!DENT!F!CATION OF ELECTROLYTIC CELL PARAMETERS 
USJNG A SELF-TUNING PREDICTOR 
by 
FREDERICK LLOYD COHEN 
B.S.E.E. TUFTS UNIVERISTY 
1974 
Submitted in Partial Fulfillment of 
the Requirements .. for- the Degree of 
Master of Science 
at the 
Massachusetts Institute of Technology 
May 1978 
/ j 
Sfgn~ture of Author. ~ Signature redacted ~ -- • - '! • ' - • '' • ~. ~ ! ~ - ! ~ • • • • 
• .. • • t • • • • 
Depart,IT)ent c;,f Electrical Engineering, May 26, 1978 
Signature redacted 
Certified by. . ,. -r~~~:-:· / .- . --'/~·~· > .· • ~ / • rhe~ i ~ Su~e~v j s~r. 
/ Signature redacted --· 
Accepted ~-MASSACH~SE;SffiS~llf.iil~Dep·-{rt~;~~~ t c~mfui ~t;e o~ Grad~a~e • S~ude~t~ 
OF TECHNOU)GV"' A h' . 
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I I
IDENTIFICATION OF ELECTROLYTIC CELL PARAMETERS
USING A SELF-TUNING PREDICTOR
by
FREDERICK LLOYD COHEN
Submitted to the Department of Electrical Engineering and Computer Science
on May26, 1978 in parital fulfillment of the requirements for the Degree
of Master of Science.
ABSTRACT
Electrode measurements of bioelectric signals are corrupted by noise,
offset, and parameter variations. The confidence on observer has that the
measurements accurately reflect the bioelectric potential determines the
extent to which analysis may be conducted. A self-tuning predictor is
proposed to remove electrode-induced noise and offset from the observation
signal, taking into account effects of parameter variations inthe biological
medium and at the electrode junctions.
Thesis Supervisor: Timothy L. Johnson
Title; Associate Professor of
Electrical Engineering
F--i-
-3-
ACKNOWLEDGEMENT
I take this opportunity to express my gratitude to my advisor, Profes-
sor Tim Johnson, working with him has been a rewarding and educational
experience.
Particular thanks goes to Wolf Kohn. Though we worked together for
a short time he was most helpful in introducing me to MIT's adage computer
facility.
Lou Dadok was extremely helpful in preparing the electrolytic cell and
obtaining the impedence vs. frequency plots.
The work done in this thesis has been partially supported by funds
from the National Science Foundation under Grant NSF ENG 77-05200.
DEDICATION
To my mother and step-father
Lillian and Joseph Goff
I I
--r
-5-
TABLE OF CONTENTS
ABSTRACT
ACKNOWLEDGEMENT
DEDICATION
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
Introduction to the Problem
1.1 Introduction
1.2 Summary of Thesis Results
1.3 Organization of the Thesis
Model ing
2.1 Introduction
2.2 The Model
2.3 The Optimal Predictor
2.4 The Self-Tuning Predictor
Simulation Studies
3.1 Introduction
3.2 Implementation
3.3 Analysis of Simulation Results
Electrode Studies
4.1 Introduction
4.2 Implementation
4.3 Discussion of Results
Summary and Conclusions
5.1 Summary and Conclusions
5.2 Recommendations for Future Research
Page
2
3
4
7
7
8
9
11
16
18
23
23
23
28
31
31
31
33
37
37
38
I I
-6-
Page
APPENDIX ho
FIGURES 46
TABLES 57
REFERENCES 63
-7'-
CHAPTER I
INTRODUCTION TO THE PROBLEM
1.1 INTRODUCTION
Galvani demonstrated, through his third experiment on contraction the
existence of a biolectric potential [1]. Since this discovery during the
last decade of the eighteenth century, the use of electrodes to measure
bioelectric events has progressed significantly. Until recently, careful
attention to electrode selection and application has been the means of mini-
mizing electrode artifacts [2]. As experimental situations become more de-':
manding, requiring greater refinement of the measured signal, this empirical
method, basic band-pass filtering methods, and fixed parameter linear fil-
ters, are ineffective in selectively removing electrode effects. The
adaptive predictor, introduced by Wittenmark and Astrom [4],[8] takes ad-
vantage of modern statistical filtering theory to directly estimate para-
meters of a minimum mean square error predictor from input-output data.
The tuned parameters are then utilized for the prediction of future obser-
vations. The optimal predictor parameters are not independent of the plant
paramters: we investigate the possibility of deriving the plant parameters
from the estimated predictor parameters.
The usual experiment involves placement of metallic electrodes in
contact with an electrolyte (electrode paste or biological material such
as the skin) in order to make measurements of bioelectric potential either
in response to some underlying bioelectric event or to a stimulating cur-
rent pulse. The potential actually measured at the terminals of the elec-
trodes is a result of the stimulating event and electrode-induced distor-
I I
-8-
tions. Geddes points out [2] that even when procedural precautions are
taken to minimize the electrochemical potential difference between elec-
trodes, there often exists a residual potential difference which may be
unstable and randomly varying. The origin of this residual voltage is
hypothesized as slight differences in electrode metal or surface contamin-
at ion of the electrodes. There are other sources of disturbance; for
example, mechanical vibrations in the electrode-electrolyte interface gene-
rate electrical artifacts in the frequency spectrum of the bioelectric
source so that simple filtering can not be employed without loss of the
desired signal component. Environmental changes such as temperature con-
centration and pressure may result in parameter drift that would be dif-
ficult to remove with a fixed parameter filter. The self-tuning predictor
learns from experience and is capable of tracking these types of parameter
fluctuat ions.
1.2 SUMMARY OF RESULTS
The self-tuning predictor has been applied to a simulated plant, gene-
rating input-output data digitally, and to data obtained from experiments
with an electrolytic cell. The self-tuning predictor is compared to var-
ious non-adaptive predictors, including the optimal predictors.
The results of the simulation studies presented in tables one through
four, lead us to the conclusion that the self-tuning predictor is very
good at reducing the prediction error to a minimum. However there exist
parameter sets, beside the calculated optimal sets, that are comparable in
their prediction capabilities.
The experimental results utilizing the cell, presented in tables 5
-9-
and 6, indicate the estimated parameters of the predictor converge to
values that upon implementation in a non-adaptive predictor have predic-
tion capabilities comparable to the self-tuning predictor.
In conclusion, the self-tuning predictor is an appropriate means of
filtering random noise from observation signals, with the added feature of
convergence to parameter sets that are nearly optimal. It is also shown
that the self-tuning predictor is capable of following trend variations in
the plant parameters.
1.3 ORGANIZATION OF THE THESIS
This thesis deals with the applicability of a particular filtering
scheme to the problem of estimating the true plant output given noisy
observations, and the indentification of the underl ying plant parameters.
In the following chapter on modelling, a continuous time state space model
is developed from an equivalent circuit model of the electrode-subject
system. The continuous time model of the plant is then discretized and
expressed in auto-regressive form. The optimal predictor assuming known
and constant plant parameters is derived to obtain the minimum square error.
A type of extended Kalman filter for the predictor gains is then imple-
mented for the case when the plant parameters are not a priori known or
include some time variation.
The following twochapters describe the implementation of the self-tuning
predictor. A simulation study was conducted to help monitor proper coding
of the algorithm along with presenting the opportunity to study the filter
response in a controlled environment.
The fourth chapter concerns the electrolytic cell study. A cell simi-
-10-
Var to the one studied by Johnson and Salzsieder [3] was used. The elec-
trodes in this application performed both the stimulation and recording
functions. Time series data was processed to determine estimates of the
predictor parameters. The estimates are compared to those derived manu-
ally from impedence measurements on the cell.
The last chapter concludes the thesis with a summary and suggestions
for further research.
