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Approximate identification of process dynamics in computer controlled adaptive systems
Approximate Identification of Process Dynamics in COlnputer Controlled Adaptive Systems L. BRA UN, Jr., E. MISHKIN and J. G. TRUXAL Introduction A knowledge of the dynamics of the controlled process is a pre-requisite to the intelligent design of any feedback control system. Although considerable effo.rt has been expend~d over a period of many years in developlOg various analytIcal and experimental means for identifying. the dynamIcs of a process, none of the techniques developed IS dIrectly applIcable to the solution of the identification problem in adaptive control systems. Adaptive control syst~ms are a class in whic? the controller is automatically adjusted to maIntaIn desIrable performance in the presence of random variatio~s ofyroc.ess parametersl,2. In adaptive control systems, process IdentIficatIon must be carried out automatically, during normal system operation, so that solutions of the identification problem must be amenable to simple computer realization if they are to be applicable to adaptive control systems. An elegant and concise solution of this identification pr?blem exists, in the case of linear lumped-parameter systems, 10 the form of the well known pole-zero configurations 3 which result from application of transformation tec?niques from th~ time domain to the frequency domain. Non-lInear processes with no memory can be described uniquely . by their input-outp~t characteristic in analytical or graphIcal form , or by theIr response to a sinusoidal forcing function with varying amplitude which can be obtained analytically or experimentally4-6. A method of characterizing complex non-linear systems with non-zero memory in terms of Hermite and Laguerre functions has been described by Wiener et al. 8- lo . In such cases the nonlinear system is described by an infinite s~t of coefficients. It is possible to equip a computer to determlOe a finIte number of these coefficients. The references cited above describe two classes of solution of the identification problem: a closed form solution, in the case of a pole- zero configuration for a linear,. lu~ped-paramet~r system; and a solution in the form of an InfinIte series, as 10 Wiener's approach to the theory of non-llOear systems. In this paper, a method is described fO.r identifying a process by approximating its input and output SIgnals by .means of rapl~ly convergent series. In order to ensure finalIty of the series coefficients, as they are determined, attention is focused on expansions in series of orthogon~l functions . T~e particular choice of a set of orthogonal functIOns IS dIctated, In each case, by the interval of definition of the signal to be expanded, by the \veighting factor associated with the e.rror arising ~rom the finite series used to approximate the SIgnal (the finIteness IS dictated by practical considerations), and by the character of the signal being approximated . Once a particula~ set of or.thogonal functions has been chosen, the SIgnal beIng considered IS characterized by the infinite set of coefficients of the expansionthese coefficients being called the spectrum of the analysed signal. In practice, a finite number of terms from the spectrum
suffices to describe the signal to within a specified accuracy. The choice of the particular set of orthogonal functions is motivated by the above considerations and, in addition, by the requirement (for adaptive control systems, ~t least) that ~he spectrum must be obtainable by means of sImple computIng circuits. Expansion in a Sum of Orthonormal Functions In cases, such as adaptive control systems, where it is impossible or inconvenient to express the process dynamics in closed form, an identification in terms of a finite portion of an infinite series becomes useful. If a function f(t), representing a system signal, is to be approximated by the finite sum faCt), of n functions Cir/>;(t), then I/.
f(t) ~ fa (t) =
L Ciep;(t)
(I)
i= 1
The approximation problem here has two aspect~: the ch?ice of the functions epi(t) should be made to obtaIn the deSIred degree of accuracy with the smallest number of terms, and, having once selected the functions epi(t), the coefficients Ci must be chosen to achieve the best approximation to f(t) in some sense. Should there arise the necessity to improve the approximation by the addition of another term [the (n + I) term], the preceding n coefficients should remain unchanged. The~e ar~ a number of criteria by which the goodness of an apprOXimatIOn may be measured, each of which is appropriate i? ~ .certain class of problems. The criterion used here for opttmlZlng the process of approximation is that of minimizing the integral of the weighted squared error; i.e. the coefficients Ci are to be chosen so that the integral E =
f'
[let) - .fcJt)j2 wet) dt
(2)
is minimized . The infinite integration interval permitsf(t) to be defined over the whole range of t. Iff(t) is defined only over a finite range, the range of integration may be adjusted, or the weight function wet) may be chosen to emp~asize the interval of interest. It may be shown that the condItIOn of finalIty of the selection of the coefficients Ci is ensured if the functions epi(t) are chosen to satisfy the relation
C
ep i(t)ep/t)I\'(t) = 0
i
~O j
(3)
~O
The relationship expressed in equation 3 is called the orthogonality condition by analogy with the geometric concept of the orthogonality of two vectors. It may also be shown that , if the ep i(t) satisfy equation 3 and, in addition , satisfy the normalizing condition
f:
ep/(t)w(t) dt = I
596
606
for all i
(4)
APPROXIMATE IDENTIFICATION OF PRCCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
the values of the coefficients equation 2 are
Ci
which minimize the integral in
(5) If the functions epi(/) satisfy equations 3 and 4, they are said to be orthonormal with respect to the weight function w(/). The
weight function must be chosen to emphasize the portion of the range of I which is of interest in the particular application, without introducing any undue analytical difficulty into equations 3- 5. In the two following sections, two examples of orthonormal functions of particular interest in adaptive control systems are described. These are the Laguerre functions and the orthogonalized exponential functions, both of which permit simple analogue computer realization.
response is composed of a sum of exponentials and damped sine waves. A natural set of functions to use in such a case, for a series expansion of the system impulse response, is a set of exponential functions. Exponential functions are not orthogonal in any interval, so that the ISE criterion is not easily applicable to such a set of functions. It is, however, possible to generate a set of orthogonal functions from appropriate linear combinations of exponential functions. One procedure for effecting this result is the Gram-Schmidt procedure l5 , which is used to generate a set of orthonormal functions, ep;(t), from a set of component functions, jj(t), in such a way that epl(t) ep2(t) ep3(t)
= al/I(t) = a2[/I(t) + aaH/)] = alf/ t ) + a 4 Ht) + asfa(t)]
(10)
Laguerre Functions A given function f(/) may be expanded in a series of orthonormal functions composed of polynomials in I. The exact form of the polynomials depends upon the range of definition of f(t) and upon the choice of the weight function w(t). When the range of definitionf(t) is 0 < t < 00 and the weight function is chosen to be w(t) = e- t, one obtains the familiar Laguerre polynomialsl l . The exponential character of W(/) emphasizes errors occurring for small t. The Laguerre expansion of f(t) is en
f(t) =
L ciLlt)
(6)
i~O
where Lo = 1
L1 = -t
+1
12
L2 = - - 2t 2
and, in general Ln(t)
=
~ :~ Et
1
+1
where the ak are selected to satisfy the relations expressed in equations 3 and 4. The weight function, w(/), is chosen to emphasize the error in the region of interest. In adaptive control systems, interest is usually concentrated in a finite time interval, 0 t < T, and errors throughout this interval are of equal importance. This indicates that the rectangular weight function shown in Figure I is suitable for use in the design of adaptive control systems. A rectangular weight function, unfortunately, introduces considerable analytical difficulty. An exponential weight function, w(t) = E- 2at , also emphasizes the error over a small time interval, and has the distinct advantage over a rectangular weight function of analytical convenience.
(7)
wet)
J
[tne- t ]
The Laguerre polynomials given in equation 7 are orthonormal with respect to e-t, i.e.
