--
dr
Cl
Kl -1'1
(6)
MZlK c,K 1 K 2
... K,-1
K y -1',
M [ZlKJ
cnK I K 2
K n- 1 Kn M M [ZlKJ"
K
v=-o-e
-
M
.. •
(7)
(8)
(9)
With the new variables the block diagram is shown in Figure 2. Xo is the control error (disregarding scale factor) [see eqn (5)]; Xv Cv = 1 ... n) is the vth derivative of Xo with regard to T; and the limiting value for X Il is unity. 211
L -_ _ _ _ _ _ _ _ _
With very large signals the non-linear function representing the limitation in Figure 3 is greatly overloaded with the output Xn = ± 1 dependent on whether y (the switching function) is positive or negative. If it is now assumed that it is possible to determine the nonlinear feedback functions f, (xV> so that the response will be that desired for a certain large signal level, e. g. for a certain initial value of x o, it is natural to require the same response for other large signal levels, though with altered amplitude and time scales. Which types of non-linear functions fulfil this requirement are investigated here. A further time transformation is now made for this purpose as follows:
r' = (3r d dr,=q
(10)
where fJ is a positive constant and q is a new operator determined from (11) p=(3q
E. V. PERSSON
The block diagram in Figure 3, due to the transformation (11), becomes that shown in Figure 4, which in turn can be redrawn by means of amplitude transformations giving that shown in Figure 5. Comparing Figure 5 with Figure 3, it is apparent that the systems are exactly the same except that Xv is replaced by (3n-v Xv and that the switching function y is replaced by y'. If the non-linear feedback functions are now selected so that y' will also be a positive constant times y (the constant may be a function of (3), then (3"-V XV as a function of T' and Xv as a function of T will be indentical for the systems shown in Figures 5 and 3.
An attempt is now made, however, to determine the constants kv so that the response will approach the bang-bang response as
closely as possible. Determining the Constants k v to Approach Bang-bang Response The bang-bang response comprises a series of time intervals with the duration T I , T2 ... n according to Figure 6, with Xn varying between + 1 and - 1. The numbering has been chosen so that TI refers to the last interval. At the end of this interval (T = Ts) the state of equilibrium, characterized by the fact that all the Xv values are zero, is attained. In order to simplify the calculations a new normalized, but reversed time {}, is introduced according to (14)
which means that the final state for state for {} = 0, namely:
XO=Xl = ...
=X n -1
T =
Ts
becomes the initial
(15)
=0 for .9=0
::~T2T'
Figure 4. System with modified time-scale 1'
'I
: :
:
I
11 I
I I
-+-o-+---h: :...+---+--: ...-+---1f--~: :
r---
-1
I
I 11 .'--+1_..LI_ _-II '-I..J......_ _~,
"
1Jn
1Jn _,
"1Jm
'
c-'
_ L -_ _ :I
' 1 1p
11
Ts ,
1J2 lJ,
T
I I
0
Figure 6. Bang-bang response
The variables
Xv
are now related by the following equations:
xv= -
Figure 5. System with modified time- and amplitude-scale
f:
xv+ 1 d.9
(16)
1,'=O ... n-l The condition for this is consequently
y' =
cp (/3) Y
Iv (f3n-v
xJ =
cp(fJ) Iv (xv)
(12)
where cp «(3) is an arbitrary, positive function of (3. The solution of (12) is
Starting with Xn according to Figure 6, interpreted as a sum of unit step functions, and applying successively eqn (16) gives the following value for Xv at {} = {}m: X vm =(_l)n-,,(
n
~ ),[.9~n-v+/=I-1 (-I Y (.9 m-.9p )"-v] v.
p= 1
(17)
Since Iv (x,,) must be an odd function of xv, the final solution will be
(13) In eqn (13) kv is a constant that can be freely selected for each l' value, whereas B is an arbitrary constant that must have the same value for all values of v. Now that the form of the function is given according to eqn (13), the practical synthesis work can be accomplished without difficulty by aid of an analogue computer. This work may be limited to one large amplitude, and suitable values for the n constants kv can be determined. When a desired response has been obtained, the same response for other large amplitudes will automatically be obtained, although in another time scale. 212
At this instant Xn should change sign, which means that the switching function y should be equal to zero. The switching function has the form: (18)
Y=-
Inserting for Xv in eqn (18) values xv'" according to eqn (17) and setting y equal to zero, yields the following condition for bang-bang response:
L
v=n-1
v=o
kv( _1)"-v
[
1
-~-,
(n-v).
]
B
n-v·
cfJ ~.9) B m ·!cfJ m(.9)!n-v=O (19) !cfJ m(.9)!
111=1,2, ... ,11-1 where p=m-1
cfJ mC.9) = .9~,- v+ 2
L (- lY (.9 m-.9p)"- v
p=l
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
Equations (19) must be satisfied, by selecting the constants kv and B, for all values of {fl' {f2 •.. {f n-l if the system is to operate bang-bang irrespective of the initial condition. This is not possible, however, except in the simple case n = 2. One is therefore restricted to requiring bang-bang response only for a step input. In this case the following applies, according to Burmeister 2 , for the relative duration of the intervals in Figure 6:
9r
. 2 rn -=Sln -
9"
bang-bang for B :s; 2, but not for B > 2. When B > 2, the first switching is correct, as'is also the beginning of the second one. The second switching is abnormal, however, with a number of rapid oscillations about zero taking place over a short interval. If the results are to be of practical use, it is naturally also necessary for the system to be stable; this is not certain even if the system has a bang-bang step response.
n=4 -;:-6 B/ 3 {[4(J2+ 1)4 -lJ8/4 + 1} (J2-1)B+2( -} y/4
2n
for r = 1,2 ... n - 1. From this:
ko (20) fl
~p=
9 . n
Sln
2
pn
211
l
2B/2 [4 (J2+ 1)4 _1]8/4 -1
Using the relationships (20) and (21) and introducing the following function A of n, v and m: p=m-l
I
A(n,v,m)=1+2
(-1t 1-
p=1
sin
~nJn-v
n sin 2 mn 2n
I
v=o
kv(-lf
_ v
_ ( ')' n-v
n
j;.