I I
-11-
CHAPTER 2
MODELLING
2.1 INTRODUCTION
In order to apply modern filtering theory it is necessary to compose
the problem in an appropriate manner. This chapter is intended to form a
bridge between the empirical knowlege that exists concerning application of
electrodes and the structured form of the theory. We begin with a linear
circuit representation of the electrodes and their environment, generalize
for variation in the environment and then develope a discrete model whose
output may be considered an observation of the underlying system parameters.
2.2 THE MODEL
To facilitiate conceptualization of the relationship between the bio-
electric source, medium, and the electrodes refer to Figure la. Electrodes
are placed on the surface of a subject in order to make observations of
either the underlying source signal or response to stimulation from the
electrodes themselves. A slight modification of Geddes' [2] approximate
equivalent circuit for the electrode arrangement of Figure la is shown in
Figure lb.
Johnson and Salzsieder [3] have investigated the effects of variation
of certain silver/silver cloride cell parameters on cell impedence. Johnson
[9] has suggested the generalization of the circuit model of Figure lb by
incorporating functional dependences of the model elements on the cell para-
meters. Let p be the set of time varying parameters:
p,= temperature of the medium
P2 = bulk solution concentration of ion(s) participating
wimppomppomEpw
I I
-12-
in the electrode reaction
p3  bulk solution concentration of non-participating ions
p =pl9,p 2,p3
The following functional dependencies may be derived using exper-
imental data obtained by methods presented by Geddes Il].
RI= R(p1 ,p2 'p3  vei = vei(p 1 p 2 p3 ,vs), i=1,2
R2 = R2(pl'p2 ,s) C = C(p 1,p2 vs
R3 = constant
Analysis of the circuit of Figure lb results in the following set
of equations:
x (t) = ax1 (t) + S(vs (t) + ve(t)) la
y(t) = Yx1 (t) + g(vs(t) + Ve (t)) lb
ve(t) = ve (t) + ve2(t) Ic
a = -(2(R + R2) + R3)/CR2(2R + R3)
$ = 1/(C(2RI + R3))
y = -2R3/(2R 1 + R3)
= R3/(2R] + R3)
The total junction potential, v (t), may be decomposed as:
e
v (t) = W'(p)v_ (t) + v0 (p)
The first term on the right hand side represents fluctuations in the
half cell potential due to the driving source voltage. The second
-13-
term shows the implicit dependence of the half cell potential on the
parameters p.
Substituting for v (t) in equations la,b:
x (t) = tx(t) + M(1 + S'(p))v (t) + Sv"(p) 2a1s e -
y(t) = YxI (t) + (1 + 61p))vs(t) + Ev(p) 2b
- e
In general, p is dependent on time, but substantial simplification
results when we assume parameter variation is significantly slower than
the response time of the biological system. We then make the approx-
imat ion:
p(t) = p0
and
v0(p) = v0(_p 0 ) for ali't.
Recognizing
d o "
- v
0(p) = 0
we can augment the state equation 2a setting x2 = v0 (p0) and x =0:2 e - 2
.t)a(t))0 B x(t0)
(t) = = o t]+ L vs(t) 3a
and the observation equation is:
y(t) = fY E] Lx(t) + yl + s'(p)) Vs (t)
3b
x2(t)
The system equations are now in the matrix form:
I -
I I
-14-
k(t) = A x(t) + B vs(t) 4a
y(t) = C x(t) + D vs(t) 4b
whe re
y(t) and v s(t) are scalars
x(t) is aC2xl) vector
A is a(2x2) matrix
B is a(2x1) matrix
C is a(lx2) matrix
D is a(lx1) matrix
The auto-regressive representation of the system may be derived
by first obtaining the sampled data equivalent to equations 4a,b.
x(t i+) = x(t 1) + Bu(t.) 5a
y(t ) = tx(t.) + Du(ti 1) 5b
where
W=exp(AA) A=t i+1 - t 6a
B expfA (t - T) ] BdT 6b
C = C 6c
0 = D 6d
U(ti ) = Vs(t = t ) for t < t < t i+l
u(t1 ) is the piecewise constant sampled data equivalent of the
continuous input vs(t).
The following notational convention has been assumed:
-15-
t = continuous time
.th
t. = discrete instant of time representing the i
sampl ing instant.
For constant sampling interval A, t. = iA.
The sampled data transfer function matrix is:
A+
G(z) = Y(z)/U(z)
G(z) may be written as G'(z ) = Y(z)/U(z). G(z ) is a
-1 -1
ratio of polynomials in z . Taking z to be the backward shift
(in time) operator [5] the system equations may be expressed in
the autoregressive form:
y(t ) = -aly(t il)-a2y(ti-2)-..-.-ay(ti-n)+bu(t 1)+b2 u(ti-2+...+
b u( t1 ).bm ( -m -
The coefficients in this equation inherit the dependence on p,
which is assumed to vary slowly with time. We will incorporate effects
of nonlinearities and uncertainty in the source u(t.) as a single
disturbance term v(t1) acting on the output y(t ) 18]. v(t1 ) is taken
to be stationary, white and zero mean, with covariance E[v(t.)v(t.)] =
V6(t.-t.).
For convenience with later equation manipulation the autoregres-
sive equation is represented in operator form:
A(z ) y(t.) = B(z-I) u(t1 ) + C (z ) v(t) 7
I I
-16-
where:
-1 -1 -nA(Z ) = 1 + a1 z + *.. + az
B(z) bz + b2Z + ... + bmz
C(z ) = 1 + c 1z + ... + cnZ
z = The Backward Shift Operator
We will concentrate on the case C(z~) = 1.
2.3 THE OPTIMAL PREDICTOR
Consider the stochastic process introduced in the last section:
A(z 1 ) Y(t.) = B(z ) u(t.) + C(z1 ) v(t.) 8
where
Tv(t ) ,i = 0,1,2,...] is the output sequence
Iu(t.) , i = 0,1,2,...] is the input sequence, and
fv(t ), i = 0,1,2,...] is the noise sequence.
For the moment A(z ), B(z ), C (z ) are assumed to be known
and constant. The k step-ahead prediction of the output signal, given
observations through time t. is denoted y(ti+k/t i The predictor
error is defined as:
E(t i+k) =Y(ti+k) - y(t i+k/tI). 9
The loss function is taken to be:
L = EE2(t i+k) ] . 10
,- F ------ lF --
-17-
We now wish to find the predictor which minimizes the loss
function. Astrom 18] has solved the prediction problem using the
i'denti ty:
C(z )=A(z )F(z ) + z-k G(z ) 11
where
F(z ) = 1 + f z + .. k + fkl-k+1
G(z-) = g0 + g 1Z- + .. + g 1n-z-n+1
The process (8) may be written as:
y(t ) = fB(z )/A(z )] u(t.) + IC(z -)/A(z )] v(t.) 12
The prediction error equation (9) becomes:
E(ti+k) = zk(B(z I)/A(z )) u(t.) + zk C(z )/A(z )) v(t.) -
Y(ti+k/t;)
It can be shown that
(ti+k i G(z 1 ) y(t1) + (zk _ G(z ) B(z) u(t.) 13y -i~iC (Z-. C(z )I A(z 1)
reduces the prediction error (9) to
E(ti+k -I (C(z'1 )zk - G(z1)) V(t )
A(z )
Applying the identity (11) in the form:
zk C(z I= zk A(z ) F(z ) + G(z )
(9) becomes
-18-
k -
(ti+k) = zk F(z ) v(t.).
Thus the predictor (13) makes the error, z, a moving average of v.
Since there is no "energy storage" this gives a minimum variance.
The k-step ahead prediction (13) is a linear combination of past pre-
dictions, present and past observations, and future, present and past in-
puts. The inclusion of future input values may present a problem depending
on the experimental situation. If the electrodes are used for both stimu-
lating and recording it is likely the experimenter knows the input series
a priori or can implement the identification off line with measured values
of the input s ignal. However, where the electrodes are employed for recor-
ding only; and the underlying source signal is not known, estimation of the
input series must be conducted. This thesis focuses on the experimental
configuration using one pair of electrodes for stimulation and recording.