l
oo
o
L;(t)Llt)e- t dt
=
{O '
i "* j i
I,
=j
o
(8)
T
F(ljll/'e 1. Reclanglllar we(ljhl jllncliOll
The spectrum of f(t) is given by Ci
=
i "'f(t)Li(t)e- t dt
(9)
When H'(t) = e- 2nt , the integrals of equations 3 and 4 may be evaluated in the complex frequency domain using Parseval's theorem which states that
If, for example, the functionf'(t) = e- t is to be expanded in the first four terms of a series of Laguerre functions !Cl)
= e- t
~ coLo(t)
From equation 9, the coefficients off(t) are
so that f(t) becomes (after re-arrangement of terms) !Ct) ~ I - t
Orthogonalized Exponential
(11)
+ c 1L l (t) + c 2 L 2(t) + caLa(t)
+ }t 2 -
This theorem permits utilization of the powerful methods of the theory of functions of a complex variable in the evaluation of the integrals of equations 3 and 4. Application of Parseval's theorem (equation II to equations 3 and 4) when 11'(1) = e- 2at yields
ita
Functionsl~ - U
In many cases of practical significance the dynamic performance of a process may be described by a set of linear ordinary differential equations, so that the process impulse
where
i'J ij
=
0
= I 597
607
i "* j i =j
L. BRA UN, JR., E. MISHKIN AND J. G. TRUXAL
If the component functions /;(/) in equation 10 are exponential functions of the form aj
rjJ (s) = (24)1 /2 (s
3
positive real for all J
4
Application of equation 12 to this case yields a set of orthonormal functions
-
+ 0(1)]1 /2 _I_ s + 0:1 + 0: . )]1/2 (s - 2a - 0(1) (s + 0(1)(5 + 0(2)
x
+ 0:,,)]1/2 j
If the orthonormal functions, (/);(s), are constructed from pairs of complex poles, then (/) k(S) will contain k /2 complex pole pairs at s = -0: 1 +if3l> -0:2 + Jf32 ... , -O:k/2 +jf3k/ 2' Evaluation of equation 12 in this case gives
+ a )]112
1
1
where IS112 = (a
I
(s - a + S ll) (s + a1)2 + f312
+ 0(1)2 + f312 + 0:1)]1/2 ( (s a
no. ( ) = [2(
'1"2 S
S
- a - hi) + 0:1)2 + f3 12
and, in general
1(14)
(/)2V - l(S)} _
(/)2v(S)
I
1/2
- [2(a +o:v)] [Cs - 2a - 0(1)2
X
+ f312 ] ... [(s
- 2a - O:v_ 1)2
sv I) -;-;--~--::o-~-;;-n.-;-;--....,.+-,-f3t",-.: . 1. : . ](,; -S...-;-a_+--!I....:..!C. [Cs + 0(1)2 + f312 ][(S + 0(2)2 + f32 2] .. . [Cs + 0:,,)2
+ f3,,2]
where 11 =
1,2 ... ,
n
2;
n = number of poles in expansion
ISvl2 = (a
+ O:v)2 + f3v2
In equation 14, the (+) sign is associated with Is. 1 in (/)2" - 1(S) and the (-) sign is associated with Isvl in (/)2v(S). It is noted that in the special case a = 0, which corresponds to a unity weight function wet) = I, equations 13 and 14 reduce to the orthonormal exponential functions described by Huggins12 and by Kautz 13 . As an example of the application to equations 13 and 14, a set of orthonormal functions is constructed from two pairs of complex poles and a single real pole. These poles are at
where
OAs
+
(S2
+
OAs
+
1·04)(s2
+
+ 409·04)
+ 409'04)(S2 I '04)(S2
+
+
4s
4s
104)
- 44s + 584) + 104)(s + 2)
/1/(t) = 0
(15)
1< 0
hk(T) = L "' /II(T)g k(T - T) dT
(16)
In equation 15 it is assumed that /II(t) = 0, for I < O. This implies that there is no energy stored in the process. If this is not the case, the stored energy may be removed by applying one of the techniques described in reference 2, and then equation 15 is valid. Comparison of equation 16 with equation 5 shows that identification of 111(/) with I(t) andgk(T - t) with CPloC/)W(/) produces the result " k(T) = Ck; that is, if
s = -2
The weight function in this case is chosen to be w(/) = £- 20 1. The five orthonormal functions corresponding to this set of poles and this weight function are (/) (s) = (20A)1 /2 • S + 0·249 I s- + OAs + 1·04
2
+
At the instant I = T, hk(t) is
s = -2 ±jlO
= (20A)1 /2
(S2
" k(t) = L "" /II(T)gk(1 - T) dT
s = -0·2 ±jl
rf> (s)
- 40As + 409·04) 1'04)(s2 + 4s + 104)
Solution of the identification problem by expansion of a function in a set of orthonormal functions, as described above, has two advantages over the methods mentioned in the introduction: an easily calculated error measure is availab le ; and the effect on the approximation error of an additional single term in the expansion is easily determined, so that the advantage of including this term may be balanced against the added system complexity required by the addition of this term. For a specified set of orthonormal functions, solution of the identification problem requires a determination of the coefficients of the expansion; that is, the spectrum of the signal being characterized by the expansion must be determined. The term 'spectrum' as used here refers to the coefficients of the expansion relative to a prescribed orthogonal coordinate system. So as to make the technique of expansion in a set of orthonormal functions a useful technique in solving the identification problem for adaptive control systems, methods for obtaining the spectrum of a signa l experimentally, in a time which is short compared to parameter drift times, must be developed. In this section an approach is described which is a modification of one originally presented by Huggins12 and which is more readily applicable to adaptive control systems than Huggins' approach. In order to obtain the spectrum of the signals which exist in adaptive systems, means must be developed for operating directly upon the signal of interest to obtain the coefficients Ci , without recording the signal and playing it back, as required in reference 12. For a linear, time-invariant process (or, for a piece-wise linear, piece-wise time-invariant process, as in adaptive systems), the relationship among the process input m(/), the process output hk(l) and the process impulse response g k(t) is described by the convolution integral
( 13)
(5 - 2a - O:I)(S - 2a - 7.2) ... (s - 2a - 0:"_1) (s + O:I)(S + 0(2) . . . (s + O:"_I)(S + 0( 11 )
(/) (s) = [2(a
+
Spectrum Analyser for Experimental Identification of Process Dynamics
and in general (/) n(s) = [2(a
OAs
rjJ (s) = (24)1 /2 (S2 - 40As 5
(/).(s) = [2(a
+ 5·62)(S2 +
rjJ (s) = (24)1 /2 (s - 25'62)(S2 - 40As
(/) k(S) will contain k real poles at s = -0:1' -0:2"', -7.k'
q\(s) = [2(a
(S2
(s - 20·249) S2 + OAs + 1·04
m(t) =/(1)
(17)
/Jk(T - t) = CPk(/)lI'(/)
(I S)
and
598
608
APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
then equation 16 becomes "k(T)
=
{YJ(t)cp,,(t)l\.(t) dt
=
Ck
The relationships expressed in equations 22 indicate that the choice of a set of orthogonal functions should be made in such a way that the individual blocks in Figure 3 are easily realized. Two sets of orthogonal functions which satisfy this requirement
(19)
Equation 19 is seen to be identical with 5 which is the defining equation for Cl." By changing variables in equation 18 it is possible to determine g,,(t) in terms of the weight function wet) and the orthonormal function cp,,(t). This change of variables gives (20)
_T1(s) t T2(s) t T3(S). -- - - - _Tn(S) t-- ~~
Cl
Selection of g,,(t) according to equation 20 results in a spectrum analyser which is useful in obtaining the spectrum of a signa\. If a signal f(t) is applied at the input of a network with impulse response defined by equation 20, the coefficient c" is obtained by measuring the value of the system response "k(t) at the instant t = T, as indicated in equation 19. A word of caution is in order at this point. Physical networks cannot respond until the excitation has been applied: in other words, when 111(1) = 0 for t < 0, then g,,(t) = 0 for t < O. If this observation is applied to equation 19, hilT) may be written as
C2
C3
Cn
Figure 3. Block diagram of spectrum analyser
are the Laguerre functions and the orthogonalized exponential functions described previously. Examples of spectrum analysers employing these two types of functions are presented in the following sections. Laguerre Spectrum Analyser 1G If the orthonormal functions are chosen to be the Laguerre
(21)
Equation 21 indicates that the coefficient c" which is obtained from the spectrum analyser is not the kth coefficient of the expansion of f(t), but is, instead, the kth coefficient of a functionfI(t) defined by Ht) =f(t) for 0 < t < T and for t> T Ht) = 0
Figure 4. Block diagram of Laguerre spectrum analyser
The relationship between f(t) and fl(t) is shown for a typical signal in Figure 2. This conclusion is not surprising, since it is 1M.Cl
f(t)
1M.o.
1I-J.F
t,lt)
1MD. 1MO
1Mfi
Ez(s)
Ez(s) _ 1
o
s
(b) EI(S) =~
(a) El(s) -s-l
T
Ez
El
Ez
El
Figure 2. Relation between f(t) and j;(t)
Figure 5. Analogue complller realization of 1/(s - 1) and sits - 1)
necessary to make measurements over all time to obtain complete information about f(t) so that no measurements over a finite interval will be sufficient. In adaptive systems this is no serious difficulty, since an approximation over only a finite interval is desired. In the general block diagram of a spectrum analyser shown in Figure 3, the transmittances I , TIl ... , are selected to satisfy the relations
functions, the impulse response given in equation 2 becomes gk(t) = Lk(T - t)c(1'-1) k = 0, I, 2. . . (23) The Laplace transform of equation 23 for the first four Laguerre functions is s-1'
Go(s) = - - 1 s -
T T2 ... ,
I
GI(S) = TI(s)
G 2(s) = T I (s)T2(s)
G 2(s) = GI(s) s _S 1
(22) G3(s) = G2(s) s
n
Gn(s) =
IT Ti(s) ;~ l
(T2) +2' - T Go(s)
r
:3 +
J
~ 1 +(-
T2 - T) Go(s)
(24)
Figure 4 shows a block diagram which realizes the transfer
599
609
L. BRAUN, JR . , E. MISHKIN AND J. G. TRUXAL
functions in equation 24. An analogue computer realization of the transfer functions I/(s - 1) and s/(s - 1) is shown in Figure 5. The multiplying factor ET associated with each output in Figure 4 is a constant and may be considered a scaling factor. Spectrum Analyser using Orthogonalized
=c
0·0142
Gl(s)
Exponentials 14 G 2(s)
where
Rki
=
= residue of Wk(s) in pole at s =
where
T 2(s) = -4-45 s - 13 ·9 s - 12 Gis ) = 0.1
-rxi
T3(S)
+ . .. +
2a
= -1.58 (s - 12'9)(s - 18'3) (s - 13)(s - 13'9)
+ ... + R kkE - (rxk + 2a )T E('Y.k + 2a )t
E-(al + 2a)1'
kl s - rxl -
= T 3 (s)T2(s)T1(s)
where
(28) R
(31)
(s - 13)(s - 13 '9)
(27)
or Gk(s), the Laplace transform of g k(t), becomes
Gls) =
(s - 12'9)(s - 18 ·3) (s -11)(s -12)(s -13)
= _1 '58(s -12 ·9)(s -18'3)G 2(s)
If equations 26 and 27 are applied to 20, gk(t) becomes gk(t) = R kl E-(rx 1 + 2a)1'E (rx 1 + 2a )1
= -0'0632 (s _ II)(s - 12)
2a t
+ . . . + RkkE-"k t
R kl E- rx l t
T 1(s)
s - 13 ·9 -4-45 s _ 12 T 1(s) = T 2(s)T1(s)
then the orthonormal function CP.-(t) may be written as cfk(t)
= s _ 11 =
s - 13·9
When the orthonormal functions cp,(t) are orthogonalized exponential functions and wet) is an exponential function of the form wet) = c 2at , the orthogonal spectrum analyser is easily realized using analogue computers. If, in particular, the functions cp,(t) are constructed from a set of real poles at (25) n = 1,2,3 ... S = -:.J. n and if w(t)
If these results are inserted in expression 29, the spectrum analyser for the coefficients of CPl(t), CP 2(t) and cf3(t) has the respective impulse responses
The block diagram and the analogue computer diagram for the spectrum analyser described by equations 31 are shown in Figures 6 and 7 respectively.