IA
(24)
n=3
ko k2 ko
28/2 11 8/ 3 +1 ----68/ 3 2
1 11 8/ 3 _1 68/3 2
24
8 4 /
2[(
y/4
J~ily/3 +(J2-1)n]
Small Signals
If the system in Figure 3 is to operate linearly within the small-signal range, the following must apply here:
(27) where a v are constants selected by means of synthesis methods for linear systems. The following applies for large signals [see eqn (13)]:
..lL.
x
fv(x.)=kv~IIXvln-v
(28)
IX v
The constants kv are selected according to eqns (24)-(26) or from computer studies. Depending on whether Bin - ~, is less than, equal to, or larger than unity, the functions will have the appearance shown in Figure 7 (a), (b) or (c). For the case Bin - v = 1, kv has been selected equal to a v' In this case the function is the same in both ranges. For the other cases, where Bin - v oj= 1, the full-drawn curves in Figure 7 are selected.
n=2
kl
ko
-2(4-
(23)
The system of eqns (23) contains n equations, which must be satisfied by selection of the n constants ko, kl ... k n - 1 and the constant B, and this is always possible. B and one of the constants kv can be arbitrarily selected. The system of eqns (23) determines, however, for a given value of B the relationship between the n constants kv' The calculation has been performed for n = 2, 3 and 4, the result being as follows:
~:=( ~) B/2
_1_{[4(J2+ 1)4_1]8/4 + 1}(7 Y / 3
Modifying the Non-linear Functions to give Linear Operation for
n, v, m)
m = 1, 2, ... , n -1
k3
(26)
]..lL. A ((n, v, m).IIA(n,V,I11)ln-v=o ..lL.
1
[
2
(22)
then eqn (19) can be transformed into v=n-1
2[(J~!i1y/3 +(J2-1)8]
84 /
24B/4
(21)
2
24
(25)
When synthesizing according to eqns (24) to (26), the switching function will be zero at all those instants necessary for bang-bang step response. This does not necessarily mean, however, that the system must operate bang-bang, since it is possible that the switching function will pass through zero at other instants as well. An investigation of the derivatives of the switching function at the zero points suggests that the step response should be bang-bang for n = 2 and n = 3 irrespective of the value of B. For n = 4, the step response should be 213
...I!(a) n-v
< 1
(b)
n~v
=1
(c)
n~v
Figure 7. Proposed non-linear feedback function
>1
E. V. PERSSON
When making this choice of thc functions, the system will operate linearly for small signals and approximately bang-bang for such large signals that the linear range will be negligibly small. The transition between these two modes of operation takes place gradually without any need for switching devices. When selecting the non-linear functions according to eqn (28), the constant B in the exponent can be given any positive value. In the first place, however, a whole number ought to be selected within the range 1 :s; B :s; n. With this choice, one of the functions will be linear, namely fn-B (Xn-B) which is equal to kn-B Xn-B. The constant kn-B is determined by
Case n
=
2
In this case the system operates bang-bang for all initial conditions disregarding the small deviation due to the linear range in the non-linear functions. Some responses are presented in Figure 8. The feedback functions are as follows. In the linear range:
In the non-linear range: For practical reasons B = 1 is preferred, provided that this is consistent with satisfactory response. The choice of appropriate values for B and for the constants kv is best made with the aid of an analogue computer. Such an investigation has been started but not yet completed, and it is thus only possible to report the results obtained from preliminary investigations, which were intended to establish what possibilities exist to improve a system by introducing a non-linear controller of the type proposed. Preliminary Analogue Computer Studies Preliminary investigations have been performed on systems with two, three and four integrators (n = 2, 3 and 4). Some of the results are presented here. The synthesis in the linear range (selection of the constants a v ) has been made according to a method described in the work by Kessler 3 • This synthesis method will yield a system with a step response overshoot of about 5 per cent.
B=l
The step response is shown in Figure 8 (b) and (d) for two different amplitudes of the step. For comparison the corresponding step responses with a linear controller are shown in Figure 8 (a) and (c). As is apparent, the non-linear controller operates almost ideally bang-bang for large steps. Case n
=
3
In this case the system is not stable with B = 1 and the constants kv selected according to eqn (25). With B = 3, however, a system operating practically bang-bang is obtained. A few responses are shown in Figures 9 and 10 with the following feedback functions [not true for Figure 9 (d)].
-=rl·~ .' -10
c
b
Figure 8. Step response for system with n
=
7:10
,'
I
.
d
2; (a) Linear controller; (b) non-linear controller; (c) linear controller; (d) non-linear controller
214
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
, ~o . :.'
10 1 --1
" \ ~ .~
r - - - - •..
y
;'0
. .
" +!-
,\
\/--'
-
-f
r----
I
;t1V--' :
L.~
.
'-----;w~ ,
- f
~I VjT= 10
I .
7=10
a
Figure 9, Step response for system with n
2
T
b e d
3: Ca) Linear controller; Cb) non-linear controller; Cc) nail-linear controller; Cd) nail-linear con-
=
troller, but on a system of higher order than third
f~·
9.~Xl
..J.
e-
- :c!~ ~
l J~
i 1
i
~
,
osL , ( , \
~l-/
,~--
7"=10
a Figure 10. Ramp response for system with n
b =
3: Ca) Linear call trailer; Cb) non-linear control/er
215
E. V. PERSSON
These values of the constants k v correspond to:
In the linear range:
a o =8
10 (xo)=8 Xo
al =8
11 (x 1)=8x 1
a 2 =4
12 (x2)=4x 2
whereas eqns (26) give for B = 6:
In the non-linear range:
kl = 14'5 ko
B=3 10 (xo)= 8 Xo kl=16.)2 [from eqn (25)J 11 (Xl) = 16.)2 1:: Ilx 1 13 / 2 20 /(2=3"
20 3 [from eqn(25)J I2(x2)=3"x2
Figure 9 (a) shows the step response with a linear controller for such a large step that the stability has started to deteriorate. For an insignificantly increased step the system is quite unstable. Responses with the non-linear controller are shown in Figure 9 (b) and (c) for two steps, both larger than the largest step that the linear controller can handle. As is apparent these responses are almost ideal bang-bang responses, which is not surprising, since the system has been designed for bang-bang step response. Figure 10 (b) shows, however, that the system operates practically bang-bang even for other inputs. In this case, Xl has been given an initial value, approximately equal to 1 (corresponding to a ramp input). Under the same conditions a system with a linear controller is unstable. The response with a linear controller and an initial value for Xl immediately below the stability limit is shown in Figure 10 (a).