2.4 THE SELF-TUNING PREDICTOR
The coefficients in (8) are taken to be time varying and imperfectly
known as a means of approximating the effect variation of p has on the
equivalent model elements. Considering that the relation between the coef-
ficients and p is very complicated we make the assumption that the coeffi-
cients are themselves a random process generated by:
q(ti+ 1) = p(t.) + Y(t.)
where t()=[a a ... a b b ... b c c ...'c]T1 2 n 1 2 ml 1 n
and Y(-) = LY Y2.....'.'' 
''................' '2n+m
{y(t )} is a zero-mean white noise sequence.
I I
Since A(z ) B(z ) and C(z ) are now time varying theperformance of the
non-adaptive filter presented in the previous section will be suboptimal.
The filter can be made adaptive by recognizing (8) as an observation on the
process generating $(-). Kalman filtering can be applied to obtain mini-
mum-variance estimates of C(t.) given observations through time t.. The
new estimates of A(-), B(-), and C(-) are then used to derive the optimal
predictor. This procedure requires three steps: parameter estimation and
then calculation of the optimal predictor parameters via the factorization
(11) and solving of the predictor equation (13).
These latter steps are time consuming and thus not conducive to on-
line implementation. It has been suggested in the literature [4,9,10] that
the form of the predictor (13) may be adopted, allowing the parameters to
be identified directly, as follows.
For C(z )=l, the optimal predictor takes one of the following two
forms:
(i) Multiplication of both sides of equation (13) by the term A(z )
-1 -1 -1 k -1 -1A(z )y(tk/t.)=G(z )A(z )y(t )+(z -G(z ))B(z )u(t.); (15)i+k i I
(ii) Recognize that the identity (11) may be expressed in the form
(zk-G(z )) = zk A(z )F(z )
the predictor may be written as:
A -l k -l -lY(ti+k/t.) = G(z )y(t.) + z F(z )B(z )u(t.) (16)
In the case C(z1 )#l, (15) will be the optimal predictor. For either
form of the predictor in the previous case, the predictor equation may be
-20-
written explicitly as:
y(t ik/t.) = -q I'y(t. /t. Ai-)-...-q ny(t ikn/t. )n+k i+k-l - n i+k-n i-n
+ q Yn+ y(t)+...+q3n y(ti-2n+l + q3n+lu(t i+k-1
+. .+ u (t . ). (17)
. q4n+m+k- i-m-n+l
This equation may be factored into the following form:
y(ti+k/ti) = H(t )Q (18)
hre Q=[q q q ]T
where [q=q1 2 4n+m+k-l
and
H(t ) = -y(t i k /t. -I)...-y(t i k n/t. n)y(t.)..
Y~t u~t ) .u(t.
y(ti-2n+l)u(t i+k-) i-m-n+l
Each element of the vector Q is derived from the plant parameters by apply-
ing (11) and (13). As stated earlier, if the plant parameters are known
and constant the predictor parameters derived in this manner will repre-
sent the optimal predictor; in the case where the plant parameters are un-
known or time varying the predictor derived from our initial estimates of
the plant parameters will be suboptimal.
The time variation of the plant parameters and thus the predictor
parameters is due to the dependence of the equivalent model elements on p.
The assumption of slow time variation of p in relation to the response time
of the subject, biological preparation, and electrodes, allows the assump-
tion that the unkown coefficients Q are generated by the random process:
-- -I----
I I
-21-
Q(t.) = Q(t._1 ) + w(tq) q(0) = Q(19)
where w(t.) is zero mean, white sequence with correlation matrix
E[w(t.)w(t )] = W 6(t.-t )j m j m
We interpret (19) as the state equation of a linear dynamical system
with the unknown predictor parameters as state variables. We view the
observation given by (18), as an observation of the parameters, Q, where
the observation matrix H('), is time varying, -consisting of past predic-
tions, and measured input and output values.
In order to apply the well known method of Kalman filtering it is
necessary to make general statistical assumptions. We assume the initial
estimate of Q, the state vector, is a Gaussian random variable:
Q(0) ' N(Q ,P(O/-l))
We have already stated the assumptions concerning the state driving
noise and the observation noise.
The Kalman filter equations may be given as:
Q(t.) = Q(t.i) + K(t.) (y (t.) - y(t.It.k)) (20a)
i -1i i i -k
T T -1K(t ) = P(t./t. )H (ti k)(V+H(tik)P(ti/t. 1)H (t.k)) (20b)
P(t /t.) = P(t .i/t . -I)+W-K(t ) (V+H(t 
. -k)P(t 
./t. i -)(2 cI i -k ))K-k t 1-
(20c)
HT (t.k))KJ(ti)
y(t 1/ti-k) = H(ti-k)Q(ti-k)
-22-
A block diagram for this filter is presented in figure (3). The state
vector Q(t), is interpreted as the parameter estimate incorporating obser-
vations through time t.. The predict ion 9 (t /t-k) is the predicted plant
output using the observat ions and parameter update through time t.k.i -k
T
I I
-23-
CHAPTER 3
SIMULATION STUDY
3.1 INTRODUCTION
The simulation phase of this research was benficial in that it allowed
controlled examination of the filter characteristics. The tests were
designed to permit explicit determination of the capabilities of the self-
tuning predictor when initial conditions, noise level, and design parameters
were varied. The plant structure was also variable. Two cases are includ-
ded here: constant plant parameters, and time-varying random parameters.
3.2 IMPLEMENTATION
The simulation study was implemented as follows. A plant of the form
(7) was selected. The choice was based on consideration of stability and
computational simplicity. The input time series was previously determined
and stored on disk. The noise sequence was generated digitally to allow
for variability in thestarting "seed" value of the pseudorandom noise gene-
rator. After certain initial conditions were established the observations
were calculated recursively by application of the auto-regressive equation
(7). With each new observation the self tuning predictor parameters were
updated by application of the filter (20). The simulation ends when either
the input series is exhausted or a specified number of iterations has
occured.
We now elaborate upon each of the above steps. Two plant structures
were used for simulation. They differed in that one involved constant
plant parameters, and the other a time series trend superimposed upon a
nominal value. In general, the plant was of the following form:
y(t.) = -a y(t._ )-a 2 y (t 9i-2)+b Iu(t. 1 )+v(t.) (21)
The first set of tests involved constant plant parameters. In parti-
cular they were:
a = .25
a2 = .5
b,= 1.0
The operators are identified to be
A(z ) = 1 + .25z + .5z-2
Bz-l -1
B(z) = z
C(z ) =
The stability of this plant is determined by ascertaining the location
on the complex plane of the roots of the characteristics equation. The
characteristic equation is:
s2 + .25s + .5 = 0
Applying the quadratic formula the roots, a complex conjugate pair are
obtained:
= -.125 + .696j
s2 = -.125 - .696j
s19,21 = .707
I I
-25-
Since Fs.l<1 for i=1,2 the plant is stable. [5]
The second set of tests involved trend variation in the plant para-
meters. In this case the initial a.'s were taken to be the same as above.j
The a.'s would then vary in time according to the following:
a. (t )+l = a.(t.) + 6
where 6 was chosen to be a constant, .0001. The "poles" of this time-
varying system move with time. Initially they are the same as in the
case with constant plant parameters. At the end of these tests a,=.458
and a2=.708, and the new poles are
s = -.23 .81.
1,2
Is1,21 = .842
The system is still stable.
The input series was taken as a sum of sinusoids. Mehra [12] presents
an algorithm for calculating optimal design inputs. We have chosen the
input series as
NF
U(t.)= Z v.sin(2Trf.t.)j=l i
where t. = iA i=1,2,...
A = sampling period
.th
v. = magnitude of j sinusoid
f.= frequency of j hsinusoid
.= 1,2,...NF
NF = number of frequencies superimposed.