RkkE-('Y.k + 2a) l'
s - rxk -
2a
(29)
As an example, if rxn = a =
n 5 T = 0·5
n = 1, 2 ... Note : hi(T) :Ci . for all i
then, from equation 13, the first three orthonormal functions become
Figure 6. Block diagram of spectrum analyser
W (s) = (12)112_1_ s
1
m.
_
'l-'2(S) - (14)
1/2
The three transfer functions Gk), G 2(s) and G3 (s) have poles in the right half of the s plane and are, consequently, unstable. This result is not unexpected since in the development of equation 16 it was necessary to 'flip' g k(t) in contrast to the flipping of the input signaI(t), as suggested by Huggins. Since the spectrum analyser is unstable, the output of each block must be clamped until the instant t = 0, at which time l(t) is applied and, simultaneously, the outputs are unclamped. At some convenient instant f c ;> T, the outputs must again be clamped in order to re-set the system for the next measurement.
+1
S - 11 _ [13 12 ] ( 1)( 2) - 14 --2 - - - (30) s+ s+ s+ s+1
W (s) = (16)1/2 (s - 11)(s - 12) 3
(s + 1)(s + 2)(s + 3) - 16[ 78 182 105 ] s+l-s+2+:;:-+3
00091
0077
00833
0072
Note All resistances In megohms All capacitances in IJ.F h , (T): C, , for all i
F(f{ure 7. Ana/agile computer reali:atioll of spectrum ana(vser of equation 31
600
610
APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
where T = length of the control interval
An Example of Experimental Identification U It is possible to instrument a relatively simple solution of the identification problem using the orthonormal spectrum analyser developed earlier, when the orthonormal functions used to establish a coordinate system are properly chosen. If the orthonormal functions are a set of orthogonalized exponen-
c",(t) becomes c",(t)
=
(37)
Moa(t)
where (38)
is the unit step response of the process. The unit step response may be expanded in a set of orthonormal functions cPi(t). This expansion takes the form a(t) ~ aa(t)
tials or a set of Laguerre functions, then the spectrum analyser is easily constructed using R- C networks and operational amplifiers to realize the desired transfer functions. In this section, the system forcing function is restricted to be a staircase function of the form shown in Figure 8. This restriction of the form of lII(t) permits utilization of the unit step response as a system characterization. Since met) is a staircase function, the component of the output due to present excitation may be characterized by the amplitude of the present input step, and by the same set of poles and zeros as is used to characterize the unit step response. The system response may be expressed in terms of the forcing function met), and the system impulse response get), by the convolution integral
= f~XJm(T)g(t
alcPl(t)
+ a2cP2(t) + . . . + ancPr.(t)
(32)
- T) dT
c",(t) ~ c,,/(/(t)
=
CnrlcPI(t)
+ C,.,2cP2(t) + ... + Cm ncPn(t) (40)
for 0 <
t
< T
If aa(t) in equation 39 is substituted for a(t) in equation 37, and the resulting series is compared term-by-term with that in 40, the coefficients of these two series must satisfy the relations
C ml C m2
= MOa l = M oa 2
=
cJt)
+
C",(t)
= f~'lJ III(T)g(t
- T) dT
+.c
Equation 41 may be solved for the coefficients a, of the approximate step response in terms of the quantities Mo and C mi . These coefficients are
1 I
I
III(T)g(t - T) dT
I
where
f~xlll(T)g(t
- T) dT
an
= output due to stored energy (34)
"
c ",Ct) = /1II(T)g(t - T) = output due to present excitation •0
Modern control systems are complex and must meet stringent performance requirements . There is an obvious need in such systems for development of design procedures for computercontrolled systems which make use of the highly developed computer art. This paper presents an attempt to apply computers in the solution of one class of such systems. The realization of the novel computer circuits required is outlined, and the overall design logic is presented. The designer versed in the art will have no difficulty integrating the individual circuits. It appears that analogue circuits interconnected by means of
- cJt)
It will be assumed in the following development that the signal c",(t) is available directly. If lII(t) is a staircase function of the
form lII(t)
=
MOI/I(T)
+ MII'-l(t
- T)
+ M 2u_ l (t
- 2T)
II
)
Conclusions
Cit) may be determined from measurements of the process output as described 2 , and may then be used to determine c m(r) from the relation
= c(t)
Cmn Mo
(42)
The quantity Mo is available by direct measurement of 1II(t) and the coefficients Cm ' are available at the output of a spectrum analyser, to that equations 42 represent a realizable solution of the identification problem .
(35)
C ",(t)
(41)
J
(33)
dt) =
1 r
The output c(t) in equation 32 may be expressed as the sum of two components c(t)
(39)
where n = number of terms in the expansion. Function c",(t) has the same form as the unit step response, during any single control interval, except for a difference in level depending upon the amplitude of the step applied at the start of the interval; so that c",(t) may be approximated by an expansion in the same set of orthonormal functions used in the expansion of a(t). The expansion of c ,,.et) takes the form
F{I{I/re 8. Staircase forcing jilllctioll
c(t)
=
+ ... (36)
601
611
L.
BRAU , JR., E. MISHKIN AND J. G. TRUXAL ZADEH L. A . On the identification problem. TrailS. IlIst. Radio Dtijrs. 011 Circuit Theory Dec. (1956) 277 7 WIENER, N. Mathematical Problems of Communicatioll Theory. J 954. Massachusetts I nstitute of Technology unpublished notes (Summer session lecture) 8 BOSE, A. G . A theory of no n-linear systems. Mass. illst. Tech. Res. Lab. Electroll. Tech. Rep. No. 309 (1956) 9 CAMERON, R. H. and MARTIN, W. T. The orthogonal development of non-linear functions in series of Fourier-Hermite Functionals . Alln. Math ., Princetoll 48, No. 2 (1947) 10 MISHKIN, E. and BRA UN , L. JR . Adaptive Control Systems, Chap. 8. 1960. New York ; McGraw-Hill 11 COURANT, R. and HILBERT, D. M ethods of Mathematical Physics, Chap. 2. J 953. New York; lnterscience 12 HUGGINS, W . H. Representation and analysis of signals, Part I, the use of orthogonalized exponentials. Johlls Hopkins Univ. Tech. Rep. AFCKC TR-57-357 Sept. (J 957) 13 KAUTZ, W. H. Network synthesis for specified transient response. Mass. Inst. Tech. Rep . Lab. Electron. Tech. Rep. No. 209 (1952) " BRAUN, L. JR. On adaptive control systems Doct. Dissert . Polyt. illst. Brooklyn , June (1959) 15 HALMOS P. R. Finite-Dimensional Vector Spaces. Princeton University Press 16 MISHKIN, E. On the computer realization of an orthonormal spectrum of a given signal function. Proc. Inst. Radio ElIgrs, N. Y. May (1959) 1003
switching devices will be useful in the realization of the required computing circuits. The required computer facility seems to be reasonable and commensurate with the control problem. More work is needed in the study of the effects of noise, measurement error, computation error, etc., and in the performance evaluation of computer-controlled adaptive systems.
6
The work described in this report was sponsored by the Office of Ordnance R esearch under Contracts DA-30-069-0 RD-2646 and DA-30-069-0RD-1560.