Case n
=
4
In the case of four integrators, it became apparent that the design on the basis of eqns (26) does not yield a satisfactory response. The system is unstable for B = 2. For B = 4, limitcycle oscillations were experienced. The same applies for B = 6, although the amplitudes of the oscillations were smaller than for B = 4. It is possible that a further increase in B would have made the system stable, but the investigation was directed instead towards finding a combination of constants k v giving acceptable response for B = 6. The following feedback functions were selected: In the linear range:
ao=4
Io(xo)= 4x o
al =8
11 (Xl)= 8x l
a z =8
I2(x 2)=8x 2
Q3=4
13 (X3)= 4X 3
/(2=53'5 ko
Although eqns (26) do not provide kv values that can be applied, they nevertheless yield values that do not differ from those selected more than by a factor of about 2. Equations (26) are therefore of value for selecting reasonably starting values when determining the constants k v in an analogue computer. A couple of step responses with a linear controller are shown in Figure 11 (a) and (b), the first one for a step that the system can just handle, the second one with a slightly increased step where the system becomes unstable. Corresponding step responses with the non-linear controller are given in Figure 11 (c) and (d). Figure 11 (e) and (f) show a couple of step responses with a considerably larger amplitude. A comparison of Figure 11 (e) and (f) reveals that the responses have almost exactly the same form. In Figure 11 (f) the ideal bang-bang response has been plotted as the dotted line in the recording of the controlled output Col M. Even if the system investigated for large amplitudes yields six switchings, while in an ideal bang-bang system only four switchings occur, the settling time is only about 50 per cent longer than the shortest possible. No investigations have been made on systems with limitations in a higher derivative than the fourth one. This does not mean, however, that the application is limited to fourth order systems. On the contrary, applications have been made on higher order systems with additional linear transfer functions introduced immediately ahead of the non-linear function representing the limitation (at e in Figure 1). For example with three integrators behind the limitation function (n = 3), practically ideal bang-bang response is obtained even in this case, provided that the linear system is synthesized according to Kessler3 and that a small correction is made to the constants kv' (This correction was obtained simply by assuming the limitation level M to be smaller than the real one by a factor of \'4.) A step response for such a system is shown in Figure 9 (d). Thus the proposed type of controller can be used for a system of arbitrary order, as long as there are no limitations in derivatives of higher order than the fourth, and the transfer functions between the limitation and the controlled output are integrators. Conclusions
In the non-linear range:
k o =4·8 Xl
11 (xJ=30'5TXJlxl1 12 (x2)=440x~ I3(x3)=64-lx31Ix316 X3
2
The investigation reported in this paper shows that it is possible to design a non-linear controller, operating linearly for small signals and also having a satisfactory performance for large signals when used on a process, comprising a number of integrators, and in which one of the derivatives of the controlled output is limited. The controller is simple, since the only difference in comparison with a conventional linear controller is that the feedbacks already existing in the latter are made non-linear, which can easily be achieved by means of biased diodes or nonlinear resistors. The transition between linear and non-linear operation takes place smoothly without any need for switching devices. 216
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
---;-.-
1
1 '
y
-" -l'-. .
:tt--W r " :"' l"7 ~""'- · .. ··1··:·:· .. ·· ···· 5. 0..
I
-$0
.
'
.. .
'
.
. ..
. .~ .. .
)0 . .•.
I
. :'
I
..
;."
~. -, ~r-.
L \.·· _ · . .
.
..."
4-"
. .• .
.
...... ,'*. '
!"'j
t-: , '; wT•...'......; '...... . ,'
.
i
"
}
, .
. ..•1
·
0"
,
iI
!
• :-'
,, ~.
j
:
.. .. .
.'
..:
'
.......• .. ... :
-2,
,
.
.
,
: ...• i
. ..
r
~
'
, ...
'.
. .. ,
.
.
-
.
Zl j ' .. .
lLJ · · ···~ :
-jtV '
_~ ~
I
.
>------t
7: 10
a
b
T
c
e
d
Figure 11. Step response for system with n = 4: (a) Linear controller; (b) linear controller; (c) non-linear controller; (d) nOli-linear controller; (e) non-linear controller; (I) non-linear controller; Dotted curve : ideal bang-bang response
An almost exact bang-bang response can be obtained for systems with limitation in the second or third derivative of the controlled output. In the case where the fourth derivative is limited, the response is sufficiently good not to justify a changeover to the extremely complicated controller required to achieve exact bang-bang response. By introducing the non-linear controller, improvements can also be achieved in the linear range, since the controller can be designed here for the best small-signal response without regard to the large-signal response. Even though the investigation presented here is confined to systems comprising solely integrators in the control loop, the results may be applied to cases where the integrators are replaced by tra nsfer functions with poles sufficiently close to the origin in the s plane. It should therefore be possible to apply controllers of this type to a large number of processes.
(a) Further investigations with n = 3 and 4, but with B = I and 2 (b) Investigation of systems with more than one limited deriv-
ative. (c) Investigations with other types of transfer functions than
integrators. (d) Investigation of how far it is possible to a pproximate the non-linear functions without losing the main advantages. The author expresses his thanks to Mr . HI Sorensen who per/armed the investigations on the analogue computer_
References 1
2
Remaining Work 3
The studies are being continued along the following lines : 217
CHANDAKET, P., and LEONDES, C. T. Synthesis of quasi-stationary optimum control system. Pt I. AIEE Trans. Applications and Industry , No. 58 (1962) 313-319 BURMEISTER, H. L. Zeitoptimale Ubergangsvorgiinge mit beschriinkter n-ter Ableitung. Z. Messen-Steuern- Regeln 4, 10 (1961) 407-409 KESSLER, C. Ein Beitrag zur Theorie mehrschleifiger Regelungen_ Regelungstechnik 8,8 (1960) 261-266
E. V. PERSSON
DISCUSSION M.
ATHANS,
E. V.
M.I.T. Lincoln Laboratory, Lexington, Mass. U.S.A.