I I
-26-
Although different input groups were employed in the course of the
research, one group was selected to impart some standar~rdizat ion to the
results presented here. The input series was generated by recursive
appliation of the following.
u(t.) = .6 sin(lO0wiA)-.5sin(150iA)+.2sin(196TriA)
with a sampling period of A=.00117 seconds.
A random number generator was designed exclusively for this project.
Because of some difficulty in obtaining a sequence with a variance equal
to theone desired, a standard design variance was chosen. For the random
number sequence generated, a mean and variance were calculated. Correction
for bias was unnecessary since the mean was zero to 5 decimal places. To
obtain sequences with various magnitudes and variances a scale factor was
introduced.
2Let r be a random number sequence with mean vi and variance a2. The
multiplication of r by a constant a results in a new random variable r'=ar.
2 2It is well known the mean of r' is cp and the variance of r' is a a2. The
distribution, mean, and variance for the standard noise sequence is pre-
sented in figure (4).
The system was assumed to be initially at rest. Specifically, u(t)=O
for t.<O and y(t )=O for t.<O. The initial guess of the predictor para-
meters was desi-gnated Q(-l).
The initial parameter estimate correlation matrix, P(O/-l), the corre-
lation matrix for the noise driving the predictor parameters, W, and the
covariance of the observation noise,V, were taken as filter design para-
meters. This seems reasonable since we may have difficutly in quantifying
-27-
1) How accurately we feel our initial estimate of the predictor para-
meters truly reflect the optimal parameters; 2) the noise process that is
driving the plant parameters and;3) the characteristics of the noise sig-
nal corrupting the observations.
If the plant .parameters are known and constant the optimal predictor
(13) for the simulated plant (21) may be determined by first applying the
identity (11).
For k=l,
-1 -2 -1 -l
= (l+aIz +a2Z )(l)+z (g0 +g1 z )
G(z ) = -a1 -a2 z
F(z 1) 1.
Since C(z )=1 there are two forms to the optimal one step-ahead pre-
dictor. We specify a predictor by its parameter vector Q. These vectors
for the optimal predictor are designated Qoptl and Qopt2'
Solving (13) the optimal predictor takes the form
y(ti 1/t) = H(t)QOptj j=1,2
with
TQ =[0 0 -a -a2  0 0 b 0 01 (22a)optl1 21
Qopt2 = [a a2-a - (a12+a2) -aIa2 a22 b ab a2b] (22b)
and H = [-y(tI/t )-y'(t /ti-2)y(t.)y(t. 1)y(ti- 2)y(t -)u(t)
U(t 
- )u(t i-2
-28-
The above equations show the explicit dependence of the optimal pre-
dictor parameters on the plant parameters. It is clear that if the plant
parameters vary the desired optimal predictor parameters will change. The
non-adaptive filter will not be able to follow this change and will thus
be sub-optimal.
In the case where the plant parameters are constant and a priori known,
the prediction error for the optimal predictor is given by (14). For k=l,
this becomes:
e(ti+l) = z F(z -I)v(t)
Since F(z )=l, for our simulated plant
E(t ) = v(t. 1)
We expect the optimal predictor to work very well under these condi-
tions of perfect knowledge of the plant parameters.
3.3 ANALYSIS OF SIMULATION RESULTS
We will evaluate two capabilities of the self-tuning predictor: First,
the proficiency of the self-tuning predictor as a filter; Second, our abil-
ity to infer information about the plant from the parameter estimate vector
Q(). For comparison of prediction capability a non-adaptive predictor is
used. The optimal predictor, for a plant with constant parameters, is a
non-adaptive predictor whose parameters are those given in equations 22a or
b. Non-optimal parameter sets are also implemented. The criteria for com-
paring prediction capabilities will be:
tf
Accumulated Absolute Error = Z Ie(ti)I
i=l
--T
I I
-29-,
tf 2
Accumulated Square Error = E s (t.).
Test results are summarized in Tables 1,2,3, and 4. All tests uti-
lized the same input series and initial conditions. The noise sequence
differed by a multiplication factor thus resulting in different noise
variances. The accumulated loss and error is over a period of 2081 iter-
ations.
Analysis of the results leads to the following conclusions concerning
predictor capabilities. When the plant is known a priori and the parameters
are known to be constant, the non-adaptive optimal predictor is excellent.
However, if the predictor parameters are poorly known the self-tuning pre-
dictor learns very quickly. For example, Test 25 on Table 2 involves an
initial parameter estimate that is grossly distant from the optimal set,
Figure ( ) shows both the accumulated loss for the self-tuning predictor
(curve ii) and the accumulated loss for the optimal predictor (curve i).
From the graph one sees that most of the loss is incurred within the first
twenty iterations of the filter. In fact, the average loss per iteration
over the last half of the test is almost identical for the self-tuning pre-
dictor and the optimal predictor. Figures 6 , 7 , 8 , plot the parameter
estimates for this same test. The estimates approximate the optimal values
rather quickly, but final adjustments take a considerable amount of time.
In fact, it is well-known that the estimates may be biased and may not
converge to the optimal values.
One may also deduce from the results presented in Tables 1 and 2 that
although Qtl and Qt2 are the theoretical optimal parameters, there are
other sets that offer comparable prediction capability. In particular, a
I I
-30-
parameter set whose elements were determined by taking the numerical average
of the respective optimal elements was comparable in prediction capability
to the two optimal sets. Applied in a non-adaptive predictor the accumu-
lated loss for this new parameter set (Tests 3, 7, and 11) is almost iden-
tical to the accumulated loss of either optimal set (Tests 1,2,5,6,9,10).
When this parameter set is taken as an initial estimate for the self-tuning
predictor (Tests 15, 20 and 24) there is no tendency for the parameter up-
dates to diverge from their initial values. We ask the immediate question
of uniqueness. Although Qoptl and Qopt2 are derived from the unique F(z1)
and G(z ) that solve (11), it is not clear that (11) is the only means of
reducing the prediction error to a moving average of the noise. This possi-
bility has not been fully explored in this work, but represents an area for
future investigation.
Tables 3 and 4 summarize the tests involving a time series trend in
the simulated plant parameters. In all cases the self-tuning predictor
performed superior to the non-adaptive predictor. There is a significant
effect upon prediction and estimation of the filter design parameter W.
In the case W=0 (Tests 44, 46) the predictor parameter estimates diverge
from the proper set. However when W is non zero (Tests 45, 47) the pre-
diction and estimation performance is enhanced. This is partially explained
by the integration of W via the estimate covariance equation (20c) and its
effect on the Kalman gain K(-).
The simulation tests that were conducted were not meant to be exhaus-
tive in their examination of the self-tuning predictor. They were chosen
to be representative of the type of situations to which the predictor would
be applied.
-31-
CHAPTER 4
ELECTRODE STUDIES
4.1 INTRODUCTION
The application phase of the research focuses on indentification of
electrolytic cell impedence parameters. In general, the theory of the
self-tuning predictor presented in this thesis is applicable to any single-
input, single-output system that can be characterized by continuous or
discrete state-space models or an auto-regressive equation. We have chosen
a pair of silver-silver chloride electrodes in a saline soultion as the
plant to be identifed. Two silver disc electrodes form the end pieces of
a cylindrical volume containing sodium chloride solution.
Johnson and Salzsieder [3] have conducted extensive testing to deter-
mine the dependence of this cell's impedence on various experimental vari-
ables (e.g. temperature, concentration), We have applied the self-tuning
predictor to cells containing different concentrations of salt in order to
determrine the adaptability of the predictor.
4.2 IMPLEMENTATION
Figures 9 , 10 present the circuit and flow chart for experimental data
collection. A single pair of electrodes is used both to ''stimulate'' the
medium and record the response. A previously calculated input sequence is
selected and read into memory from the disk. This input is applied as a
piecewise constant voltage via the digital to analogue converter. The two
analogue-to-digital channels sample the voltage across the known resistance
R s. For identification purposes the cell current is taken as the stimulus
and the cell voltage is the responce. The current may be calculated from
the time series voltage measurements as follows:
C(t ) 
-d(t.)
uc(tt=
s
The different time subscripts are the result of our sampling the volt-
tages c(') and d(-) at the end of each sampling interval; u(t 1),) is the
current applied to the cell over the time interval (t._1 ,t.]. The cell
voltage response, y(t.) is of course d(t.). The time series measurements
are stored on disk.