References ASELTINE, J. A., MANCINI, A. R. and SARTURE C. W . A survey of adaptive control systems. Trans. inst. Radio Engrs . on AUlOmatic Control, Dec. (1958) 2 MISHKIN, E. and BRAUN, L. Jr. Adaptive Control Systems. 1960. New York ; McGraw-Hill 3 TRuXAL J. G. Automatic Feedback Control System Synthesis, Chap. I. 1955. New York; McGraw-Hill 4 MALYUSHINETZ, G. D . On the dependence between the amplitude characteristic and the characteristic of a non-linear element. C. R .Acad. Sci. U.S.S.R. 54, No. 6 ( 1946) 491 , MATHEWS, M. V. A method for evaluating non-linear servomechanisms. Trails . Amer. inst. elect. Eng rs. (Appl. & Ind.) May (1955) J 17 1
Summary The need for an identification of the dynamics of the controlled process in the analysis a nd synthesis of control systems is clear. In cases where the dynamics of the process vary with environment, it is frequently impossible to predict the system behaviour from a priori knowledge. In such cases, the process identification must be redetermined periodically during the course of system operation. This requirement has indicated a need to obtain rapidly an experimental identification of the process dynamics. In the subject paper, a technique is described for obtaining a
process identification in terms of a set of orthogonal functions . The method of expansion of a function in a set of orthogonal functions is briefly reviewed. The requirements of a spectrum analyser to obtain the expansion coefficients of a signal in terms of a pre-determined set of orthogonal functions are developed . Examples are presented of spectrum a nalysers, using a set of Laguerre functions and a set of orthogonalized exponentials. The paper concludes with a description of an application of the spectrum ana lyser to adapt ive control systems.
Sommaire Le besoin est clair d'une identification de la dynamique des processus commandes, dans I'a nalyse en la synthese des systemes de commande. Dans les cas oll la dynamique du processus varie avec les conditions ambiantes, il est frt:quemment impossible de prevoir le comportement du systeme en partant d'une connaissance a priori. Dans de tels cas, I' identification du process us doit etre redeterminee periodiquement pendant le cours de I'operation du systeme. Cette necessite a montre le besoin d'obtenir rapidement une identification experimentale de la dynamique du processus. Dans le rapport sur ce sujet on decrit une technique pour obtenir
une identification du processus , en fonction, d'un ensemble de fonctions orthogonales. On passe brievement en revue la methode de developpement d'une fonction en series de fonctions orthogonales. On developpe les qualites requises d'un analyseur de spectre pour obtenir les coefficients du developpement d'un signal en fonction d'un ensenible predetermine de fonctions orthogonales. On presente des exemples d'a nalyseurs de spectre deve loppant en series de fonctions de Laguerre et en series d'exponentielles orthogonales. Le rapport se termine par la description d'une application de l'analyseu r de spec tre aux systemes de controle adaptif.
Zusammenfassung Die Notwendigkeit fUr die Bestimmung der dyna mi sche n Eigenschaften einer Regel strecke fUr die Analyse und Synthese des Regelsystems ist selbstverstandlich. D o n wo sich die dynamischen Eigenschaften der Strecke mit den Umgebungsbedingungen iindern, kann das Verhalten des Systems nur se lten von vo rnherein bestimmt werden. In solchen Fiillen mussen die Eigenschaften der Strecke wahrend des Betriebes in periodischen Abstanden neu erfar.lt werden. D er Beitrag beschreibt ein Verfahren zur Bestimmung der Eigenschaften der Strecke in einer Gruppe ort hogonaler Funktionen . Auf
die Meth ode der Entwicklung einer Funktion in eine Gruppe orthogona ler Funktionen wird kurz eingegangen. Die Forderungen, die an einen Spektralanalysator zur Bestimmung der Entwicklungskoeffizienten eines Signals in Gliedern einer vo rgegebenen Gruppe von Orthogonalfunktionen zu stelle n sind, werden abgeleitet. Als Beispielewe rden Spektralanalysatoren behandelt, die eine Gruppe vo n Laguerre-Funkti onen und eine Gruppe von Orthogonalexponenten verwenden. Der Beitrag sch lie r.lt mit der Besch reibung eines Spektralanalysators fUr ein adaptives Regelsystem.
DISCUSSION
A. M.
LETO V
(U.S.s.R.)
Please formulate, in rigorous mathematical form, the problem of stability of a dapti ve systems.
J. TR UXAL, ill reply. The person who has put th is problem knows more than J. We , and o ther investigators in the U.S.A., consider
that a direct formulation of this problem is very difficult. The difficulty being first that the process itself is complex, and , seco ndly that in our universities we usually like to consider second- and thirdorder systems, whereas adaptive systems are of higher order. In addition there are difficulties connected with the presence of nonlinear elements. The method we employ consists in considering the stability forms. Li 's syste m has been stud ied in various forms.
602
612
APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS Evidently, this circumstance is the cause of substantial error in determining the expansion coefficients. At the same time existing circuits, for example, for storing discrete values of system transfer functions, are stable. The latter method has the advantage of clarity, while the corresponding circuits are fairly simple. (3) The most important question in determining dynamic characteristics of processes is the elimination of the effects of random noise. In the well-known work of Goodman and Reswick, and a number of articles by other authors, the effects of noise uncorrelated with the input of the controlled process are eliminated by calculating the autocorrelation function of the input and the cross-correlation function between the input and output of the controlled process. At the same time, the method proposed in the paper does not, in itself, permit eliminating the influence of low-frequency noise. Only averaging of the observation results over several intervals would seem to reduce its effect. However, the circuit of the spectrum analyser would have to be more complex and the operation of the system slower. In what way is it proposed to take into account the influence of random noise" This question could not be touched on in the paper; I would like to obtain some explanation of this.
Unfortunately we have no general approach to this problem. I assume, considering previous experience, that this general formulation will be obtained in the Soviet Union.
A. M.
BATKOV
(U.S.5.R.)
How do you select the observation interval T'
J. TRuxAL, in reply. Various methods were used, but, I have not discussed any of them in detail. We take the time Tequal to a quarter or a half of the duration of a transient. In the system mentioned the time T was taken equal to half the transient.
A. M.
BATKOV
(U.S.S.R.)
Have any experiments been conducted with spectrum analysers" What is the opinion of the authors on the reliability of stable transfer function realization"? Reply. I have written several articles on quantitative analysis of such systems, but I have not brought them with me. KAPENOVICH (U.S.5.R.) Do you not think that the direct determination of dynamic characteristics, using the convolution integral, is better than finding them by means of statistical methods:
J. TRUXAL, in reply. This problem was posed several times in the States several years ago. There was some competition between various organizations using different methods for the solution.
W. K. TAYLOR (U.K.) It seems to me that the stability conditions of an adaptive control
system depend on the input signal and therefore, without determining the input characteristics, it is not possible to formulate stability conditions. The method of Professor Truxal is apparently limited to laboratory investigations, and I would like to know if it can be modified in any way for use in systems whose operation cannot be interrupted for test pulses"?
J. TRUXAL, in reply. I think this is valid since real systems respond to a definite class of signals. Non-linear elements are chosen with respect to the input signals. Li's system relates to this category. Here the problem arises whether you can ensure stability for a defined class of input signals. Knowing this class of signals in advance, we can, however, construct a completely non-adaptive system, so that it is necessary to decide first if we are to build an adaptive system.
A. A. FELDBAUM (U.S.5.R.) The problem of automatic determination of the dynamic characteristics of objects presents great interest. Their determination is necessary to study the objects before designing the system and to develop self-adjusting systems which include automatically corrected models of the object. The schemes proposed in the paper are mainly intended for the solution of the second of these problems. The following remarks can be made about the paper: (I) The study was limited to a linear system with constant or slowly varying parameters. It was assumed that the control signal is a stepwise function of time ; in each interval the coefficients for expansion of the process transfer function in a series of orthogonal functions are determined. However, in which way can this method be generalized to the case of certain classes of non-linear systems, or even to linear systems with control signals varying arbitrarily: If this method has , .in principle, no possibility of being generalized, what is its advantage in comparison with other existing methods, for example, with simple storage of discrete values of the transfer function " It would be of value to know the opinion of the authors on this question. (2) The spectrum analyser circuit given in the paper is unstable because the transfer function poles are located in the right half-plane.