It should be emphasized that the system of Figure 3 is not a timeoptimal one. Since the word ' bang-bang' now implies time-optimal control it is somewhat confusing in this paper. [ would like to ask the following questions. Is the system of Figure 3 stable for all possible initial conditions? How is B chosen? How can it be proved that the system is stable? Has the author investigated the relation of the response time to the minimum time?
F.
MESCH ,
Ins/itut [fir Regeiul1gstechnik, T. H. Darms/adt, Germany
At the end of the paper it is stated that the results obtained for the integrators might be extended to systems with poles sufficiently close to the origin of the s plane. I do not believe that this is true for complex poles. Furthermore, I feel it to be somewhat unrealistic to assume that the output signals of all the integrators can be measured in a practical system. E. PAVLIK, Siemens, Lassallestr. 9, Karlsruhe, Germany
According to Figure 3 of the paper, the non-linear controller feedback functions are taken from the outputs of the integrators of the controlled system. Normally, in technical control systems, these signals are not available. What means exist for approximated bang-bang control, when these feedback signals are not available? M.
HAMzA,
ETH Zurich, Switzerland
I would like to know how far it is possible to approximate the nonlinear funCtions without losing the main advantages of this method? At the end of this paper it was mentioned that such studies were being performed. I feel that this method will have many limitations for most systems in practice. From the previous discussion, I understand that studies were performed, using a system having complex poles. Would Mr. Persson please indicate how he would apply his method if the open-loop system transfer function is of the form
218
PERSSON,
in reply
In reply to Mr. Athans' remark concerning the word 'bang-bang', I refer to the title and the introduction of my paper, from which I hope it is clear that I am aware of the fact that the system is not a true bang-bang system (except for the case n = 2). In my opinion this is less important than the fact that the system can be made stable without having to compromise between small and large signal response . Investigations have indicated that the system of Figure 3 can be designed to be stable for all initial conditions, even if this has not been strictly proved. The system n = 3 and n = 4 were checked for stability over a large domain of initial conditions, as well as for harmonic response over a large frequency range. The choice of B is a compromise between practical considerations and the desire to approach bang-bang response as closely as possible. In my opinion B = 1 or 2 should be preferred, even if the ratio between the response time and the minimum time will increase for a fourth-order system from \'5, as obtained in Figure 11 in my paper, to approximately 2. Mr. Mesch and Dr. Hamza have drawn attention to systems with complex poles in the s plane. Systematic studies of such systems have not been carried out, but the method has been successfully applied to a fourth-order practical system of this kind . Briefly, the method was applied by selecting the exponents in the non-linear functions from the analogue computer diagram simulating the system, neglecting the feedbacks existing between the integrators. Then the constants kv were determined by means of computer studies. Mr. Mesch and Dr. Pavlik remarked that normally all the quantities necessary for feedback are not available in practical systems. This is true even with a linear controller. To be specific, let us assume a process, consisting of three integrators, where no derivatives of the output are available. In order to make the linear system stable, one then has to produce the first and second derivative by means of lead networks. There are no additional difficulties in applying the method propmed in my paper. In fact, the method has been applied to a practical system of this kind. Finally, replying to Dr. Hamza's first question, the non-linear functions can be approximated by three straight lines, without losing the main advantages.
Introduction
A non-linear controller operating linearly in the small-signal range and approximately 'bang-bang' in the large-signal range is proposed. Its application to a process comprising n integrators with limited nth derivative of the controlled output is treated. The non-linear feedback functions of the controller are given such a form that the response in principle is independent of the signal amplitude. The possibilities of giving the system a bang-bang character for large control deviations are investigated. Analogue-computer studies have been made on systems with limited second, third and fourth derivatives. These studies show that very satisfactory results are achieved for the large-signal response, which deviates only insignificantly from the bang-bang response. The proposed type of controller is very simple in application, since it is only necessary to make non-linear the normal feedbacks in a conventional controller.
Conventional control systems designed as linear systems operate satisfactorily within the limits of linearity, and the desired response and accuracy can be achieved by means of correct design for which effective methods of synthesis exist. If, however, a system of this kind is subjected to such large changes in the command input or such large disturbances that limiting is reached, the response may be unsatisfactory and many systems will even become unstable. Thus a suitable compromise must be made between large-signal and small-signal response, a compromise which may mean that neither will be fully satisfactory. If, on the other hand, one designs the system for the fastest possible response with the control effort available, this will lead to a bang-bang system. Such a system will be very complicated, however, and difficult to apply if the process is described by a differential equation of higher order than the second. Furthermore the bang-bang response is not desirable for small control errors, since very high-frequency oscillations will appear at steady state. It has been proposed 1 that the non-linear controller should be disconnected when the error has been corrected and that the steady state should be maintained by connecting up a new controlling device. It is hardly likely, however, that such a system would operate satisfactorily for small disturbances occurring frequently. This paper presents a system which operates mainly bangbang for large control. errors but linearly for smaJl errors. The transition between the two modes of operation takes place smoothly without any switching devices. The author has aimed at a system which is simple to apply, and for this reason the requirements as to an exact bang-bang response have been waived. This paper is limited to the case where the control loop comprises a number (11) of integrators and where the input to the first integrator is limited to its absolute value, i.e. the nth derivative of the controlled output is limited.