The identification routine is applied to the stored data. Figure
presents a flow chart for the identification procedure. Filter design
parameters and the initial estimates of the predictor parameters are the
program inputs. The stimulus and response series are selected from those
stored on disk and read into memory. The routine then iterates through the
2080 sample pairs. The block diagram for the filter is presented in figure
3. If the sampling interval was extended or a more efficient implementation
of the algorithm were used it would be possible to implement the self-
tuning predictor in real time.
The optimal predictor prameters are derived from estimated plant para-
meters (see the appendix). The plant parameters are assessed by matching a
transfer function of the form:
zz 2zn
Z(s)=K Z2  Z (23)
( + 1 ( + .. ( +
p2  n
to the log impedence vs log frequency plot for the cell (Figure 11 a,b)
I I
-33-
The auto regressive form for the stimulus-response relationship is obtained
by determining a continuous time state-space model realization for Z(s).
This model is discretized via the transformation equations (6); and the
sampled data transfer function is obtained. The auto regressive form is
the result of taking z as the backward shift (in time) operator. The
optimal parameters are found by solving the identity (11) for G(z ) and
then (13) for the predictor parameters. These parameter sets are utilized
in the non-adaptive "optimal" predictors and as initial estimator for the
self-tuning predictor. The estimate covariance matrix P(O/-l) was chosen
to reflect our uncertainty in these estimates. W and V were varied as fil-
ter design parameters.
4.2 ANALYSIS OF RESULTS
The results of selected tests are presented in Tables 5 and 6 for
saline solutions of .01 and .1 normality respectively. As in the simula-
tion study we wish to analyze the self-tuning predictor as both a filter
and a parameter estimator.
Strict comparison of the accumulated square error can be misleading.
Both tables offer examples of wide difference in accumulated square error,
but closer examination indicates the predictors to be almost indentical.
Tests 2 and 4 of Table 5 and Tests 2 and 3 of Table 6 are indicative of
this situation. Both pairs of tests begin with different initial estimates
of the predictor parameters. In both cases the self-tuning predictor para-
meters converge to similar values. Calculations of the average square
error over the last 208 iterations indicates that the predictors are almost
indentical in performance. This is evidence of the robustness of the algor-
I I
ithm and the predictors' ability to learn to track the cell parameters.
For the .01 normal cell initial predictor parameters estimator were
derived form plant estimates assuming a transfer function with one pole
and one zero. The plot of log impedence vs log frequency for the .01 N
cell is presented in Figure 1a . The asymtotic Bode plot of the proposed
transfer function is superimposed- over the laboratory results. The optimal
predictor for a plant with one pole and one zero is of order 5 and is
derived in the appendix. In table 5 we compare two 5th order self-tuning
predictors, one of which takes the calculated optimal predictor as its
initial estimate (test 1) and the other begins with zero's (test 3). The
estimates of the predictor parameters converged in both cases. We compare
the average square error of these self-tuning predictors with the average
square error the calcualted "optimal" predictor (test 5) and the non-adap-
tive implementation of the converged parameters (test 6). The "optimal"
predictor has an average square error over the enitre test on the order of
3 times greater than theself-tuning predictors. The non-adaptive implemen-
tation of the converged parameters results in an average square error over
the entire test that is approximately 1.6 times as great as the self-tuning
predictor.
Simialr results were obtained for tests 2,4,7 and 8 of table 5. The
order of the self-tuning predictor was increased to nine to investigate
the effect of increasing the degree's'of freedom of the predictors. Initial
predictor gain estimates were again the "optimal" set (assuming one pole
and one zero in the transfer function) based on prior experimental data
and the zero set. The estimates again converge but to values that are
slightly different from those of the lower order predictor. Prediction via
F -F-
-35-
the converged parameter set (test 7) is significantly better than the cal-
culated optimal predictor and not quite as good as the self-tuning pre-
dictor.
The cell containing the .1 Normal solution was hypothesised to be of
higher order than the cell with the lower concentration of ions. A trans-
fer function with two poles and two zeros was used to match the data points.
The derivation of the optimal predictor is in the appendix. Table 6 presents
the results of selective tests. Three self-tuning predictors were applied,
one of order 5 and the other two of order eleven. The inital estimates are
presented in the table under the column Q. The two eleventh order pre-
dictors were initiated with parameter estimates of the zero vector (test 2)
and the calculated "optimal" values (test 3). The estimated parameters
for both predictors converge to similar values. The time series date
indicates the parameters themselves are not constant, but the two pre-
dictors converge to this time-varying parameters set within 500 iterations.
The time variation of the cell parameters is believed to be a result of
the input series applied to the cell. The time series data for the test
results of the .01 normal cell show a convergence to constant parameters.
The non-adaptive predictors performed similarly to those of table 5. The
converged parameters sets were best, with the fifth order predictor having
a lower accumulated square error but a slightly higher average over the
last 208 iterations. The calculated "optimal" predictor performed signi-
ficantly worse.
The question of deciding which parameter set to invert to obtain esti-
mated plant parameters is non-trivial. One possible procedure involves
implementing self-tuning predictors of various orders to obtain estimated
-36-
parameters, use these parameters in the non-adaptive model and choose
the set that minimizes the accumulated square error. Attempts to ob-
tain the plant parameters via inversion of the procedure in Section 2.3
may prove futile; in particular we can not solve for a G(z ) with the
parameters from test 7 table 5. However we can solve for the plant
parameters for the predictor of test 6, table 5 and test 6 table 6. The
inversion is simple and the auto regressive equations for the plants are
y(t.)= .62y(t i) - 240u(t.- ) + 440u(t . ) (24)
y(t.) = .238y(t.) + .02y(ti-2) - 18 0u(t ) + 24 0u(t)i-2
(25)
for test 6 and table 5 and test 6 and table 6 respectively.
In the above equations y(-) represents the response voltage across
the cell and u(-) isthe stimulating current. The coefficients of the
last two terms on the right hand side thus aquire the physical dimension
of resistance. The results are consistent with our qualitative expectation
that the cell with the higher concentration should have the lower resistance.
The test results have shown that the self-tuning preditor does very
well in reducing theprediction error of the apriori - ''optimal'' predictor
We have also found that the self-tuning predictors initialized with dif-
ferent parameter sets converge to the same values. This set of parameters
when implemented as a non-adaptive predictor performs comparably to the self-
tuning predictor and significantly better than the apriori - "optimal"
predictor. The self-tuning predictor does not always reduce the accumu-
lated square error, however it does reduce the average square error after
an initial period of tuning.
I I
-37-
CHAPTER 5
SUMMARY AND CONCLUSIONS
5.1 SUMMARY AND CONCLUSIONS
We have applied the self-tuning predictor to simulated data and to
an actual laboratory application. The simulation results, for conditions
of known and constant plant parameters, indicate that the prior calculated
otpmal predictor does in fact perform best. However there are sets of
parameters other than the optimal sets that perform almost identically.
The appl ication results show that parameter estimates converge asymptotical-
ly to nearly identical (but generally time-varying) values for different
initializations. Since the parameters describing the cell may not be
constant, we do not expect the predictor parameters to converge to some
constant value, and the time series data indicates that this is the case.
The time dependence of the plant parameters is more of a factor in the cell
with the .1 Normal solution than with the cell containing the solution with
the lower concentration.