J. TRUXAL, in reply. The remarks which have just been made are very desirable and apposite. Since the particular comments are similar to each other, it appears to be possible to give a combined answer, without considering each comment individually. With respect to the comments of Professor Feldbaum, it should be mentioned that we apply the method, of course, only to cases where the process dynamics are very slow in comparison with the measurement time and the response time, where they make discrete but infrequent jumps. The limitations in this respect are similar to those in other methods of determining characteristics. Further, the proposed method is not limited to a narrow class of input signals such as, for example, a stepwise signal or a well-formed pulse perturbatIOn signal. Concerning the crucial problem of stability, it can be said that we have been particularly concerned with the stability of self-adjusting systems consisting of a characteristic determining system and a control system, which has no relationship to the stability of ordinary feedback systems to which the self-adjusting elements are connected. Stability questions in connection with realization of self-adjusting controllers are frequently considered in the U.S.A. However, if the system operates successfully, further development of stability questions is not emphasized. The published circuits and deSigns of self-adjusting systems are relatively simple and their parameters do not change rapidly . . In the future we can expect increasing importance to be given to questions connected with the stability of such systems, which of course depends on the input signals due to the presence of non-linear elements, although for other input signals it is known in advance that the system may be linear. . Concerning the choice of T, there are a number of examples In recent papers on our experimental studies of adaptive systems, based both on representation of the signals in the form of orthogonal functions and on the repre, entltion of the initial portion of the pulse transfer characteristic in the form of a Taylor series. The values of T of the order of 0·5 system time constants is satisfactory. Obviously the choice of T depends also on the required precision of determining the system response characteristics. Consideration of noise effects, as well as the comparison of this method with correlational methods , must be carried out in connection with specific problems, and depends on the relative possibilities of determining the time intervals necessary to find the characteristics and the precision of determining them. Concerning the realization of 'unstable elements,' the degree of difficulty depends on the choice of T. For reasonable values of T the usual difficulties connected with the realization of transfer functions with poles in the right half-plane do not arise since the pulse transfer function is stable and, of course, may be approximated to any desired degree of precision without special difficulties, using passive networks. The solution of this problem, using operational amplifiers, is given in the paper only for concreteness. The authors appreciate all the comments and remarks.
603
613
suffices to describe the signal to within a specified accuracy. The choice of the particular set of orthogonal functions is motivated by the above considerations and, in addition, by the requirement (for adaptive control systems, ~t least) that ~he spectrum must be obtainable by means of sImple computIng circuits. Expansion in a Sum of Orthonormal Functions In cases, such as adaptive control systems, where it is impossible or inconvenient to express the process dynamics in closed form, an identification in terms of a finite portion of an infinite series becomes useful. If a function f(t), representing a system signal, is to be approximated by the finite sum faCt), of n functions Cir/>;(t), then I/.
f(t) ~ fa (t) =
L Ciep;(t)
(I)
i= 1
The approximation problem here has two aspect~: the ch?ice of the functions epi(t) should be made to obtaIn the deSIred degree of accuracy with the smallest number of terms, and, having once selected the functions epi(t), the coefficients Ci must be chosen to achieve the best approximation to f(t) in some sense. Should there arise the necessity to improve the approximation by the addition of another term [the (n + I) term], the preceding n coefficients should remain unchanged. The~e ar~ a number of criteria by which the goodness of an apprOXimatIOn may be measured, each of which is appropriate i? ~ .certain class of problems. The criterion used here for opttmlZlng the process of approximation is that of minimizing the integral of the weighted squared error; i.e. the coefficients Ci are to be chosen so that the integral E =
f'
[let) - .fcJt)j2 wet) dt
(2)
is minimized . The infinite integration interval permitsf(t) to be defined over the whole range of t. Iff(t) is defined only over a finite range, the range of integration may be adjusted, or the weight function wet) may be chosen to emp~asize the interval of interest. It may be shown that the condItIOn of finalIty of the selection of the coefficients Ci is ensured if the functions epi(t) are chosen to satisfy the relation
C
ep i(t)ep/t)I\'(t) = 0
i
~O j
(3)
~O
The relationship expressed in equation 3 is called the orthogonality condition by analogy with the geometric concept of the orthogonality of two vectors. It may also be shown that , if the ep i(t) satisfy equation 3 and, in addition , satisfy the normalizing condition
f:
ep/(t)w(t) dt = I
596
606
for all i
(4)
APPROXIMATE IDENTIFICATION OF PRCCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
the values of the coefficients equation 2 are
Ci
which minimize the integral in
(5) If the functions epi(/) satisfy equations 3 and 4, they are said to be orthonormal with respect to the weight function w(/). The
weight function must be chosen to emphasize the portion of the range of I which is of interest in the particular application, without introducing any undue analytical difficulty into equations 3- 5. In the two following sections, two examples of orthonormal functions of particular interest in adaptive control systems are described. These are the Laguerre functions and the orthogonalized exponential functions, both of which permit simple analogue computer realization.
response is composed of a sum of exponentials and damped sine waves. A natural set of functions to use in such a case, for a series expansion of the system impulse response, is a set of exponential functions. Exponential functions are not orthogonal in any interval, so that the ISE criterion is not easily applicable to such a set of functions. It is, however, possible to generate a set of orthogonal functions from appropriate linear combinations of exponential functions. One procedure for effecting this result is the Gram-Schmidt procedure l5 , which is used to generate a set of orthonormal functions, ep;(t), from a set of component functions, jj(t), in such a way that epl(t) ep2(t) ep3(t)
= al/I(t) = a2[/I(t) + aaH/)] = alf/ t ) + a 4 Ht) + asfa(t)]
(10)
Laguerre Functions A given function f(/) may be expanded in a series of orthonormal functions composed of polynomials in I. The exact form of the polynomials depends upon the range of definition of f(t) and upon the choice of the weight function w(t). When the range of definitionf(t) is 0 < t < 00 and the weight function is chosen to be w(t) = e- t, one obtains the familiar Laguerre polynomialsl l . The exponential character of W(/) emphasizes errors occurring for small t. The Laguerre expansion of f(t) is en
f(t) =
L ciLlt)
(6)
i~O
where Lo = 1
L1 = -t
+1
12
L2 = - - 2t 2
and, in general Ln(t)
=
~ :~ Et
1
+1
where the ak are selected to satisfy the relations expressed in equations 3 and 4. The weight function, w(/), is chosen to emphasize the error in the region of interest. In adaptive control systems, interest is usually concentrated in a finite time interval, 0 t < T, and errors throughout this interval are of equal importance. This indicates that the rectangular weight function shown in Figure I is suitable for use in the design of adaptive control systems. A rectangular weight function, unfortunately, introduces considerable analytical difficulty. An exponential weight function, w(t) = E- 2at , also emphasizes the error over a small time interval, and has the distinct advantage over a rectangular weight function of analytical convenience.
(7)
wet)
J
[tne- t ]
The Laguerre polynomials given in equation 7 are orthonormal with respect to e-t, i.e.
l
oo
o
L;(t)Llt)e- t dt
=
{O '
i "* j i
I,
=j
o
(8)
T
F(ljll/'e 1. Reclanglllar we(ljhl jllncliOll
The spectrum of f(t) is given by Ci
=
i "'f(t)Li(t)e- t dt
(9)
When H'(t) = e- 2nt , the integrals of equations 3 and 4 may be evaluated in the complex frequency domain using Parseval's theorem which states that
If, for example, the functionf'(t) = e- t is to be expanded in the first four terms of a series of Laguerre functions !Cl)
= e- t
~ coLo(t)
From equation 9, the coefficients off(t) are
so that f(t) becomes (after re-arrangement of terms) !Ct) ~ I - t
Orthogonalized Exponential
(11)
+ c 1L l (t) + c 2 L 2(t) + caLa(t)
+ }t 2 -
This theorem permits utilization of the powerful methods of the theory of functions of a complex variable in the evaluation of the integrals of equations 3 and 4. Application of Parseval's theorem (equation II to equations 3 and 4) when 11'(1) = e- 2at yields
ita
Functionsl~ - U
In many cases of practical significance the dynamic performance of a process may be described by a set of linear ordinary differential equations, so that the process impulse
where
i'J ij
=
0
= I 597
607
i "* j i =j
L. BRA UN, JR., E. MISHKIN AND J. G. TRUXAL