Sommaire
L'auteur propose un n!gulateur non-lineaire qui fonctionne lineairement dans la gamme des petits signaux et approximativemcnt «bangbang» dans la gamme des grands signaux. II traite de son application a un processus qui comprend n integrateurs avec limitation de la nieme derivee de la grandeur reglee. On donne aux asservissements non lineaires du regulateur une forme telle qu'en principe, la n!ponse soit independante de I'amplitude du signal. On examine les possibilites de donner au systemc un caractere «bang-bang» pour de grands ecarts de reglage. Des etudes au calculateur analogique ont ete faites sur des systemes avec limitation de la seconde, troisieme et quatrieme derivees. Ces etudes montrent qu'on obtient une reponse tres satisfaisante pour de grands signaux, reponse qui ne differe que d'une fa~on insignifiante de la reponse bang-bang. Le type de regulateur propose est tres simple a realiser en pratique de fait qu'il est seulement necessaire de rendre non-lineaires les reactions normales d'un regulateur classiqlle. Zusammenfassung
Es wird ein nichtlinearer Regler vorgeschlagen, der bei kleinen Signalamplituden ein lineares Verhalten, bei groBen Signalamplituden angenahert Zweipllnktverhalten aufweist. Seine Anwendung fur eine Regelstrecke, die alls n Integratoren mit beschrankter lit er Ableitung der RegelgroBe besteht, wird behandelt. Die nichtlinearen Ruckflihrungen sind so ausgeflihrt, daB die Ubergangsfunktion im Prinzip von der Signal amplitude unabhangig is!. Die Moglichkeiten, dem Regelsystem bei groBen Regelabweichungen Zweipunktverhalten zu geben, werden betrachtet. Untersuchungen von Systemen mit beschrankter zweiter, dritter und vierter Ableitung am Analogrechner zeigen, daB sich sehr befriedigende Ergebnisse erzielen lassen. Die Ubergangsfunktion bei groBen Signalamplituden weicht nur geringfligig vom Zweipunktverhalten ab. Der vorgeschlagene Regler laBt sich in der Praxis leicht verwirklichen, da es nur notwendig ist, die normalen Ruckfuhrungen eines konventionellen Reglers nichtlinear zu machen. 2\0
The Problem Let us start with a non-stabilized control system according to Figure 1, where it is assumed that the process is represented by 11 integrators.
Figure 1.
Non-stabilized system under studI'
ro = command input; Co = controlled output; M = limitil1g leeel for C n
SYNTHESIS OF
CO~rROL
SY5TE\B OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARG E SIGNALS
The following notation is used: = Command input ry = d y fold t y = vth derivative of the command input Co = Controlled output Cv = Variable equal to constant· dYco/d (' (see Figure 1) Cn = Input to first integrator M = Limiting value for Cn according to: - M < Cn < M e = Input to the non-linear function representing the limitation K,(l' = 1 ... n) = Gain of the integrators fa
l!~n
K
=
n K,
"~ I
Ko
=
s
= Time = Laplace operator
Gain of the non-linear function within the linear range
Figure 2. Normalized system Xo = control error in relative scale = Co - ro/M; x"
=
d" xo/dT";
n
where T =
yk
t
Since d n ro/dt = 0 according to eqn (2), all the variables x, at steady state will assume the value 0, and the behaviour of the system, according to Figure 1, can be studied instead in the system shown in Figure 2 by allowing the variable x, to assume the initial value
(1'.)'=0
It is now desired to design a practicable non-linear controller, operating linearly for small signals and functioning satisfactorily even for large signals. The bang-bang response is taken as the ideal for large-signal response. The command input is assumed to have the form
(1) where bo ... bn - 1 are constants. As is apparent, it has been assumed that:
r=dnro=o n dt n
M [ZlKJ Introduction of a Non-linear Controller A non-linear controller is now introduced into the system shown in Figure 2. The variable y is determined by the control error Xo and its derivatives Xl' X 2 ••. X n - l with regard to the normalized time T. In order to make this controller practicable, y is assumed to be the sum of the single-variable non-linear functions fa (xo), fl (Xl) .. .fn-1 (X"-l) as shown in Figure 3.
(2)
I
Normalizing the Equations to Reduce the Number of Parameters The following variables are now introduced instead of the original ones:
I ~f~o~(X~o~)------~~~
r=ZI K· t
(3)
dId 1 p=-=- - = - s dr ZlK dt ZlK
(4)
Figure 3. Introduction of /lon-linear control/er with feedback functions fv
(5)
The Large-signal Response of the System. Form of the Non-linear Feedback Functions
CO-fO
.\ 0
= -----;.:;rdx o
.\1=--
dr
Cl
Kl -1'1
(6)
MZlK c,K 1 K 2
... K,-1
K y -1',
M [ZlKJ
cnK I K 2
K n- 1 Kn M M [ZlKJ"
K
v=-o-e
-
M
.. •
(7)
(8)
(9)
With the new variables the block diagram is shown in Figure 2. Xo is the control error (disregarding scale factor) [see eqn (5)]; Xv Cv = 1 ... n) is the vth derivative of Xo with regard to T; and the limiting value for X Il is unity. 211
L -_ _ _ _ _ _ _ _ _
With very large signals the non-linear function representing the limitation in Figure 3 is greatly overloaded with the output Xn = ± 1 dependent on whether y (the switching function) is positive or negative. If it is now assumed that it is possible to determine the nonlinear feedback functions f, (xV> so that the response will be that desired for a certain large signal level, e. g. for a certain initial value of x o, it is natural to require the same response for other large signal levels, though with altered amplitude and time scales. Which types of non-linear functions fulfil this requirement are investigated here. A further time transformation is now made for this purpose as follows:
r' = (3r d dr,=q
(10)
where fJ is a positive constant and q is a new operator determined from (11) p=(3q
E. V. PERSSON
The block diagram in Figure 3, due to the transformation (11), becomes that shown in Figure 4, which in turn can be redrawn by means of amplitude transformations giving that shown in Figure 5. Comparing Figure 5 with Figure 3, it is apparent that the systems are exactly the same except that Xv is replaced by (3n-v Xv and that the switching function y is replaced by y'. If the non-linear feedback functions are now selected so that y' will also be a positive constant times y (the constant may be a function of (3), then (3"-V XV as a function of T' and Xv as a function of T will be indentical for the systems shown in Figures 5 and 3.
An attempt is now made, however, to determine the constants kv so that the response will approach the bang-bang response as
closely as possible. Determining the Constants k v to Approach Bang-bang Response The bang-bang response comprises a series of time intervals with the duration T I , T2 ... n according to Figure 6, with Xn varying between + 1 and - 1. The numbering has been chosen so that TI refers to the last interval. At the end of this interval (T = Ts) the state of equilibrium, characterized by the fact that all the Xv values are zero, is attained. In order to simplify the calculations a new normalized, but reversed time {}, is introduced according to (14)
which means that the final state for state for {} = 0, namely:
XO=Xl = ...