To explain the above results we hypothesis the existence of points
in parameter space representing the optimal parameter sets. (Refer to figure
13 ). If the underlying plant parameters are constant these optimal points
are also constant; if the plant parameters are time-dependent then the opti-
mal parameters will follow some time dependent trajectory through parameter
space. For the simulated plant with constant parameters we calculated two
distinct ''optimal'' predictors. These predictors represent distinct points
in the parameter space. There exists a region connecting these points where
the capability of non-adaptive predictors with parameter sets from this
-38-
region will be nearly optimal. Self-tuning predictors initialized with
these parameter sets will show little if anytendency to converge to opti-
mal values inthe face of plant -driving noise. Self-tuning predictors ini-
tialized with parameter sets outside this region will converge to apriori
optimal parameter sets. Further investigation is necessary to determine
whether the apriori "optimal" parameter sets represent extreme points with-
in the''nearly optimal" set or central values of an "optimal" volume. The
square error accumulated by the predictors will depend on some measure
of their deviation from the optimal. Experiments have shown that most
of the square error accumulated by theself-tuning predictor is incurred
during the first few interations of the algorithm. The self-tuning pre-
dictor learns quickly to reduce the prediction error to a minimum.
In conclusion, the self-tuning predictor is well suited for applica-
tions of the nature described herein. It has been shown that the self-
tuning predictor reduces mean square error to a minimum, and its parameters
converge to sets that are nearly optimal.
The inversion of the relationships between the plant parameters and
the predictor parameters is not always possible. However we can always
reduce the order of the predictor such that the inversion is simple
5.2 AREAS FOR FUTURE RESEARCH
During the course of this research many areas of investigation had to
be curtailed due to time constraints. Specifically we were unable to deter-
mine the effect different input groups would have onthe cell parameters and
on the identification of predictor parameters. In the interest of standard-
ization for comparative purposes we restricted ourselves to one standard
m ~1~~
I I
-39-
input group (actually two different input groups were used; one for simu-
lation and another for the electrode studies). The cell parameters are
dependent on the input applied. In fact the dynamics of the input may cause
time variation inthe underlying plant parameters. Present theory on optimal
input synthesis for identification is inadequate under these circumstances.
We studied the effect of a step change inthe concentration.of the salt
solution in the cell. Future researchers may wish to study the system under
dynamic variation, such as continuous change of solution concentration, or
temperature.
The experimetntal configuration used to collect data employed the
electrodes to both stimulate and record. Other arrangements, more closely
realizing the pattern in figure one should also be considered. Three cells
could be placed in series. The stimulating signal would be applied to the
electrodesoifthe middle cell; and the response measured at the terminal
of the outside cells.
The self-tuning predictor could also be used for identifying para-
meters of the Hall cell. Proper control of electrolytic reduction of
aluminum, for instance, could be enhanced with identification of impedence
parameters.
APPENDIX
In this appendix we derive the optimal predictors for electrolytic
cells with salt solutions of .01 and .1 normality given the impedence vs.
frequency plots. We first derive optimal predictor for the electrolytic
cell containing the .01 normal solution.
The first step is to match a transfer function of the form (23) to
the data points. Figure la illustrates the match of the asympotic Bode
plot for a transfer function with one pole and one zero. The pole and zero
frequencies of Z(s) are:
f = 1Hz
p
f = 4Hz
z
( S+.-)
Z(s) = K K = 2000 (Al)
(--L 1)
6.3
The transfer function, Z(s), is the ratio of the Laplace transforms of
the voltage response to the stimulating current.
Y(s)
U5s) = Z(s)
The Laplace transform of the differential equation representing this
system is
( + 1) Y(s) = 2000(-- + 1l) U(s)
6.3 25
or
(s + 6.3)V(s) = (500s + 12500) U(s) (A2)
I --
I I
-441--
A state space representation may be obtained using the procedure proposed
by DeRousso [13].
x(t) = 6.3x(t) + 9350u(t) (A3a)
y(t) = x(t) + 500u(t) (A3b)
The sampled data version of (A3) is obtained by the transformation (6).
x (n+l) = A x (n) + 'B u(n-1) (k
y(n) = t x(n) + [i u(n-1)
Note: u(n-1) is used since the current calculated at sample time n is the
current that was applied to the cell over the interval [t. 1 ,t.].
AT
A = e = 1.0074
Tft rTAT
B = e B dT = 11
0
C =Cft
D =D
T = .0012
G(z) = C(zI-A) B+
= (z - 1.0074) (11) + 500
y(n) - 11 +50]Z- "-1
un1) z-l.0074 +- G(z )
z
ll007Iz I+ 500
1-1.0074z
-42-
The estimated auto-regressive representation of the plant is;
y(n)[1-1.0074z1] = lizI + 500[1-1.0074z I] u(n-1)+v(n)
where v is added to incorporate the observation noise.
y(n) = 1.0074y(n-1) + 500u(n-1) - 500u(n-2) + (n) (A)
A(z ) = (1-1.0074z )
-l -l -2B(z ) =(500z -500z )
C(z ) =
Since k = 1
F(z ) = 1
The identity (11) is approximately
1= (-z ) + z 9g0  go=
The optimal predictbr is given by (13):
yA t0 - -I 500z -2
(t I/t.) = y (t.) + (z-1) (S:z- 5o z ) u(t )
1-lz
Cancelling common factors of the last term on be right hand side:
y(ti+]/t.) = y(t.) + 500u(t.) - 500u(t.1)
The H vector for the one step predictor is
H(t ) ^ [- (t./t. )y(t )y(t. )u(t.)u(t. ) .
I 
i -I  -lu
The optimal Q vector based on theprior experimental data is then:
r--
I I
Q = [0 1 0 500 -500] (A5)
Calculation of the optimal predictor for the cell with .1 normal solu-
tion follows the same procedure. Z(s) in this case is proposed to have two
poles and two zeros:
f = .127 H f = 8 H
p1  z z z
f = 35 H f = 40H
Z(s) = K + l)(251+
.98 220
K=5620
The match of the asymptot ic Bode plot, for Z (s), to the
illustrated in figure lib.
The state space representation is derived to be:
x Lt) = [
-2253
y(t) = [
4.16
_ -3.7
data points is
x 1041
x 06 u(t)
x 10j
0] x (t) + 495 u (t) + v(t)
Since the ranks of [B AB] and [C] are both two the system is both con-
trollable and observable-.
The sampled data version of A6 is obtained by the transformation (6).
% AT 1  .0012]A =e F=
L-1.16 .773j
T AT -13
B= f e B dT=
0L-3.43 x 103
(A6)
-44-
C = [1 0]
0 = 495
The sampled data transfer function is found to be
G(z) = t(zi-') i +
= [1 01 z-1 -.0012 -- 13 + [495
1.16 z -.773_, -3.43 x 10 3
This can be reduced to
y(z) - 495z2 - 877z + 396
u(z) z2 - 1.773z + .8
and in terms of z
(z- y(z1 ) _495 - 8 77z + 396z 2
-1 
-1 -2
u(z ) 1 - 1.773z + .8z
The form A(z )y(z ) = B(z )u(z ) + v(z ) is obtained recognizing
-l-1 - -2A(z ) = (1 - 1.773z + .8z )
-1 -1 -2B(z ) = 495 - 877z + 396z
and incorporat'ing an addit ive noise term, C(z ) = 1.
Solving the identity (11)
F(z ) = 1
G(z ) 1.773 - .8z.
The optimal predictor is now derived by solving (13).