If the component functions /;(/) in equation 10 are exponential functions of the form aj
rjJ (s) = (24)1 /2 (s
3
positive real for all J
4
Application of equation 12 to this case yields a set of orthonormal functions
-
+ 0(1)]1 /2 _I_ s + 0:1 + 0: . )]1/2 (s - 2a - 0(1) (s + 0(1)(5 + 0(2)
x
+ 0:,,)]1/2 j
If the orthonormal functions, (/);(s), are constructed from pairs of complex poles, then (/) k(S) will contain k /2 complex pole pairs at s = -0: 1 +if3l> -0:2 + Jf32 ... , -O:k/2 +jf3k/ 2' Evaluation of equation 12 in this case gives
+ a )]112
1
1
where IS112 = (a
I
(s - a + S ll) (s + a1)2 + f312
+ 0(1)2 + f312 + 0:1)]1/2 ( (s a
no. ( ) = [2(
'1"2 S
S
- a - hi) + 0:1)2 + f3 12
and, in general
1(14)
(/)2V - l(S)} _
(/)2v(S)
I
1/2
- [2(a +o:v)] [Cs - 2a - 0(1)2
X
+ f312 ] ... [(s
- 2a - O:v_ 1)2
sv I) -;-;--~--::o-~-;;-n.-;-;--....,.+-,-f3t",-.: . 1. : . ](,; -S...-;-a_+--!I....:..!C. [Cs + 0(1)2 + f312 ][(S + 0(2)2 + f32 2] .. . [Cs + 0:,,)2
+ f3,,2]
where 11 =
1,2 ... ,
n
2;
n = number of poles in expansion
ISvl2 = (a
+ O:v)2 + f3v2
In equation 14, the (+) sign is associated with Is. 1 in (/)2" - 1(S) and the (-) sign is associated with Isvl in (/)2v(S). It is noted that in the special case a = 0, which corresponds to a unity weight function wet) = I, equations 13 and 14 reduce to the orthonormal exponential functions described by Huggins12 and by Kautz 13 . As an example of the application to equations 13 and 14, a set of orthonormal functions is constructed from two pairs of complex poles and a single real pole. These poles are at
where
OAs
+
(S2
+
OAs
+
1·04)(s2
+
+ 409·04)
+ 409'04)(S2 I '04)(S2
+
+
4s
4s
104)
- 44s + 584) + 104)(s + 2)
/1/(t) = 0
(15)
1< 0
hk(T) = L "' /II(T)g k(T - T) dT
(16)
In equation 15 it is assumed that /II(t) = 0, for I < O. This implies that there is no energy stored in the process. If this is not the case, the stored energy may be removed by applying one of the techniques described in reference 2, and then equation 15 is valid. Comparison of equation 16 with equation 5 shows that identification of 111(/) with I(t) andgk(T - t) with CPloC/)W(/) produces the result " k(T) = Ck; that is, if
s = -2
The weight function in this case is chosen to be w(/) = £- 20 1. The five orthonormal functions corresponding to this set of poles and this weight function are (/) (s) = (20A)1 /2 • S + 0·249 I s- + OAs + 1·04
2
+
At the instant I = T, hk(t) is
s = -2 ±jlO
= (20A)1 /2
(S2
" k(t) = L "" /II(T)gk(1 - T) dT
s = -0·2 ±jl
rf> (s)
- 40As + 409·04) 1'04)(s2 + 4s + 104)
Solution of the identification problem by expansion of a function in a set of orthonormal functions, as described above, has two advantages over the methods mentioned in the introduction: an easily calculated error measure is availab le ; and the effect on the approximation error of an additional single term in the expansion is easily determined, so that the advantage of including this term may be balanced against the added system complexity required by the addition of this term. For a specified set of orthonormal functions, solution of the identification problem requires a determination of the coefficients of the expansion; that is, the spectrum of the signal being characterized by the expansion must be determined. The term 'spectrum' as used here refers to the coefficients of the expansion relative to a prescribed orthogonal coordinate system. So as to make the technique of expansion in a set of orthonormal functions a useful technique in solving the identification problem for adaptive control systems, methods for obtaining the spectrum of a signa l experimentally, in a time which is short compared to parameter drift times, must be developed. In this section an approach is described which is a modification of one originally presented by Huggins12 and which is more readily applicable to adaptive control systems than Huggins' approach. In order to obtain the spectrum of the signals which exist in adaptive systems, means must be developed for operating directly upon the signal of interest to obtain the coefficients Ci , without recording the signal and playing it back, as required in reference 12. For a linear, time-invariant process (or, for a piece-wise linear, piece-wise time-invariant process, as in adaptive systems), the relationship among the process input m(/), the process output hk(l) and the process impulse response g k(t) is described by the convolution integral
( 13)
(5 - 2a - O:I)(S - 2a - 7.2) ... (s - 2a - 0:"_1) (s + O:I)(S + 0(2) . . . (s + O:"_I)(S + 0( 11 )
(/) (s) = [2(a
+
Spectrum Analyser for Experimental Identification of Process Dynamics
and in general (/) n(s) = [2(a
OAs
rjJ (s) = (24)1 /2 (S2 - 40As 5
(/).(s) = [2(a
+ 5·62)(S2 +
rjJ (s) = (24)1 /2 (s - 25'62)(S2 - 40As
(/) k(S) will contain k real poles at s = -0:1' -0:2"', -7.k'
q\(s) = [2(a
(S2
(s - 20·249) S2 + OAs + 1·04
m(t) =/(1)
(17)
/Jk(T - t) = CPk(/)lI'(/)
(I S)
and
598
608
APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
then equation 16 becomes "k(T)
=
{YJ(t)cp,,(t)l\.(t) dt
=
Ck
The relationships expressed in equations 22 indicate that the choice of a set of orthogonal functions should be made in such a way that the individual blocks in Figure 3 are easily realized. Two sets of orthogonal functions which satisfy this requirement
(19)
Equation 19 is seen to be identical with 5 which is the defining equation for Cl." By changing variables in equation 18 it is possible to determine g,,(t) in terms of the weight function wet) and the orthonormal function cp,,(t). This change of variables gives (20)
_T1(s) t T2(s) t T3(S). -- - - - _Tn(S) t-- ~~
Cl
Selection of g,,(t) according to equation 20 results in a spectrum analyser which is useful in obtaining the spectrum of a signa\. If a signal f(t) is applied at the input of a network with impulse response defined by equation 20, the coefficient c" is obtained by measuring the value of the system response "k(t) at the instant t = T, as indicated in equation 19. A word of caution is in order at this point. Physical networks cannot respond until the excitation has been applied: in other words, when 111(1) = 0 for t < 0, then g,,(t) = 0 for t < O. If this observation is applied to equation 19, hilT) may be written as
C2
C3
Cn
Figure 3. Block diagram of spectrum analyser
are the Laguerre functions and the orthogonalized exponential functions described previously. Examples of spectrum analysers employing these two types of functions are presented in the following sections. Laguerre Spectrum Analyser 1G If the orthonormal functions are chosen to be the Laguerre
(21)
Equation 21 indicates that the coefficient c" which is obtained from the spectrum analyser is not the kth coefficient of the expansion of f(t), but is, instead, the kth coefficient of a functionfI(t) defined by Ht) =f(t) for 0 < t < T and for t> T Ht) = 0
Figure 4. Block diagram of Laguerre spectrum analyser
The relationship between f(t) and fl(t) is shown for a typical signal in Figure 2. This conclusion is not surprising, since it is 1M.Cl
f(t)
1M.o.
1I-J.F
t,lt)
1MD. 1MO
1Mfi
Ez(s)
Ez(s) _ 1
o
s
(b) EI(S) =~
(a) El(s) -s-l
T
Ez
El
Ez
El
Figure 2. Relation between f(t) and j;(t)
Figure 5. Analogue complller realization of 1/(s - 1) and sits - 1)
necessary to make measurements over all time to obtain complete information about f(t) so that no measurements over a finite interval will be sufficient. In adaptive systems this is no serious difficulty, since an approximation over only a finite interval is desired. In the general block diagram of a spectrum analyser shown in Figure 3, the transmittances I , TIl ... , are selected to satisfy the relations
functions, the impulse response given in equation 2 becomes gk(t) = Lk(T - t)c(1'-1) k = 0, I, 2. . . (23) The Laplace transform of equation 23 for the first four Laguerre functions is s-1'
Go(s) = - - 1 s -
T T2 ... ,
I
GI(S) = TI(s)
G 2(s) = T I (s)T2(s)
G 2(s) = GI(s) s _S 1
(22) G3(s) = G2(s) s
n
Gn(s) =
IT Ti(s) ;~ l
(T2) +2' - T Go(s)
r
:3 +
J
~ 1 +(-
T2 - T) Go(s)
(24)
Figure 4 shows a block diagram which realizes the transfer
599
609
L. BRAUN, JR . , E. MISHKIN AND J. G. TRUXAL
functions in equation 24. An analogue computer realization of the transfer functions I/(s - 1) and s/(s - 1) is shown in Figure 5. The multiplying factor ET associated with each output in Figure 4 is a constant and may be considered a scaling factor. Spectrum Analyser using Orthogonalized
=c
0·0142
Gl(s)
Exponentials 14 G 2(s)
where
Rki
=
= residue of Wk(s) in pole at s =
where
T 2(s) = -4-45 s - 13 ·9 s - 12 Gis ) = 0.1
-rxi
T3(S)
+ . .. +
2a
= -1.58 (s - 12'9)(s - 18'3) (s - 13)(s - 13'9)
+ ... + R kkE - (rxk + 2a )T E('Y.k + 2a )t
E-(al + 2a)1'
kl s - rxl -
= T 3 (s)T2(s)T1(s)
where
(28) R
(31)
(s - 13)(s - 13 '9)
(27)
or Gk(s), the Laplace transform of g k(t), becomes
Gls) =
(s - 12'9)(s - 18 ·3) (s -11)(s -12)(s -13)
= _1 '58(s -12 ·9)(s -18'3)G 2(s)
If equations 26 and 27 are applied to 20, gk(t) becomes gk(t) = R kl E-(rx 1 + 2a)1'E (rx 1 + 2a )1
= -0'0632 (s _ II)(s - 12)
2a t
+ . . . + RkkE-"k t
R kl E- rx l t
T 1(s)
s - 13 ·9 -4-45 s _ 12 T 1(s) = T 2(s)T1(s)
then the orthonormal function CP.-(t) may be written as cfk(t)
= s _ 11 =
s - 13·9
When the orthonormal functions cp,(t) are orthogonalized exponential functions and wet) is an exponential function of the form wet) = c 2at , the orthogonal spectrum analyser is easily realized using analogue computers. If, in particular, the functions cp,(t) are constructed from a set of real poles at (25) n = 1,2,3 ... S = -:.J. n and if w(t)
If these results are inserted in expression 29, the spectrum analyser for the coefficients of CPl(t), CP 2(t) and cf3(t) has the respective impulse responses
The block diagram and the analogue computer diagram for the spectrum analyser described by equations 31 are shown in Figures 6 and 7 respectively.