=X n -1
T =
Ts
becomes the initial
(15)
=0 for .9=0
::~T2T'
Figure 4. System with modified time-scale 1'
'I
: :
:
I
11 I
I I
-+-o-+---h: :...+---+--: ...-+---1f--~: :
r---
-1
I
I 11 .'--+1_..LI_ _-II '-I..J......_ _~,
"
1Jn
1Jn _,
"1Jm
'
c-'
_ L -_ _ :I
' 1 1p
11
Ts ,
1J2 lJ,
T
I I
0
Figure 6. Bang-bang response
The variables
Xv
are now related by the following equations:
xv= -
Figure 5. System with modified time- and amplitude-scale
f:
xv+ 1 d.9
(16)
1,'=O ... n-l The condition for this is consequently
y' =
cp (/3) Y
Iv (f3n-v
xJ =
cp(fJ) Iv (xv)
(12)
where cp «(3) is an arbitrary, positive function of (3. The solution of (12) is
Starting with Xn according to Figure 6, interpreted as a sum of unit step functions, and applying successively eqn (16) gives the following value for Xv at {} = {}m: X vm =(_l)n-,,(
n
~ ),[.9~n-v+/=I-1 (-I Y (.9 m-.9p )"-v] v.
p= 1
(17)
Since Iv (x,,) must be an odd function of xv, the final solution will be
(13) In eqn (13) kv is a constant that can be freely selected for each l' value, whereas B is an arbitrary constant that must have the same value for all values of v. Now that the form of the function is given according to eqn (13), the practical synthesis work can be accomplished without difficulty by aid of an analogue computer. This work may be limited to one large amplitude, and suitable values for the n constants kv can be determined. When a desired response has been obtained, the same response for other large amplitudes will automatically be obtained, although in another time scale. 212
At this instant Xn should change sign, which means that the switching function y should be equal to zero. The switching function has the form: (18)
Y=-
Inserting for Xv in eqn (18) values xv'" according to eqn (17) and setting y equal to zero, yields the following condition for bang-bang response:
L
v=n-1
v=o
kv( _1)"-v
[
1
-~-,
(n-v).
]
B
n-v·
cfJ ~.9) B m ·!cfJ m(.9)!n-v=O (19) !cfJ m(.9)!
111=1,2, ... ,11-1 where p=m-1
cfJ mC.9) = .9~,- v+ 2
L (- lY (.9 m-.9p)"- v
p=l
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
Equations (19) must be satisfied, by selecting the constants kv and B, for all values of {fl' {f2 •.. {f n-l if the system is to operate bang-bang irrespective of the initial condition. This is not possible, however, except in the simple case n = 2. One is therefore restricted to requiring bang-bang response only for a step input. In this case the following applies, according to Burmeister 2 , for the relative duration of the intervals in Figure 6:
9r
. 2 rn -=Sln -
9"
bang-bang for B :s; 2, but not for B > 2. When B > 2, the first switching is correct, as'is also the beginning of the second one. The second switching is abnormal, however, with a number of rapid oscillations about zero taking place over a short interval. If the results are to be of practical use, it is naturally also necessary for the system to be stable; this is not certain even if the system has a bang-bang step response.
n=4 -;:-6 B/ 3 {[4(J2+ 1)4 -lJ8/4 + 1} (J2-1)B+2( -} y/4
2n
for r = 1,2 ... n - 1. From this:
ko (20) fl
~p=
9 . n
Sln
2
pn
211
l
2B/2 [4 (J2+ 1)4 _1]8/4 -1
Using the relationships (20) and (21) and introducing the following function A of n, v and m: p=m-l
I
A(n,v,m)=1+2
(-1t 1-
p=1
sin
~nJn-v
n sin 2 mn 2n
I
v=o
kv(-lf
_ v
_ ( ')' n-v
n
j;.
IA
(24)
n=3
ko k2 ko
28/2 11 8/ 3 +1 ----68/ 3 2
1 11 8/ 3 _1 68/3 2
24
8 4 /
2[(
y/4
J~ily/3 +(J2-1)n]
Small Signals
If the system in Figure 3 is to operate linearly within the small-signal range, the following must apply here:
(27) where a v are constants selected by means of synthesis methods for linear systems. The following applies for large signals [see eqn (13)]:
..lL.
x
fv(x.)=kv~IIXvln-v
(28)
IX v
The constants kv are selected according to eqns (24)-(26) or from computer studies. Depending on whether Bin - ~, is less than, equal to, or larger than unity, the functions will have the appearance shown in Figure 7 (a), (b) or (c). For the case Bin - v = 1, kv has been selected equal to a v' In this case the function is the same in both ranges. For the other cases, where Bin - v oj= 1, the full-drawn curves in Figure 7 are selected.
n=2
kl
ko
-2(4-
(23)
The system of eqns (23) contains n equations, which must be satisfied by selection of the n constants ko, kl ... k n - 1 and the constant B, and this is always possible. B and one of the constants kv can be arbitrarily selected. The system of eqns (23) determines, however, for a given value of B the relationship between the n constants kv' The calculation has been performed for n = 2, 3 and 4, the result being as follows:
~:=( ~) B/2
_1_{[4(J2+ 1)4_1]8/4 + 1}(7 Y / 3
Modifying the Non-linear Functions to give Linear Operation for
n, v, m)
m = 1, 2, ... , n -1
k3
(26)
]..lL. A ((n, v, m).IIA(n,V,I11)ln-v=o ..lL.