-U-
I I
-45-
y(t /t.) = H(t.)Q
where
H (t.i) = [-y(t.i/t. )-y(t. /t. A i - )y (t.i)y(t. I)y(t.i-2)y (t. )II 1--1 g-l-2 ii-1 1-2 1-3
U(t.)u(t )u(t.)u(t. 3 )u(t.Q]a -[ -28 -3 i-4
and Q =(0 0 1. 8 -8 0 0 495 -8 77 4oo. o 0] T
-46-
subject electrodes
observations, y(t)
2
bioelectric
source
vs(Il biological transmission
medium
Figure la
R 2
Vel
v SC t)
RR
1v
Ve2
2
Figure Ib
resistance of biological medium
resistance of electrode junction
capacitence of electrode junction
amplifier input resistance
bioelectric source signal
v e2(t) half-cell potentials due to electrode -
electrolyte interface
measured voltage
R-
R 2
C
R 3
vs(t)
v (t)vel t
yt)
"-r
I I
, 1
IR 3 Y(t)
-47-
System Block Diagram
v k(t) y(t)
source -stimulating znbiological receiving
electrode me di um e e ectrode
A/D A/D
uy(t) Y/t
predictor 0 y (t /t)
f + K i
parameter
estimator
Figure 2
Filter Block Diagram
+ + Q(t )
y(t.) jK (t.)
AA
A delay
k-step H(t)
y~t./tt)
A
Y~ i-k Y(ti+k/ti)
Figure 3
I I . I
-.03 -. 027 -. 02 1
I I
-.015 -.009 -. 003 0 .003 .009 .015
Figure 4
Distribution of Standard Random Noise Sequence
?LE SIZE: 2000
LE MEAN: 0000000
?LE VARIANCE: 0.00008
- 250
- 200 SAME
SAME
SAME
- 150
- 100
-50
.021 a027 .03
1. I I I I I
co
I2
25
20
15
500 1000
TIME
Figure 5
Accumulated Square Error
Test 25, Table II
1500 2000
NOTE: LOSS OF OPTIMAL PREDICTOR IS
IDENTICAL TO THE SUMATION OF
THE SQUARE OF THE NOISE
1 2080
2 v2(i) = 8.11 x 10
1040 1041
2080
I F-2(i = 8.12 x 10-3
1040 1041
t 2
i= 1
12
Ln
0
0
LU
-4D
D2
Z)
u
4
10
5
0
a
KEY: q ~x
q5 ~0
1.0 u-
.15
.15
0-
-.05
-. 1-
-. 15
-.2
-.25
-.3
-. 35
-.4-
-.45
-.5
-.55
K A K A A A A
000 0000 0 00 0 0000
Io
0
0
0
Lu
o 0 0 0 0 0 0 0 0 0 0 0
1000 1500 2000
TIME (ITERATIONS)
Figure 6
Parameter Estimates vs. Time
Test 25, Table II
0
x 0
A
0 x x xx x K K K
~o 0
0 100
0
500
II
K K K K K
KEY: q7 ~K"
q2 
~0
q6 0
.11
.05 I-
I'
Cn
0
0
o 0
0 0
00
C
0 00 0 0 0
0 00
1000
TIME (ITERATIONS)
Figure 7
Parameter Estimates vs. Time
Test 25, Table II
1.05
,95
K
K
K
K
'Kx K K K K K
K
-.05 V
100 500
0 0 0 0 0 0 0 0 0
1500
I I
2000
.9
.85
I I I
KEY: q3 
~
q 8 ~o
9-~ 0
0
0
0 00
0
0 0 0 0 0 
' 00 0-0-0-0-0--0
0
0
- 0
K
Xx* K
500
Xx x - - x xX- X -a--x-- -x
1000
TIME (ITERATIONS)
Figure 8
Parameter Estimates vs. Time
Test 25, Table II
1.0 *-
.3
.25
.2
.15
.1
0 4
10o
0
0
K
0 0
0 0
0
0 0
.05
0
-. 05
-.1
-. 15
-.25
-.3
0 0
Il
Hn
0-
100 - 1500 2000
I I I I
D/A
(t 
.)1: /D
DISK C PU 0 1 C( t. )
MEMORY d ( T) A/DElectrolytic Cell
Figure 9
Electrolytic Cell Implementation Circuit
-53-
READ INPUT SERIES FROM
DISK TO MEMORY, i = 1
GET
AND APPLY TO
D/A CHANNEL
SAMPLE A/D CHANNELS
C (t );d(t )
CALCULATE STIMULUS
c(t ) -d(t )
Rs
NO
ii > 2080
YES
WRITE STIMULUS SERIES u(-)
AND RESPONSE SERIES y(-) = d(-)
ON DISK
Figure 10
Electrolytic Cell Implementation Flow Chart
T
I I
-54-
10000
1000.
Izl -
(ohms)
1007-
10I
1 1 10 100 1000
FREQUENCY (Hz)
(a)
.01 Normal Solution
10000
1000
IzI
(ohms) .
0
100-
0 0 0 0 0OoooO
10
110 100 1000
FREQUENCY (Hz)
(b)
.1 Normal Solution
Figure 11
Log Impedence vs. Log Frequency
-55-
INITIALIZE FILTER DESIGN PARAMETERS,
ESTIMATE OF PREDICTOR PARAMETERS,i= 1
READ STIMULUS AND RESPONSE
SERIES FROM DISK TO MEMORY
CALCULATE PREDICTOR
Y2(ti+ 1/tWH(t.)Q (t)
Y = y(t +)
U= u(t + 1
CALL IDENTIFICATION ROUTINE
ESTIMATE Q(iagra
UPDATE H( t)
NO4
i > 2080
4YES
ENDI
Figure 12
identification Block Diagram
T-
I I
-56-
Parameter Space
*Q4
convergence
trajectory
Q3 I -
region of near.
optimal perforrmance
Q2
91 and Q2 are optimal parameter sets assuming
constant plant parameters
Q3 faIIs within region of near optimal performance
no convergence to optimal sets
Q4 initial estimate outside region of near optimal
performance... estimates of parameters converge
to an optimal set
Figure 13
Relation of predictor parameters
in parameter space
Actual
Observation Non-Adaptive Accumulated Accumulated
Test Noise Predictor Absolute Square
Number Variance Parameters Error Error
1 0 [o -.25 -.5 0 0 1 0 0 1 0.00000 0.00000
2 0 [ .25 .5 -.25 -.563 -.25 -.25 1 .25 .5 1 0.00052 0.00000
3 0 [ .125 .25 -.25 -.532 -.125 -.125 1 .125 .25 1 0.13353 0.00001
4 0 [ 1 1 1 I 1 1 1 I 1228.7 1090.4
5 8 x 10 7  [0 0 -.25 -.5 0 0 1 0 0 1 1.55089 0.00171
-7
7 8 x 10 [ .125 .25 -.25 -.532 -.125 -.125 1 .125 .25 ] 1:55441 0.00172
8 8 x 107  [1 1 1 1 1 1 1 1 1 ] 1228.8 1090.7
9 .008 [0 0 -.25 -.5 0 0 1 0 0 1 155.1 17.13
10 .008 [ .25 .5 -.25 -.563 -.25 -.25 1 .25 .5 ] 155.1 17.13
11 .008 [ .125 .25 -.25 -.532 -.125 -.125 1 .125 .25 1 155.1 17.133
12 .008 [1 1 1 1 1 1 1 1 ] 1510.6 1678.4
Note:
Accumulated Accumulated
Absolute Square
Noise Noise
Tests 1-4 0 0
Tests 5-8 1.551 0.00171
Tests 9-12 155.1 17.13
Table I
Non Adaptive Predictor
Simulated Plant/Constant Parameters
Ul
Initial Estimate of Self-Tuning Predictor Paramters Final Estimate of Parameters
____ - -- + r 1-
[0 0
1 .25 .5
1 .125 .25
[1 1
[I I
-. 25
-. 25
-. 25
-. 5
-. 563
-. 562
0
-.25
-.125
0
-.25
-.125
0 0 1
.25 .5 ]
.125 .25 1
I I ]
[0
1 .25
.125
.013
..073
0
.5
.25
:014
.078
-. 25 -. 5
-. 25 -. 563
-. 25 -. 532
.167 .083
.258 .069
0
-.25
-.125
.319
.322
0
-.25
-. 125
.235
.238
0
.25
.125
.999 .402
.951 -. 315
0 ]
.5 1
.25 1
-. 470]
-. 486]
* $
(0 0
[ .25 .5
1 .125 .25
[0 0
[ .25 .5
[ .125 .25
(1 I
[0 0
-. 25
-.25
-.25
1
-. 5
-.563
-. 532
-. 25 -.5
-. 25 -. 563
-. 25 -. 532