RkkE-('Y.k + 2a) l'
s - rxk -
2a
(29)
As an example, if rxn = a =
n 5 T = 0·5
n = 1, 2 ... Note : hi(T) :Ci . for all i
then, from equation 13, the first three orthonormal functions become
Figure 6. Block diagram of spectrum analyser
W (s) = (12)112_1_ s
1
m.
_
'l-'2(S) - (14)
1/2
The three transfer functions Gk), G 2(s) and G3 (s) have poles in the right half of the s plane and are, consequently, unstable. This result is not unexpected since in the development of equation 16 it was necessary to 'flip' g k(t) in contrast to the flipping of the input signaI(t), as suggested by Huggins. Since the spectrum analyser is unstable, the output of each block must be clamped until the instant t = 0, at which time l(t) is applied and, simultaneously, the outputs are unclamped. At some convenient instant f c ;> T, the outputs must again be clamped in order to re-set the system for the next measurement.
+1
S - 11 _ [13 12 ] ( 1)( 2) - 14 --2 - - - (30) s+ s+ s+ s+1
W (s) = (16)1/2 (s - 11)(s - 12) 3
(s + 1)(s + 2)(s + 3) - 16[ 78 182 105 ] s+l-s+2+:;:-+3
00091
0077
00833
0072
Note All resistances In megohms All capacitances in IJ.F h , (T): C, , for all i
F(f{ure 7. Ana/agile computer reali:atioll of spectrum ana(vser of equation 31
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APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS
where T = length of the control interval
An Example of Experimental Identification U It is possible to instrument a relatively simple solution of the identification problem using the orthonormal spectrum analyser developed earlier, when the orthonormal functions used to establish a coordinate system are properly chosen. If the orthonormal functions are a set of orthogonalized exponen-
c",(t) becomes c",(t)
=
(37)
Moa(t)
where (38)
is the unit step response of the process. The unit step response may be expanded in a set of orthonormal functions cPi(t). This expansion takes the form a(t) ~ aa(t)
tials or a set of Laguerre functions, then the spectrum analyser is easily constructed using R- C networks and operational amplifiers to realize the desired transfer functions. In this section, the system forcing function is restricted to be a staircase function of the form shown in Figure 8. This restriction of the form of lII(t) permits utilization of the unit step response as a system characterization. Since met) is a staircase function, the component of the output due to present excitation may be characterized by the amplitude of the present input step, and by the same set of poles and zeros as is used to characterize the unit step response. The system response may be expressed in terms of the forcing function met), and the system impulse response get), by the convolution integral
= f~XJm(T)g(t
alcPl(t)
+ a2cP2(t) + . . . + ancPr.(t)
(32)
- T) dT
c",(t) ~ c,,/(/(t)
=
CnrlcPI(t)
+ C,.,2cP2(t) + ... + Cm ncPn(t) (40)
for 0 <
t
< T
If aa(t) in equation 39 is substituted for a(t) in equation 37, and the resulting series is compared term-by-term with that in 40, the coefficients of these two series must satisfy the relations
C ml C m2
= MOa l = M oa 2
=
cJt)
+
C",(t)
= f~'lJ III(T)g(t
- T) dT
+.c
Equation 41 may be solved for the coefficients a, of the approximate step response in terms of the quantities Mo and C mi . These coefficients are
1 I
I
III(T)g(t - T) dT
I
where
f~xlll(T)g(t
- T) dT
an
= output due to stored energy (34)
"
c ",Ct) = /1II(T)g(t - T) = output due to present excitation •0
Modern control systems are complex and must meet stringent performance requirements . There is an obvious need in such systems for development of design procedures for computercontrolled systems which make use of the highly developed computer art. This paper presents an attempt to apply computers in the solution of one class of such systems. The realization of the novel computer circuits required is outlined, and the overall design logic is presented. The designer versed in the art will have no difficulty integrating the individual circuits. It appears that analogue circuits interconnected by means of
- cJt)
It will be assumed in the following development that the signal c",(t) is available directly. If lII(t) is a staircase function of the
form lII(t)
=
MOI/I(T)
+ MII'-l(t
- T)
+ M 2u_ l (t
- 2T)
II
)
Conclusions
Cit) may be determined from measurements of the process output as described 2 , and may then be used to determine c m(r) from the relation
= c(t)
Cmn Mo
(42)
The quantity Mo is available by direct measurement of 1II(t) and the coefficients Cm ' are available at the output of a spectrum analyser, to that equations 42 represent a realizable solution of the identification problem .
(35)
C ",(t)
(41)
J
(33)
dt) =
1 r
The output c(t) in equation 32 may be expressed as the sum of two components c(t)
(39)
where n = number of terms in the expansion. Function c",(t) has the same form as the unit step response, during any single control interval, except for a difference in level depending upon the amplitude of the step applied at the start of the interval; so that c",(t) may be approximated by an expansion in the same set of orthonormal functions used in the expansion of a(t). The expansion of c ,,.et) takes the form
F{I{I/re 8. Staircase forcing jilllctioll
c(t)
=
+ ... (36)
601
611
L.
BRAU , JR., E. MISHKIN AND J. G. TRUXAL ZADEH L. A . On the identification problem. TrailS. IlIst. Radio Dtijrs. 011 Circuit Theory Dec. (1956) 277 7 WIENER, N. Mathematical Problems of Communicatioll Theory. J 954. Massachusetts I nstitute of Technology unpublished notes (Summer session lecture) 8 BOSE, A. G . A theory of no n-linear systems. Mass. illst. Tech. Res. Lab. Electroll. Tech. Rep. No. 309 (1956) 9 CAMERON, R. H. and MARTIN, W. T. The orthogonal development of non-linear functions in series of Fourier-Hermite Functionals . Alln. Math ., Princetoll 48, No. 2 (1947) 10 MISHKIN, E. and BRA UN , L. JR . Adaptive Control Systems, Chap. 8. 1960. New York ; McGraw-Hill 11 COURANT, R. and HILBERT, D. M ethods of Mathematical Physics, Chap. 2. J 953. New York; lnterscience 12 HUGGINS, W . H. Representation and analysis of signals, Part I, the use of orthogonalized exponentials. Johlls Hopkins Univ. Tech. Rep. AFCKC TR-57-357 Sept. (J 957) 13 KAUTZ, W. H. Network synthesis for specified transient response. Mass. Inst. Tech. Rep . Lab. Electron. Tech. Rep. No. 209 (1952) " BRAUN, L. JR. On adaptive control systems Doct. Dissert . Polyt. illst. Brooklyn , June (1959) 15 HALMOS P. R. Finite-Dimensional Vector Spaces. Princeton University Press 16 MISHKIN, E. On the computer realization of an orthonormal spectrum of a given signal function. Proc. Inst. Radio ElIgrs, N. Y. May (1959) 1003
switching devices will be useful in the realization of the required computing circuits. The required computer facility seems to be reasonable and commensurate with the control problem. More work is needed in the study of the effects of noise, measurement error, computation error, etc., and in the performance evaluation of computer-controlled adaptive systems.
6
The work described in this report was sponsored by the Office of Ordnance R esearch under Contracts DA-30-069-0 RD-2646 and DA-30-069-0RD-1560.
References ASELTINE, J. A., MANCINI, A. R. and SARTURE C. W . A survey of adaptive control systems. Trans. inst. Radio Engrs . on AUlOmatic Control, Dec. (1958) 2 MISHKIN, E. and BRAUN, L. Jr. Adaptive Control Systems. 1960. New York ; McGraw-Hill 3 TRuXAL J. G. Automatic Feedback Control System Synthesis, Chap. I. 1955. New York; McGraw-Hill 4 MALYUSHINETZ, G. D . On the dependence between the amplitude characteristic and the characteristic of a non-linear element. C. R .Acad. Sci. U.S.S.R. 54, No. 6 ( 1946) 491 , MATHEWS, M. V. A method for evaluating non-linear servomechanisms. Trails . Amer. inst. elect. Eng rs. (Appl. & Ind.) May (1955) J 17 1
Summary The need for an identification of the dynamics of the controlled process in the analysis a nd synthesis of control systems is clear. In cases where the dynamics of the process vary with environment, it is frequently impossible to predict the system behaviour from a priori knowledge. In such cases, the process identification must be redetermined periodically during the course of system operation. This requirement has indicated a need to obtain rapidly an experimental identification of the process dynamics. In the subject paper, a technique is described for obtaining a
process identification in terms of a set of orthogonal functions . The method of expansion of a function in a set of orthogonal functions is briefly reviewed. The requirements of a spectrum analyser to obtain the expansion coefficients of a signal in terms of a pre-determined set of orthogonal functions are developed . Examples are presented of spectrum a nalysers, using a set of Laguerre functions and a set of orthogonalized exponentials. The paper concludes with a description of an application of the spectrum ana lyser to adapt ive control systems.