1
[
2
(22)
then eqn (19) can be transformed into v=n-1
2[(J~!i1y/3 +(J2-1)8]
84 /
24B/4
(21)
2
24
(25)
When synthesizing according to eqns (24) to (26), the switching function will be zero at all those instants necessary for bang-bang step response. This does not necessarily mean, however, that the system must operate bang-bang, since it is possible that the switching function will pass through zero at other instants as well. An investigation of the derivatives of the switching function at the zero points suggests that the step response should be bang-bang for n = 2 and n = 3 irrespective of the value of B. For n = 4, the step response should be 213
...I!(a) n-v
< 1
(b)
n~v
=1
(c)
n~v
Figure 7. Proposed non-linear feedback function
>1
E. V. PERSSON
When making this choice of thc functions, the system will operate linearly for small signals and approximately bang-bang for such large signals that the linear range will be negligibly small. The transition between these two modes of operation takes place gradually without any need for switching devices. When selecting the non-linear functions according to eqn (28), the constant B in the exponent can be given any positive value. In the first place, however, a whole number ought to be selected within the range 1 :s; B :s; n. With this choice, one of the functions will be linear, namely fn-B (Xn-B) which is equal to kn-B Xn-B. The constant kn-B is determined by
Case n
=
2
In this case the system operates bang-bang for all initial conditions disregarding the small deviation due to the linear range in the non-linear functions. Some responses are presented in Figure 8. The feedback functions are as follows. In the linear range:
In the non-linear range: For practical reasons B = 1 is preferred, provided that this is consistent with satisfactory response. The choice of appropriate values for B and for the constants kv is best made with the aid of an analogue computer. Such an investigation has been started but not yet completed, and it is thus only possible to report the results obtained from preliminary investigations, which were intended to establish what possibilities exist to improve a system by introducing a non-linear controller of the type proposed. Preliminary Analogue Computer Studies Preliminary investigations have been performed on systems with two, three and four integrators (n = 2, 3 and 4). Some of the results are presented here. The synthesis in the linear range (selection of the constants a v ) has been made according to a method described in the work by Kessler 3 • This synthesis method will yield a system with a step response overshoot of about 5 per cent.
B=l
The step response is shown in Figure 8 (b) and (d) for two different amplitudes of the step. For comparison the corresponding step responses with a linear controller are shown in Figure 8 (a) and (c). As is apparent, the non-linear controller operates almost ideally bang-bang for large steps. Case n
=
3
In this case the system is not stable with B = 1 and the constants kv selected according to eqn (25). With B = 3, however, a system operating practically bang-bang is obtained. A few responses are shown in Figures 9 and 10 with the following feedback functions [not true for Figure 9 (d)].
-=rl·~ .' -10
c
b
Figure 8. Step response for system with n
=
7:10
,'
I
.
d
2; (a) Linear controller; (b) non-linear controller; (c) linear controller; (d) non-linear controller
214
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
, ~o . :.'
10 1 --1
" \ ~ .~
r - - - - •..
y
;'0
. .
" +!-
,\
\/--'
-
-f
r----
I
;t1V--' :
L.~
.
'-----;w~ ,
- f
~I VjT= 10
I .
7=10
a
Figure 9, Step response for system with n
2
T
b e d
3: Ca) Linear controller; Cb) non-linear controller; Cc) nail-linear controller; Cd) nail-linear con-
=
troller, but on a system of higher order than third
f~·
9.~Xl
..J.
e-
- :c!~ ~
l J~
i 1
i
~
,
osL , ( , \
~l-/
,~--
7"=10
a Figure 10. Ramp response for system with n
b =
3: Ca) Linear call trailer; Cb) non-linear control/er
215
E. V. PERSSON
These values of the constants k v correspond to:
In the linear range:
a o =8
10 (xo)=8 Xo
al =8
11 (x 1)=8x 1
a 2 =4
12 (x2)=4x 2
whereas eqns (26) give for B = 6:
In the non-linear range:
kl = 14'5 ko
B=3 10 (xo)= 8 Xo kl=16.)2 [from eqn (25)J 11 (Xl) = 16.)2 1:: Ilx 1 13 / 2 20 /(2=3"
20 3 [from eqn(25)J I2(x2)=3"x2
Figure 9 (a) shows the step response with a linear controller for such a large step that the stability has started to deteriorate. For an insignificantly increased step the system is quite unstable. Responses with the non-linear controller are shown in Figure 9 (b) and (c) for two steps, both larger than the largest step that the linear controller can handle. As is apparent these responses are almost ideal bang-bang responses, which is not surprising, since the system has been designed for bang-bang step response. Figure 10 (b) shows, however, that the system operates practically bang-bang even for other inputs. In this case, Xl has been given an initial value, approximately equal to 1 (corresponding to a ramp input). Under the same conditions a system with a linear controller is unstable. The response with a linear controller and an initial value for Xl immediately below the stability limit is shown in Figure 10 (a).
Case n
=
4
In the case of four integrators, it became apparent that the design on the basis of eqns (26) does not yield a satisfactory response. The system is unstable for B = 2. For B = 4, limitcycle oscillations were experienced. The same applies for B = 6, although the amplitudes of the oscillations were smaller than for B = 4. It is possible that a further increase in B would have made the system stable, but the investigation was directed instead towards finding a combination of constants k v giving acceptable response for B = 6. The following feedback functions were selected: In the linear range:
ao=4
Io(xo)= 4x o
al =8
11 (Xl)= 8x l
a z =8
I2(x 2)=8x 2
Q3=4
13 (X3)= 4X 3
/(2=53'5 ko
Although eqns (26) do not provide kv values that can be applied, they nevertheless yield values that do not differ from those selected more than by a factor of about 2. Equations (26) are therefore of value for selecting reasonably starting values when determining the constants k v in an analogue computer. A couple of step responses with a linear controller are shown in Figure 11 (a) and (b), the first one for a step that the system can just handle, the second one with a slightly increased step where the system becomes unstable. Corresponding step responses with the non-linear controller are given in Figure 11 (c) and (d). Figure 11 (e) and (f) show a couple of step responses with a considerably larger amplitude. A comparison of Figure 11 (e) and (f) reveals that the responses have almost exactly the same form. In Figure 11 (f) the ideal bang-bang response has been plotted as the dotted line in the recording of the controlled output Col M. Even if the system investigated for large amplitudes yields six switchings, while in an ideal bang-bang system only four switchings occur, the settling time is only about 50 per cent longer than the shortest possible. No investigations have been made on systems with limitations in a higher derivative than the fourth one. This does not mean, however, that the application is limited to fourth order systems. On the contrary, applications have been made on higher order systems with additional linear transfer functions introduced immediately ahead of the non-linear function representing the limitation (at e in Figure 1). For example with three integrators behind the limitation function (n = 3), practically ideal bang-bang response is obtained even in this case, provided that the linear system is synthesized according to Kessler3 and that a small correction is made to the constants kv' (This correction was obtained simply by assuming the limitation level M to be smaller than the real one by a factor of \'4.) A step response for such a system is shown in Figure 9 (d). Thus the proposed type of controller can be used for a system of arbitrary order, as long as there are no limitations in derivatives of higher order than the fourth, and the transfer functions between the limitation and the controlled output are integrators. Conclusions
In the non-linear range:
k o =4·8 Xl
11 (xJ=30'5TXJlxl1 12 (x2)=440x~ I3(x3)=64-lx31Ix316 X3
2
The investigation reported in this paper shows that it is possible to design a non-linear controller, operating linearly for small signals and also having a satisfactory performance for large signals when used on a process, comprising a number of integrators, and in which one of the derivatives of the controlled output is limited. The controller is simple, since the only difference in comparison with a conventional linear controller is that the feedbacks already existing in the latter are made non-linear, which can easily be achieved by means of biased diodes or nonlinear resistors. The transition between linear and non-linear operation takes place smoothly without any need for switching devices. 216
SYNTHESIS OF CONTROL SYSTEMS OPERATING LINEARLY FOR SMALL SIGNALS AND APPROXIMATELY 'BANG-BANG' FOR LARGE SIGNALS
---;-.-
1
1 '
y
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:tt--W r " :"' l"7 ~""'- · .. ··1··:·:· .. ·· ···· 5. 0..