-. 25 -. 5
0 0
-.25 -.25
-. 125 -.125
0 0
-. 25 -. 25
-.125 -.125
0 0
0 0 ]
.25 .5 1
.125 .25
S 1 ]
0 01
.25 .5 1
.125 .25 1
S 1
0 01l
[-.005 -.005
.2 38 .488
-.117 .241
[ .011 .012
[-.064 .057
1 .128 .6
1 .034 .310
.018 .064
[-.039 -. 009
-.256 -.512 -.002
-.259 -.576 -.240
-.257 -.544 -. [25
.059 -.082 .237
-. 267
-. 269
-. 268
-. 254
-. 263
-. 517
-. 569
-. 545
-. 518
-. 523
-. 009
-. 252
-.127
-. 037
-.01
-. 003 1
-.251 1
-.127 1
.169 .997
-. 012
-. 303
-. 148
-. 002
.008
.975
.978
.975
.964
.983
.002 .006]
.246 .501)
.125 .25 1
-. 790 -. 343]
.016
.183
.106
.107
.024
.035]
.6171
.305]
.0031
.001]
Accumulated Accumultated
Absolute Square
Error Error
0.0
.00095
.00125
1.50167
17.02
1.57371
1.57073
1.5698
3.08
156.47
155.79
156.28
163.39
155.92
5 . _________________~.~~~~-t.-
0.0
0.0
0.0
.23674
4.5
.00176
.00177
.00176
.2377
17.42
17.29
17.39
24.83
17.29
Notes:
1. Tests run with following design parameters:
W- [o.o]I
P (0/-1)-. 51
V-. 0009
* V-. 09
P-.051
2. For accumulated noise loss see note Table I
Self-Tuning Predictor
Simulated Plant/Constant Parameters
Test
Number
Actual
Observation
Noise
Var iance
13
14
15
16
17*
18
19
20
21
22
23
24
25
26**
0.0
0.0
0.0
0.0
0.0
8 x 10-7
8 x 107
8 x o
8 x 107
.008
.008
.008
.008
,008
InU-OD
I
Table I I
-r
-.- L- I
I
1
i
i
i
Actual
Observation Accumulated Accumulated
Test Noise Non Adaptive Predictor Parameters Absolute Square
Number Variance Error Error
4o 0.0 [0 0 -.25 -.5 0 0 1 0 0 1 110.67 11.44
41 0.0 [ .25 .5 -.25 -.563 -.25 -.25 1 .25 .5 1 110.67 11.44
42 0.0 [ .125 .25 -.25 -.562 -.125 -.125 1 .125 .5 1 111.64 11.50
48 8 x 10-7  [0 0 -.25 -.5 0 0 1 0 0 1 110.72 11.44
43 .008 [0 0 -.25 -.5 0 0 1 0 0 1 198.73 29.43
TABLE I I I
Non Adaptive Predictor
Simulated Plant/Time Series Trend
Ip
'-0
4Table ,IV
Self-Tuning Predictor
Simulated Plant/Time Series Trend
ActualT
Observac ion Accumulated Accumulated
Test Noise Initial Estimate of Self-Tuning Predictor Parameters Final Estimate of Parameters Absolute Square
Number Variance Error Error
46* 0.0 [0 0 -.25 -.5 0 0 1 0 0 .0 .204 2.59 -.838 .601 -.467 .917 -145 .83 9.295 .06503
45'' 0.0 [0 0 -.25 -.5 0 0 1 0 0 1 .07 .07 -.32 -.57 .05 .02 1.0 -.07 -.03 .275 .00005
8 1x 07
I________ J ____________ I. -
C'
a
Note:
i. For design per meters see Table 2.
-. 25 -. 5 0 0 1 0 0 1
.02 .01 -.49 -.67 -.03 -.03 1.0 -.04 -.03 2.07756 .00335
Table V
Electrode Studies
.01 Normal Solution
2080 2080
Tes t LI,-O
Number P(0/-1 I V2080 iIi1872
Self-
Tuning
Predictors
P 0dio 001].00091[0 1 0 500 -500 1 [0 .62 0 -239 435 .09285 4.16S6 105[
2 1 6  001 .0009 [0 0 1 0 0 0 500 -500 0 j .5 .13 .84 .16 .25 .11 -293 463 .33 1 11.29 3.8 x 10 -5
3 106 .001 .0009,[0 0 0 0 0 1 [0 .625 .001 -242 438 1 .09072 4.17 x 10'
4o 10 .001 .0009 [0 0 0 0 0 0 0 0 0 3 -55 .14 .84 .19 .25 .11 -301 448 .37 1 .07856 3.7 x 10-
Non
Adaptive
Predictors I
-5
5 [0 1 0 500 -5001 .39419 14.6 x 10
6 [0 .6 0 -240 440 1 .14984 4.19 x 10-
7 [ .55 .13 .84 .17 .25 .11 -300 450 .351 .09623 4-63 x 10
8 [0 0 1 0 0 0 500 -500 0 . 39419 14.6 x 10-5
[0 0 0 0 0 1
[0 0 0 0 0 0 0 0 0 0 0
[0 0 1.8 -.8 0 0 h95 -877 400 0 0
[0 0 1.8 -.8 0 0 495 -877 400 0 0
1 .19 .03 -.01 -.12 .27 -.003 -243 120 1.13 115 32.8 I
[o .238 .02 -180 239.2 1
Q2080
2080
2 .0 ii i- 1872
[0 .24 .02 -180 239 1 1.99 1.24 x 10-3
1 .19 .0) -. 01 -.12 .27 0 -243 125 1.5 165 32.5 1 855.5 1.06 x io0
..19 .03 -.01 -.12 .27 -.003 -243 126 1.13 166 32.8 1 2.30573 1.1 x 10-3
21.2 19.9 10-3
2-55128 1.04 10-3
2.236 91 1.12 10-o 3
Table VI
Electrode Studies
.1 Normal Solution
P w I 0
Terc
Number
5e If-
Tun Ing
Predictors
2
3
Non-adapcI ve
Predictors
4
6
0
10 6
10 6
.001
.001
.001
.0009
.0009
.0009
0~'
I I
-63-
REFERENCES
1. Geddes, L.A., Electrodes and the Measurement of Bioelectric Events,
Wiley-tnterscience, New York, 1972.
2. Geddes, L.A. and Hoff, H.E., "The discovery of Bioelectricy and
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3. Johnson, T.L. and Salzsieder, B., "Physical Properties of Silver-
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4. WiHenmark, B., "A Self-tuning Predictor," IEEE Trans. Auto.
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5. Bishop, A.B., Introduction To Discrete Linear Controls, Academic
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6. Schweppe, F.C., Uncertain Dynamic Systems, Prentice-Hall, 1973.
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9. Johnson, T.L., "Inverse Digital Filtering of Electrode-Produced
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12. Mehra, Ramon, K., System Identification Advances and Case Studi es,
Academic Press, New York, 1976.
13. DeRousso, P.M., State Variable for Engineers, Wiley and Sons, New
1~
-64-
York, 1967.
14. Plonsey, R. and Fleming, D.G., Bioelectric Phonomena, McGraw Hill,
New York, 1969.
15. Astrom, KJ. and WiHenmark, B., "On Self-Tuning Regulators," Auto-
matica, Vol. 9, pp. 185-199, 1973.
16. Wieslander, J. and WiHenmark, B., "An Approach to Adaptive Control
Using Real Time Identification," Automatica, vol. 7, pp. 211-
217, 1971.
17. Astrom, K.J. and Eykhoff, P., "System Identification - A Survey,"
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18. Aoki, M. and Staley, R.M., "On input Signal Synthesis in Parameter
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