Sommaire Le besoin est clair d'une identification de la dynamique des processus commandes, dans I'a nalyse en la synthese des systemes de commande. Dans les cas oll la dynamique du processus varie avec les conditions ambiantes, il est frt:quemment impossible de prevoir le comportement du systeme en partant d'une connaissance a priori. Dans de tels cas, I' identification du process us doit etre redeterminee periodiquement pendant le cours de I'operation du systeme. Cette necessite a montre le besoin d'obtenir rapidement une identification experimentale de la dynamique du processus. Dans le rapport sur ce sujet on decrit une technique pour obtenir
une identification du processus , en fonction, d'un ensemble de fonctions orthogonales. On passe brievement en revue la methode de developpement d'une fonction en series de fonctions orthogonales. On developpe les qualites requises d'un analyseur de spectre pour obtenir les coefficients du developpement d'un signal en fonction d'un ensenible predetermine de fonctions orthogonales. On presente des exemples d'a nalyseurs de spectre deve loppant en series de fonctions de Laguerre et en series d'exponentielles orthogonales. Le rapport se termine par la description d'une application de l'analyseu r de spec tre aux systemes de controle adaptif.
Zusammenfassung Die Notwendigkeit fUr die Bestimmung der dyna mi sche n Eigenschaften einer Regel strecke fUr die Analyse und Synthese des Regelsystems ist selbstverstandlich. D o n wo sich die dynamischen Eigenschaften der Strecke mit den Umgebungsbedingungen iindern, kann das Verhalten des Systems nur se lten von vo rnherein bestimmt werden. In solchen Fiillen mussen die Eigenschaften der Strecke wahrend des Betriebes in periodischen Abstanden neu erfar.lt werden. D er Beitrag beschreibt ein Verfahren zur Bestimmung der Eigenschaften der Strecke in einer Gruppe ort hogonaler Funktionen . Auf
die Meth ode der Entwicklung einer Funktion in eine Gruppe orthogona ler Funktionen wird kurz eingegangen. Die Forderungen, die an einen Spektralanalysator zur Bestimmung der Entwicklungskoeffizienten eines Signals in Gliedern einer vo rgegebenen Gruppe von Orthogonalfunktionen zu stelle n sind, werden abgeleitet. Als Beispielewe rden Spektralanalysatoren behandelt, die eine Gruppe vo n Laguerre-Funkti onen und eine Gruppe von Orthogonalexponenten verwenden. Der Beitrag sch lie r.lt mit der Besch reibung eines Spektralanalysators fUr ein adaptives Regelsystem.
DISCUSSION
A. M.
LETO V
(U.S.s.R.)
Please formulate, in rigorous mathematical form, the problem of stability of a dapti ve systems.
J. TR UXAL, ill reply. The person who has put th is problem knows more than J. We , and o ther investigators in the U.S.A., consider
that a direct formulation of this problem is very difficult. The difficulty being first that the process itself is complex, and , seco ndly that in our universities we usually like to consider second- and thirdorder systems, whereas adaptive systems are of higher order. In addition there are difficulties connected with the presence of nonlinear elements. The method we employ consists in considering the stability forms. Li 's syste m has been stud ied in various forms.
602
612
APPROXIMATE IDENTIFICATION OF PROCESS DYNAMICS IN COMPUTER CONTROLLED ADAPTIVE SYSTEMS Evidently, this circumstance is the cause of substantial error in determining the expansion coefficients. At the same time existing circuits, for example, for storing discrete values of system transfer functions, are stable. The latter method has the advantage of clarity, while the corresponding circuits are fairly simple. (3) The most important question in determining dynamic characteristics of processes is the elimination of the effects of random noise. In the well-known work of Goodman and Reswick, and a number of articles by other authors, the effects of noise uncorrelated with the input of the controlled process are eliminated by calculating the autocorrelation function of the input and the cross-correlation function between the input and output of the controlled process. At the same time, the method proposed in the paper does not, in itself, permit eliminating the influence of low-frequency noise. Only averaging of the observation results over several intervals would seem to reduce its effect. However, the circuit of the spectrum analyser would have to be more complex and the operation of the system slower. In what way is it proposed to take into account the influence of random noise" This question could not be touched on in the paper; I would like to obtain some explanation of this.
Unfortunately we have no general approach to this problem. I assume, considering previous experience, that this general formulation will be obtained in the Soviet Union.
A. M.
BATKOV
(U.S.5.R.)
How do you select the observation interval T'
J. TRuxAL, in reply. Various methods were used, but, I have not discussed any of them in detail. We take the time Tequal to a quarter or a half of the duration of a transient. In the system mentioned the time T was taken equal to half the transient.
A. M.
BATKOV
(U.S.S.R.)
Have any experiments been conducted with spectrum analysers" What is the opinion of the authors on the reliability of stable transfer function realization"? Reply. I have written several articles on quantitative analysis of such systems, but I have not brought them with me. KAPENOVICH (U.S.5.R.) Do you not think that the direct determination of dynamic characteristics, using the convolution integral, is better than finding them by means of statistical methods:
J. TRUXAL, in reply. This problem was posed several times in the States several years ago. There was some competition between various organizations using different methods for the solution.
W. K. TAYLOR (U.K.) It seems to me that the stability conditions of an adaptive control
system depend on the input signal and therefore, without determining the input characteristics, it is not possible to formulate stability conditions. The method of Professor Truxal is apparently limited to laboratory investigations, and I would like to know if it can be modified in any way for use in systems whose operation cannot be interrupted for test pulses"?
J. TRUXAL, in reply. I think this is valid since real systems respond to a definite class of signals. Non-linear elements are chosen with respect to the input signals. Li's system relates to this category. Here the problem arises whether you can ensure stability for a defined class of input signals. Knowing this class of signals in advance, we can, however, construct a completely non-adaptive system, so that it is necessary to decide first if we are to build an adaptive system.
A. A. FELDBAUM (U.S.5.R.) The problem of automatic determination of the dynamic characteristics of objects presents great interest. Their determination is necessary to study the objects before designing the system and to develop self-adjusting systems which include automatically corrected models of the object. The schemes proposed in the paper are mainly intended for the solution of the second of these problems. The following remarks can be made about the paper: (I) The study was limited to a linear system with constant or slowly varying parameters. It was assumed that the control signal is a stepwise function of time ; in each interval the coefficients for expansion of the process transfer function in a series of orthogonal functions are determined. However, in which way can this method be generalized to the case of certain classes of non-linear systems, or even to linear systems with control signals varying arbitrarily: If this method has , .in principle, no possibility of being generalized, what is its advantage in comparison with other existing methods, for example, with simple storage of discrete values of the transfer function " It would be of value to know the opinion of the authors on this question. (2) The spectrum analyser circuit given in the paper is unstable because the transfer function poles are located in the right half-plane.
J. TRUXAL, in reply. The remarks which have just been made are very desirable and apposite. Since the particular comments are similar to each other, it appears to be possible to give a combined answer, without considering each comment individually. With respect to the comments of Professor Feldbaum, it should be mentioned that we apply the method, of course, only to cases where the process dynamics are very slow in comparison with the measurement time and the response time, where they make discrete but infrequent jumps. The limitations in this respect are similar to those in other methods of determining characteristics. Further, the proposed method is not limited to a narrow class of input signals such as, for example, a stepwise signal or a well-formed pulse perturbatIOn signal. Concerning the crucial problem of stability, it can be said that we have been particularly concerned with the stability of self-adjusting systems consisting of a characteristic determining system and a control system, which has no relationship to the stability of ordinary feedback systems to which the self-adjusting elements are connected. Stability questions in connection with realization of self-adjusting controllers are frequently considered in the U.S.A. However, if the system operates successfully, further development of stability questions is not emphasized. The published circuits and deSigns of self-adjusting systems are relatively simple and their parameters do not change rapidly . . In the future we can expect increasing importance to be given to questions connected with the stability of such systems, which of course depends on the input signals due to the presence of non-linear elements, although for other input signals it is known in advance that the system may be linear. . Concerning the choice of T, there are a number of examples In recent papers on our experimental studies of adaptive systems, based both on representation of the signals in the form of orthogonal functions and on the repre, entltion of the initial portion of the pulse transfer characteristic in the form of a Taylor series. The values of T of the order of 0·5 system time constants is satisfactory. Obviously the choice of T depends also on the required precision of determining the system response characteristics. Consideration of noise effects, as well as the comparison of this method with correlational methods , must be carried out in connection with specific problems, and depends on the relative possibilities of determining the time intervals necessary to find the characteristics and the precision of determining them. Concerning the realization of 'unstable elements,' the degree of difficulty depends on the choice of T. For reasonable values of T the usual difficulties connected with the realization of transfer functions with poles in the right half-plane do not arise since the pulse transfer function is stable and, of course, may be approximated to any desired degree of precision without special difficulties, using passive networks. The solution of this problem, using operational amplifiers, is given in the paper only for concreteness. The authors appreciate all the comments and remarks.
603
613