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-
.
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a
b
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c
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Figure 11. Step response for system with n = 4: (a) Linear controller; (b) linear controller; (c) non-linear controller; (d) nOli-linear controller; (e) non-linear controller; (I) non-linear controller; Dotted curve : ideal bang-bang response
An almost exact bang-bang response can be obtained for systems with limitation in the second or third derivative of the controlled output. In the case where the fourth derivative is limited, the response is sufficiently good not to justify a changeover to the extremely complicated controller required to achieve exact bang-bang response. By introducing the non-linear controller, improvements can also be achieved in the linear range, since the controller can be designed here for the best small-signal response without regard to the large-signal response. Even though the investigation presented here is confined to systems comprising solely integrators in the control loop, the results may be applied to cases where the integrators are replaced by tra nsfer functions with poles sufficiently close to the origin in the s plane. It should therefore be possible to apply controllers of this type to a large number of processes.
(a) Further investigations with n = 3 and 4, but with B = I and 2 (b) Investigation of systems with more than one limited deriv-
ative. (c) Investigations with other types of transfer functions than
integrators. (d) Investigation of how far it is possible to a pproximate the non-linear functions without losing the main advantages. The author expresses his thanks to Mr . HI Sorensen who per/armed the investigations on the analogue computer_
References 1
2
Remaining Work 3
The studies are being continued along the following lines : 217
CHANDAKET, P., and LEONDES, C. T. Synthesis of quasi-stationary optimum control system. Pt I. AIEE Trans. Applications and Industry , No. 58 (1962) 313-319 BURMEISTER, H. L. Zeitoptimale Ubergangsvorgiinge mit beschriinkter n-ter Ableitung. Z. Messen-Steuern- Regeln 4, 10 (1961) 407-409 KESSLER, C. Ein Beitrag zur Theorie mehrschleifiger Regelungen_ Regelungstechnik 8,8 (1960) 261-266
E. V. PERSSON
DISCUSSION M.
ATHANS,
E. V.
M.I.T. Lincoln Laboratory, Lexington, Mass. U.S.A.
It should be emphasized that the system of Figure 3 is not a timeoptimal one. Since the word ' bang-bang' now implies time-optimal control it is somewhat confusing in this paper. [ would like to ask the following questions. Is the system of Figure 3 stable for all possible initial conditions? How is B chosen? How can it be proved that the system is stable? Has the author investigated the relation of the response time to the minimum time?
F.
MESCH ,
Ins/itut [fir Regeiul1gstechnik, T. H. Darms/adt, Germany
At the end of the paper it is stated that the results obtained for the integrators might be extended to systems with poles sufficiently close to the origin of the s plane. I do not believe that this is true for complex poles. Furthermore, I feel it to be somewhat unrealistic to assume that the output signals of all the integrators can be measured in a practical system. E. PAVLIK, Siemens, Lassallestr. 9, Karlsruhe, Germany
According to Figure 3 of the paper, the non-linear controller feedback functions are taken from the outputs of the integrators of the controlled system. Normally, in technical control systems, these signals are not available. What means exist for approximated bang-bang control, when these feedback signals are not available? M.
HAMzA,
ETH Zurich, Switzerland
I would like to know how far it is possible to approximate the nonlinear funCtions without losing the main advantages of this method? At the end of this paper it was mentioned that such studies were being performed. I feel that this method will have many limitations for most systems in practice. From the previous discussion, I understand that studies were performed, using a system having complex poles. Would Mr. Persson please indicate how he would apply his method if the open-loop system transfer function is of the form
218
PERSSON,
in reply
In reply to Mr. Athans' remark concerning the word 'bang-bang', I refer to the title and the introduction of my paper, from which I hope it is clear that I am aware of the fact that the system is not a true bang-bang system (except for the case n = 2). In my opinion this is less important than the fact that the system can be made stable without having to compromise between small and large signal response . Investigations have indicated that the system of Figure 3 can be designed to be stable for all initial conditions, even if this has not been strictly proved. The system n = 3 and n = 4 were checked for stability over a large domain of initial conditions, as well as for harmonic response over a large frequency range. The choice of B is a compromise between practical considerations and the desire to approach bang-bang response as closely as possible. In my opinion B = 1 or 2 should be preferred, even if the ratio between the response time and the minimum time will increase for a fourth-order system from \'5, as obtained in Figure 11 in my paper, to approximately 2. Mr. Mesch and Dr. Hamza have drawn attention to systems with complex poles in the s plane. Systematic studies of such systems have not been carried out, but the method has been successfully applied to a fourth-order practical system of this kind . Briefly, the method was applied by selecting the exponents in the non-linear functions from the analogue computer diagram simulating the system, neglecting the feedbacks existing between the integrators. Then the constants kv were determined by means of computer studies. Mr. Mesch and Dr. Pavlik remarked that normally all the quantities necessary for feedback are not available in practical systems. This is true even with a linear controller. To be specific, let us assume a process, consisting of three integrators, where no derivatives of the output are available. In order to make the linear system stable, one then has to produce the first and second derivative by means of lead networks. There are no additional difficulties in applying the method propmed in my paper. In fact, the method has been applied to a practical system of this kind. Finally, replying to Dr. Hamza's first question, the non-linear functions can be approximated by three straight lines, without losing the main advantages.